chapter 1 nonlinear optics

1

Nonlinear Opt5. Nonlinear Optics

This chapter provides a brief introduction into thebasic nonlinear-optical phenomena and discussessome of the most significant recent advances andbreakthroughs in nonlinear optics, as well as novelapplications of nonlinear-optical processes anddevices.

Nonlinear optics is the area of optics thatstudies the interaction of light with matter inthe regime where the response of the materialsystem to the applied electromagnetic field isnonlinear in the amplitude of this field. At lowlight intensities, typical of non-laser sources, theproperties of materials remain independent ofthe intensity of illumination. The superpositionprinciple holds true in this regime, and light wavescan pass through materials or be reflected fromboundaries and interfaces without interacting witheach other. Laser sources, on the other hand, canprovide sufficiently high light intensities to modifythe optical properties of materials. Light wavescan then interact with each other, exchangingmomentum and energy, and the superpositionprinciple is no longer valid. This interaction of lightwaves can result in the generation of optical fieldsat new frequencies, including optical harmonics ofincident radiation or sum- or difference-frequencysignals.

5.1 Nonlinear Polarizationand Nonlinear Susceptibilities ............... 3

5.2 Wave Aspects of Nonlinear Optics ........... 4

5.3 Second-Order Nonlinear Processes ......... 55.3.1 Second-Harmonic Generation........ 55.3.2 Sum- and Difference-Frequency

Generation and ParametricAmplification ............................... 7

5.4 Third-Order Nonlinear Processes ............ 85.4.1 Self-Phase Modulation ................. 95.4.2 Temporal Solitons......................... 105.4.3 Cross-Phase Modulation ............... 105.4.4 Self-Focusing............................... 115.4.5 Four-Wave Mixing ........................ 13

5.4.6 Optical Phase Conjugation ............. 135.4.7 Optical Bistability and Switching .... 145.4.8 Stimulated Raman Scattering......... 165.4.9 Third-Harmonic Generation

by Ultrashort Laser Pulses .............. 17

5.5 Ultrashort Light Pulsesin a Resonant Two-Level Medium:Self-Induced Transparencyand the Pulse Area Theorem .................. 225.5.1 Interaction of Light

with Two-Level Media .................. 225.5.2 The Maxwell and Schrödinger

Equations for a Two-Level Medium 225.5.3 Pulse Area Theorem ...................... 245.5.4 Amplification

of Ultrashort Light Pulsesin a Two-Level Medium ................ 25

5.5.5 Few-Cycle Light Pulsesin a Two-Level Medium ................ 27

5.6 Let There be White Light:Supercontinuum Generation.................. 295.6.1 Self-Phase Modulation,

Four-Wave Mixing,and Modulation Instabilitiesin Supercontinuum-GeneratingPhotonic-Crystal Fibers ................. 29

5.6.2 Cross-Phase-Modulation-InducedInstabilities ................................. 31

5.6.3 Solitonic Phenomena in Mediawith Retarded Optical Nonlinearity. 33

5.7 Nonlinear Raman Spectroscopy .............. 375.7.1 The Basic Principles ...................... 385.7.2 Methods of Nonlinear Raman

Spectroscopy ............................... 405.7.3 Polarization Nonlinear Raman

Techniques .................................. 435.7.4 Time-Resolved Coherent

Anti-Stokes Raman Scattering........ 45

5.8 Waveguide Coherent Anti-StokesRaman Scattering ................................. 465.8.1 Enhancement of Waveguide CARS

in Hollow Photonic-Crystal Fibers ... 46

PartA

5

2 Part A Basic Principles and Materials

5.8.2 Four-Wave Mixing and CARSin Hollow-Core Photonic-CrystalFibers ......................................... 49

5.9 Nonlinear Spectroscopywith Photonic-Crystal-Fiber Sources....... 535.9.1 Wavelength-Tunable Sources

and Progressin Nonlinear Spectroscopy ............. 53

5.9.2 Photonic-Crystal Fiber FrequencyShifters ....................................... 54

5.9.3 Coherent Anti-Stokes RamanScattering Spectroscopywith PCF Sources .......................... 55

5.9.4 Pump-Probe NonlinearAbsorption Spectroscopyusing Chirped Frequency-ShiftedLight Pulsesfrom a Photonic-Crystal Fiber ........ 57

5.10 Surface Nonlinear Optics, Spectroscopy,and Imaging ........................................ 60

5.11 High-Order Harmonic generation........... 635.11.1 Attosecond Metrology –

Historical Background ................... 635.11.2 High-Order-Harmonic Generation

in Gases ...................................... 645.11.3 Microscopic Physics ...................... 665.11.4 Macroscopic Physics ...................... 69

5.12 Attosecond Pulses:Measurement and Application ............... 715.12.1 Attosecond Pulse Trains

and Single Attosecond Pulses......... 715.12.3 Basic Concepts

for XUV Pulse Measurement ........... 715.12.3 The Optical-Field-Driven XUV Streak

Camera Technique ........................ 745.12.4 Applications of Sub-femtosecond

XUV Pulses: Time-ResolvedSpectroscopy of Atomic Processes ... 78

References .................................................. 80

Although the observation of most nonlinear-opticalphenomena requires laser radiation, some classes ofnonlinear-optical effects were known long before theinvention of the laser. The most prominent examples ofsuch phenomena include Pockels and Kerr electroopticeffects [5.1], as well as light-induced resonant absorp-tion saturation, described by Vavilov [5.2, 3]. It was,however, only with the advent of lasers that systematicstudies of optical nonlinearities and the observation ofa vast catalog of spectacular nonlinear-optical phenom-ena became possible.

In the first nonlinear-optical experiment of the laserera, performed by Franken et al. in 1961 [5.4], a ruby-laser radiation with a wavelength of 694.2 nm wasused to generate the second harmonic in a quartz crys-tal at the wavelength of 347.1 nm. This seminal workwas followed by the discovery of a rich diversityof nonlinear-optical effects, including sum-frequencygeneration, stimulated Raman scattering, self-focusing,optical rectification, four-wave mixing, and many others.While in the pioneering work by Franken the efficiencyof second-harmonic generation (SHG) was on the or-der of 10−8, optical frequency doublers created by early

1963 provided 20%–30% efficiency of frequency con-version [5.5, 6]. The early phases of the developmentand the basic principles of nonlinear optics have beenreviewed in the most illuminating way in the classi-cal books by Bloembergen [5.7] and Akhmanov andKhokhlov [5.8], published in the mid 1960s.

Over the following four decades, the field of nonlin-ear optics has witnessed an enormous growth, leadingto the observation of new physical phenomena and giv-ing rise to novel concepts and applications. A systematicintroduction into these effects along with a comprehen-sive overview of nonlinear-optical concepts and devicescan be found in excellent textbooks by Shen [5.9],Boyd [5.1], Butcher and Cotter [5.10], Reintjes [5.11]and others. One of the most recent up-to-date reviews ofthe field of nonlinear optics with an in-depth discussionof the fundamental physics underlying nonlinear-opticalinteractions was provided by Flytzanis [5.12]. Thischapter provides a brief introduction into the mainnonlinear-optical phenomena and discusses some of themost significant recent advances in nonlinear optics, aswell as novel applications of nonlinear-optical processesand devices.

PartA

5

Nonlinear Optics 5.1 Nonlinear Polarization and Nonlinear Susceptibilities 3

5.1 Nonlinear Polarization and Nonlinear Susceptibilities

Nonlinear-optical effects belong to a broader class ofelectromagnetic phenomena described within the gen-eral framework of macroscopic Maxwell equations. TheMaxwell equations not only serve to identify and classifynonlinear phenomena in terms of the relevant nonlinear-optical susceptibilities or, more generally, nonlinearterms in the induced polarization, but also govern thenonlinear-optical propagation effects. We assume theabsence of extraneous charges and currents and writethe set of Maxwell equations for the electric, E(r, t),and magnetic, H(r, t), fields in the form

∇ × E = −1

c

∂B∂t

, (5.1)

∇ × B = 1

c

∂D∂t

, (5.2)

∇ · D = 0 , (5.3)

∇ · B = 0 . (5.4)

Here, B = H +4πM, where M is the magnetic dipolepolarization, c is the speed of light, and

D = E+4π

t∫−∞

J(ζ )dζ , (5.5)

where J is the induced current density. Generally, theequation of motion for charges driven by the electromag-netic field has to be solved to define the relation betweenthe induced current J and the electric and magneticfields. For quantum systems, this task can be fulfilledby solving the Schrödinger equation. In Sect. 5.5 of thischapter, we provide an example of such a self-consistentanalysis of nonlinear-optical phenomena in a modeltwo-level system. Very often a phenomenological ap-proach based on the introduction of field-independentor local-field-corrected nonlinear-optical susceptibilitiescan provide an adequate description of nonlinear-opticalprocesses.

Formally, the current density J can be representedas a series expansion in multipoles:

J = ∂

∂t(P −∇ · Q)+ c (∇ × M) , (5.6)

where P and Q are the electric dipole and electricquadrupole polarizations, respectively. In the electricdipole approximation, we keep only the first term onthe right-hand side of (5.6). In view of (5.5), this givesthe following relation between the D, E, and P vectors:

D = E+4π P. (5.7)

We now represent the polarization P as a sum

P = PL + PNL , (5.8)

where PL is the part of the electric dipole polarizationlinear in the field amplitude and PNL is the nonlinearpart of this polarization.

The linear polarization governs linear-optical phe-nomena, i. e., it corresponds to the regime where theoptical properties of a medium are independent of thefield intensity. The relation between PL and the electricfield E is given by the standard formula of linear optics:

PL =∫

χ(1)(t − t′)E(t′)dt′, (5.9)

where χ(1)(t) is the time-domain linear susceptibilitytensor. Representing the field E and polarization PL inthe form of elementary monochromatic plane waves,

E = E (ω) exp (ikr −ωt)+ c.c. (5.10)

and

PL = PL(ω) exp(ikr −ωt

)+ c.c. , (5.11)

we take the Fourier transform of (5.9) to find

PL(ω) = χ(1)(ω)E(ω) , (5.12)

where

χ(1)(ω) =∫

χ(1)(t) exp(iωt)dt . (5.13)

In the regime of weak fields, the nonlinear part of thepolarization PNL can be represented as a power-seriesexpansion in the field E:

PNL =∫∫

χ(2)(t − t1, t − t2) : E(t1)E(t2)dt1 dt2

+∫∫∫

χ(3)(t − t1, t − t2, t − t3)

...E(t1)E(t2)E(t3)dt1 dt2 dt3 + . . . , (5.14)

where χ(2) and χ(3) are the second- and third-ordernonlinear susceptibilities.

Representing the electric field in the form of a sumof plane monochromatic waves,

E =∑

i

Ei (ωi ) exp(ikir −ωi t)+ c.c. , (5.15)

we take the Fourier transform of (5.14) to arrive at

PNL(ω) = P(2)(ω)+ P(3)(ω)+ . . . , (5.16)

PartA

5.1


where

P(2)(ω) = χ(2)(ω; ωi , ω j ) : E(ωi )E(ω j ) , (5.17)

P(3)(ω) = χ(3)(ω; ωi , ω j , ωk)...E(ωi )E(ω j )E(ωk) ,

(5.18)

χ(2)(ω; ωi, ω j ) = χ(2)(ω = ωi +ω j )

=∫∫

χ(2)(t1, t2) exp[i(ωi t1 +ω j t2)]dt1 dt2 (5.19)

is the second-order nonlinear-optical susceptibility and

χ(3)(ω; ωi, ω j , ωk) = χ(3)(ω = ωi +ω j +ωk)

=∫∫

χ(3)(t1, t2, t3)

exp[i(ωi t1 +ω j t2 +ωkt3)]dt1 dt2 dt3 (5.20)

is the third-order nonlinear-optical susceptibility.The second-order nonlinear polarization defined

by (5.17) gives rise to three-wave mixing processes,optical rectification and linear electrooptic effect.In particular, setting ωi = ω j = ω0 in (5.17) and(5.19), we arrive at ω = 2ω0, which corresponds tosecond-harmonic generation, controlled by the nonlin-ear susceptibility χ

(2)SHG = χ(2)(2ω0; ω0, ω0). In a more

general case of three-wave mixing process withωi = ω1 �= ω j = ω2, the second-order polarization de-fined by (5.17) can describe sum-frequency generation(SFG) ωSF = ω1 +ω2 Fig. 5.1 or difference-frequencygeneration (DFG) ωDF = ω1 −ω2, governed by the

� (2)

ω 1, k 1

ω 2, k 2

ω 3, k 3

Fig. 5.1 Sum-frequency generation ω1 +ω2 = ω3 in a me-dium with a quadratic nonlinearity. The case of ω1 = ω2

corresponds to second-harmonic generation

nonlinear susceptibilities χ(2)SFG = χ(2)(ωSF; ω1, ω2) and

χ(2)DFG = χ(2)(ωDF; ω1,−ω2), respectively.

The third-order nonlinear polarization defined by(5.18) is responsible for four-wave mixing (FWM),stimulated Raman scattering, two-photon absorption,and Kerr-effect-related phenomena, including self-phase modulation (SPM) and self-focusing. For theparticular case of third-harmonic generation, we setωi = ω j = ωk = ω0 in (5.18) and (5.20) to obtainω = 3ω0. This type of nonlinear-optical interaction, inaccordance with (5.18) and (5.20), is controlled bythe cubic susceptibility χ

(3)THG = χ(3)(3ω0; ω0, ω0, ω0).

A more general, frequency-nondegenerate case can cor-respond to a general type of an FWM process. Theseand other basic nonlinear-optical processes will be con-sidered in greater details in the following sections.

5.2 Wave Aspects of Nonlinear Optics

In the electric dipole approximation, the Maxwell equa-tions (5.1–5.4) yield the following equation governingthe propagation of light waves in a weakly nonlinearmedium:

∇ × (∇ × E)− 1

c2

∂2E∂t2 − 4π

c2

∂2 PL

∂t2 = 4π

c2

∂2 PNL

∂t2 .

(5.21)

The nonlinear polarization, appearing on the right-handside of (5.21), plays the role of a driving source, inducingan electromagnetic wave with the same frequency ω asthe nonlinear polarization wave PNL(r, t). Dynamics ofa nonlinear wave process can be then thought as a resultof the interference of induced and driving (pump) waves,controlled by the dispersion of the medium.

Assuming that the fields have the form of quasi-monochromatic plane waves propagating along the z-

axis, we represent the field E in (5.21) by

E (r, t) = Re[eA (z, t) exp (ikz −ωt)

](5.22)

and write the nonlinear polarization as

PNL (r, t) = Re[ep PNL (z, t) exp

(ikpz −ωt

)],

(5.23)

where k and A(z, t) are the wave vector and the envelopeof the electric field, kp and PNL(z, t) are the wave vectorand the envelope of the polarization wave.

If the envelope A(z, t) is a slowly varying func-tion over the wavelength, |∂2 A/∂z2| � |k∂A/∂z|, and∂2 PNL/∂t2 ≈ −ω2 PNL, (5.21) is reduced to [5.9]

∂A

∂z+ 1

u

∂A

∂t= 2πiω2

kc2PNL exp (iΔkz) , (5.24)

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5.2

Nonlinear Optics 5.3 Second-Order Nonlinear Processes 5

where u = (∂k/∂ω)−1 is the group velocity andΔk = kp − k is the wave-vector mismatch.

In the following sections, this generic equationof slowly varying envelope approximation (SVEA)

will be employed to analyze the wave aspects ofthe basic second- and third-order nonlinear-opticalphenomena.

5.3 Second-Order Nonlinear Processes

5.3.1 Second-Harmonic Generation

In second-harmonic generation, a pump wave with a fre-quency of ω generates a signal at the frequency 2ω

as it propagates through a medium with a quadraticnonlinearity (Fig. 5.1). Since all even-order nonlinearsusceptibilities χ(n) vanish in centrosymmetric me-dia, SHG can occur only in media with no inversionsymmetry.

Assuming that diffraction and second-order dis-persion effects are negligible, we use (5.24) fora quadratically nonlinear medium with a nonlinear SHGsusceptibility χ

(2)SHG = χ(2) (2ω; ω,ω) to write a pair of

coupled equations for the slowly varying envelopes ofthe pump and second-harmonic fields A1 = A1(z, t) andA2 = A2(z, t):

∂A1

∂z+ 1

u1

∂A1

∂t= iγ1 A∗

1 A2 exp (iΔkz) , (5.25)

∂A2

∂z+ 1

u2

∂A2

∂t= iγ2 A2

1 exp (−iΔkz) , (5.26)

where

γ1 = 2πω21

k1c2 χ(2) (ω; 2ω,−ω) (5.27)

and

γ2 = 4πω21

k2c2 χ(2)SHG (5.28)

are the nonlinear coefficients, u1 and u2 are the groupvelocities of the pump and second-harmonic pulses,respectively, and Δk = 2k1 −k2 is the wave-vector mis-match for the SHG process.

If the difference between the group velocities of thepump and second-harmonic pulses can be neglected fora nonlinear medium with a given length and if the in-tensity of the pump field in the process of SHG remainsmuch higher than the intensity of the second-harmonicfield, we set u1 = u2 = u and |A1|2 = |A10|2 = const.in (5.25) and (5.26) to derive in the retarded frame of

reference with z′ = z and η = t − z/u

A2 (L) = iγ2 A210

sin(

ΔkL2

)ΔkL

2

L exp

(iΔkL

2

), (5.29)

where L is the length of the nonlinear medium.The intensity of the second-harmonic field is then

given by

I2 (L) ∝ γ 22 I2

10

(sin

(ΔkL

2

)ΔkL

2

)2

L2, (5.30)

where I10 is the intensity of the pump field.Second-harmonic intensity I2, as can be seen from

(5.30) oscillates as a function of L Fig. 5.2 with a periodLc = π/|Δk| = λ1(4|n1 −n2|)−1, where λ1 is the pumpwavelength and n1 and n2 are the values of the refractiveindex at the frequencies of the pump field and its secondharmonic, respectively. The parameter Lc, defining thelength of the nonlinear medium providing the maximumSHG efficiency, is referred to as the coherence length.

0 6

0.4

0.3

0.2

0.1

0.01 2 3 4 5

Second-harmonic intensity (arb. units)

L/Lc

Lc2 = 2Lc1

Lc1

Fig. 5.2 Second-harmonic intensity as a function of thelength L of the nonlinear medium normalized to the coher-ence length Lc for two values of Lc: (dashed line) Lc1 and(solid line) Lc2 = 2Lc1

PartA

5.3


Although the solution (5.29) describes the simplestregime of SHG, it is very instructive as it visualizesthe significance of reducing the wave-vector mismatchΔk for efficient SHG. Since the wave vectors k1 andk2 are associated with the momenta of the pump andsecond-harmonic fields, p1 = �k1 and p2 = �k2, with� being the Planck constant, the condition Δk = 0,known as the phase-matching condition in nonlinearoptics, in fact, represents momentum conservation forthe SHG process, where two photons of the pump fieldare requited to generate a single photon of the secondharmonic.

Several strategies have been developed to solve thephase-matching problem for SHG. The most practi-cally significant solutions include the use of birefringentnonlinear crystals [5.13, 14], quasi-phase-matching inperiodically poled nonlinear materials [5.15, 16] andwaveguide regimes of nonlinear interactions with thephase mismatch related to the material dispersion com-pensated for by waveguide dispersion [5.7]. Harmonicgeneration in the gas phase, as demonstrated by Milesand Harris [5.17], can be phase-matched through anoptimization of the content of the gas mixture. Fig-ure 5.3 illustrates phase matching in a birefringentcrystal. The circle represents the cross section of therefractive-index sphere n0(ω) for an ordinary wave atthe pump frequency ω. The ellipse is the cross sec-tion of the refractive-index ellipsoid ne(2ω) for anextraordinary wave at the frequency of the second har-monic 2ω. Phase matching is achieved in the directionwhere n0ω = ne(2ω), corresponding to an angle θpmwith respect to the optical axis c of the crystal inFig. 5.3.

When the phase-matching condition Δk = 0 is satis-fied, (5.29) and (5.30) predict a quadratic growth of the

c

θ ηo(ω)

ηe(2ω)

Fig. 5.3 Phase-matching second-harmonic generation ina birefringent crystal

4

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.03210

Pump amplitude(arb. units)

z/zsh

Second-harmonic amplitude(arb. units)

Fig. 5.4 The amplitudes of the pump and second-harmonicfields as functions of the normalized propagation distancez/zsh with zsh = [γρ10(0)]−1

second-harmonic intensity as a function of the lengthL of the nonlinear medium. This scaling law holdstrue, however, only as long as the second-harmonicintensity remains much less than the pump intensity.As |A2| becomes comparable with |A1|, depletion ofthe pump field has to be taken into consideration.To this end, we introduce the real amplitudes ρ j andphases ϕ j of the pump and second-harmonic fields,A j = ρ j exp

(iϕ j

), with j = 1, 2. Then, assuming that

u1 = u2 = u and γ1 = γ2 = γ , we derive from (5.25)and (5.26)

ρ1 (η, z) = ρ10 (η) sech [γρ10 (η) z] , (5.31)

ρ2 (η, z) = ρ10 (η) tanh [γρ10 (η) z] . (5.32)

The solutions (5.31) and (5.32) show that theentire energy of the pump field in the phase-matching regime can be transferred to the secondharmonic. As the pump field becomes depletedFig. 5.4, the growth of the second-harmonic fieldsaturates.

Effects related to the group-velocity mismatch be-come significant when the length of the nonlinearmedium L exceeds the length Lg = τ1/|u−1

2 −u−11 |,

where τ1 is the pulse width of the pump field. The lengthLg characterizes the walk-off between the pump andsecond-harmonic pulses caused by the group-velocitymismatch. In this nonstationary regime of SHG, theamplitude of the second harmonic in the constant-pump-

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5.3

Nonlinear Optics 5.3 Second-Order Nonlinear Processes 7

field approximation is given by

A2 (z, t) = iγ2

z∫0

A210

[t − z/u2 + ξ

(u−1

2 −u−11

)]

×exp (−iΔkξ) dξ . (5.33)

Group-velocity mismatch may lead to a considerableincrease in the pulse width of the second harmonicτ2. For L Lg, the second harmonic pulse width,τ2 ≈ |u−1

2 −u−11 |z, scales linearly with the length of the

nonlinear medium and is independent of the pump pulsewidth.

5.3.2 Sum- and Difference-FrequencyGeneration and ParametricAmplification

In sum-frequency generation Fig. 5.1, two laser fieldswith frequencies ω1 and ω2 generate a nonlinearsignal at the frequency ω3 = ω1 +ω2 in a quadrati-cally nonlinear medium with a nonlinear susceptibilityχ

(2)SFG = χ(2) (ω3; ω1, ω2). In the first order of dis-

persion theory, the coupled equations for slowlyvarying envelopes of the laser fields A1 = A1(z, t) andA2 = A2(z, t) and the nonlinear signal A3 = A3(z, t) arewritten as

∂A1

∂z+ 1

u1

∂A1

∂t= iγ1 A3 A∗

2 exp (iΔkz) , (5.34)

∂A2

∂z+ 1

u2

∂A2

∂t= iγ2 A3 A∗

1 exp (iΔkz) , (5.35)

∂A3

∂z+ 1

u3

∂A3

∂t= iγ3 A1 A2 exp (−iΔkz) , (5.36)

where

γ1 = 2πω21

k1c2 χ(2) (ω1; ω3,−ω2) , (5.37)

γ2 = 2πω22

k2c2χ(2) (ω2; ω3,−ω1) , (5.38)

γ3 = 2πω23

k3c2 χ(2)SFG (5.39)

are the nonlinear coefficients, u1, u2, and u3 and k1, k2,and k3 are the group velocities and the wave vectors ofthe fields with frequencies ω1, ω2, and ω3, respectively,and Δk = k1 + k2 − k3 is the wave-vector mismatch forthe SFG process.

As long as the intensity of the sum-frequency fieldremains much less than the intensities of the laser fields,the amplitudes of the laser fields can be can be as-sumed to be given functions of t, A1(z, t) = A10(t) and

A2(z, t) = A20(t), and the solution of (5.36) yields

A3 (z, t) = iγ3

z∫0

A10

[t − z/u3 + ξ

(u−1

3 −u−11

)]

× A20

[t − z/u3 + ξ

(u−1

3 −u−12

)]×exp (−iΔkξ) dξ. (5.40)

The efficiency of frequency conversion, as can be seenfrom (5.40) is controlled by the group delays Δ21 ≈|u−1

2 −u−11 |L , Δ31 ≈ |u−1

3 −u−11 |L , and Δ32 ≈ |u−1

3 −u−1

2 |L between the pulses involved in the SFG process.In particular, the laser fields cease to interact with eachother when the group delay Δ21 starts to exceed thepulse width of the faster laser pulse.

In difference-frequency generation (DFG), two inputfields with frequencies ω1 and ω2 generate a nonlinearsignal at the frequency ω3 = ω1 −ω2. This process is ofconsiderable practical significance as it can give rise tointense coherent radiation in the infrared range. In thelimiting case of ω1 ≈ ω2, this type of nonlinear-opticalinteraction corresponds to optical rectification, whichhas been intensely used over the past two decades forthe generation of terahertz radiation.

If the field at the frequency ω1 is strong and re-mains undepleted in the process of nonlinear-opticalinteraction, A1(z, t) = A10(t), the set of coupled equa-tions governing the amplitudes of the remaining twofields in the stationary regime is written as

∂A2

∂z+ 1

u2

∂A2

∂t= iγ2 A1 A∗

3 exp (iΔkz) , (5.41)

∂A3

∂z+ 1

u3

∂A3

∂t= iγ3 A1 A∗

2 exp (−iΔkz) , (5.42)

where,

γ2 = 2πω22

k2c2 χ(2) (ω2; ω1,−ω3) , (5.43)

γ3 = 2πω23

k3c2χ(2) (ω3; ω1,−ω2) (5.44)

are the nonlinear coefficientsm and Δk = k1 −k2 −k3 isthe wave-vector mismatch for the DFG process.

With no signal at ω3 applied at the input of the non-linear medium, A3(0, t) = 0, the solution to (5.41) and(5.42) in the stationary regime is given by [5.12]

A2 (z) = A2 (0)[cosh (κz)+ i

Δk

2κsinh (κz)

],

(5.45)

A3 (z) = iA2 (0) sinh (κz) , (5.46)

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5.3


where

κ2 = 4γ2γ∗3 |A1|2 − (Δk)2 . (5.47)

Away from the phase-matching condition, the amplifica-tion of a weak signal is achieved only when the intensityof the pump field exceeds a threshold,

I1 > Ith = n1n2n3c3 (Δk)2

32π3∣∣∣χ(2)

DFG

∣∣∣2 ω2ω3

, (5.48)

where we took

χ(2)(ω2; ω1,−ω3) ≈ χ(2)(ω3; ω1,−ω2) = χ(2)DFG .

Above, this threshold, the growth in the intensity I2of a weak input signal is governed by

I2 (z) = I2 (0)

[γ2γ

∗3 |A10|2κ2 sin2 (κz)+1

]. (5.49)

This type of three-wave mixing is often referred to asoptical parametric amplification. A weak input field, re-ferred to as the signal field (the field with the amplitudeA2 in our case), becomes amplified in this type of pro-cess through a nonlinear interaction with a powerfulpump field (the undepleted field with the amplitude A1in the case considered here). In such a scheme of opticalparametric amplification, the third field (the field withthe amplitude A3) is called the idler field.

We now consider the regime of optical paramet-ric amplification ω1 = ω2 +ω3 where the pump, signaland idler pulses are matched in their wave vectors andgroup velocities. Introducing the real amplitudes ρ jand phases ϕ j of the pump, signal, and idler fields,A j = ρ j exp

(iϕ j

), where j = 1, 2, 3, assuming that

γ2 = γ3 = γ in (5.35) and (5.36), A1(z, t) = A10(t) andA3(0, t) = 0, we write the solution for the amplitude ofthe signal field as [5.18]

A2 (η, z) = A20 (η) cosh [γρ10 (η) z] . (5.50)

The idler field then builds up in accordance with

A3 (η, z) = A∗20 (η) exp [iϕ10 (η)] sinh [γρ10 (η) z] .

(5.51)

As can be seen from (5.50), optical parametric ampli-fication preserves the phase of the signal pulse. Thisproperty of optical parametric amplification lies at theheart of the principle of optical parametric chirped-pulseamplification [5.19], allowing ultrashort laser pulses tobe amplified to relativistic intensities. It also suggestsa method of efficient frequency conversion of few-cyclefield waveforms without changing the phase offset be-tween their carrier frequency and temporal envelope,making few-cycle laser pulses a powerful tool for theinvestigation of ultrafast electron dynamics in atomicand molecular systems.

In the nonstationary regime of optical paramet-ric amplification, when the pump, signal, and idlerfields propagate with different group velocities, usefuland important qualitative insights into the phase rela-tions between the pump, signal, and idler pulses canbe gained from energy and momentum conservation,ω1 = ω2 +ω3 and k1 = k2 +k3. These equalities dictatethe following relations between the frequency deviationsδω j in the pump, signal, and idler fields ( j = 1, 2, 3):

δω1 = δω2 + δω3 (5.52)

and

δω1/u1 = δω2/u2 + δω3/u3 . (5.53)

In view of (5.52) and (5.53), we find

δω2 = q2δω1 (5.54)

and

δω3 = q3δω1, (5.55)

where q2 = (u−11 −u−1

3 )/(u−12 −u−1

3 ), q3 = 1−q2.In the case of a linearly chirped pump,

ϕ1(t) = α1t2/2, the phases of the signal and idlerpulses are given by ϕm(t) = αmt2/2, where αm = qmα1,m = 2, 3. With qm 1, the chirp of the signal and idlerpulses can thus considerably exceed the chirp of thepump field.

5.4 Third-Order Nonlinear Processes

Optical nonlinearity of the third order is a univer-sal property, found in any material regardless of itsspatial symmetry. This nonlinearity is the lowest-order nonvanishing nonlinearity for a broad class ofcentrosymmetric materials, where all the even-ordernonlinear susceptibilities are identically equal to zero

for symmetry reasons. Third-order nonlinear processesinclude a vast variety of four-wave mixing processes,which are extensively used for frequency conversion oflaser radiation and as powerful methods of nonlinearspectroscopy. Frequency-degenerate, Kerr-effect-typephenomena constitute another important class of third-

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5.4

Nonlinear Optics 5.4 Third-Order Nonlinear Processes 9

order nonlinear processes. Such effects lie at the heart ofoptical compressors, mode-locked femtosecond lasers,and numerous photonic devices, where one laser pulse isused to switch, modulate, or gate another laser pulse. Inthis section, we provide a brief overview of the mainthird-order nonlinear-optical phenomena and discusssome of their practical applications.

5.4.1 Self-Phase Modulation

The third-order nonlinearity gives rise to an intensity-dependent additive to the refractive index:

n = n0 +n2 I (t) , (5.56)

where n0 is the refractive index of the medium in theabsence of light field, n2 = (2π/n0)2χ(3)(ω; ω,ω,−ω)is the nonlinear refractive index, χ(3)(ω; ω,ω,−ω) isthe third-order nonlinear-optical susceptibility, referredto as the Kerr-type nonlinear susceptibility, and I(t)is the intensity of laser radiation. Then, the nonlinear(intensity-dependent) phase shift of a pulse at a distanceL is given by

Φ (t) = ω

cn2 I (t) L . (5.57)

Due to the time dependence of the radiation inten-sity within the light pulse, the nonlinear phase shift isalso time-dependent, giving rise to a generally time-dependent frequency deviation:

Δω (t) = ω

cn2L

∂I

∂t. (5.58)

The resulting spectral broadening of the pulse can beestimated in the following way:

Δω = ω

cn2L

I0

τ, (5.59)

where I0 is the peak intensity of the light pulse and τ isthe pulse duration.

The first-order dispersion-theory equation for theslowly varying envelope A(t, z) of a laser pulse prop-agating in a medium with a Kerr-type nonlinearity iswritten as [5.9]

∂A

∂z+ 1

u

∂A

∂t= iγ |A|2 A , (5.60)

where u is the group velocity of the laser pulse and

γ = 3πω

2n20c

χ(3) (ω; ω,−ω,ω) . (5.61)

is the nonlinear coefficient.

In the retarded frame of reference, z′ = z andη = t − z/u, the solution to (5.60) is written as

A (η, z) = A0 (η) exp[iγ |A0 (η)|2 z

], (5.62)

where A0(η) is the initial field envelope.Since the group-velocity dispersion was not included

in (5.60), the shape of the pulse envelope remains un-changed as the pulse propagates through the nonlinearmedium. The intensity-dependent change in the refrac-tive index gives rise to a nonlinear phase shift

ϕNL (η, z) = γSPM I0 (η) z , (5.63)

where γSPM = 2πn2/λ and I0(η) is the initial intensityenvelope.

The deviation of the instantaneous frequency of thepulse is given by

δω (η, z) = −∂ϕNL (η, z)

∂t= −γSPM

∂I0 (η)

∂ηz .

(5.64)

A quadratic approximation of the pulse envelope,

I0 (η) ≈ I0(0)

(1− η2

τ20

), (5.65)

where τ0 is the pulse width, which is valid around themaximum of the laser pulse, gives a linear chirp of the

–5 5

1.0

0.8

0.6

0.4

0.2

0.0–4 –3 –2 –1 0 1 2 3 4

Pump spectra (arb. units)

Ω (arb. units)

432

1

Fig. 5.5 Self-phase-modulation-induced spectral broaden-ing of a a sech-shaped laser pulse with an initial pulsewidth of 30 fs in a fused-silica optical fiber with n2 =3.2×10−16 cm2/W. Curve 1 presents the input spectrumof the pulse. The input pulse energy (2) 0.1 nJ, (3) 0.2 nJ,and (4) 0.3 nJ

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5.4


pulse

δω (η, z) ≈ −2γSPMI0 (0)

τ20

ηz . (5.66)

The spectrum of a self-phase-modulated pulse is givenby

S (ω) =∣∣∣∣∣∣

∞∫0

I (η) exp [iωη+ iϕNL (η)] dη

∣∣∣∣∣∣2

. (5.67)

Figure 5.5 illustrates SPM-induced spectral broadeningof a sech-shaped laser pulse with an initial pulse widthof 30 fs and different initial pulse energies in a fusedsilica optical fiber with n2 = 3.2 × 10−16 cm2/W.

Thus, self-phase modulation results in spectralbroadening of a light pulse propagating through a hol-low fiber. This effect allows compression of light pulsesthrough the compensation of the phase shift acquiredby the pulse in a hollow fiber. Compensation of a lin-ear chirp, corresponding to a linear time dependence ofthe instantaneous frequency, is straightforward from thetechnical point of view. Such a chirp arises around themaximum of a light pulse, where the temporal pulse en-velope can be approximated with a quadratic functionof time [see (5.65, 5.66)].

It is physically instructive to consider the compres-sion of chirped light pulses in the time domain. Since thefrequency of a chirped pulse changes from its leadingedge to its trailing edge, dispersion of our compressorshould be designed in such a way as to slower downthe leading edge of the pulse with respect to the trailingedge of the pulse. In other words, the group velocitiesfor the frequencies propagating with the leading edgeof the pulse should be lower than the group velocitiesfor the frequencies propagating with the trailing edgeof the pulse. This can be achieved by designing a dis-persive element with the required sign of dispersion andappropriate dispersion relation. Systems of diffractiongratings and, recently, multilayer chirped mirrors [5.20]are now widely used for the purposes of pulse compres-sion. In certain regimes of pulse propagation, self-phasemodulation and pulse compression may take place in thesame medium.

5.4.2 Temporal Solitons

The nonlinear phase shift acquired by a laser pulse prop-agating through a medium with a Kerr nonlinearity canbe balanced by group-velocity dispersion, giving riseto pulses propagating through the nonlinear dispersive

medium with an invariant or periodically varying shape:optical solitons.

Optical solitons is a special class of solutions to thenonlinear Schrödinger equation (NLSE)

i∂q

∂ξ+ 1

2

∂2q

∂τ2 +|q|2 q = 0 . (5.68)

The NLSE can describe the evolution of optical wavepackets including the dispersion β(ω) of optical wavesin a bulk material or in a waveguide structure throughthe power series expansion

β (ω) ≈ β (ω0)+ 1

u(ω−ω0)

+ 1

2β2 (ω−ω0)

2 + . . . , (5.69)

where ω0 is the central frequency of the wavepacket, u = (∂β/∂ω|ω=ω0 )−1 is the group velocity,and β2 = ∂2β/∂ω2|ω−ω0 . Thus, with the NLSE (5.68)projected on laser pulses propagating in a nonlinearmedium, q is understood as the normalized pulse en-velope, q = A/(P0)1/2, with ξ being the normalizedpropagation coordinate, ξ = z/Ld, Ld = τ2

0 /|β2| be-ing the dispersion length, P0 and τ0 defined as thepulse width and the pulse peak power, respectively, andτ = (t − z/u)/τ0.

The canonical form of the fundamental soliton solu-tion to (5.68) is [20]

q (ξ, τ) = sech (τ) exp

(iξ

2

). (5.70)

The radiation peak power required to support such a soli-ton is given by

P0 = |β2| /(γτ20 ) . (5.71)

Solitons retain their stable shape as long as theirspectrum lies away from the spectrum of dispersivewaves that can propagate in the medium. High-orderdispersion perturbs solitons, inducing Cherenkov-typewave-matching resonances between solitons and disper-sive waves [5.21, 22]. Under these conditions solitonstend to lose a part of their energy in the form of blue-shifted dispersive-wave emission. For low pump-fieldpowers, the generic wave-matching condition for suchsoliton–dispersive wave resonances is written [5.22]Ω = 1/2ε, where Ω is the frequency difference betweenthe soliton and the resonant dispersive wave and ε isthe parameter controlling the smallness of perturbationof the nonlinear Schrödinger equation, which can berepresented as ε = |β3/6β2| for photonic-crystal fibers

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5.4


(PCFs) with second-order dispersion β2 = ∂2β/∂ω2 andthird-order dispersion β3 = ∂3β/∂ω3. This dispersive-wave emission of solitons is an important part ofsupercontinuum generation in nonlinear optical fibers,including photonic-crystal fibers.

5.4.3 Cross-Phase Modulation

Cross-phase modulation (XPM) is a result of nonlinear-optical interaction of at least two physically distinguish-able light pulses (i. e., pulses with different frequencies,polarizations, mode structures, etc.) related to the phasemodulation of one of the pulses (a probe pulse) due tothe change in the refractive index of the medium inducedby another pulse (a pump pulse).

The cross-action of a pump pulse with a frequencyω1 on a probe pulse with a frequency ω2 gives rise toa phase shift of the probe pulse, which can be writtenas [5.23].

ΦXPM (η, z) = 3πω22

c2k2χ(3) (ωs; ωs, ωp,−ωp

)z∫

0

∣∣∣∣Ap

(η− ζ

σ, 0

)∣∣∣∣2 dζ , (5.72)

whereχ(3)(ωs; ωs, ωp,−ωp) is the third-order nonlinear-optical susceptibility of the medium; 1/σ = 11/u1 −1/u2; u1 and u2 are the group velocities of the pump andprobe pulses, respectively; and k2 is the wave number ofthe pump pulse. Taking the time derivative of the non-linear phase shift, we arrive at the following expressionfor the frequency deviation of the probe pulse

δωXPM (η, z) = −3πω22

c2k2χ(3) (

ωs; ωs, ωp,−ωp)

×σ

[∣∣Ap (η, 0)∣∣2 −

∣∣∣Ap

(η− z

σ, 0

)∣∣∣2] . (5.73)

Similarly to self-phase modulation, cross-phasemodulation can be employed for pulse compression.The dependence of the chirp of the probe pulse on thepump pulse intensity can be used to control the parame-ters of ultrashort pulses [5.24]. Cross-phase modulationalso opens the ways to study the dynamics of ultrafastnonlinear processes, including multiphoton ionization,and to characterize ultrashort light pulses through phasemeasurements on a short probe pulse [5.25].

5.4.4 Self-Focusing

Self-focusing is a spatial counterpart of self-phase mod-ulation. While SPM originates from the time-dependent

change in the refractive index induced by a laser pulsewith an intensity envelope I(t) varying in time, self-focusing is related to a nonlinear lens induced by a laserbeam with a spatially nonuniform intensity distribu-tion I(r). Given a transverse intensity profile I(r), thenonlinear additive to the refractive index is written as

n (r) = n0 +n2 I (r) . (5.74)

If the field intensity peaks at the center of the beam atr = 0, the nonlinear change in the refractive index alsoreaches its maximum at r = 0, yielding a focusing ordefocusing lens, depending on the sign of n2.

The stationary regime of self-focusing is governedby [5.9]

2ik∂A

∂z+Δ⊥ A = −2k2 Δn

n0A , (5.75)

where Δn = n2 I = n2|E|2, Δ⊥ is the transverse part ofthe Laplacian.

We consider a Gaussian beam and assume that thisbeam retains its profile as it propagates through thenonlinear medium,

A (r, z) = A0

f (z)exp

[− r2

2a20 f 2 (z)

+ iψ (z)

],

(5.76)

where a0 is the initial beam size, f (z) characterizesthe evolution of the beam size along the propagationcoordinate z [ f (0) = 1], and the function ψ(z) describesthe spatial phase modulation of the field.

In the paraxial approximation, r � a0 f (z), (5.75)and (5.76) give [5.18]

d2 f

dz2=

(L−2

df − L−2NL

)f 3 (z)

, (5.77)

where Ldf = 2πa20/λ and LNL = a0[2n0/(n2|A|2)]1/2

are the characteristic diffraction and nonlinear lengths,respectively.

Solving (5.77), we arrive at

f 2 (z) = 1+(

z

Ldf

)2 (1− P0

Pcr

), (5.78)

where P0 is the total power of the laser beam and

Pcr = cλ2

16π2n2(5.79)

is the critical power of self-focusing. The focal lengthof the nonlinear lens is given by

Lsf = Ldf(P0Pcr

−1)1/2 . (5.80)

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With P0 > Pcr, the nonlinear lens leads to a beam col-lapse. In reality, beam collapse can be arrested by thesaturation of optical nonlinearity occurring at high fieldintensities.

Beyond the paraxial approximation, the scenario ofself-focusing is much more complicated. The beam doesnot collapse as a whole, as the focal length of the non-linear lens for peripheral beams differs from the onefor paraxial beams. In the quasistationary regime, i. e.,when the pulse duration τ0 is much larger than the char-acteristic response time of optical nonlinearity τnl, thelength of self-focusing is a function of time, giving riseto moving foci [5.26]. In the nonstationary regime, i. e.,on time scales less than τnl, the leading edge of thepulse experiences no focusing, but induces a nonlinearlens that focuses the trailing edge of the pulse. As a re-sult, the beam becomes distorted, evolving to a hornlikepattern [5.27].

The equation of self-focusing (5.75) allows a wave-guide solution [5.28], which corresponds to the regimewhere the nonlinear lens exactly compensates for thediffraction of the laser beam. This solution is, however,unstable with respect to infinitely small fluctuations,which either give rise to a diffraction divergence orlead to a beam collapse. Such nonlinear waveguidescan be stabilized, as shown by Fibich and Gaeta [5.29],by reflections of light from guiding boundaries in opti-cal waveguides. To illustrate this regime of nonlinearbeam dynamics, we consider a cylindrical gas-filledhollow waveguide and define nondimensional cylin-drical coordinates r and z as r = R/R0 (R is thedimensional radial coordinate and R0 is the inner ra-dius of the hollow fiber) and z = Z/Ldf (Z is thedimensional longitudinal coordinate). The radial profilesQβ(r) of light intensity distribution in the waveguidesolutions ψ ∝ exp(iβz)Qβ(r) to the NLSE governingself-focusing (i. e., waveguides induced through the Kerrnonlinearity along the z-axis on a bounded domain witha circular symmetry) are described [28, 29] by so-lutions to the ordinary differential equation Δ⊥Qβ −βQβ + Q3

β = 0 [Δ⊥ = ∂2/∂r2 + (1/r)∂/∂r], subject tothe boundary conditions dQβ(0)/dr = 0, Qβ(1) = 0.Although this model neglects the fields outside the fibercore (e.g., radiation modes), it provides useful physi-cal insights into the spatial self-action in hollow PCFs.This differential equation has an infinite number ofsolutions Q(n)

β , n = 0, 1, 2, . . .. The Hamiltonians forall the guided modes Q(n)

β are positive, preventing theblowup of the profiles Q(n)

β in the presence of smallfluctuations, thus stabilizing the nonlinear waveguidesin a hollow fiber. The ground-state solution Q(0)

β is

a monotonically decreasing function of r, tending toa zeroth-order Bessel function in the case of low fieldamplitudes Q(0)

β ∝ εJ0(2.4r), where ε is a small param-eter controlled by the field intensity and the nonlinearityof the gas filling the fiber.

Although the modes Q(n)β are stable with respect

to small perturbations, these solutions are centers,rather than attractors, in a conservative system [5.30].However, mode solutions corresponding to Kerr-nonlinearity-induced waveguides may become attractorsin systems with dissipation, e.g., in hollow waveguides

� (3)

ω 1, k 1

ω 3, k 3

ω 4, k4

a)

ω 2, k 2

FWM

� (3)

3ω, kTH

b)

ω, kTHG

� (3)

ω, k 2ω, k 1

DFWM

d)

ω, k 3ω, k 4

ω 1, k 1

ω 2, k 2

ωCARS, kCARS

ω 3, k 3

� (3)CARS

Ω

ω 1

ω 2

ω 3

ω 4

c)

Fig. 5.6a–d Four-wave mixing: (a) general-type FWM,(b) third-harmonic generation, (c) coherent anti-Stokes Ra-man scattering, and (d) degenerate four-wave mixing

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5.4


with losses. The circularly symmetric field distributionis then formed at the output of the hollow wave-guide regardless of the initial beam profile [30]. Asdemonstrated by Moll et al. [5.31], collapsing lightbeams tend to form universal, Townesian profiles [5.28]while undergoing self-focusing on an infinite domainin bulk materials. Unlike Townesian beam profiles,which are known to be unstable in free space, theground-state waveguide modes observed in hollowphotonic-crystal fibers [5.32] remain stable with respectto small fluctuations, in agreement with the theory ofself-focusing on bounded domains [5.29, 30], resultingin no blowup until the critical power of beam collapseis reached.

5.4.5 Four-Wave Mixing

In general-type four-wave mixing (Fig. 5.6a), three laserfields with frequencies ω1, ω2, and ω3 generate thefourth field with a frequency ωFWM = ω1 ±ω2 ±ω3.In the case when all three laser fields have the samefrequency (for example, when all the three pump pho-tons are taken from the same laser field), ωFWM = 3ω

(Fig. 5.6b), and we deal with third-harmonic gener-ation (THG), which is considered in greater detailfor short-pulse interactions in Sect. 5.4.9. If the fre-quency difference ω1 −ω2 of two of the laser fieldsis tuned to a resonance with a Raman-active mode ofthe nonlinear medium (Fig. 5.6c), the FWM processωFWM = ω1 −ω2 +ω3 = ωCARS is refereed to as coher-ent anti-Stokes Raman scattering (CARS). An FWMprocess involving four fields of the same frequency(Fig. 5.6d) with ωDFWM = ω = ω−ω+ω correspondsto degenerate four-wave mixing (DFWM). The FWMfield in this nonlinear-optical process is the phase-conjugate of one of the laser fields, giving rise to anothername of this type of FWM – optical phase conjugation.

For the general-type FWM ωFWM = ω1 +ω2 +ω3,we represent the pump fields as

E j = A j exp[i(k j r−ω j t

)]+ c.c. , (5.81)

where j = 1, 2, 3 and k j = k′j + iα j are the complex

wave vectors of the pump fields.The FWM signal is written as

EFWM = AFWM exp [i (kFWMr−ωFWMt)]+ c.c.(5.82)

where kFWM = k′FWM + iαFWM is the complex wave

vector of the FWM fields.

The third-order nonlinear polarization responsiblefor the considered FWM process is

P(3)FWM = χ(3) (ωFWM; ω1, ω2, ω3)

...E (ω1) E (ω2) E (ω3) . (5.83)

With no depletion of the pump fields, the SVEA equa-tions give the following expression for the envelope ofthe i-th Cartesian component of the FWM field [5.9]:

[AFWM (z)]i = −2πω2FWM

kFWMc2χ(3) (ωFWM; ω1, ω2, ω3)

× A1 j A2k A3l exp

(iΔkz

2

)

×exp (−αFWMz)sin

(Δkz

2

)Δkz

2

z ,

(5.84)

where

Δk = k′1 + k′

2 + k′3 − k′

FWM

+ i (α1 +α2 +α3 −αFWM) (5.85)

is the z-component of the wave-vector mismatch.Phase matching, as can be seen from (5.84), is the key

requirement for high efficiency of frequency conversionin FWM. The fields involved in FWM can be phasematched by choosing appropriate angles between thewave vectors of the laser fields and the wave vector ofthe nonlinear signal or using the waveguide regime withthe phase mismatch related to the material dispersioncompensated by the waveguide dispersion component.

5.4.6 Optical Phase Conjugation

Optical phase conjugation is generally understood as thegeneration of an optical field with a time-reversed wavefront, or with a conjugate phase. This effect can be usedto correct aberrations in certain types of optical problemsand systems [5.33]. Suppose that a light beam with aninitially plane wave front propagates through an aberrat-ing medium, such as, for example a turbulent atmosphereor a material with inhomogeneities of the refractiveindex. The wave front of the light beam transmittedthrough such a medium is distorted. We now use an op-tical phase-conjugate process to generate the field witha wave front time-reversed with respect to the wave frontof the beam transmitted through the aberrating system.As the phase-conjugate beam now propagates throughthe aberrating medium in the backward direction, itswave front becomes restored.

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5.4


A phase-conjugate field can be produced through thedegenerate four-wave mixing of light fields

E j (r, t) = A j (r, t) exp[i(k jr −ωt

)]+ c.c. ,(5.86)

where j = 1, 2, 3, 4.The phase-conjugate geometry of DFWM is shown

in Fig. 6d. In this scheme, two strong counterpropagat-ing pump fields E1 and E2 with the same frequency ω

and wave vectors k1 and k2 = −k1 illuminate a mediumwith a cubic nonlinearity χ

(3)DFWM = χ(3)(ω; ω,−ω,ω).

The DFWM interaction of these two pump fields witha weak signal of the same frequency ω and an arbitrarywave vector k3 gives rise to a field with a frequency ω

that propagates in the opposite direction to the signalbeam and that is phase-conjugate of the signal field. Thephase-conjugate field generated through DFWM can beinstructively thought of as a result of scattering of theforward pump field off the grating induced by the back-ward pump and the signal field or as a result of scatteringof the backward pump from the grating induced by theforward pump and the signal field.

The nonlinear polarization responsible for phaseconjugation in DFWM is

P(3)DFWM = 6χ

(3)DFWM

...E1E2E∗3 . (5.87)

If the depletion of the pump fields is negligible, thenonlinear propagation equation (5.24) for the consid-ered DFWM process is reduced to the following twoequations for the amplitude of the signal field and itsphase-conjugate [5.1]

dA3

dz= iκ3 A3 + iκ4 A∗

4, (5.88)

dA4

dz= −iκ3 A4 − iκ4 A∗

3, (5.89)

where

κ3 = 12πω

cnχ

(3)DFWM

(|A1|2 +|A2|2

), (5.90)

κ4 = 12πω

cnχ

(3)DFWM A1 A2 . (5.91)

By introducing

A3 = B3 exp (iκ3z) , (5.92)

A4 = B4 exp (−iκ3z) , (5.93)

we reduce the set of equations (5.88) and (5.89) to

dB3

dz= iκ4 B∗

4 , (5.94)

dB4

dz= −iκ4 B∗

3 . (5.95)

The solution can now be written as

B∗3 (z) = − i |κ4|

κ4

sin (|κ4| z)

cos (|κ4| L)B4 (L)

+ cos [|κ4| (z − L)]

cos (|κ4| L)B∗

3 (0) , (5.96)

B4 (z) = cos (|κ4| z)

cos (|κ4| L)B4 (L)

− iκ4

|κ4|sin [|κ4| (z − L)]

cos (|κ4| L)B∗

3 (0) , (5.97)

where B3(0) and B4(L) are the boundary conditions forthe signal and DFWM fields. With B4(L) = 0, (5.97)gives

B4 (0) = iκ4

|κ4| tan (|κ4| L) B∗3 (0) . (5.98)

This expression visualizes the structure of the phase-conjugate field generated through DFWM. The intensityreflection coefficient of the DFWM medium serving asa phase-conjugate mirror is given by

RDFWM = tan2 (|κ4| L) . (5.99)

As can be seen from (5.99), the reflectivity ofthe DFWM-based phase-conjugate mirror can exceed100%. This becomes possible due to the energy suppliedby the strong pump fields.

5.4.7 Optical Bistability and Switching

In optical bistability or multistability, an optical systemhas two or more than two stable states, most often rep-resented by the intensity at the output of the system asa function of the input intensity. With the level of theoutput intensity determined by a certain operation ona light beam, a bi- or multistable system makes a deci-sion in which state it will operate, acting as a switch foroptical communications or optical data processing.

As an example of an optical bistable system, weconsider a Fabry–Perot cavity with a Kerr-nonlinearmedium inside. We assume that the cavity mirrors areidentical and have an amplitude reflectance r and trans-mittance t. Let A1, A2 and A3 be the amplitudes of theincident, reflected, and transmitted fields, respectively.The amplitudes of forward and backward waves insidethe cavity will be denoted as B1 and B2, respectively.The amplitudes of the fields inside and outside the cavityare related by the expressions [5.1]

B2 = rB1 exp (2ikL −αL) , (5.100)

B1 = tA1 +rB2 , (5.101)

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5.4


where k is the wave number and α is the intensityabsorption coefficient.

Solving (5.100) and (5.101), we arrive at the relation

B1 = tA1

1−r2 exp (2ikL −αL), (5.102)

which is known as the Airy equation. When k or α

is a strongly nonlinear function of the light intensity,(5.102) leads to a bistable behavior of the system in thetransmitted field.

Let us assume that the absorption is negligible andrewrite (5.102) as

B1 = tA1

1− R exp (iδ), (5.103)

where R = ρ2 exp(−iϕ) is the intensity reflectance ofthe cavity mirrors and the phase shift δ, correspondingto a full round trip around the cavity, is given by

δ = δ0 + δNL , (5.104)

where

δ0 = ϕ = 2n0ω

cL (5.105)

is the linear phase shift and

δNL = 2n2 Iω

cL (5.106)

is the nonlinear phase shift, with I = [cn/(2π)](|B1|2 +|B2|2) ≈ [cn/(π)]|B1|2

The ratio of the field intensity inside the cavity I2 =[cn/(2π)]|B1|2, to the intensity of the incident field I1 =[cn/(2π)]|A1|2 is now given by

I2

I1= F (I2) , (5.107)

where

F (I2) = 1

T + 4RT sin2

(δ2

) , (5.108)

where T is the intensity transmittance of the cavitymirrors, and

δ = δ0 + 4n2ωL

cI2 . (5.109)

In Fig. 5.7, we present the function F(I2) and the ratioI2/I1, i. e., the left- and right-hand sides of (5.107), plot-ted as functions of the intensity I2 for different input fieldintensities I1. The points where the straight lines repre-senting the left-hand side of (5.107) cross the plot ofthe Airy-type function F(I2) define the operation points

2

1

0

2

1

00.60.40.20.0

F I2/ I1

1

2

3

I2/ I0

Fig. 5.7 The function F(I2) and the ratio I2/I1, appearingon the right- and left-hand sides of (5.107), plotted as func-tions of the intensity I2 normalized to I0 = λ/(n2L) fordifferent input field intensities I1: (1) I1 = I ′, (2) I1 = I ′′,and (3) I1 = I ′′′, I ′′′ > I ′′ > I ′. The circles in the Airy-function curve show the range of input intensities wherethe operation of the Fabry–Perot cavity is unstable

of the nonlinear Fabry–Perot cavity. The circles in theAiry-function curve show the range of input intensitieswhere the operation is unstable. As the input intensity I1is increased, the system displays a bistable behavior andhysteresis. More hysteresis loops and a multistable be-havior are observed, as can be seen from Fig. 5.7, witha further increase in I1.

Control

Signal

Output 1

Output 2

n = n0 + n2l

n = n0

Φnl

Φnl = 0

M2

M1

BS1

BS2

Fig. 5.8 Mach–Zehnder interferometer with a nonlinearmedium in one of its arms, functioning as an all-opticalswitching device: BS1, BS2, beam splitters; M1, M2, mir-rors

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5.4


Optical switching can now be implemented [5.1] byincluding a nonlinear medium into one of the arms ofa Mach–Zehnder interferometer Fig. 5.8. The intensitiesat the output ports 1 and 2 are determined by the interfer-ence of light fields transmitted through the arms of theinterferometer. If there is only one input beam applied tothe system with symmetric beam splitters BS1 and BS2,output intensities display an oscillating behavior as func-tions of the nonlinear phase shift ΦNL acquired by thelight field in one of the arms of the interferometer. WhenΦNL = 0, the intensity at output port 1 is at its maximum,while the intensity at output port 2 is at its minimum.An opposite relation between the output intensities isachieved with ΦNL = π. The requirement ΦNL = π istypical of a broad class of all-optical switching devices.

5.4.8 Stimulated Raman Scattering

Vibrations or rotations of molecules, electronic motionsin atoms or collective excitations of matter can inter-act with light, shifting its frequency through an inelasticscattering process by the frequency Ω of Raman-activemotions (as shown in Fig. 5.6c). This phenomenon wasdiscovered by Raman and Krishnan [5.34] and almostsimultaneously by Mandelstam and Landsberg [5.35] in1928. In an intense laser field, pump laser photons andfrequency-shifted photons act coherently to resonantlydrive molecular motions, leading to the amplificationof the Raman-shifted signal. This effect is called stim-ulated Raman scattering (SRS). In the SRS regime,Raman-active modes of a material function as opticalmodulators, forcing the driving laser field to oscillate atnew frequencies. An intense laser field under these con-ditions not only gives rise to photons at new frequenciesthrough interaction with Raman-active modes, but alsoamplifies the light consisting of those photons.

In the continuous-wave regime, the interaction be-tween the pump field and the frequency-shifted (Stokes)signal is governed by the following set of coupled equa-tions for the intensities of the pump Ip and Stokes Isfields [5.23]:

dIs

dz= gR Ip Is −αs Is , (5.110)

dIp

dz= −ωp

ωsgR Ip Is −αp Ip . (5.111)

Here, αp and αs are the losses at the frequencies of thepump and Stokes fields, ωp and ωs are the frequencies ofthe pump and Stokes fields, and gR is the Raman gain,which is related to the imaginary part of the relevant

third-order susceptibility [5.9],

gR ∝ Im[χ(3) (ωs; ωp,−ωp, ωs

)]. (5.112)

Neglecting the action of the Stokes field on the pump,i. e., omitting the first term on the right-hand side of(5.111), we arrive at the following solution for the in-tensity of the Stokes field at the output of a Raman-activemedium with a length L:

Is (L) = Is (0) exp (gR I0Leff −αsL) , (5.113)

where I0 is the incident pump intensity at z = 0 and

Leff = 1

αp

[1− exp

(−αpL)]

. (5.114)

Thus, pump absorption restricts the nonlinear interactionlength to Leff.

In the absence of the Stokes field at the input ofa Raman-active medium, Is(0) = 0, the Stokes fieldbuilds up from spontaneous Raman scattering inside themedium. The power of the Stokes signal at the output ofa medium with a length L is then given by

Ps (L) = P0 exp [gR (ωs) I0Leff −αsL] , (5.115)

where

P0 = �ωs Beff

Beff = (2π)1/2 [|g2(ωs)|I0Leff]−1/2 ,

and

g2(ωs) = (∂2gR/∂ω2)|ω=ωs .

With an assumption of a Lorentzian Raman gainband, the critical pump power corresponding to thethreshold of SRS effect is given by an approximateformula

Pcr ≈ 16Seff

gRLeff, (5.116)

where Seff is the effective mode area of the pump field.In Sect. 5.6.3, we will consider in greater detail the

influence of SRS on the propagation of ultrashort laserpulses in a nonlinear medium. It will be shown, in par-ticular, that the Raman effect in fibers gives rise toa continuous frequency downshift of solitons propagat-ing in optical fibers.

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5.4


5.4.9 Third-Harmonic Generationby Ultrashort Laser Pulses

Third-harmonic generation (THG) is one of the basicnonlinear-optical processes, which has been intenselystudied and employed for numerous applications sincethe early days of nonlinear optics [5.7–9]. The semi-nal work by Miles and Harris [5.17] has demonstrateda tremendous potential of direct THG related to the cubicoptical nonlinearity χ(3) of gases for efficient frequencyconversion of laser radiation and for the diagnostics ofthe gas phase. Solid-state strategies of frequency con-version, on the other hand, mainly rely on the quadraticnonlinearity χ(2) of non-centrosymmetric crystals,with frequency tripling conventionally implementedthrough cascaded second-order nonlinear-optical pro-cesses, phase matched by crystal anisotropy [5.13, 14]or periodic poling of nonlinear materials [5.16].

Photonic-crystal fibers (PCFs) [5.36, 37] – opticalfibers of a new type where the solid continuous cladding

d)

a) b)

c)

d)

Fig. 5.9a–d SEM images of photonic-crystal fibers: (a) periodic- and (b) double-cladding fused-silica PCFs, (c) high-index-step birefringent PCF, and (d) hollow-core PCF

of standard fibers is replaced by a microstructure withan array of air holes running along the fiber paral-lel to its core Fig. 5.9 – have opened a new phasein nonlinear optics [5.38, 39]. Controlled dispersion ofguided modes [5.40] and large interaction lengths pro-vided by these fibers for light fields strongly confined ina small fiber core [5.41] result in a radical enhance-ment of nonlinear-optical frequency conversion andspectral transformation of laser radiation through self-and cross-phase modulation (SPM and XPM) [5.38],supercontinuum generation [5.42–44], four-wave mix-ing (FWM) [5.45], third-harmonic generation [5.42–52],modulation instabilities [5.53], and soliton frequencyshifting [5.54]. Third-order nonlinear-optical processesenhanced in PCFs now offer a useful alternative tofrequency-conversion schemes using χ(2) nonlinearcrystals.

Highly efficient THG has recently been observedin fused silica [5.46–50] and multicomponent glassPCFs [5.51], as well as in tapered fibers [5.52]. These

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5.4


experiments not only demonstrated the significance ofTHG for efficient, guided-wave frequency tripling offemtosecond laser pulses, but also revealed severalnew interesting nonlinear-optical phenomena. The third-harmonic signal has been shown to display asymmetricspectral broadening [5.51, 52] or even a substantialfrequency shift. We will demonstrate here that sucha behavior is a universal intrinsic feature of multi-mode guided-wave THG. Based on the arguments ofthe slowly varying envelope approximation (SVEA),we will show that the sign and the absolute value ofthe third-harmonic frequency shift, observed in manyPCF experiments, is controlled by the phase- and group-index mismatch for the interacting pair of pump andthird-harmonic modes. The possibility to tune the fre-quency of the main spectral peak in the spectrum of thethird harmonic by varying the group-velocity mismatchis a unique property of THG-type processes, which is nottypical of standard parametric FWM processes, wherethe first-order dispersion terms cancel out of the bal-ance of the field momenta. New regimes of THG willbe identified with no signal produced at the central fre-quency of the third harmonic 3ω0 and with the pumpenergy efficiently converted to spectrally isolated nar-rowband frequency components, which can be tunedwithin a spectral range of several tens of terahertz fromthe 3ω0 frequency.

We start with qualitative arguments illustrating phasematching for third-harmonic generation generalized toinclude the phase and group-velocity mismatch of thepump and third-harmonic fields, as well as the Kerr ef-fect, broadening the pump spectrum through SPM. Werepresent the wave numbers (or the propagation con-stants of guided modes in a waveguide regime) kp andkh at the frequencies of the pump field and the thirdharmonic as

kp (ω) ≈ k (ω0)+v−1p Ω/3+κSPM P , (5.117)

kh (3ω) ≈ k (3ω0)+v−1h Ω +2κXPM P , (5.118)

where ω0 is the central frequency of the pumpfield; vp,h = (∂k/∂ω)−1

ω0,3ω0are the group velocities

of the pump and its third harmonic; Ω = 3ω−3ω0;κSPM = ω0n2/cSeff and κXPM = 3ω0n2/cSeff are theSPM and XPM nonlinear coefficients (Seff is the ef-fective beam, or mode, area and n2 and n2 are thenonlinear refractive indices at ω0 and 3ω0, respectively);and P is the power of the pump field. In writing (5.117)and (5.118), we neglect group-velocity dispersion andhigher-order dispersion effects, as well as the SPM of thethird-harmonic field. With n2 ≈ n2, the phase mismatch

is then given by

Δk = kh −3kp ≈ Δk0 + ξΩ +3κSPM P , (5.119)

where Δk0 = k(3ω0)−3k(ω0) is the phase mismatchof the pump and third-harmonic wave numbers at thecentral frequencies of these fields and ξ = v−1

h −v−1p is

the group-velocity mismatch.As can be seen from (5.119), the group delay of the

pump and third-harmonic pulses is an important fac-tor in THG momentum conservation. In this respect, thebalance of momenta for THG radically differs from stan-dard phase-matching conditions for parametric FWMprocesses [5.23], where the first-order dispersion orderterms cancel out, reducing the FWM momentum balanceto group-velocity dispersion (GVD)-related issues.

The phase-matching condition (5.119) suggests thepossibility of substantially frequency shifting the max-imum in the spectrum of the third harmonic. However,the amplitude of an Ω-shifted spectral component inthe spectrum of the third harmonic and, hence, the ef-ficiency of Ω-shifted peak generation is determined bythe spectrum of the pump field. To specify this depen-dence, we proceed with an SVEA analysis of THG in thefield of SPM-broadened pump field by writing SVEAcoupled equations for the envelopes of the pump andthird-harmonic fields, A(t, z) and B(t, z):(

∂

∂t+ 1

νp

∂

∂z

)A = iγ1 A |A|2 , (5.120)(

∂

∂t+ 1

νh

∂

∂z

)B = iβ (A)3 exp (−iΔk0z)

+2iγ2 B |A|2 , (5.121)

where vp and vh are the group velocities of the pump andthird-harmonic pulses, respectively, and γ1, γ2 and β arethe nonlinear coefficients responsible for SPM, XPM,and THG, respectively; and Δk = kh −3kp is the phasemismatch (or the difference of propagation constants inthe guided-wave regime) in the absence of the nonlinearphase shifts of the pump and third-harmonic fields.

Solution of (5.120) and (5.121) yields [5.24, 55]

A(tp, z

) = A0(tp

)exp

[iϕSPM

(tp, z

)], (5.122)

B (th, z) = iσ

z∫0

dz′ A30

(th + ξz′)

×exp[−iΔβ0z′ +3iϕSPM

(th + ξz′, z′)

+ iϕXPM(th, z′, z′ )

], (5.123)

where tl = (t − z/vl) with l = p, h for the pump and thefield, respectively; A0(t) is the initial-condition envelope

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5.4


of the pump pulse;

ϕSPM(tp, z

) = γ1∣∣A0

(tp

)∣∣2 z (5.124)

is the SPM-induced phase shift of the pump field; and

ϕXPM(th, z′, z

) = 2γ2

z∫z′

∣∣A0(th + ξz′)∣∣2 dz′

(5.125)

is the XPM-induced phase shift of the third-harmonicfield.

In the regime where the nonlinear phase shifts, givenby (5.124) and (5.125), are small, the Fourier trans-form of (5.123) yields the following expression for thespectrum of the third-harmonic intensity:

I (Ω, z) ∝ β2 sin2[(Δk0 +Ωξ) z

2

](Δk0 +Ωξ)2

×

∣∣∣∣∞∫

−∞

∞∫−∞

A(Ω −Ω′) A

(Ω′ −Ω′′)

× A(Ω′′) dΩ′ dΩ′′

∣∣∣∣2

, (5.126)

where A(Ω) is the spectrum of the input pump field.Analysis of the regime of small nonlinear phase shifts isthus very instructive as it allows phase-matching effectsto be decoupled from the influence of the spectrum ofthe pump field. While the phase matching is representedby the argument in the exponential in the first factor onthe right-hand side of (5.126), the significance of thepump spectrum is clear from the convolution integralappearing in this expression. Depending on the signs ofthe phase and group-velocity mismatch, Δk0 and ξ , thepeak in the spectrum of the third harmonic can be eitherred- or blue-shifted with respect to the frequency 3ω0.The spectral width of this peak, as can be seen from(5.126), is given by δ ≈ 2π(|ξ|z)−1, decreasing as z−1

with the growth in the propagation coordinate z (seeinset in Fig. 5.10).

With low pump powers, the generalized phase-matching condition (5.119), as can be seen also fromthe exponential factor in (5.126), defines the centralfrequency 3ω0 +Ωmax, Ωmax = −Δk0/ξ , of the peakin the spectrum of the third harmonic. The ampli-tude of this peak, as shown by (5.126), is determinedby the amplitudes of the pump field components withfrequencies ω1 = ω0 +Ωmax −Ω′, ω2 = ω0 +Ω′ −Ω′′and ω3 = ω0 +Ω′′, which can add up to transfer the en-ergy to the 3ω0 +Ωmax component in the spectrum of the

1.0

0.5

0.0–4 –2 0 2 4

2

0.4

0.2

0.010–1

Third-harmonic intensity(arb. units)

Third-harmonic intensity(arb. units)

Ω τ

3

1

2

3

1 2

Ω τ

Fig. 5.10 Spectra of the third harmonic generated in theregime of weak self-phase modulation of the pump fieldwith Δk0τ/ξ = −2 (curve 1), −3 (2) and 0 (3). The insetillustrates narrowing of the main peak in the spectrum ofthe third harmonic with Δk0τ/ξ = −1 and ξL/τ = 2 (1), 4(2), and 10 (3)

third harmonic through the ω1 +ω2 +ω3 = 3ω0 +Ωmaxprocess. The spectrum of the pump field should there-fore be broad enough to provide a high amplitude of thispeak. In the regime when the SPM-induced broadeningof the pump spectrum is small, the tunability range of thethird harmonic (i. e., the range of frequency offsets Ω)is mainly limited by the bandwidth of the input pumpfield. This regime of THG is illustrated in Fig. 5.10.As the ratio Δk0τ/ξ changes from −2 (curve 1) to −3(curve 2), the peak in the spectrum of the third harmonicis bound to shift from Ωmax = 2/τ to 3/τ . A similarspectral shift limited by a field-unperturbed pump spec-trum has been earlier predicted by Akhmanov et al. forsecond-harmonic generation [5.56]. For large |Δk0/ξ|,however, the pump power density at the wings of thespectrum is too low to produce a noticeable peak withΩmax = −Δk0/ξ in the spectrum of the third harmonic(cf. curves 2 and 3 in Fig. 5.10).

In the general case of non-negligible nonlinearphase shifts, phase-matching effects cannot be decou-pled from the influence of the pump spectrum, and weresort to approximate integration methods to identifythe main features of the THG process. For a pump pulsewith a Gaussian envelope, A0(tp) = A exp[−t2

p/(2τ2)],where A and τ are the amplitude and the initial duration

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5.4


of the pump pulse, the spectrum of the third harmonic,in view of (5.123), is given by

B (Ω, z) = iβ A3

∞∫−∞

dth

z∫0

dz′

×exp

[−3

(th + ξz′)2

2τ2− iΔk0z′ +3iγ1

∣∣ A∣∣2

×exp

(−

(th + ξz′)2

τ2

)z′

]

×exp

⎡⎣2iγ2

∣∣ A∣∣2 z∫

z′dz′′

× exp

(−

(th + ζ z′′)2

τ2

)+ iΩth

](5.127)

Changing the order of integration in (5.127), we applythe saddle-point method to estimate the integral in dth :

B(Ω, z) ∝ iβ A3τ

z∫0

dz′

×exp{−i

(Δk0 +Ωξ −3γ1| A|2

)z′

+2iγ2| A|2 τ

ξΦ

[(z − z′) ξ

τ

]}, (5.128)

where Φ(x) =x∫0

exp(−x2)dx.

With ζ z/τ 1 and Φ[(z − z′)ζ ] constant, the phasematching is controlled by the factor

F (Ω, z) =sin2

[(Δk +Ωξ +3γ1

∣∣ A∣∣2) z

2

](Δk +Ωξ +3γ1

∣∣ A∣∣2)2 .

(5.129)

Thus, the saddle-point estimate of the SVEA integralfor the third-harmonic field recovers the generalizedphase-matching condition in the form of (5.119). Thefrequency offset Ωmax providing the maximum effi-ciency of THG is now determined by the dispersion ofthe material, its nonlinear properties, and the intensityof the pump field.

Spectral broadening of the pump field due to SPMradically expands the tunability range of the third har-monic (Fig. 5.11). With γ I0L = 1 (L is the interactionlength and I0 is the pump-field peak intensity), the spec-trum of the pump field is broad enough to producea high-amplitude peak at Ωmax = −6/τ . The peak be-

10–2

10–3

10–4

10–5

–5 0 5 10Ω τ

1

2

3

Third-harmonic intensity (arb. units)

6.0 x 10–4

4.0 x 10–4

2.0 x 10–4

00–10–20

th/τ

12

Fig. 5.11 Spectra of the third harmonic generated with noSPM of the pump (curve 1) and with an SPM-broadenedpump (curves 2, 3) with γ I0L = 1 (1) and 2 (2); Δk0τ/ξ =−6, ξL = 20. The insets show the temporal structure of thethird harmonic for γ I0L = 1 (1) and 2 (2) with ξL/τ = 20

comes narrower as the length of the nonlinear mediumincreases (cf. curves 2 and 3 in Fig. 5.11), suggestinga convenient way for an efficient generation of short-wavelength radiation with a well-controlled spectrumfor spectroscopic and metrological applications. In thetime domain, the third harmonic tends to break upinto two pulses, as shown in the inset to Fig. 5.11 forγ I0L = 1 (curve 1) and 2 (curve 2) with ξLτ = 20. Thefirst peak at thτ = −20 represents a third-harmonic pulsethat propagates with the pump pulse and that is phase-matched to the pump field in the sense of (5.119). Thesecond pulse propagates with the group velocity of thethird harmonic and is group-delayed in our case with re-spect to the pump field. Due to the phase matching, theratio of the amplitudes of the first and second peaks in-creases with the growth in γ I0L (cf. curves 1 and 2 inthe inset to Fig. 5.11).

Examples of frequency-shifted THG can be foundin the recent literature on nonlinear optics of PCFsand tapered fibers [5.48–52]. Interesting collinearand Cherenkov-type intermode phase-matching op-tions [5.50, 51] have been highlighted. The nature ofthe frequency shift has been identified in [5.57, 58].Generation of an asymmetrically broadened and spec-trally shifted third harmonic is the most frequentlyencountered situation, observed in nonlinear-optical ex-

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5.4


0.30 0.45

2

1

0

–1

0.35 0.40

Δk (μm–1); (fs/μm); Δk + Ω (μm–1)

λ (μm)

3

2

1

1 2

1300λ (nm)

1

0.1

0.01

Pump intensity(arb. units)

12001100100010–3

450λ (nm)

1.2

0.8

0.4

0.0

Third-harmonicintensity (arb. units)

400350

Fig. 5.12 The phase mismatch Δk0(1), group-velocity mis-match ξ (2), and the effective phase mismatch Δk0 + ξΩ

(3) for third-harmonic generation in an air-clad fused-silicafiber with a core radius of 0.9 μm. The pump wavelength is1.25 μm. The vertical dashed line shows the wavelength λ

= 383 nm, where Δk0 + ξΩ = 0. The insets show a typicalexperimental spectrum of blue visible emission (1) and thespectrum of the Cr:forsterite laser pump field (2) at the out-put of a photonic-crystal fiber with a length of 10 cm (after[5.57])

periments with PCFs [5.49–51] and tapered fibers [5.52].The two most striking recent experimental findingsinclude the observation of a 260 nm spectral compo-nent generated by Ti:sapphire laser pulses in high-deltaPCFs [5.49] and the generation of a spectrally isolatedfrequency component at 380 nm by femtosecond pulsesof 1.25 μm Cr:forsterite laser radiation [5.59]. In theformer case, the central wavelength of the ultraviolet(UV) component remained stable [5.49] as the cen-tral pump wavelength was tuned from 770 to 830 nm[which seems to agree well with the phase-matchingcondition of (5.119)]. In the latter case, generation ofthe blue-shifted third harmonic allowed a highly effi-cient frequency conversion of pump radiation over thefrequency interval exceeding 540 THz [5.59]. The spec-tral peak of visible blue emission was shifted from thefrequency 3ω0 in these experiments by 34 nm (a typical

spectrum is presented in the inset to Fig. 5.12), with nosignal produced at 3ω0.

To identify the main features of the dramatic fre-quency shift in the spectrum shown in the inset toFig. 5.12, we consider the dispersion of guided modesin an air-clad fused-silica thread as a generic exam-ple of PCF dispersion. In Fig. 5.12, we plot the phaseand group-velocity mismatch, Δk0 and ξ , as well as theeffective phase mismatch Δkeff = Δk0 + ξΩ for inter-modal THG in an air-clad fused silica fiber with a coreradius of 0.9 μm. We assume that the pump field is cou-pled into the fundamental mode of the fiber, while thethird harmonic is generated in one of high-order HE13-type modes. The effective phase mismatch Δkeff passesthrough zero at the wavelength of 383 nm, correspond-ing to a frequency shift of about 63 THz relative to3ω0, which agrees well with the typical spectrum ofblue emission from a PCF [5.59] presented in inset 1 toFig. 5.12. The main features and dominant tendencies ofthe third-harmonic spectrum are adequately describedby (5.119) and (5.127). Comparison of the pump (inset2 to Fig. 5.12) and third-harmonic spectra presented inFig. 5.11 also suggests that a blue shifting of the mainpeak in the spectrum of the third harmonic by more than60 THz becomes possible due to a strong broadening ofthe pump field, which is, of course, not only due to SPMin realistic conditions of PCF experiments.

We have shown that third-harmonic generation inthe field of spectrally broadened short pump pulsescan display interesting and practically significant newfeatures. A short-pulse pump field broadened due to self-phase modulation can generate its third harmonic withina broad spectral range. However, the phase-matchingcondition generalized to include the phase and group-velocity mismatch of the pump and third-harmonicfields, as well as the Kerr-nonlinearity-induced spectralbroadening of the pump field, tends to select a narrowspectral region of efficient THG. The possibility to tunethe frequency of the main spectral peak in the spec-trum of the third harmonic by varying the group-velocitymismatch is a unique property of THG-type processes,which is not typical of standard parametric FWM pro-cesses, where the first-order dispersion terms cancel outof the balance of the field momenta. For pump fieldswith large nonlinear frequency deviations, this spec-tral region may lie several tens of terahertz away fromthe central frequency of the third harmonic 3ω0. Thisnon−3ω THG process, leading to no signal at 3ω0, isshown to result in interesting and practically significantspectral-transformation phenomena in photonic-crystalfibers.

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5.5 Ultrashort Light Pulses in a Resonant Two-Level Medium:Self-Induced Transparency and the Pulse Area Theorem

5.5.1 Interaction of Lightwith Two-Level Media

The phenomenological approach based on nonlinear-optical susceptibilities does not include nonstationaryphenomena related to a dynamic modification of the non-linear medium by the laser field. Interaction of a resonantlaser field with a two-level system is a physically inter-esting and methodologically important regime wherethe equations governing the evolution of the laser fieldin a nonlinear medium can be solved self-consistentlywith the equations of motion for the quantum systeminteracting with the field. Such a self-consistent anal-ysis reveals the existence of a remarkable regime ofnonlinear-optical interactions. A resonance laser pulsewhose amplitude and pulse width are carefully matchedto the two-level system can propagate in a two-levelmedium with no absorption-induced attenuation of thepulse amplitude, a phenomenon known as self-inducedtransparency.

The interaction of laser radiation with a two-levelsystem is a classic problem of laser physics [5.60]. It hasbeen extensively studied with the use of various approx-imate analytical approaches and numerical proceduresover four decades, revealing important aspects of theinteraction of laser radiation with a two-level systemand, generally, laser–matter interactions. The standardapproach based on the slowly varying envelope approx-imation (SVEA) and rotating-wave approximation givesMaxwell–Bloch equations in the case of a two-levelatom [5.60, 61]. These equations provide an adequatedescription of laser radiation propagating in resonantmedia within a broad range of parameters and give thekey to understanding such fundamental resonant op-tical phenomena as the formation of 2π solitons andself-induced transparency [5.60–63].

As shown in the classic works by McCall andHahn [5.62, 63], light pulses with pulse areas multi-ple of π can propagate in a two-level medium with nochanges in their shape, while pulses with other pulseareas tend to change their areas during the propaga-tion in a two-level medium evolving to pulses withareas multiple of p. Much analytical and numericalwork has been done over the past decade to extendthis SVEA result to ultrashort pulses. Eberly [5.64] hasrederived the area theorem for the case of short lightpulses, modifying this theorem to include pulse chirp-ing and homogeneous damping. Ziolkowski et al. [5.65]

applied the finite-difference time-domain (FDTD) tech-nique [5.66] to solve the semiclassical Maxwell–Blochequations numerically. This approach revealed severalimportant features of short-pulse propagation in a two-level medium and allowed a more detailed analysisof self-induced transparency effects. Hughes [5.67, 68]has employed the FDTD approach to demonstrate thepossibility of generating sub-femtosecond transients ina two-level medium.

Tarasishin et al. [5.69] have applied FDTD tech-nique to integrate jointly the Maxwell and Schrödingerequations for an ultrashort light pulse propagating ina two-level medium. Below in this section, we explainthe FDTD-based algorithm [5.69] solving the Maxwelland Schrödinger equations to model the interactionof ultrashort laser pulses with an ensemble of two-level atoms. FDTD simulations presented below revealinteresting regimes of short-pulse propagation and am-plification in two-level media, including the evolution ofthe pulse to a 2π soliton, amplification of a single-cyclepulse in a medium with a spatially modulated distri-bution of dipole moments of resonant transitions, andamplification of chirped light pulses.

5.5.2 The Maxwell and SchrödingerEquations for a Two-Level Medium

We shall start with the extension of the standard FDTDprocedure to the case of short pulses propagating ina two-level medium, when the Maxwell equation forthe fields and the Schrödinger equation for the wavefunctions should be solved without any assumptions thatare usually employed in the SVEA approach. In the one-dimensional case, the FDTD algorithm involves step-by-step integration of two Maxwell curl equations

∂Dz(x, t)

∂t= ∂Hy(x, t)

∂x, (5.130)

∂Hy(x, t)

∂t= ∂Ez(x, t)

∂x. (5.131)

To perform this integration, we have to define the re-lation between the components of the electromagneticinduction and the electromagnetic field. This can be donethrough the equation for the polarization of the medium.In our case of a two-level medium, this involves the solu-tion of the Schrödinger equation for the wave functionsof the energy levels.

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5.5

Nonlinear Optics 5.5 Ultrashort Light Pulses: Self-Induced Transparency and the Pulse Area Theorem 23

We will consider an ensemble of noninteracting two-level atoms or molecules whose wave functions can berepresented as superpositions of two basis states 1 and 2:

ψ(t) = a(x, t)ψ1 +b(x, t)ψ2, (5.132)

where ψ1 and ψ2 are the eigenfunctions of an unper-turbed system corresponding to the states with energiesE1 and E2 (we assume for definiteness that E1 > E2),respectively, and a(x, t) and b(x, t) are complex coef-ficients. Then, the Schrödinger equation for the wavefunction yields the following set of differential equa-tions:

i�da(x, t)

dt= E1a(x, t)−μEz(x, t)b(x, t), (5.133)

i�db(x, t)

dt= E2b(x, t)−μEz(x, t)a(x, t), (5.134)

where μ is the dipole moment of transition betweenthe levels 1 and 2. Following Feynman et al. [5.68], weintroduce real combinations of the complex quantitiesa(x, t) and b(x, t):

r1(x, t) = a(x, t)b∗(x, t)+a∗(x, t)b(x, t) , (5.135)

r2(x, t) = i[a(x, t)b∗(x, t)−a∗(x, t)b(x, t)] , (5.136)

r3(x, t) = a(x, t)a∗(x, t)−b(x, t)b∗(x, t) , (5.137)

r4(x, t) = a(x, t)a∗(x, t)+b(x, t)b∗(x, t) . (5.138)

The physical content of the parameters defined by(5.135–5.138) is well known from classic textbooks oncoherent optics [5.60,61]. It can be easily verified usingthe set (5.133) and (5.134) that r4(x, t) is independent oftime and can be interpreted as the probability to find thesystem in either state 1 or state 2. The quantities r1(x, t)and r3(x, t) play an especially important role. Dependingon the sign of r3(x, t), resonant electromagnetic radia-tion can be either amplified or absorbed by a two-levelsystem. The quantity r1(x, t) controls the polarization ofthe medium in the case of a linearly polarized light field:

Pz(x, t) = 4πμr1(x, t)N , (5.139)

where N is the volume density of two-level atoms ormolecules. Thus, the quantity r1(x, t) defines the soughtrelation between the components of the electromagneticinduction and the electromagnetic field.

Using (5.133) and (5.134), we arrive at the followingset of equations for the quantities defined by (5.135–5.137):

dr1(x, t)

dt= −ω0r2(x, t) , (5.140)

dr2(x, t)

dt= ω0r1(x, t)+2 (μ/�) Ez(x, t)r3(x, t) ,

(5.141)

dr3(x, t)

dt= −2 (μ/�) Ez(x, t)r2(x, t) , (5.142)

where ω0 = (E1 − E2)/�. Differentiating (5.141), us-ing (5.140) for dr1(x, t)/dt, and taking into con-sideration that r1(x, t) = Pz(x, t)/(μN) = [Dz(x, t)−Ez(x, t)]/(4πμN), we find that

d2 Dz(x, t)

dt2− d2 Ez(x, t)

dt2+ω2

0 [Dz(x, t)− Ez(x, t)]

+24πμ2ω0 N

�Ez(x, t)r3(x, t) = 0 , (5.143)

dr3(x, t)

dt= 2

4π�ω0 NEz(x, t)

×

(dDz(x, t)

dt− dEz(x, t)

dt

). (5.144)

The FDTD approach involves difference approxima-tion [5.66] of time and spatial derivatives involved in(5.130) and (5.131):

Dn+1z,i = Dn

z,i +Δt

Δx(Hn+1/2

y,i+1/2 − Hn+1/2y,i−1/2) , (5.145)

Hn+3/2y,i+1/2 = Hn+1/2

y,i+1/2 + Δt

Δx(En+1

z,i+1 − En+1z,i ) , (5.146)

where i and n indicate the values of discrete spatialand temporal variables, respectively, x = iΔx, t = nΔt,Δx and Δt are the steps of discretization in spatial andtemporal variables. This approach yields the followingset of difference equations:

En+1z,i = Dn+1

z,i + Dn−1z,i − En−1

z,i

+(4En

z,i −4Dnz,i +4Δt2 4πμ2ω0 N

�En−1

z,i rn−13,i

)2+Δt2ω2

0

,

(5.147)

rn+13,i = rn−1

3,i +0.52

4π�ω0 N

(En+1

z,i + Enz,i

)×

(Dn+1

z,i − Dn−1z,i − En+1

z,i + En−1z,i

).

(5.148)

Thus, we arrive at the following closed algorithm ofnumerical solution:

1. substitute Dn+1z,i determined from (5.145) for the cur-

rent value of the discrete-time variable into the set

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5.5


of (5.147) and (5.148) and determine the values ofrn+13,i and En+1

z,i ,2. substitute the quantities thus determined into (5.146)

and determine the values of the magnetic fieldHn+3/2

y,i ,3. substitute these values of the magnetic field into

(5.145) and repeat the procedure for the next valueof the discrete-time variable.

5.5.3 Pulse Area Theorem

To test the FDTD-based procedure of simulations de-scribed in Sect. 5.5.2, we model the propagation of lightpulses

E(x, t) = A(x, t)eiφ+ikz−iωt + c.c. , (5.149)

where A(x, t) is the pulse envelope, in a two-levelmedium and compare the results of these simulationswith the predictions of the McCall–Hahn area the-orem [5.62, 63]. This plan can be accomplished bykeeping track of the pulse area

θ(x) =∞∫

−∞Ω(x, τ)dτ , (5.150)

where Ω(x, t) = 2μ�

A(x, t) is the real Rabi frequency.The SVEA equation for the pulse envelope A(x, t)

can be represented as

∂A(x, t)

∂x+ n

c

∂A(x, t)

∂t= i

2πω

ncP(x, t) , (5.151)

where n is the refractive index of the medium, P(x, t) isthe slowly varying amplitude of the polarization inducedin the medium. As shown by McCall and Hahn [5.62,63], on exact resonance, P(x, t) = iμN sin[θ(x, t)], with

θ(x, t) =t∫

−∞Ω(x, τ)dτ . Hence, (5.151) yields

∂A(x, t)

∂x+ n

c

∂A(x, t)

∂t= −2πωμN

ncsin[θ(x, t)] .

(5.152)

The celebrated solution to (5.152) is a pulse witha hyperbolic-secant shape, which propagates in a reso-nant two-level medium with no changes in its envelope:

A(x, t) = �

μτsech

(t − x/V

τ

), (5.153)

where V is the pulse velocity in the medium,

V =(

4πμ2ωNτ2

�nc+ n

c

)−1

. (5.154)

In the regime of weak absorption (when the pulsearea only slightly deviates from its stable values), theevolution of the pulse area is governed by the equation

∂θ(x)

∂x= −2πωμN

ncE0θ(x) sin θ(x), (5.155)

where E0 is the initial pulse amplitude. With θ ≈ 2π,we arrive at

∂θ(x)

∂x= −α

2sin θ(x), (5.156)

where α = 8π2ωμNncE0

.According to (5.156), pulses whose areas θ(x0) are

multiples of π propagate in a two-level medium withno changes in their envelopes (the soliton propagationregime). However, solitons with pulse areas equal to π,3π, 5π, . . . are unstable. Thus, a pulse with an arbitraryinitial area changes its waveform propagating in a two-level medium until its pulse area becomes multiple of2π. The characteristic spatial scale of this process isestimated as α−1.

Numerical simulations [5.69] have been performedfor hyperbolic-secant pulses:

E(x0, t) = E02 cos [ω(t − t0)]

exp[−(t − t0)/T ]+ exp[(t − t0)/T ] .(5.157)

The parameters of the medium and the pulse were cho-sen as follows: Ω = 2μE0/2� = 0.0565ω, 4πμNr3(0)= −0.12E0, and ω = ω0 (exact resonance).

In accordance with the area theorem, a pulse witha duration T = 5.631 2π/ω should propagate in the two-level medium with these specified parameters valueswithout any changes in its envelope. FDTD simula-tions [5.69] reveal no changes in the waveform and theamplitude of such a pulse within a distance of 100 λ

with an accuracy better than 0.1%.Now, we examine how an arbitrary light pulse

evolves to a 2π pulse in a two-level medium. FDTD sim-ulations were performed for the propagation of a pulse(5.157) with a duration T = 7.0392π/ω in a two-levelmedium with these specified parameters values. Whilethe amplitude of such a pulse increases by 15% as itpropagates through the medium, the duration of thepulse decreases by a factor of about two. The pulseareas FDTD-simulated for these values of the propaga-tion coordinate are equal to 2.5π, 2.33π, 2.24π, and2.1π, while the pulse areas for the same values of x cal-culated from (5.156) are equal to 2.5π, 2.35π, 2.22π,

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5.5


and 2.09π, respectively. Thus, the results of FDTD sim-ulations agree very well with the predictions of thearea theorem, indicating the adequacy of the numericalapproach.

5.5.4 Amplificationof Ultrashort Light Pulsesin a Two-Level Medium

Since the numerical algorithm based on the FDTD tech-nique is intended to simulate the evolution of very shortpulses, it enables one to explore many important aspectsof short-pulse amplification in a two-level medium, pro-viding a deeper understanding of the problems arising inthe amplification of short pulses and the ways that canbe employed to resolve these problems.

Propagation of π pulses seems to provide optimalconditions for amplification in a two-level system, sincesuch pulses transfer atoms (or molecules) in an ini-tially inverted medium to the lower state. However, theRabi frequency increases as a light pulse propagatesthrough the medium and its amplitude increases due tothe amplification. Because of this change in the Rabifrequency, the pulse now cannot transfer inverted atomsor molecules to the lower state, but leaves some excita-tion in a medium, which reduces the gain and leads topulse lengthening due to the amplification of the pulsetrailing edge in a medium with residual population inver-sion. Thus, some precautions have to be taken to keepthe area of the pulse constant in the process of pulseamplification in the medium. Below, we use the FDTDtechnique to explore two possible ways to solve thisproblem: (i) modulation of the spatial distribution of thedipole moments of resonant transitions in a two-levelsystem and (ii) amplification of a frequency-detunedchirped pulse.

Modulation of the Spatial Distributionof the Dipole Moment

First, let us explain why we expect that the modula-tion of the spatial distribution of the dipole momentmay improve the efficiency of amplification of an ultra-short light pulse in a two-level medium. To do this,we multiply (5.152) by A(x, t) and integrate the re-sulting expression in time. Then, introducing the pulseenergy, Φ(x) = ∫ ∞

−∞ A2(x, t)dt, and taking into con-sideration that the pulse energy remains constant ifθ(x0) = 0, 2π, . . . , we arrive at the following equationrelating the pulse energy and area:

∂Φ(x)

∂x= 2πωN�

nc[cos θ(x)−1] . (5.158)

Thus, if the pulse area is kept equal to π, as the pulsepropagates through an inverted two-level medium, thenthe energy of such a pulse would grow linearly asa function of the distance:

Φ(x) = (1+βx)Φ(0) , (5.159)

where β = 4πωN�Φ(0)nc and Φ(0) is the initial energy of the

pulse.By looking at (5.150) and (5.159) and taking

into consideration that Φ(0) = 2τ E20 = 2E0�/μ(0) for

a hyperbolic-secant pulse (5.157), we find that the pulsearea can be kept equal to π by modulating the spatialdistribution of the dipole moment in accordance with

μ(x) = μ0√1+βx

. (5.160)

Such a modulation of the spatial distribution of dipolemoments can be achieved, for example, by preliminarilyorienting molecules in the medium.

In accordance with the area theorem, the evolutionof the amplitude of a π pulse propagating in an invertedtwo-level medium with a spatial profile of the dipolemoment described by (5.160) is governed by

E(x) = E0√

1+βx . (5.161)

This growth of the pulse amplitude is exactly compen-sated by the decrease in the dipole moment of transitionsin the two-level medium. The net effect is that the pulsearea remains constant and equal to π.

Figures 5.13a–d present the results of FDTD simula-tions for the amplification of a light pulse with a durationT = 2π/ω corresponding to a single optical cycle in aninverted two-level medium with a uniform distribution ofthe dipole moment (Figs. 5.13a,b) and in an inverted two-level medium where the spatial distribution of dipolemoments of resonant transitions is modulated in accor-dance with (5.160) (Figs. 5.13c,d). Simulations wereperformed for the case when Ω = 2μ0 E0/2�= 0.159ω,4πμNr3(0) = 0.12E0, and ω = ω0. The time in theseplots is measured from the moment when the center ofthe pulse passes through the entrance boundary of themedium.

Figures 5.13a,b display the evolution of a π pulse inan inverted uniform two-level medium and the popula-tion difference in this medium at the distances 0, 3β−1

and 6β−1. These plots show that a two-level mediumwith a uniform distribution of the dipole moment oftransitions cannot ensure an efficient amplification ofa transform-limited resonant π pulse. Figures 5.13c,dpresent the results of FDTD simulation for a π pulse

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5.5


propagating through a medium with a spatial evolutionof the dipole moment described by (5.160). These simu-lations illustrate the possibility of efficient amplificationof a single-cycle pulses in such a medium. Note that thepulse amplitude in Figs. 5.13c and 5.13d increases lin-

3

2

1

0

–1

–2

–3–4–8 0 4 8

1.0

0.5

0.0

–0.5

–1.0–4–8 0 4 8

3

2

1

0

–1

–2

–3–4–8 0 4 8

1.0

0.5

0.0

–0.5

–1.0–4–8 0 4 8

4

3

2

1

0

–1

–2

–3

–4–8 –1–6 –4 –2 0 2 4 6 8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4

3

2

1

0

–1

–2

–3

–4–7 –5 –3 –1 1 3 5 7

E (arb. units)

t (2π/ω)

r3

t (2π/ω)

E (arb. units)

t (2π/ω) t (2π/ω)

a) b)

c) d)

E (arb. units)

t (2π/ω)

E (arb. units)

t (2π/ω)

e) f)

3

6

3

6

3

6

3

6

X (� –1)X (� –1)

X (� –1)X (� –1) r3

early with the propagation coordinate x (the deviationfrom the linear dependence does not exceed 10−4). Thisresult agrees very well with the predictions of the areatheorem and can also be considered as another test ofthe reliability of the developed algorithm.

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5.5


Frequency-Detuned Chirped PulsesThe second idea of keeping the area of the amplifiedpulse unchanged is to use chirped pulses whose fre-quency is detuned from the exact resonance. Such a pulseexperiences compression as it propagates in the consid-ered two-level medium. With an appropriate choice ofpulse parameters, this decrease in pulse duration mayserve to preserve the area of the light pulse undergoingamplification.

The Rabi frequency in this case is written

Ω = 1

2

√Δ2 +

(2μ0 E0

�

)2

, (5.162)

where Δ = ω−ω0 is the detuning of the central fre-quency of the pulse from the resonance.

Figures 5.13e,f present the results of FDTD simu-lations for a single-cycle pulse with a quadratic initialchirp

E(0, t) = E02 cos

(ωt + kt2/T 2

)exp (−t/T )+ exp (−t/T )

, (5.163)

with k = 1.4, propagating in a uniform two-levelmedium with the same parameters as in Figs. 5.13a,b.The distance covered by the pulse in the medium is 6β−1,and the frequency detuning is Δ = ω−ω0 = −0.02ω0.As can be seen from Figs. 5.13e,f, although higher-orderdispersion effects noticeably distort the waveform of thepulse, the peak intensity of an initially chirped pulse atthe output of an amplifying uniform medium is nearlytwice as high as the output energy of a transform-limitedpulse. Furthermore, comparison of Figs. 5.13a,b,e,fshows that, chirping a light pulse, one can avoid un-desirable pulse-lengthening effects.

Fig. 5.13 (a)–(d) The evolution of (a), (c) the pulse wave-form and (b), (d) the population difference for a single-cyclelight pulse propagating in an inverted two-level mediumwith (a), (b) a uniform distribution of the dipole momentand (c), (d) a spatial distribution of dipole moments of res-onant transitions modulated in accordance with (5.160)simulated with the use of the FDTD technique [5.69]for Ω = 2μ0 E0/2� = 0.159ω, 4πμNr3(0) = 0.12E0, andω = ω0. The time in these plots is measured from the mo-ment when the center of the pulse passes through the inputboundary of the medium. (e), (f) Amplification of a single-cycle chirped laser pulse (5.163) in a two-level medium:(e) the input pulse and (f) the output pulse simulated with theuse of the FDTD technique for Ω = 2μ0 E0/2� = 0.159ω,4πμNr3(0) = 0.12× 100 , and ω = ω0�

5.5.5 Few-Cycle Light Pulsesin a Two-Level Medium

The results of FDTD simulations, as shown by Tara-sishin et al. [5.70], typically agree very well with thepredictions of McCall and Hahn theory [5.62, 63] forlight pulses propagating in a two-level medium until thepulse duration T becomes less than the duration T0 ofa single optical cycle. In agreement with the general pre-dictions of McCall and Hahn, a 2.9π pulse with T = T0,2μE0πT/� = 2.9π, μN/E0 = 0.00116, for example, istransformed until its area becomes equal to 2π. The peakamplitude of the pulse increases by a factor of 1.31 underthese conditions, while its duration decreases to 0.55T0.

Noticeable deviations from the McCall–Hahnregime were observed for pulses with durations shorterthan the duration of a single field cycle. In particular,half-cycle 2π pulses become asymmetric as they prop-agate through a two-level medium (Fig. 5.14a), and thecharacteristic length corresponding to the phase shift(5.9) equal to π estimated on the basis of FDTD sim-ulations for such pulses was equal to 9.4 λ, whichappreciably differs from the SVEA estimate for the char-acteristic length corresponding to the π phase shift,L = (�nc)/(4ω2μ2T 2 N) = (E2

0nc)/(4ω2 N�). Quarter-cycle 2π pulses display noticeable distortions andlengthening in the process of propagation through a two-level medium (Fig. 5.14b).

Deviations observed in the behavior of very short2π pulses from the McCall–Hahn scenario are due tothe fact that, although, formally, such pulses have anarea of 2π, the cycle of interaction between light anda two-level system remains incomplete in this case, as thepulses do not even contain a full cycle of the field (Figs.5.14a,b). As a result, such pulses leave some excitationin a two-level medium (Fig. 5.14d) instead of switchingexcited-state population back to the ground state, as inthe case of longer 2π pulses (Fig. 5.14c).

The amplitude of the leading edge of the pulse be-comes higher than the amplitude of its trailing edge,and the pulse waveform becomes noticeably asymmetric(Fig. 5.14a). The group velocity of such very short pulsesincreases due to this incompleteness of the light-two-level-system interaction cycle, leading to a discrepancybetween the SVEA estimate and the FDTD result for thecharacteristic length corresponding to the π phase shift.The residual population in the medium and the asymme-try of the pulse waveform increase with pulse shortening.

The results of FDTD simulations presented in thissection thus show that the general predictions of Mc-Call and Hahn for the evolution of the amplitude and the

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5.5


1.0

0.5

0.0

–0.5

–1.0–5 0 5

1.0

0.8

0.6

0.4

0.2

0.0–5 0 5

1.0

0.5

0.0

–0.5

–1.0–5 0 5

1.0

0.8

0.6

0.4

0.2

0.0–5 0 5

E (arb. units)

t (2π/ω)

E (arb. units)

t (2π/ω)

Population

t (2π/ω)

Population

t (2π/ω)

a) b)

c) d)

1

2

0

1

1

2

2

0

1

2r1r2

r1r2

X (L)X (L)

X (L) X (L)

Fig. 5.14 (a) Evolution of a half-cycle 2π pulse in a two-level medium: T = 0.5T0, 2μE0πT/�= π, and μN/E0 = 0.0016,(b) Evolution of a quarter-cycle 2π pulse in a two-level medium: T = 0.25T0, 2μE0πT/� = 2π, and μN/E0 = 0.0032.(c), (d) Evolution of the excited- and ground-state populations in a two-level medium under the action of (c) a half-cycle 2π

pulse and (d) a quarter-cycle 2π pulse: (dashed line) ground-state population r1 and (solid line) excited-state population r2.Simulations were performed with the use of the FDTD technique [5.70]

phase of short pulses in a two-level medium generallyagree reasonably well with the results of numerical sim-ulations until the pulse duration becomes less than theduration of a single optical cycle. Numerical analysisreveals several interesting physical features in the for-mation of 2π solitons produced as a result of splitting ofsingle-cycle pulses propagating in a two-level medium.In particular, the resulting pulses may have differentamplitudes, durations, and group velocities, allowingthe formation of sub-femtosecond pulses and slowingdown of the light in two-level media. Noticeable devi-ations from the McCall–Hahn regime can be observed

for pulses with durations shorter than the duration ofa single field cycle. Half-cycle 2π pulses become asym-metric as they propagate through a two-level medium,while quarter-cycle 2π pulse display considerable dis-tortions and lengthening in the process of propagationthrough a two-level medium. Deviations observed in thebehavior of very short 2π pulses from the McCall–Hahnscenario are due to the fact that the cycle of interactionbetween light and a two-level system remains incom-plete in this case, and light pulses leave some excitationin a two-level medium instead of switching excited-statepopulation back to the ground state.

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5.5

Nonlinear Optics 5.6 Let There be White Light: Supercontinuum Generation 29

5.6 Let There be White Light: Supercontinuum Generation

Supercontinuum (SC) generation – a physical phe-nomenon leading to a dramatic spectral broadening oflaser pulses propagating through a nonlinear medium– was first demonstrated in the early 1970s [5.71, 72](see [5.73] for an overview of early experiments on su-percontinuum generation). Presently, more than threedecades after its discovery, supercontinuum generationis still one of the most exciting topics in laser physicsand nonlinear optics [5.44], the area where high-field

a)

b)

c)

Fig. 5.15a–c Spectral transformations of ultrashort pulsesin photonic-crystal fibers: (a) supercontinuum generation,(b) third-harmonic generation, and (c) frequency shifting

science meets the physics of low-energy unamplified ul-trashort pulses in the most amazing way. The advent ofphotonic-crystal fibers [5.36, 37], capable of generatingsupercontinuum emission with unamplified, nano- andeven sub-nanojoule femtosecond pulses, has resulted inrevolutionary changes in frequency metrology [5.74–77]opened new horizons in ultrafast science [5.78, 79] andallowed the creation of novel wavelength-tunable andbroadband fiber-optic sources for spectroscopic [5.80]and biomedical [5.81] applications. The rainbow of col-ors produced by a laser beam Fig. 5.15 has become anoptical instrument and a practical tool.

As a physical phenomenon, supercontinuum gen-eration involves the whole catalog of classicalnonlinear-optical effects, such as self- and cross-phasemodulation, four-wave mixing, stimulated Raman scat-tering, solitonic phenomena and many others, which addup to produce emission with an extremely broad spectra,sometimes spanning over a couple of octaves. Below,we discuss the basic physical processes contributingto supercontinuum generation in greater detail, withspecial emphasis made on self-phase modulation, four-wave mixing, and modulation instabilities (Sect. 5.6.1),cross-phase modulation (Sect. 5.6.2), as well as thesolitonic phenomena and stimulated Raman scattering(Sect. 5.6.3).

5.6.1 Self-Phase Modulation,Four-Wave Mixing,and Modulation Instabilitiesin Supercontinuum-GeneratingPhotonic-Crystal Fibers

Propagation of laser pulses in PCFs is always ac-companied by SPM-induced spectral broadening. Thebasic features of SPM are discussed in Sect. 5.4.1.For very short laser pulses and broadband field wave-forms, SPM can be thought of as a four-wave mixingωp1 +ωp2 = ω3 +ω4 with frequency components ωp1and ωp2 from the spectrum of the laser field serving aspump photons generating new frequency componentsω3 and ω4. In the case of a frequency-degenerate pump,ωp1 = ωp2 = ωp, the new frequency components gener-ated through FWM appear as Stokes and anti-Stokessidebands at frequencies ωs and ωa in the spectrumof the output field. Such FWM processes become es-pecially efficient, as emphasized in Sect. 5.4.4, whenphase matching is achieved for the fields involved inthe nonlinear-optical interaction. Under certain condi-

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tions, the pump fields modify the effective refractiveindices of the guided modes involved in the FWM pro-cess, inducing the phase matching for the FWM andleading to a rapid growth of spectral sidebands repre-senting the Stokes and anti-Stokes fields. This regimeof four-wave mixing, referred to as modulation in-stability (MI), plays an especially important role innonlinear-optical spectral transformations of ultrashortpulses in PCFs.

In the simplest, scalar regime of modulation instabil-ity, two photons of the pump field with the frequency ωpgenerate Stokes and anti-Stokes sidebands with frequen-cies ωs = ωp −Ω and ωa = ωp +Ω. To illustrate thisregime of MI, we represent the propagation constantsof the Stokes and anti-Stokes sidebands as Taylor-seriesexpansions around ωp,

β(ωp +Ω

) ≈ β0(ωp

)+ 1

u pΩ

+ 1

2β2

(ωp

)Ω2 +2γ P , (5.164)

β(ωp −Ω

) ≈ β0(ωp

)− 1

u pΩ

+ 1

2β2

(ωp

)Ω2 +2γ P , (5.165)

where P is the peak power of the pump field,β0(ωp) is the Kerr-effect-free propagation constantof the pump field mode (i. e., the propagation con-stant of the pump field corresponding to the regimewith P = 0), u p = (∂β/∂ω|ω=ωp )−1 is the group ve-locity of the pump pulse, β2(ωp) = ∂2β/∂ω2|ω = ωp,γ = (n2ωp)/(cSeff) is the nonlinear coefficient, andSeff=[∫ ∞

−∞∫ ∞−∞ |F(x, y)|2 dx dy]2/ ∫ ∞

−∞∫ ∞−∞ |F(x, y)|4

dx dy is the effective area of a guided mode with thetransverse field profile F(x, y).

With the propagation constant of the pump fieldwritten as

β(ωp

) = β0(ωp

)+γ P , (5.166)

the mismatch of the propagation constants of the fieldsinvolved in the FWM process is given by

ΔβFWM = β(ωp +Ω

)+β(ωp +Ω

)−2β(ωp

)≈ β2

(ωp

)Ω2 +2γ P . (5.167)

The phase matching can thus be achieved for this typeof FWM at

Ω = ±(

2γ P∣∣β2(ωp

)∣∣)1/2

(5.168)

only when the central frequency of the pump field liesin the range of anomalous group-velocity dispersion,where β2(ωp) < 0.

Figure 5.16 illustrates typical features of the scalarMI in PCFs observed by Fedotov et al. [5.82]. In thoseexperiments, unamplified 50 fs pulses of 790–810 nmTi:sapphire laser radiation with a repetition rate of10 MHz and an energy of 0.1–1.4 nJ were coupled intomicro-waveguide channels off the central core of thePCF (the inset in Fig. 5.16). The zero group-velocitydispersion (GVD) wavelength for the micro-waveguidechannel used to observe MI is estimated as λ0 ≈ 720 nm,providing an anomalous GVD for the pump field. Witha special choice of the pump pulse power, it is thenpossible to use the SPM phase shift to induce phasematching for efficient four-wave mixing (Fig. 5.16). Thisprocess can be understood as SPM-induced modulationinstability, resulting in an exponential growth of spec-tral sidebands phase-matched with the pump field. In theoutput spectrum presented in Fig. 5.16, the 795 nm pumpfield generates sidebands centered at 700 and 920 nm.To understand this result, we use the standard result ofMI theory (5.168) for the frequency shift correspond-ing to the maximum MI gain. With γ ≈ 50 W−1km−1,D ≈ 30 ps/(nm · km), pulse energy E ≈ 0.5 nJ, ini-tial pulse duration τ ≈ 50 fs, we find Ωmax/2π ≈50 THz, which agrees well with the frequency shifts ofthe sidebands observed in the output spectra presented

600 1000

1.0

0.5

0.0700 800 900

Intensity (arb. units)

λ (nm)

Fig. 5.16 Sideband generation through modulation insta-bility in the spectrum of an ultrashort pulse transmittedthrough a photonic-crystal fiber (shown in the inset). Theinput energy of laser pulses is 0.5 nJ. A scanning electronmicroscopy (SEM) image of the photonic-crystal fiber isshown in the inset

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in Fig. 5.16. SPM-induced broadening sufficient to seedthe considered MI-type FWM process is achieved withina fiber length z ≈ (2LdLNL)1/2, where Ld = τ2/|β2| andLNL = (γ P)−1 are the dispersion and nonlinear lengths,respectively. For the above-specified parameters of laserpulses and a PCF, we have Ld ≈25 cm and LNL ≈0.2 cm. SPM-induced broadening can thus provide seed-ing for sideband generation at Ωmax with PCF lengthsexceeding 3.2 cm. This condition was satisfied in our ex-periments, where the PCF length was equal to 8 cm. Ourexperimental results thus demonstrate efficient regimesof SPM-induced MI in microchannel waveguides ofPCFs, which offer much promise for parametric fre-quency conversion and photon-pair generation.

5.6.2 Cross-Phase-Modulation-InducedInstabilities

Self-induced MI in PCFs has been shown [5.53, 82] toprovide a convenient means for broadband paramet-ric amplification, permitting the creation of compactand convenient all-fiber optical parametric oscillators.Cross-phase-modulation-induced MI, on the other hand,does not require anomalous dispersion [5.23], sug-gesting convenient and practical control knobs for thefrequency conversion of ultrashort pulses in PCFs, al-lowing the frequency shifts and amplitudes of sidebandsin output spectra of the probe field to be tuned by varyingthe amplitude of the pump field.

1.8

200

0

–200

–400

–600

2.04

2.02

2.00

0.6 0.9 1.2 1.5

GVD (ps/(nm × km))

λ (μm)

νg (1010cm/s)

Fig. 5.17 The group velocity (the solid curve and the right-hand axis) and the group-velocity dispersion (the dashedcurve and the left-hand axis) calculated as functions of theradiation wavelength λ for the fundamental mode of thePCF with a cross-sectional structure shown in the inset

Experiments [5.83] were performed with high-index-step fused-silica PCFs having a cross-sectionalgeometry shown in the inset to Fig. 5.17 with a corediameter of 4.3 μm. Figure 5.17 displays the groupvelocity and GVD calculated for this PCF usingthe polynomial expansion technique [5.84]. The stan-dard theory of XPM-induced MI, as presented byAgrawal [5.85], was used to analyze the main featuresof this phenomenon for fundamental-wavelength andsecond-harmonic femtosecond pulses of a Cr:forsteritelaser co-propagating in a PCF with the above-specifiedstructure. This theory predicts that stationary solutionsto slowly varying envelope approximation equations forthe pump and probe fields including dispersion up to thesecond order become unstable with respect to a smallharmonic perturbation with the wave vector K and thefrequency Ω if K has a nonzero imaginary part. The do-mains of this instability can be found by analyzing thedispersion relation[

(K −Ωδg/2)2 −h1

] [(K +Ωδg/2)2 −h2

]= C2 ,

(5.169)

where

h j = β22 jΩ

2(Ω2 +4γ j Pj g/β2 j

)/4 , (5.170)

C = 2Ω2√

β21β22γ1γ2 P1 P2 , (5.171)

γ j = n2ω j/(cS j ) are the nonlinear coefficients,δ = (vg2)−1 − (vg1)−1, β2 j = (d2β j/dω2)ω=ω j , Pj , ω j ,vg j and β j are the peak powers, the central frequencies,the group velocities, and the propagation constants of thepump ( j = 1) and probe ( j = 2) fields, n2 is the nonlin-ear refractive index, and S j are the effective mode areasfor the pump and probe fields. The gain of instabilitieswith a wave number K is given by G(Ω) = 2 Im(K ).

Analysis of the dispersion properties of the PCFsemployed in our experiments (Fig. 5.17) yields β21 ≈−500 fs2/cm, β22 ≈ 400 fs2/cm, and δ = 150 fs/cm.The dimensionless frequency shift of the probe field,f = Ω/Ωc [where Ωc = (4γ2 P2/|β22|)1/2] changesfrom approximately 3.3 up to 3.8 as the γ1 P1/γ2 P2ratio is varied from 0.3 to 2.5. As highlighted byAgrawal [5.85], such a weak dependence of the fre-quency shift of the probe field on the pump power istypical of XPM-induced MI in the regime of pump–probe group-velocity mismatch. With γ2 P2 ≈ 1.5 cm−1,the frequency shift f ≈ 3.8 gives sidebands shifted byΩ/2π ≈ 74 THz with respect to the central frequencyω2 of the second harmonic (which corresponds to ap-proximately 90 nm on the wavelength scale). As will

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be shown below, this prediction agrees well with ourexperimental results.

The laser system used in experiments [5.83] con-sisted of a Cr4+:forsterite master oscillator, a stretcher,an optical isolator, a regenerative amplifier, and a com-pressor. The master oscillator, pumped with a fiberytterbium laser, generated 30–60 fs light pulses of radi-ation with a wavelength of 1.23–1.25 μm at a repetitionrate of 120 MHz. These pulses were then transmittedthrough a stretcher and an isolator, to be amplified ina Nd:YLF-laser-pumped amplifier and recompressed tothe 170 fs pulse duration with the maximum laser pulseenergy up to 40 μJ at 1 kHz. A 1 mm-thick β bariumborate (BBO) crystal was used to generate the sec-ond harmonic of amplified Cr:forsterite laser radiation.Fundamental-wavelength, 1235 nm radiation of a fem-tosecond Cr:forsterite laser and its second harmonicwere used as pump and probe fields, respectively. Ascan be seen from Fig. 5.16, the pump wavelength fallswithin the area of anomalous dispersion for the funda-mental mode of the PCF, while the second-harmonicprobe lies in the range of normal dispersion. The fasterpump pulse Fig. 5.17 was delayed in our experimentswith respect to the slower probe pulse at the input of thePCF by a variable delay time of τ .

Figure 5.18 presents the results of experimentalmeasurements performed with 170 fs pump pulses (thefundamental radiation of the Cr:forsterite laser) withan energy ranging from 0.2 up to 20 nJ and 3 nJ,180 fs probe pulses (the second-harmonic output ofthe Cr:forsterite laser) transmitted through a 5 cm PCFwith the cross-sectional structure shown in the inset toFig. 5.17. For delay times τ around zero, the slowerprobe pulse sees only the trailing edge of the faster mov-ing pump pulse. In such a situation, XPM predominantlyinduces a blue shift of the probe field. For τ ≈ δL ≈750 fs, where L = 5 cm is the PCF length, the leadingedge of the pump pulse catches up with the probe fieldcloser to the output end of the fiber, which results ina predominant red shift of the probe. To symmetrize theinteraction between the pump and probe fields with re-spect to the XPM-induced frequency shift, we choosethe delay time τ = δL/2 ≈ 375 fs. In the regime of lowpeak pump powers (less than 3 kW), the output spectrumof the probe field displays only slight broadening due toself-phase modulation (Fig. 5.18a). Pump pulses withhigher peak powers lead to radical changes in the outputspectra of the probe field, splitting the central spectralcomponent of the probe field and giving rise to intensesymmetric sidebands around the central frequency ω2(Figs. 5.18b–d).

The general tendencies in the behavior of the outputspectrum of the probe field as a function of the pumppower agree well with the prediction of the standardtheory of XPM-induced MI. In view of the splitting andslight blue-shifting of the central spectral componentof the probe field (Figs. 5.18b–d), we define the effec-tive central wavelength of the pump-broadened probespectrum as 605 nm. As the pump power changes from5 kW up to 42 kW in our experiments, the shift of theshort-wavelength sideband in the output spectrum ofthe second harmonic increases from 80 nm up to ap-proximately 90 nm. The theory predicts the wavelengthshifts of 76 nm and 90 nm, respectively, indicating thepredominant role of XPM-induced MI in the observedspectral transformations of the probe field. The ampli-

500

1.5

1.0

0.5

0.0

1

0.1

0.01600 700 500 600 700800 800

500

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a)

λ (nm)

b)Intensity (arb. units) Intensity (arb. units)

c) d)Intensity (arb. units) Intensity (arb. units)

e) f)Intensity (arb. units) Intensity (arb. units)

λ (nm)

λ (nm)λ (nm)

λ (nm)λ (nm)

Fig. 5.18a–f Output spectra of the second-harmonic fieldtransmitted through a 5 cm PCF. The power of the pumppulses is (a) 3, (b) 7, (c) 30, (d) 42, (e) 70, and (f) 100 kW.The power of the probe pulse is 8 kW

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tudes of sidebands generated by pump pulses with a peakpower of about 40 kW, as can be seen from Fig. 5.18d,become comparable or may even exceed the amplitudeof the spectral components at the central part of theprobe spectrum. The maximum frequency shift of theprobe-field sidebands achieved in our experiments with45 kW pump pulses is estimated as 80 THz, which issubstantially larger than typical frequency shifts result-ing from XPM-induced MI in conventional fibers [5.23].With pump powers higher than 50 kW, both the centralspectral components of the probe field and its side-bands featured a considerable broadening (Figs. 5.18e,f)and tended to merge together, apparently due to thecross-phase modulation induced by the pump field.

XPM-induced instabilities thus open an efficientchannel of parametric FWM frequency conversion inphotonic-crystal fibers. Fundamental-wavelength fem-tosecond pulses of a Cr:forsterite laser were used in ourexperiments as a pump field to generate intense side-bands around the central frequency of co-propagatingsecond-harmonic pulses of the same laser through XPM-induced MI in a PCF. This effect leads to efficientpump-field-controlled sideband generation in outputspectra of the second-harmonic probe field.

5.6.3 Solitonic Phenomena in Mediawith Retarded Optical Nonlinearity

Optical solitons propagating in media with noninstan-taneous nonlinear response experience reshaping andcontinuous frequency down-shifting due to the Ramaneffect phenomenon, called soliton self-frequency shift(SSFS) [5.86, 87]. Photonic-crystal fibers substantiallyenhance this nonlinear-optical process due to strong fieldconfinement in a small-size fiber core and the possibilityto tailor dispersion of guided modes by varying the fiberstructure. Liu et al. [5.54] have shown that 200 fs inputpulses of 1.3 μm laser radiation can generate sub-100 fssoliton pulses with a central wavelength tunable down to1.65 μm through the SSFS in a tapered PCF. Photonic-crystal fibers with the wavelength of zero group-velocitydispersion (GVD) shifted to shorter wavelengths havebeen used for the soliton frequency downshifting of800–1050 nm laser pulses [5.88, 89]. Abedin and Kub-ota [5.90] have employed a PCF to demonstrate a 120 nmSSFS for 10 GHz-repetition-rate picosecond pulses. Inrecent experiments [5.91, 92], PCFs with a special dis-persion profile have been shown to provide an efficientspectral transformation of chirped sub-6 fs Ti:sapphirelaser pulses through SSFS, leading to the generationof a well-resolved solitonic spectral component cen-

tered at 1.06 μm. Red-shifted soliton signals formed bysub-6 fs laser pulses in PCFs have been demonstratedto allow to a synchronized seeding of a picosecondNd:YAG pump laser, permitting a considerable simpli-fication of a few-cycle optical parametric chirped-pulseamplification (OPCPA) scheme [5.79].

With many of the key tendencies in the evolution ofultrashort pulses in PCFs analyzed in the extensive liter-ature, we focus here on the possibility of using the SSFSphenomenon for widely tunable frequency shifting offew-cycle laser pulses. Our theoretical analysis is basedon the numerical solution of the generalized nonlinearSchrödinger equation [5.93]

∂A

∂z= i

6∑k=2

(i)

k!k

β(k) ∂k A

∂τk+ iγ

(1+ i

ω0

∂

∂τ

)

×

⎡⎣A(z, τ)

∞∫−∞

R(η) |A (z, τ −η)|2 dη

⎤⎦ ,

(5.172)

where A is the field amplitude, β(k) = ∂kβ/∂ωk are thecoefficients in the Taylor-series expansion of the prop-agation constant β, ω0 is the carrier frequency, τ is theretarded time, γ = (n2ω0)/(cSeff) is the nonlinear coef-ficient, n2 is the nonlinear refractive index of the PCFmaterial,

Seff =

[ ∞∫−∞

∞∫−∞

|F (x, y)|2 dx dy

]2

∞∫−∞

∞∫−∞

|F (x, y)|4 dx dy

(5.173)

is the effective mode area [F(x, y) is the transversefield profile in the PCF mode], and R(t) is the retardednonlinear response function. For fused silica, we taken2 ≈ 3.2×10−16 cm2/W, and the R(t) function is rep-resented in a standard form [5.93, 94]:

R(t) = (1− fR)δ(t)

+ fRΘ(t)τ2

1 + τ22

τ1τ22

e− t

τ2 sin

(t

τ1

), (5.174)

where fR = 0.18 is the fractional contribution of theRaman response; δ(t) and Θ(t) are the delta and theHeaviside step functions, respectively; and τ1 = 12.5 fsand τ2 = 32 fs are the characteristic times of the Ramanresponse of fused silica.

We now apply (5.172) and (5.174) to compute theevolution of ultrashort pulses in two types of PCFs (Figs.

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5.19, 5.20). PCFs of the first type consist of a fused sil-ica core with a diameter of 1.6 μm, surrounded withtwo cycles of air holes (inset in Fig. 5.20a). To findthe parameters β(k) for these fibers, we numericallysolved the Maxwell equations for the transverse com-ponents of the electric field in the cross section ofa PCF using a modification of the method of polyno-mial expansion in localized functions [5.84]. Polynomialapproximation of the frequency dependence of the prop-agation constant β for the fundamental mode of thePCF computed with the use of this numerical proce-dure with an accuracy better than 0.1% within the rangeof wavelengths 580–1220 nm yields the following β(k)

coefficients for the central wavelength of 800 nm: β(2) ≈−0.0293 ps2/m, β(3) ≈ 9.316×10−5 ps3/m, β(4) ≈−9.666×10−8 ps4/m, β(5) ≈ 1.63×10−10 ps5/m,β(6) ≈ −3.07×10−13 ps6/m. For the fundamentalmode of such PCFs, the GVD, defined as D =−2πcλ−2β(2), vanishes at λz ≈ 690 nm. Fibers of thesecond type are commercial NL-PM-750 PCFs (fromCrystal Fibre). The core diameter for these PCFs wasequal to 1.8 μm. The parameters β(k) for these PCFswere defined as polynomial expansion coefficients forthe dispersion profile of the fundamental mode of thesefibers provided by the manufacturer. The group-velocitydispersion for PCFs of this type vanishes at λz ≈ 750 nm.

In the case studied here, the laser field at the inputof a PCF has the form of a few-cycle pulse (the upper

1.0

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10000 2000 3000 4000 5000 0.5

0.0040.000

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0.000

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0.000

0.0040.000

Intensity envelope (arb. units)a) Spectral Intensity (arb. units)b)

τ (fs)

Input

z = 2 cm

z = 4 cm

z = 8 cm

z = 12 cm

z = 24 cm

λ (μm)

Input

z = 2 cm

z = 4 cm

z = 8 cm

z = 12 cm

z = 24 cm

Fig. 5.19 Temporal (a) and spectral (b) evolution of a laser pulse with an initial energy of 0.25 nJ and an input temporalenvelope and chirp shown in Fig. 5.1b propagating through the second-type PCF (shown in the inset)

panel in Fig. 5.19a) with a broad spectrum (the upperpanel in Fig. 5.19b) and a complicated chirp [5.79, 92].For both types of PCFs, the short-wavelength part ofthe spectrum lies in the range of normal dispersion,while the wavelengths above λz experience anomalousdispersion. A typical scenario of spectral and tempo-ral evolution of a few-cycle laser pulse in PCFs ofthe considered types is illustrated by Figs. 5.19a,b.The initial stage of nonlinear-optical transformationof a few-cycle pulse involves self-phase modulation,which can be viewed as four-wave mixing of dif-ferent frequency components belonging to the broadspectrum of radiation propagating through the fiber.Frequency components lying near the zero-GVD wave-length of the PCF then serve, as shown in the classicaltexts on nonlinear fiber optics [5.23], as a pump forphase-matched FWM. Such phase-matched FWM pro-cesses, which involve both frequency-degenerate andfrequency-nondegenerate pump photons, deplete thespectrum of radiation around the zero-GVD wavelengthand transfer the radiation energy to the region of anoma-lous dispersion (spectral components around 920 nmfor z = 2 cm in Fig. 5.19b). A part of this frequency-downconverted radiation then couples into a soliton,which undergoes continuous frequency downshiftingdue to the Raman effect (Fig. 5.19b), known as solitonself-frequency shift [5.23,85,86]. In the time domain, thered-shifted solitonic part of the radiation field becomes

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delayed with respect to the rest of this field (Fig. 5.19a)because of the anomalous GVD of the fiber. As a re-sult of these processes, the red-shifted soliton becomesincreasingly isolated from the rest of the light field inboth the time and frequency domains, which reduces,in particular, the interference between the solitonic andnonsolitonic part of radiation, seen in Fig. 5.19b.

High-order fiber dispersion induces soliton instabil-ities, leading to Cherenkov-type emission of dispersivewaves [5.21,22] phase-matched with the soliton, as dis-cussed in the extensive literature (see, e.g., [5.95, 96]).

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Intensity envelope (arb. units)a)

τ (fs)

Intensity envelope (arb. units)b)Input

z = 2 cm

z = 6 cm

z = 10 cm

z = 20 cm

z = 30 cm

τ (fs)

Input

z = 1 cm

z = 2 cm

z = 3 cm

z = 4 cm

z = 5 cm

Input

z = 2 cm

z = 6 cm

z = 10 cm

z = 20 cm

z = 30 cm

Wavelength (μm)

Input

z = 1 cm

z = 2 cm

z = 3 cm

z = 4 cm

z = 5 cm

Wavelength (μm)

Spectral intensity (arb. units)c) Spectral intensity (arb. units)d)

Fig. 5.20 Temporal (a), (b) and spectral (c), (d) evolution of laser pulses with an initial energy of (a), (c) 0.15 nJ and (b),(d) 0.5 nJ and an initial pulse width of 6 fs in the first-type PCF (shown in the inset). The input pulses are assumed to betransform limited

This resonant dispersive-wave emission gives rise toa spectral band centered around 540 nm in Fig. 5.19b. Asa result of the above-described nonlinear-optical trans-formations, the spectrum of the radiation field for a PCFwith a characteristic length of 20 cm typically featuresfour isolated bands, representing the remainder of theFWM-converted pump field (the bands centered at 670and 900 nm in Fig. 5.19b), the red-shifted solitonic part(reaching 1.06 μm for z = 24 cm in Fig. 5.19b), and theblue-shifted band related to the Cherenkov emission ofdispersive waves in the visible. In the time domain, as

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can be seen from Fig. 5.19a, only the solitonic part ofthe radiation field remains well localized in the formof a short light pulse, the remaining part of the fieldspreading out over a few picoseconds.

In Figs. 5.20a–d, we illustrate tunable frequencyshifting of few-cycle laser pulses through SSFS in PCFsby presenting the results of simulations performed foran idealistic input pulse with an initial pulse width of6 fs and a Gaussian pulse shape. For the first-type PCF(shown in the inset to Fig. 5.20a), almost the entirespectrum of the input pulse falls within the range ofanomalous dispersion, and the pulse tends to form soli-tons, observed as well-resolved prominent spikes in thetime domain (Figs. 5.20a,b). In the frequency domain,the Raman effect leads to a continuous frequency down-shifting of the soliton (Figs. 5.20c,d). The rate of thisfrequency shift dν/dz, where ν is the carrier frequencyand z is the propagation coordinate, rapidly grows witha decrease in the pulse duration τ0. With a linear approx-imation of the Raman gain as a function of the frequency,the integration of the nonlinear Schrödinger equation, asshown by Gordon [5.97], yields dν/dz ∝ τ−4

0 . Althoughhigh-order dispersion and deviations of the Raman gaincurve from the linear function generally make the re-lation between dν/dz and τ in soliton dynamics muchmore complicated [5.98], the soliton pulse width re-mains one of the key parameters controlling the solitonfrequency shift for a given Raman gain profile. In thecase considered here, the short duration of 6 fs inputpulses provides a high rate of soliton-frequency shift-ing at the initial stage of pulse propagation throughthe PCF. As the spectrum of the soliton is shifted to-ward the spectral range with larger values of GVD, thepulse width increases, which slows down the frequencyshift.

It is instructive to illustrate the main tendencies in thespectral and temporal evolution of few-cycle laser pulsesin PCFs using the results of analysis of ideal solitons,i. e., solitons governed by the nonlinear Schrödingerequation [(5.68) in Sect. 5.4.2]. The NLSE (5.68) is re-covered from (5.172) by setting β(k) = 0 for k ≥ 3, takingfR = 0, and keeping only the term representing the Kerreffect, i. e., the term proportional to iγ A |A|2, in the non-linear part of the equation. In normalized, soliton units,the energy carried by a soliton j is [5.98] E j = 4ξ j ,where ξ j = W − j +0.5 is the soliton eigenvalue, con-trolled by the input pulse energy 2W2. The soliton pulsewidth is given by τ j = τ0/2ξ j , where τ0 is the input pulsewidth. The soliton pulse width can thus be reduced, lead-ing to higher SSFS rates, by increasing the energy of theinput pulse.

In the case of solitary waves evolving in fibers withhigh-order dispersion and retarded nonlinearity, the re-sults of NLSE analysis for the soliton energy and thesoliton pulse width are no longer valid. In particular,as the soliton spectrum is shifted toward larger valuesof GVD, the soliton pulse width is bound to increase,while the soliton amplitude decreases (Figs. 5.20a,b).These changes in the soliton pulse width and amplitudeare dictated by the balance between the dispersion andthe nonlinearity, necessary for the existence of the soli-ton. On the qualitative level, however, being applied toshort sections of a fiber, these simple relations provideimportant clues for the physical understanding of theevolution of Raman-shifted solitons in a PCF. Indeed, ascan be seen from the comparison of the results of simu-lations performed for input pulses with the same initialpulse width (6 fs), but different energies, the SSFS ratein the case of higher energy pulses can substantially ex-ceed the frequency-shift rate of solitons produced bylower energy pulses. A pulse with an input energy of0.15 nJ, as can be seen from Fig. 5.20c, is coupled intoa soliton, which undergoes a permanent red-shift as itpropagates through the fiber. At z = 30 cm, the spectrumof this soliton peaks at 1.06 μm. A similar input pulsethat has an initial energy of 0.5 nJ forms a soliton thatexhibits a much faster frequency downshift. The centralwavelength of this soliton reaches 1.12 μm already atz = 5 cm.

0.5

0.4

0.3

0.2

0.1

0.0

7650

0.5

0.4

0.3

0.2

0.1

0.0

7900

700

7700

750

7750

800

7800

850

7850

900 950

Intensity envelope (2)(arb. units)

Intensity envelope (1)(arb. units)Time (1)

(fs)

Time (2)(fs)

2

1

Fig. 5.21 Temporal envelopes of red-shifted solitons(close-up of the peaks labeled with boxes in Figs. 5.20a,b)generated by laser pulses with an initial pulse width of 6 fsand an initial energy of (1) 0.15 nJ and (2) 0.5 nJ in a PCFat z = 30 cm (1) and 3 cm (2)

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5.6

Nonlinear Optics 5.7 Nonlinear Raman Spectroscopy 37

To provide illustrative physical insights into theobserved behavior of red-shifted solitons in PCFs asa function of the input pulse energy, we plot in Fig. 5.21the snapshots of temporal envelopes of the solitonic partof the field corresponding to the input energies of 0.15 nJ(curve 1) and 0.5 nJ (curve 2). We take these snapshots ofsolitons, representing close-up views of intensity enve-lope sections labeled with boxes in Figs. 5.20a,b, for twodifferent values of the propagation coordinate, z = 30 cmin the case of a 0.15 nJ input pulse and z = 3 cm forthe 0.5 nJ input energy. With the spectra of red-shiftedsolitons centered around 1.06 μm in both cases (Figs.5.20c,d), these values of the propagation coordinate al-low a fair comparison of SSFS dynamics in terms ofthe dependence of the frequency shift rate on the soli-ton pulse width. As is seen from Fig. 5.21, the SSFSrate correlates well with the soliton pulse width. Whilethe soliton produced by a pulse with an initial energyof 0.15 nJ has a pulse width of about 50 fs (curve 1 inFig. 5.21), the pulse width of the soliton emerging fromthe 0.5 nJ laser pulse is about 20 fs. In qualitative agree-ment with predictions of Gordon [5.97] and Lucek andBlow [5.98], this shorter soliton in Figs. 5.20b,d dis-plays a faster downshifting as compared with the longersoliton in Figs. 5.20a,c. In the following section, this de-pendence of the SSFS rate on the energy of the pulselaunched into the fiber will be used for the experimentaldemonstration of widely tunable soliton frequency shiftof 6 fs pulses produced by a Ti:sapphire oscillator.

At higher input powers, the spectral features origi-nating from FWM, SSFS, dispersive-wave emission ofsolitons experience broadening due to SPM and XPMeffects, merging together and giving rise to a broadbandwhite-light emission (Figs. 5.15, 5.22). This supercon-

1400

0

–10

–20

–30400 600 800 1000 1200


λ (nm)

Fig. 5.22 The spectrum of supercontinuum emission pro-duced by 820 nm pump pulses with an initial duration of35 fs and an input power of 320 mW in a microstructurefiber with a length of 30 cm and the cross-section structureshown in the inset to Fig. 5.20a.

tinuum radiation generated in PCFs has been intenselyemployed through the past few years to measure and con-trol the offset between the carrier and envelope phasesof ultrashort laser pulses, as well as for the creationof novel broadband sources for nonlinear spectroscopy,biomedical applications, and photochemistry. Examplesof applications of PCF light sources based on enhancednonlinear-optical interactions of guided modes will begiven in Sect. 5.9.

5.7 Nonlinear Raman Spectroscopy

Nonlinear Raman spectroscopy is one of the most pow-erful techniques of nonlinear spectroscopy, which hasfound numerous applications in condensed- and gas-phase analysis, plasma diagnostics, investigation ofmolecular relaxation processes, temperature and con-centration measurements, condensed-phase studies, andfemtochemistry. While many of modifications of non-linear Raman spectroscopy have become a routine toolof modern optical experiments, giving rise to manysuccessful engineering applications, some of nonlinearRaman experiments carried out in the last decade haveshown that the potential of this technique for many topi-cal and sometimes interdisciplinary problems of modern

physics, chemistry, and biology is far from being com-pletely realized. Similar to frequency-tunable sourcesof coherent radiation, which revolutionized nonlinearoptics in its early days, allowing many delicate spectro-scopic experiments, including nonlinear spectroscopicstudies, to be performed, the impressive progress offemtosecond lasers in the 1990s has resulted in thebreakthrough of the nonlinear Raman spectroscopy tonew unexplored areas, giving rise to several elegantnew ideas and approaches, permitting more-complicatedsystems and problems to be attacked, and leading tothe measurements of fundamental importance. This newphase of nonlinear Raman spectroscopy also promoted

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5.7


the development of new spectroscopic concepts, includ-ing time-resolved schemes, broadband spectroscopy,polarization measurements, and CARS generalizationsbased on higher-order nonlinear processes. This concep-tual and technical progress achieved in the last decadeshows us some very important features of what nonlinearRaman spectroscopy is going to be in the nearest future,encouraging the application of new ideas, techniques,and methods in this area of spectroscopy.

This section provides a brief introduction to the mainprinciples of nonlinear Raman spectroscopy, givinga general idea of how the measurements are performedand the spectroscopic data are extracted from the resultsof these measurements. Following this plan, we will firstgive a brief introduction to the basic concepts of nonlin-ear Raman spectroscopy. Then, we will consider variousmodifications of coherent Raman four-wave mixing(FWM) spectroscopy, including the standard CARSscheme, stimulated Raman scattering, Raman-inducedKerr effect, degenerate four-wave mixing (DFWM), andcoherent hyper-Raman scattering. We will also brieflydescribe polarization techniques for nonlinear Ramanspectrometry and coherent ellipsometry, allowing se-lective investigation of multicomponent molecular andatomic systems and permitting the sensitivity of non-linear Raman spectrometry to be radically improved.Finally, in the context of the growing interest in the appli-cations of short-pulse spectroscopy for the investigationof ultrafast processes, we will provide an introductionto time-resolved nonlinear Raman spectroscopy.

5.7.1 The Basic Principles

Nonlinear Raman spectroscopy is based on nonlinear-optical interactions in Raman-active media. The nonlin-ear character of light interaction with a medium impliesthat molecular, atomic, or ionic vibrations in a mediumare no longer independent of the light field. Instead,pump light waves with frequencies ω1 and ω2 modulateRaman-active vibrations in a medium at the frequencyΩ ≈ ω1 −ω2, which can be then probed with anotherlight beam, generally having a frequency ω3 (Fig. 5.6c).This wave-mixing process involving the inelastic scatter-ing of the probe wave by molecular vibrations gives riseto coherent Stokes- and anti-Stokes-frequency-shiftedsignals, whose amplitude I , polarization (the ellipticityχ and the tilt angle ψ of the principal axis of the po-larization ellipse), and phase ϕ carry the spectroscopicinformation concerning the medium under study. Thisis the general idea of nonlinear Raman spectrometry,illustrated by Fig. 5.6c.

In practical terms, to undertake a simple three-color CARS experiment, one generally needs threelaser sources generating radiation with the frequen-cies meeting the requirements specified above. In themost popular scheme of two-color CARS, where theanti-Stokes signal is generated through the frequency-mixing scheme ωa = 2ω1 −ω2, the number of lasersrequired is reduced to two. The light beams have tobe brought into coincidence in space to excite Raman-active transitions in a medium and to generate theanti-Stokes signal. Introducing some delay time be-tween pumping and probing pulses, one can also performtime-resolved CARS measurements to keep track of thetemporal dynamics of excitation in the system underinvestigation.

While spontaneous Raman scattering often suffersfrom the low quantum yield, which eventually results inthe loss of sensitivity, coherent Raman scattering allowsmuch more intense signals to be generated, thus allowingvery high sensitivities to be achieved. Coherent Ramanscattering also offers several other very important ad-vantages that stem from the coherent character of thesignal, which is thus well collimated and generated ina precisely known direction.

Generally, the analysis of the amplitude, phase, andpolarization of the signal resulting from a nonlinearwave-mixing process in a Raman-resonant medium in-volves the calculation of the nonlinear response of themedium in terms of the relevant nonlinear susceptibili-ties and the solution of Maxwell equations for the fieldof the signal. Below, we restrict our brief introductionto the theory of nonlinear Raman processes to the de-scription of the basic notions and terminology, whichwill be employed later to explain the main concepts ofvarious nonlinear Raman methods, including frequency-and time-domain CARS, coherent ellipsometry, and theRaman-induced Kerr effect (RIKE).

Excitation of Raman ModesWithin the framework of a simple, but physically in-structive, semiclassical model of a nonlinear mediumconsisting of noninteracting molecules with a Raman-active vibration with frequency Ω, the interaction oflight with molecules (atoms or ions) can be describedin terms of the electronic polarizability of moleculesα depending on the generalized normal coordinate Q(e.g., defined as the distance between the nuclei ina molecule) [5.99]:

α (Q) = α0 +(

∂α

∂Q

)0

Q + . . . , (5.175)

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5.7


where α0 is the equilibrium polarizability of a molecule,(∂α/∂Q)0 is the derivative of the electronic polarizabilityin the normal coordinate taken for the equilibrium posi-tion of nuclei, and we restrict our consideration to termsin this expansion linear in Q. The term (∂α/∂Q)0 Qin (5.175) is responsible for the modulation of lightby molecular vibrations, as it gives rise to new fre-quency components in the induced polarization of thesystem, whose frequency shift is determined by the fre-quency of molecular vibrations. This can be seen fromthe expression for the polarization of a medium:

P = N p , (5.176)

where N is the number density of Raman-activemolecules,

p = α(Q)E = α0 E +(

∂α

∂Q

)0

QE + . . . (5.177)

is the dipole moment of a molecule and E is the electricfield strength. The energy of a molecule in a light fieldis written as

H = −pE = −α (Q) E2 . (5.178)

Thus, the light-induced force driving molecular vibra-tions is given by

F = −∂H

∂Q= ∂α

∂QE2 . (5.179)

As can be seen from (5.179) the force acting ona molecule in a light field may result in a resonant exci-tation of Raman-active vibrations with a frequency Ω ifthe field involves frequency components ω1 and ω2 withω1 −ω2 ≈ Ω.

Wave Mixing in Raman-Active MediaThe propagation of light waves in a nonlinear mediumis governed by the wave equation

ΔE− n2

c2

∂2E∂t2 = 4π

c2

∂2 Pnl

∂t2 , (5.180)

where E is the field in the light wave, n is the refractiveindex, c is the speed of light, and Pnl is the nonlinearpolarization of the medium.

Consider for example the CARS process, as an ex-ample of a nonlinear Raman process, assuming that itinvolves plane and monochromatic waves,

E (r, t) = E1 exp (−iω1t + ik1r)

+ E2 exp (−iω2t + ik2r)

+ E exp (−iωt + ikr)

+ Ea exp (−iωat + ikar)+ c.c. , (5.181)

where ω1 and ω2 are the frequencies of pump waves, ω

is the frequency of the probe wave, ωa = ω+ω1 −ω2,the field envelopes E, E1, E2, and Ea are slowly varyingfunctions of coordinates r and time t, k, k1, k2, and ka arethe wave vectors of light waves with frequencies ω, ω1,ω2, and ωa, respectively, and c.c. stands for a complexconjugate. Then, representing the nonlinear polarizationin the medium as a superposition of plane waves, we canwrite the equation for the amplitude of the anti-Stokeswave as

na

c

∂Ea

∂t+ ∂Ea

∂z= i

2πωa

cnaPnl (ωa) exp (ika z) ,

(5.182)

where Pnl(ωa), na, and ka are the amplitude of thenonlinear polarization, the refractive index, and the z-component of the wave vector at the frequency ωa. Thepump wave amplitudes will be assumed to be constant.Now, the nonlinear polarization of a medium has tobe found with the use of some model of the nonlinearmedium.

Raman-Resonant Nonlinear Polarizationand Nonlinear Susceptibilities

Within the framework of our model of a nonlin-ear medium, the third-order nonlinear polarization ofthe medium at the anti-Stokes frequency ωa is givenby [5.99]

P(3) (ωa) = N

4MD (Ω,ω1 −ω2)

(∂α

∂Q

)2

0EE1 E∗

2 ,

(5.183)

where

D (Ω,ω1 −ω2) = Ω2 − (ω1 −ω2)2

−2iΓ (ω1 −ω2) , (5.184)

derivatives are taken in the equilibrium position, Γ is thephenomenologically introduced damping constant, M isthe reduced mass of a molecule, N is the number densityof molecules, an asterisk indicates a complex conjugate,Q is the amplitude of the Raman-active molecular vi-bration, which can be expressed in terms of the densitymatrix ρ of an ensemble of molecules,

Q = Sp (ρq) = ρabqba , (5.185)

where Sp is the trace operator and q is the operator ofthe vibrational coordinate.

Introducing the third-order nonlinear-optical suscep-tibility of the medium,

χ(3)R = N

24MD (Ω,ω1 −ω2)

(∂α

∂Q

)2

0, (5.186)

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5.7


and solving (5.182), we arrive at the following expres-sion for the intensity of the CARS signal Ia:

Ia ∝∣∣∣χ(3) (ωa; ω,ω1,−ω2)

∣∣∣2

× II1 I2l2(

sin(

Δkl2

)Δkl2

)2

, (5.187)

where I , I1, I2 are the intensities of the pump and probebeams, l is the length of the nonlinear medium, and Δk =|Δk| = |ka −k−k1 +k2| is the wave-vector mismatch.As can be seen from (5.187), the anti-Stokes wave isgenerated especially efficiently in the direction of phasematching, where Δk = 0.

The cubic nonlinear-optical susceptibility ofa medium χ

(3)ijkl is a fourth-rank tensor. The knowledge

of the form of this tensor is important, in particular,for understanding the polarization properties of the sig-nal of nonlinear Raman scattering. The form of theχ

(3)ijkl tensor is determined by the symmetry properties

of a medium. For an isotropic medium, only 21 of 81tensor components of χ

(3)ijkl are nonvanishing. Only three

of these components are independent of each other, asthe relations

χ(3)1111 = χ

(3)2222 = χ

(3)3333 , (5.188)

χ(3)1122 = χ

(3)1133 = χ

(3)2211 = χ

(3)1122 = χ

(3)3311 = χ

(3)3322 ,

(5.189)

χ(3)1122 = χ

(3)1133 = χ

(3)2211 = χ

(3)1122 = χ

(3)3311 = χ

(3)3322 ,

(5.190)

χ(3)1122 = χ

(3)1133 = χ

(3)2211 = χ

(3)1122 = χ

(3)3311 = χ

(3)3322 ,

(5.191)

χ(3)1111 = χ

(3)1122 = χ

(3)1212 = χ

(3)1221 (5.192)

hold true for an isotropic medium [5.9, 99]. The formof the χ

(3)ijkl tensor for all the crystallographic classes can

be found in many textbooks on nonlinear optics [5.9,10].

Phase Matching of Focused BeamsGenerally, in the case of focused beams, phase-matchingeffects in nonlinear Raman scattering are taken intoaccount through the phase-matching integral. In partic-ular, for Gaussian beams, the expression for the overallpower of the two-color CARS signal occurring in ac-cordance with the scheme ωa = 2ω1 −ω2 is written

as [5.11, 100,101]

Pa =(

3π2ωa

c2na

)2 ∣∣∣χ(3)∣∣∣2

× P21 P2

32

π3

(b

w30

)2 ∞∫0

2πr |J |2 dr . (5.193)

Here, P1 and P2 are the powers of the pump beams withfrequencies ω1 and ω2, respectively, w0 is the waist sizeof the focused pump beams, b is the confocal parameter,

J =C2∫

−C1

exp(− r2

bH

)(1+ iξ ′) (k′′ − ik′ξ ′) H

dξ ′ , (5.194)

where −C1 and C2 are the coordinates of the boundariesof the nonlinear medium,

H = 1+ ξ ′2

k′′ − ik′ξ ′ − iξ ′ − ζ

k′ , (5.195)

k′′ = 2k1 + k2, k′ = 2k1 − k2, ζ = 2(z − f )/b is the nor-malized coordinate along the z-axis ( f is the coordinateof the beam waist along the z-axis).

Equations (5.193–5.195) show that the informationon the nonlinear cubic susceptibility in the CARS sig-nal may be distorted by phase-matching effects. Thisproblem has been analyzed both theoretically and exper-imentally for different modifications of nonlinear Ramanspectroscopy [5.102]. The influence of absorption at thewavelength of pump and probe waves and at the fre-quency of the FWM signal can be taken into accountby including the imaginary parts of the relevant wavevectors. The integral in (5.194) can be calculated in ananalytical form for several particular cases, giving a clearphysical understanding of the role of phase-matchingand absorption effects in coherent FWM spectroscopyand imaging [5.103].

5.7.2 Methodsof Nonlinear Raman Spectroscopy

In this section, we will briefly consider standard andwidely used schemes for nonlinear Raman spectroscopy(Fig. 5.2), including coherent Raman scattering, stim-ulated Raman scattering, and the Raman-induced Kerreffect and provide a brief introduction into the vast areaof DFWM.

Stimulated Raman ScatteringThe idea of using stimulated Raman scattering (SRS) asa spectroscopic technique is based on the measurement

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of the frequency dependence of the SRS small-signalgain, which is proportional to the imaginary part ofthe nonlinear cubic susceptibility of a Raman-activemedium [5.9, 99]. The power of the pump wave in suchmeasurements has to be chosen in such a way as to avoiduncontrollable instabilities and to obtain noticeable SRSgain. The SRS-based approach was successfully em-ployed, in particular, for high-resolution spectroscopyof Raman transitions [5.104].

Limitations of SRS as a spectroscopic technique aredue to instabilities arising for light intensities exceed-ing the threshold SRS intensity. These instabilities buildup under conditions when several nonlinear processes,including self-focusing and self-phase modulation, com-pete with each other, often rendering the SRS methodimpractical for spectroscopic applications.

Coherent Anti-Stokes Raman ScatteringInstead of measuring the gain of one of two waves, as isdone in SRS, Maker and Terhune [5.105] have demon-strated a spectroscopic technique based on measuringthe frequency dependence of the intensity of a new wavegenerated at the anti-Stokes frequency ωa = 2ω1 −ω2in the presence of two light waves with frequencies ω1and ω2 chosen in such a way as to meet the conditionof a Raman resonance with a Raman-active transitionin a medium: ω1 −ω2 ≈ Ω (Fig. 5.6c). This approach,called coherent anti-Stokes Raman scattering, has be-come one of the most widespread nonlinear Ramanmethods, allowing many urgent spectroscopic problemsto be successfully solved and stimulating numerous en-gineering applications of nonlinear laser spectroscopy(so-called three-color CARS with ωa = ω1 −ω2 +ω3 isshown in Fig. 5.6c). Similar to the SRS process describedin the previous section, CARS involves the stimulatedscattering of light in a Raman-active medium. However,in contrast to the standard SRS scheme, where a Stokeswave is generated or amplified, the CARS process givesrise to the appearance of a new frequency component,suggesting a spectroscopic approach that is free of insta-bilities typically arising in SRS due to the competitionof different nonlinear processes.

Due to its high spatial, temporal, and spectral res-olution, the possibilities of studying highly luminousobjects, and a rich variety of polarization methods,the CARS technique has gained a wide acceptancefor temperature and concentration measurements in ex-cited gases, combustion, and flames [5.99, 106–108],gas-phase analysis [5.99,109,110], high-resolution mo-lecular spectroscopy [5.111, 112]. Short-pulse CARSgives an access to ultrafast processes and wave-packet

dynamics in molecular systems [5.113]. The nonlinearnature and the spectral selectivity of CARS make thismethod an ideal tool for nonlinear spectroscopy [5.114,115]. The most recent advances in nonlinear Ramantechniques include coherence-controlled CARS [5.116],enhancing the potential of CARS microscopy [5.117],and CARS in photonic-crystal fibers [5.118,119].

The widely employed geometry of nonlinear Ra-man measurements implies the use of collinear focusedlaser beams. While focusing allows the intensity suf-ficient to ensure a reliable detection of the nonlinearRaman signal to be achieved, the collinear geometry ofwave mixing increases the length of nonlinear interac-tion. However, such an approach is reasonable as longas the spatial resolution along the propagation coordi-nate is not important. The scheme of nonlinear Ramanspectroscopy becomes resolvable along the propagationcoordinate as soon as collinear beams are replaced bya noncollinear one (Fig. 5.6c). In CARS, this techniqueis called the boxcars geometry [5.120]. The interactionarea is confined in this case to the region where the beamsintersect each other, allowing a high spatial resolutionto be achieved.

In the broad-beam CARS geometry Fig. 5.23, fo-cused laser beams are replaced with broad or sheetlikebeams. This approach allows the nonlinear Raman sig-nal to image the whole areas of a nonlinear medium

a)

Objectω 1

ω 1

k 1

k 1

kFWM

ωFWM

ω 2k 2

b)

k 2

kFWM

k 0

k 1

θ

�

α

Fig. 5.23a,b Broad-beam folded coherent-anti-Stokes Ra-man scattering: (a) beam arrangements and (b) wave-vectordiagram

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5.7


on a charge-coupled device (CCD) camera. The idea ofbroad-beam CARS imaging, which was discussed byRegnier and Taran [5.121] in 1973, has later provedto be efficient for the solution of many problems ofgas-phase and plasma diagnostics [5.102, 122, 123].Significant progress in extracting the data concerningparameters of a gas medium was achieved, allowingCARS signals from molecules of different types to besimultaneously detected, with the development of a dualbroadband CARS scheme [5.124, 125] and angularlyresolved CARS [5.126].

Figure 5.23 illustrates the application of the broad-beam folded CARS geometry for the investigation ofexcited and ionized gases [5.122, 123]. In this scheme,a pair of cylindrically focused coplanar broad lightbeams with frequencies ω1 and ω2 and wave vectorsk1 and k2, forming a small angle θ, irradiate a thinplasma layer in a plane parallel to the plane of the tar-get (Fig. 5.23a). A cylindrically focused or a collimatednonfocused laser beam with frequency ω3 and wave vec-tor k3, which makes an angle α with the plane of the k1and k2 vectors (Fig. 5.23b), irradiates the laser-producedspark from above. The FWM signal is generated in thedirection kFWM determined by phase-matching condi-tions, forming an angle β with the plane of the target(Figs. 5.23a,b). Imaging the one-dimensional FWM sig-nal onto a CCD array, we were able to map the spatialdistribution of resonant particles in the plasma line byline. The use of a collimated unfocused beam ω3 enablesslice-by-slice plasma imaging [5.102].

Raman-Induced Kerr EffectThe Raman-induced Kerr effect (RIKE) [5.9] is un-derstood as an optical birefringence induced in aninitially isotropic medium due to an anisotropic Raman-resonant third-order polarization of a medium. In thisscheme, a nonlinear medium is irradiated with a pair oflight beams with frequencies ω1 and ω2 whose differ-ence is tuned, in accordance with the general idea ofprobing Raman-active vibrations, to a resonance withRaman-active transitions in a medium. Then, the polar-ization of a probe wave at the frequency ω1 becomesperturbed due to the anisotropic nonlinear polariza-tion induced in the Raman-active medium, which canbe detected with the use of a polarization analyzerand a detector. The RIKE technique provides us witha convenient method for measuring the frequency depen-dence of the cubic susceptibility χ

(3)ijkl around a Raman

resonance. The transmission coefficient T of a polar-ization analyzer aligned in such a way as to block theprobe beam in the absence of the pump beam is given

by [5.99]

Tl (ω1 −ω2) ∝ sin2 2γ

∣∣∣χ(3)1122 (ω1; ω2, ω1,−ω2)

+χ(3)1221 (ω1; ω2, ω1,−ω2)

∣∣∣2 I22

(5.196)

in the case of a linearly polarized pump (I2 is the inten-sity of the pump beam and γ is the angle between thepolarization vectors of the pump and probe waves) and

Tc (ω1 −ω2) ∝∣∣∣χ(3)R

1122 (ω1; ω2, ω1,−ω2)

−χ(3)R1221 (ω1; ω2, ω1,−ω2)

∣∣∣2 I22

(5.197)

in the case of a circularly polarized pump. As can be seenfrom (5.197), the coherent background is completelysuppressed in the case of a circularly polarized pump.

Importantly, RIKE is one of the four-wave mixingprocesses where the phase-matching condition is sat-isfied automatically regardless of the arrangement ofthe wave vectors of pump and probe waves, since thephase-matching condition k1 = k2 +k1 −k2 becomes anidentity in this case.

Degenerate Four-Wave MixingAlthough, rigorously speaking, degenerate four-wavemixing does not employ Raman transitions and the mod-els used to describe DFWM may sometimes differ fromthe standard ways of CARS description [5.9, 10], it isreasonable to briefly introduce DFWM as a nonlineartechnique here, as it very frequently offers a use-ful alternative to CARS, allowing valuable data ona medium to be obtained in a convenient and phys-ically clear way. DFWM is closely related to CARSas both processes are associated with the third-ordernonlinearity of a medium. The main difference be-tween these methods is that CARS implies the use ofa two-photon Raman-type resonance (Fig. 5.6c), whileDFWM is a frequency-degenerate process (Fig. 5.6d),involving either four one-photon resonances or a pairof two-photon resonances. With modern lasers capableof generating very short pulses, having large spectralwidths, the DFWM signal can be detected simultane-ously with CSRS and CARS in the same experimentalgeometry with the same molecular system by simplytuning the detection wavelength [5.127]. The com-bination of these nonlinear-optical approaches allowsa more elaborate study of molecular relaxation and pho-tochemistry processes, providing a much deeper insight

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into the ultrafast molecular and wave-packet dynam-ics [5.127,128].

The main advantages of DFWM as a spectroscopictechnique are associated with the technical simplic-ity of this approach, which requires only one lasersource and allows phase-matching conditions to beautomatically satisfied regardless of the dispersion ofthe medium under study. Broadband DFWM [5.129]makes it possible to measure the temperature of excitedgases, including atomic gases [5.130], with a singlelaser pulse. Folded broad-beam DFWM schemes areemployed in several convenient and elegant methods fortwo-dimensional imaging of spatial distributions of gasparameters [5.131,132].

5.7.3 Polarization Nonlinear RamanTechniques

Methods of polarization-sensitive four-photon spec-troscopy provide an efficient tool for the solution ofmany problems arising in the investigation of Ramanresonances. In particular, the polarization technique isa standard method to suppress the coherent backgroundin CARS measurements [5.99, 106], which makes itpossible to considerably improve the sensitivity ofspectroscopic measurements [5.99] and improves thecontrast in CARS microscopy [5.115]. Polarization tech-niques in FWM spectroscopy can separately measurethe real and imaginary parts of the relevant third-ordernonlinear-optical susceptibility [5.133, 134], resolveclosely spaced lines in FWM spectra of molecules [5.99,135] and atoms [5.102], and improve the contrast ofcubic-susceptibility dispersion curves near Raman res-onances [5.136,137]. Polarization methods in nonlinearRaman spectroscopy [5.138–140] help to analyze theinterference of vibrational Raman resonances with one-and two-photon electronic resonances in CARS spec-tra [5.141], and can be used to determine invariants ofatomic and molecular Raman and hyper-Raman scat-tering tensors [5.102] and to perform conformationalanalysis for complex organic molecules [5.141]. A com-prehensive review of polarization techniques employedfor molecular spectroscopy was provided by Akhmanovand Koroteev [5.99].

Polarization Propertiesof the Coherent FWM Signal

When analyzing polarization properties of the FWMsignal, one has to take into account the interference ofresonant components of FWM related to various (mo-lecular or atomic) transitions in the medium and the

y

x

ab

θψ

a)

b)

ε

Φ

θμ

Pnr

Pr

Pnr

Pr

Pε

e2

e1

Fig. 5.24a,b Polarization technique of nonlinear Ramanspectrometry: (a) coherent ellipsometry and (b) polariza-tion suppression of the nonresonant background in CARSspectroscopy

nonresonant coherent background. In particular, it isthe interference of the resonant FWM component withthe nonresonant coherent background that ensures thepossibility to record complete spectral information con-cerning the resonance under study, including the data onthe phase of resonant FWM.

The polarization ellipse of a Raman-resonant FWMsignal is characterized by its ellipticity χ (which isdefined as χ = ±atan(b/a), where atan stands for thearctangent function, b and a are the small and principalsemiaxes of the polarization ellipse, respectively) andthe tilt angle ψ of its principal axis (Fig. 5.24a). Theseparameters are related to the Cartesian components ofthe third-order polarization of a medium Px and Py bythe following expressions [5.99]:

tan(2ψ) = tan(2β) cos(δ) , (5.198)

sin(2χ) = sin(2β) sin(δ) , (5.199)

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5.7


where β and δ are defined as

tan (β) =∣∣∣∣ Py

Px

∣∣∣∣ , (5.200)

δ = arg(Py

)− arg (Px) . (5.201)

The frequency dependencies of the FWM polariza-tion ellipse parameters, as can be seen from (5.198) and(5.199), provide information concerning the phase ofthe resonant FWM component, allowing a broad classof phase measurements to be performed by means ofnonlinear Raman spectroscopy.

Suppressing the Nonresonant BackgroundPolarization suppression of the nonresonant backgroundin CARS is one of the most useful, practical, and widelyemployed polarization techniques in nonlinear Ramanspectroscopy. Physically, the possibility of suppress-ing the nonresonant background in coherent Ramanspectroscopy is due to the fact that the resonant andnonresonant components of the nonlinear polarizationinduced in a Raman-active medium are generally polar-ized in different ways. Let us illustrate this technique forthe CARS process ωa = 2ω1 −ω2, where ωa is the fre-quency of the anti-Stokes signal and ω1 and ω2 are thefrequencies of pump waves, in the case of an isotropicmedium. The third-order polarization responsible for thegeneration of the signal with the frequency ωa is thenwritten as

P(3) = [Pr + Pnr] E21 E∗

2 , (5.202)

where E1 and E2 are the amplitudes of the light fields, Prand Pnr are the resonant and nonresonant components ofthe third-order polarization induced in the Raman-activemedium.

In the case of an isotropic medium, relations (5.188–5.192) are satisfied for both resonant and nonresonantcomponents of the nonlinear-optical susceptibility.However, only the nonresonant part of the cubic sus-ceptibility satisfies the Kleinman relations [5.9, 99],

χ(3)nr1111 = 3χ

(3)nr1122 = 3χ

(3)nr1221 = 3χ

(3)nr1212 , (5.203)

while the resonant part of the cubic susceptibility isusually characterized by a considerable dispersion neara Raman resonance, which implies that the resonant cu-bic susceptibility tensor components are not invariantwith respect to the permutation of their frequency ar-guments. Taking into account relations (5.203) for thenonresonant part of the cubic susceptibility, we arriveat the following expression for the nonresonant and

resonant components of the third-order polarization ofa Raman-active medium:

Pnr = χ(3)nr1111

[2e1

(e1e∗

2

)+ e∗2 (e1e1)

], (5.204)

Pr = 3χ(3)r1111

[(1− ρ

)e1

(e1e∗

2

)+ ρe∗2 (e1e1)

],

(5.205)

where ρ = χ(3)r1221/χ

(3)r1111 and e1 and e2 are the unit po-

larization vectors of the light fields with frequencies ω1and ω2, respectively (Fig. 5.24b).

Suppose that the pump fields with frequencies ω1and ω2 are linearly polarized and their polarization vec-tors e1 and e2 are oriented at an angle μ with respect toeach other, as shown in Fig. 5.24b. The vectors Pnr andPr, in accordance with (5.204) and (5.205), generallyhave different orientations in space making a nonzeroangle θ with each other. Therefore, by orienting the po-larization analyzer in such a way as to suppress thenonresonant component of the CARS signal (by settingthe angle ε measured from the direction perpendicularto the vector Pnr equal to zero, see Fig. 5.24b), one cananalyze background-free CARS spectra.

In many situations, nonlinear Raman study ofRaman-active media would be simply impossible with-out this technique. This is the case, for example, whenthe CARS signal from a resonant gas under investi-gation is too weak to be reliably detected against thenonresonant CARS signal from the windows of a gascell. Another example is the CARS spectroscopy onlow concentrations of complex biological molecules,when the coherent background due to solvent moleculesmay be so strong that it leaves no way to detectthe CARS signal from the molecules being studiedwithout polarization suppression of the nonresonantnonlinear Raman signal. Even small deviations ofthe orientation of the polarization analyzer from thebackground-suppression position may be crucial, lead-ing to dramatic changes in the signal-to-noise ratio.Another important conclusion that can be made from(5.204) and (5.205) is that, measuring the ratio of theCARS signals for different polarization arrangements isa convenient way of determining the properties of thenonlinear susceptibility tensor and, thus, characterizingthe symmetry properties of molecular transitions underinvestigation.

Coherent EllipsometryCoherent ellipsometry, i. e., the measurement of the pa-rameters of the polarization ellipse corresponding to theFWM signal, is one of widely used modifications ofpolarization-sensitive four-photon spectroscopy. Below,

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5.7


we consider the main physical principles and discuss themain ideas of coherent ellipsometry.

The possibility of reconstructing the real and imag-inary parts of the nonlinear-optical susceptibility ofa medium as functions of frequency and time is due tothe interference of the resonant FWM component withthe nonresonant coherent background, which ensuresthe recording of the phase information for the resonancebeing studied. For a broad class of problems, the non-linear polarization of a medium responsible for coherentFWM processes can be represented as a sum of the non-resonant and resonant components described by a realvector Pnr and a complex vector Pr, respectively. Choos-ing the x-axis along the vector Pnr (Fig. 5.24b), we canwrite the Cartesian components of the total polarizationof a medium cubic in the external field as

Px = Pnr + Preiϕ cos(θ), (5.206)

Py = Preiϕ sin(θ), (5.207)

where ϕ is the phase of the resonant component of thenonlinear polarization and θ is the angle between theresonant and nonresonant components.

Knowing the parameters of the polarization ellipsefrom ellipsometric measurements, we can reconstruct,with the use of (5.198), (5.199), (5.206), (5.207), thereal and imaginary parts of the resonant nonlinear po-larization as functions of frequency and time from theexperimental data of coherent ellipsometry.

In the important particular case when the resonantcomponent of the FWM signal can be considered asa small correction to the nonresonant component, thegeneral procedure of separating the real and imaginaryparts of the nonlinear polarization of a medium be-comes especially simple. One can easily verify that therelations [5.99]

ψ = β cos(ϕ) ∝ Re(P) (5.208)

χ = β sin(ϕ) ∝ Im(P) (5.209)

are satisfied in this case, showing that the spectral ortemporal dependencies of the parameters ψ and χ ofthe FWM polarization ellipse respectively reproduce thespectral or temporal dependencies of the real and imag-inary parts of the resonant component of the nonlinearpolarization of a medium.

Thus, the data obtained by means of coherent el-lipsometry enable one to extract complete informationconcerning the resonant component of the nonlinear po-larization of a medium, including information on itsphase [5.134]. Note that no assumptions regarding the

shape of the line observed in a nonlinear Raman spec-trum was made in our consideration, which means thatthis approach can be applied to a broad class of spec-tral lines. This procedure can be also extended to thetime domain, allowing not only the spectra but also thetime dependencies of the real and imaginary parts of thenonlinear polarization of a medium to be reconstructed.Finally, close molecular and atomic lines unresolvable inamplitude nonlinear Raman spectra can be also resolvedin certain cases with the use of the phase informationstored in coherent nonlinear spectra.

5.7.4 Time-Resolved Coherent Anti-StokesRaman Scattering

The method of time-resolved FWM spectroscopy im-plies that information on the parameters of atomic ormolecular systems is extracted from an impulse re-sponse of a coherently excited system rather than fromthe frequency dispersion of nonlinear susceptibilities, asis done in frequency-domain FWM spectroscopy. Theoriginal idea of time-resolved CARS is that a light pulsewith a duration shorter than the characteristic transverserelaxation time T2 induces coherent molecular vibrationswith amplitude Q(t), and the decay kinetics of these vi-brations is analyzed with the use of another, probe lightpulse, which is delayed in time with respect to the pumppulses. The complete set of equations governing the pro-cesses related to time-domain CARS includes the SVEAwave equation (5.182) and the equations for the ampli-tude of coherent molecular vibrations Q(t), defined inaccordance with (5.185), and the normalized populationdifference between the levels involved in the Ramanresonance, n = ρaa −ρaa:

∂2 Q

∂t2 + 2

T2

∂Q

∂t+Ω2 Q = 1

2M

∂α

∂QnE2 , (5.210)

∂n

∂t+ n −1

T1= 1

2�Ω

∂α

∂QE2 ∂Q

∂t, (5.211)

where T1 is the population relaxation time.In many cases, (5.182), (5.210), and (5.211) can be

simplified with the use of the slowly varying envelopeapproximation. In this approximation, the energy of theCARS signal as a function of the delay time τ of theprobe pulse in the scheme of time-resolved CARS withshort light pulses and Δk = 0 is given by [5.99]

Wa (τ) ∝∞∫

−∞|Q(t)A (t − τ)|2 dt . (5.212)

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5.7


Since the intensity of the CARS signal in such a schemeis determined by (5.212), the use of sufficiently shortprobe pulses makes it possible to measure the kineticsof Q(t). Experiments by Alfano and Shapiro [5.142]and von der Linde et al. [5.143] have demonstrated thepossibility of using the time-domain CARS technique todirectly measure the T2 time for Raman-active modes incrystals and organic liquids.

Formulas (5.185–5.187) and (5.212) show the re-lation between the information that can be obtainedby frequency- and time-domain CARS spectroscopy.In fact, this information is essentially the same, andfrequency- and time-domain CARS methods can suc-cessfully complement each other in studies of complexinhomogeneously broadened spectral bands. For exam-ple, in frequency-domain CARS spectroscopy, the phaseinformation on molecular resonances can be extracted

through polarization measurements and coherent ellip-sometry (see the discussion above), and the level ofcoherent nonresonant background can be suppressedby means of the relevant polarization technique [see(5.204), (5.205) and Fig. 5.24b]. In time-domain CARS,on the other hand, the nonresonant background ap-pears only at zero delay time between the pump andprobe pulses, having no influence on the transient sig-nal, while the impulse-response measurements maydirectly provide the information not only on the am-plitude, but also on the phase of a molecular or atomicresonance. The development of femtosecond laser sys-tems resulted in the impressive technical and conceptualprogress of time-domain FWM spectroscopy, allowingphotochemistry processes and molecular dynamics tobe monitored in real time (see [5.109, 110, 113] fora review).

5.8 Waveguide Coherent Anti-Stokes Raman Scattering

5.8.1 Enhancement of Waveguide CARSin Hollow Photonic-Crystal Fibers

The general idea of waveguide CARS [5.144–149] is toimprove the efficiency of four-wave mixing by increas-ing the interaction length and increasing the intensityof pump waves with given pump powers by reducingthe transverse size of the wave-guiding layer in planarwaveguides or the core diameter in optical fibers. CARSspectroscopy of the gas phase leaves no alternative tohollow waveguides and fibers. Since the refractive in-dex of the core in such waveguides is lower than therefractive index of the cladding, the modes guided inthe hollow core are always characterized by nonzerolosses. The magnitude of these losses scales [5.150,151]as λ2/a3 preventing fibers with small inner diametersto be used in nonlinear-optical experiments, whichlimits the waveguide CARS enhancement factors attain-able with such fibers. In hollow-core photonic-crystalfibers [5.36,152,153], which guide light due to photonicband gaps, optical losses can often be kept low even inthe case of small core diameters. Hollow PCFs withan inner diameter of ≈15 μm demonstrated by Benabidet al. [5.154] have the magnitude of optical losses on theorder of 1–3 dB/m. We will show below that this prop-erty of hollow-core microstructure fibers makes themvery attractive for waveguide CARS spectroscopy.

We start with the expression [5.11] for the power ofthe three-color CARS signal generated at the frequency

ωs = ω0 +ω1 −ω2 by pump fields with frequencies ω0,ω1 and ω2:

PCARS = 1.755×10−5 ω4s k0k1k2

c4k2s k′ D2

×∣∣∣χ(3)

eff

∣∣∣2 P0 P1 P2 F2, (5.213)

where k0, k1, k2, ks are the wave numbers of light fieldswith frequencies ω0, ω1, ω2, ωs, respectively; P0, P1,P2 are the powers of the fields with frequencies ω0, ω1,ω2, respectively; χ

(3)eff is the effective combination of cu-

bic nonlinear-optical susceptibility tensor componentscorresponding to the chosen set of polarization vectorsof pump and signal fields; D is the frequency degener-acy factor of the four-wave mixing process defined afterMaker and Terhune [5.105];

F2 = 2k′

πb

∞∫0

2π RdR

×

∣∣∣∣∣∣∣ξ∫

−ζ

dξ ′ exp(

ibΔkξ ′2

)(1+ iξ ′) (k′′ − ik′ξ ′) H

exp

(− R2

bH

)∣∣∣∣∣∣∣2

(5.214)

is the phase-matching integral, Δk = ks − (k0 + k1 − k2),k′ = k0 +k1 −k2, k′′ = k0 +k1 +k2, ξ = 2(z − f )/b, ζ =2 f /b, b = n jω jw

20/c is the confocal parameter, w0 is the

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5.8

Nonlinear Optics 5.8 Waveguide Coherent Anti-Stokes Raman Scattering 47

beam waist diameter,

H =(1+ ξ ′)2

(k′′ − ik′ξ ′)− i

ξ ′ − ξ

k′ . (5.215)

In the limiting case of tight focusing, when the con-focal parameter b is much less than the length of thenonlinear medium l, b � l, no increase in the CARSpower can be achieved by reducing the pump-beam waistradius because of the simultaneous decrease in the inter-action length. Mathematically, this well-known result isa consequence of the tight-focusing limit existing for thephase-matching integral (5.214). For small phase mis-matches, Δkl � π, the phase-matching integral can bewritten in this limiting case as

F2 = 4π2(1+ k′′

k′)2 . (5.216)

In the opposite limiting case of loosely focused pumpbeams, b l, weak absorption and negligible phasemismatches, the phase-matching integral is reduced to

F2 = k′

k′′4l2

b2 . (5.217)

Since the latter regime is exactly the case of waveguideCARS, we can use (5.216) and (5.217) to estimate theenhancement of waveguide CARS with respect to theregime of tight focusing. Phase mismatches in wave-guide CARS should be understood as difference ofpropagation constants of waveguide modes involved inthe wave-mixing process, and the mode-overlapping in-tegral should generally be included to allow for thecontribution of waveguide effects, in particular, the in-fluence of higher-order waveguide modes.

Assuming that the beam waist radius of focusedpump beams is matched to the inner radius of a hol-low fiber, a, we find from (5.216) and (5.217) that thewaveguide CARS enhancement factor scales as λ2l2/a4.The length l can be made very large in the case of fibers,but the fundamental limitation of waveguide CARS inhollow fibers comes from optical losses, whose mag-nitude scales as λ2/a3. The influence of optical lossesand phase-mismatch effects on the CARS process inthe loose-focusing regime can be included through thefactor

M ∝ exp [− (Δα+α4) l]

×

(sinh2 (

Δαl2

)+ sin2(

Δkl2

)(

Δαl2

)2 + (Δkl2

)2

)l2, (5.218)

where Δα = (α1 +α2 +α3 −α4)/2, α1, α2, α3, α4 arethe magnitudes of optical losses at frequencies ω0, ω1,ω2, ωs, respectively.

It is straightforward to see from (5.218) that the am-plitude of the CARS signal in a lossy waveguide reachesits maximum at some optimal length lCARS

opt , which isgiven by

lCARSopt = 1

Δαln

(α1 +α2 +α3

α4

). (5.219)

With α1 ≈ α2 ≈ α3 ≈ α4 = α, (5.219) yields

lCARSopt = ln 3

α. (5.220)

Then, setting Δk = 0 for phase matching and w0 =0.73a for the best matching of input beams with thefiber mode radius, assuming that the refractive index ofthe gas filling the fiber core is approximately equal tounity, and taking into consideration that M = (31/2 −3−1/2)2/(3 ln 3)2 ≈ 0.123 for Δk = 0 and l = lCARS

opt =ln 3/α, we arrive at the following expression for thewaveguide CARS enhancement factor:

μ = 1.3×10−3

(k′ + k′′)2

k′k′′λ2

α2a4 . (5.221)

We can now see from (5.221), that the waveguide CARSenhancement factor scales as λ2/α2a4 and is limited byfiber losses. We will show in the next section that, due tothe physically different mechanism behind light guiding,hollow microstructure fibers allow CARS enhancementfactors to be substantially increased with respect to stan-dard, solid-cladding hollow fibers. We will examine alsothe CARS enhancement factors as functions of the coreradius for the fibers of both types and investigate theinfluence of the phase mismatch.

We start with the case of standard, solid-claddinghollow fibers. The magnitude of optical losses for EHmnmodes in such fibers is given by [5.150]

α =(umn

2π

)2 λ2

a3

n2 +1√n2 −1

, (5.222)

where umn is the eigenvalue of the characteristic equa-tion for the relevant hollow-fiber mode (the modeparameter), n is the refractive index of the fiber cladding,and the refractive index of the gas filling the fiber coreis set equal to unity.

Plugging optical losses into the CARS enhancementfactor by substituting (5.222) into (5.221) with un = 2.4for the limiting eigenvalue of the EH11 mode of a hollowfiber, we derive the following expression for the factorof CARS enhancement in a solid-cladding hollow fiber

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relative to the tight-focusing regime in the case of exactphase matching:

ρ = 6.1×10−2

(k′ + k′′)2

k′k′′(a

λ

)2 n2 −1(n2 +1

)2 . (5.223)

Optical losses, which grow with decreasing inner ra-dius a, limit the CARS enhancement, with the factorρ rapidly lowering with decreasing a for small val-ues of the fiber inner radius. The situation radicallychanges in the case of a microstructure fiber. Themagnitude of optical losses for such fibers, as men-tioned above, may be on the order of 1–3 dB/m inthe case of fibers with a hollow core diameter ofabout 15 μm [5.154]. In the case of small inner radii,microstructure fibers provide much higher CARS en-hancement factors than solid-core hollow fibers. TheCARS enhancement factor in hollow microstructurefibers with the magnitude of optical losses equal to0.1 and 0.01 cm−1 starts to exceed the CARS enhance-ment factor in a solid-cladding hollow fiber for coreradii less than 20 and 45 μm, respectively. For hollowfibers with small core radii, the factor μ may be sev-eral orders of magnitude higher than the enhancementfactor ρ.

An additional source of radiation losses in hol-low fibers is related to radiation energy transfer tohigher-order waveguide modes. The efficiency of thisnonlinear-optical mode cross-talk process depends onradiation intensity and the mismatch Δkc of propaga-tion constants of waveguide modes involved in energyexchange. Starting with the standard expression for thepropagation constants of EHmn modes in a hollowfiber, we arrive at the following formula for the co-herence length lc = π(2|Δkc|)−1 of the mode cross-talkprocess:

lc = 2π2

λ

a2∣∣u22 −u2

1

∣∣ , (5.224)

where u2 and u1 are the parameters of cross-talkingfiber modes. The coherence length lc, as can be seenfrom (5.224), becomes very small for guided modes ofhigh orders, making the efficiencies of energy trans-fer from the fundamental to very high order modesnegligible. For the cross-talk between the lowest orderEH11 and EH12 modes, with u1 ≈ 2.4 and u2 ≈ 5.5,the coherence length can be estimated as lc ≈ 0.8a2/λ.The coherence length of such a cross-talk process istypically much smaller than the optimal length forthe wave-mixing process (5.220). However, for high-intensity pump beams, the efficiency of this cross-talk

process increases [5.155], and the energy lost fromthe fundamental mode due to the excitation of higher-order modes may become comparable with radiationenergy leakage with the characteristic length governedby (5.222).

Importantly, the scaling law of the waveguide CARSenhancement factor as a function of the magnitude of op-tical losses, fiber inner radius, and radiation wavelengthdiffers from a similar scaling law of the waveguide SRSenhancement factor [5.23], η = λ/αa2. Physically, thisdifference stems from differences in scattering mechan-isms involved in SRS and CARS, with SRS and CARSsignals building up in different fashions as functions ofthe interaction length and pump field amplitudes. Thedifference in waveguide enhancement factors for SRSand CARS suggests different strategies for optimizingfibers designed to enhance these processes.

Phase mismatch, resulting from the difference inpropagation constants of guided modes involved in theCARS process, is another important factor limiting theefficiency of CARS in a hollow fiber. In the case ofnonzero phase mismatch Δk, the optimal length forthe CARS process can be found from a transcendentalequation that immediately follows from (5.218):

Δα sinh(ΔαlCARS

opt

)+Δk sin

(ΔklCARS

opt

)+ (Δα+α4)

×[cos

(ΔklCARS

opt

)− cosh

(ΔαlCARS

opt

)]= 0 .

(5.225)

Phase mismatch reduces the maximum waveguideCARS enhancement attainable with a hollow mi-crostructure fiber, with the power of the CARS signalbecoming an oscillating function of the fiber length. Thecharacteristic period of these oscillations is determinedby the coherence length. Oscillations become less pro-nounced and then completely flatten out as optical lossesbuild up. No oscillations is observed when the attenua-tion length becomes less than the coherence length. Animportant option offered by hollow microstructure fibersis the possibility to compensate for the phase mismatchrelated to the gas dispersion with an appropriate choiceof waveguide parameters due to the waveguide disper-sion component, scaling as a−2 in the case of a hollowfiber.

We have shown in this section that hollow mi-crostructure fibers offer a unique opportunity ofimplementing nonlinear-optical interactions of wave-guide modes with transverse sizes of several micronsin a gas medium, opening the ways to improve the

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efficiency of nonlinear-optical processes, including four-wave mixing and coherent anti-Stokes Raman scattering,and suggesting the principle for the creation of highlysensitive gas-phase sensors based on nonlinear spec-troscopic techniques. Hollow-core microstructure fibershave been demonstrated to allow the waveguide CARSefficiency to be substantially increased as compared tostandard, solid-cladding hollow fibers. The theorem pre-dicting an l2/a4 enhancement for a waveguide CARSprocess in a hollow fiber with an inner radius a andlength l has been extended to include new solutionsoffered by microstructure fibers. The maximum CARSenhancement in a hollow microstructure fiber was shownto scale as λ2/α2a4 with radiation wavelength λ, radia-tion losses α, and the inner fiber radius, allowing CARSefficiency to be substantially improved in such a fiber.This λ2/α2a4 CARS enhancement factor differs fromthe λ/αa2 ratio, characterizing waveguide SRS enhance-ment in a hollow microstructure fiber, which is related tothe difference in the physical nature of SRS and CARSsignals and suggests different strategies for optimizingfibers designed to enhance CARS and SRS processes.

5.8.2 Four-Wave Mixing and CARSin Hollow-Core Photonic-CrystalFibers

Hollow-core photonic-crystal fibers (PCFs) [5.36, 152,153] offer new interesting options for high-field physicsand nonlinear optics. Waveguide losses can be radicallyreduced in such fibers relative to standard, solid-claddinghollow fibers due to the high reflectivity of a period-ically structured fiber cladding within photonic bandgaps (PBGs) [5.152, 153, 156], allowing transmissionof high-intensity laser pulses through a hollow fibercore in isolated guided modes with typical transversesizes of 10–20 μm. Due to this unique property, hol-low PCFs can substantially enhance nonlinear-opticalprocesses [5.157], including stimulated Raman scatter-ing [5.154,158,159], four-wave mixing (FWM) [5.160],coherent anti-Stokes Raman scattering (CARS) [5.118],and self-phase modulation [5.161]. Air-guided modesin hollow PCFs can support high-power optical soli-tons [5.162, 163], allow transportation of high-energylaser pulses for technological [5.68, 164] and biomedi-cal [5.165] applications.

In this section, we discuss phase-matched FWMof millijoule nanosecond pulses in hollow PCFs witha period of the photonic-crystal cladding of about 5 μmand a core diameter of approximately 50 μm. We willshow that Raman-resonant FWM in large-core hollow

PCFs enhances the potential of waveguide CARS inhollow fibers, providing a convenient sensing tool forcondensed-phase species adsorbed on the inner fiberwalls and trace-gas detection.

Large-core-area hollow PCFs employed in exper-iments [5.119, 166] were fabricated using a standardprocedure, which involves stacking glass capillaries intoa periodic array and drawing this preform at a fiber-drawing tower. Several capillaries have been omittedfrom the central part of the stack, to produce a hol-low core of the fiber. While in standard hollow PCFs,the number of omitted capillaries is seven, PCFs usedin our experiments had a hollow core in the form ofa regular hexagon with each side corresponding to fivecane diameters. The inset in Fig. 5.25 shows an image ofa hollow PCF with a period of the cladding of approxi-mately 5 mm and a core diameter of about 50 μm. Thebaking of capillaries forming the photonic-crystal struc-ture, as shown in the image, allows a hollow waveguidewith a nearly ideal 50 μm-diameter hexagonal core tobe fabricated. It is still to be explored whether this tech-nique can be scaled up to the fabrication of hollow PCFswith even larger core diameters. Transmission spectra

400

1.5

1.0

0.5

0.0500 600 700 800 900

Transmission (arb. units)

λ (nm)

Fig. 5.25 Transmission spectrum of the hollow-core PCF.The inset shows the cross-section view of the PCF witha period of the cladding structure of about 5 μm

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5.8


of hollow PCFs employed in our experiments displaywell-pronounced passbands (Fig. 5.25), indicating thePBG guidance of radiation in air modes of the fiber.

The laser system used in experiments [5.119, 166]consisted of a Q-switched Nd:YAG master oscilla-tor, Nd:YAG amplifiers, frequency-doubling crystals,a dye laser, as well as a set of totally reflectingand dichroic mirrors and lenses adapted for the pur-poses of CARS experiments. The Q-switched Nd:YAGmaster oscillator generated 15 ns pulses of 1.064 μmradiation, which were then amplified up to about30 mJ by Nd:YAG amplifiers. A potassium dihydrogenphosphate (KDP) crystal was used for the frequencydoubling of the fundamental radiation. This second-harmonic radiation served as a pump for the dye laser,generating frequency-tunable radiation within the wave-lengths ranges 540–560 and 630–670 nm, dependingon the type of dye used as the active medium forthis laser. All the three outputs of the laser system,viz., the fundamental radiation, the second harmonic,and frequency-tunable dye-laser radiation, were em-ployed as pump fields in FWM, as described below. Thefrequency dependencies of the anti-Stokes signals pro-duced through different FWM processes were measuredpoint by point by scanning the frequency of dye-laser ra-diation. The energies of these pump fields were varied inour experiments from 0.5 up to 10 mJ at the fundamen-tal wavelength, from 0.5 to 8 mJ in the second harmonic,and from 0.05 to 0.7 mJ for dye-laser radiation. To cou-ple the laser fields into the fundamental mode of thePCF, we focused laser beams into spots with a diam-eter of 35 μm at the input end of the fiber. The PCFcould withstand the energy of fundamental radiation upto 10 mJ, corresponding to a laser fluence of approxi-mately 630 J/cm2, without an irreversible degradationof fiber performance because of optical breakdown.Laser-induced breakdown on PCF walls was judgedby a dramatic irreversible reduction in fiber transmis-sion and intense sideward scattering of laser radiation,visible through the fiber cladding. While the achievedlevel of input energies was sufficient to produce reliablydetectable FWM signals in our experiments, a furtherincrease in the laser radiation energy coupled into thePCF is possible through a more careful optimization ofthe coupling geometry.

FWM processes with the CARS-type frequency-mixing scheme ωa = 2ω1 −ω2 (ω1 and ω2 are thefrequencies of the pump fields and ωa is the frequencyof the anti-Stokes signal produced through FWM) werestudied in our experiments for two different sets of pumpand signal frequencies. In the first FWM process, used

in our experiments to test phase matching and assessthe influence of waveguide losses, two waves with thewavelength λ1 = 2πc/ω1 ranging from 630 to 665 nm,provided by the dye laser, are mixed with the fixed-frequency field of the fundamental radiation at λ2 =1064 nm, to generate an anti-Stokes signal within therange of wavelengths λa from 445 to 485 nm. The sec-ond FWM process, designed to demonstrate the potentialof CARS spectroscopy with hollow PCFs, is a standardNd:YAG-laser CARS arrangement with λ1 = 532 nmand λ2 ranging from 645 to 670 nm.

To assess the influence of phase matching andradiation losses on the intensity of the FWM sig-nal generated in a hollow PCF, we use (5.213),(5.218), and (5.225) to write the power of theanti-Stokes signal as Pa ∝ |χ(3)

eff |2 P1 P22 M, where P1

and P2 are the powers of the fields with frequen-cies ω1 and ω2, respectively; χ

(3)eff is the effective

combination of cubic nonlinear-optical susceptibilitytensor components; and the factor M includes opticallosses and phase-mismatch effects: M(Δαl, αa l, δβl) =exp[(Δα+αa)l][sinh2(Δαl/2)+ sin2(δβl/2)][(Δαl/2)2

+ (δβl/δβl/2)2]−1l2, where Δα = (2α1 +α2 −αa)/2,α1, α2, and αa are the magnitudes of optical losses atfrequencies ω1, ω2, and ωa, respectively, and δβ is themismatch of the propagation constants of waveguidemodes involved in the FWM process. In order to pro-vide an order of magnitude estimate on typical coherencelengths lc = π/(2|δβ|) for FWM processes in hollowPCFs and to choose PCF lengths L meeting the phase-matching requirement L ≤ lc for our experiments, wesubstitute the dispersion of a standard hollow fiber witha solid cladding for the dispersion of PCF modes inthese calculations. As shown by earlier work on PBGwaveguides [5.167], such an approximation can providea reasonable accuracy for mode dispersion within thecentral part of PBGs, but fails closer to the passbandedges. For the waveguide FWM process involving thefundamental modes of the pump fields with λ1 = 532 nmand λ2 = 660 nm, generating the fundamental mode ofthe anti-Stokes field in a hollow fiber with a core ra-dius of 25 μm, the coherence length is estimated as lc ≈10 cm. Based on this estimate, we choose a fiber lengthof 8 cm for our FWM experiments. With such a choice ofthe PCF length, effects related to the phase mismatch canbe neglected as compared with the influence of radiationlosses.

Phase matching for waveguide CARS in the PCF wasexperimentally tested by scanning the laser frequencydifference ω1 −ω2 off all the Raman resonances (withλ1 ranging from 630 to 665 nm and λ2 = 1064 nm)

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665

12

10

8

6

4

2

0

1.0

0.5

0.0630

450

635 640 645 650 655 660

455 460 465 470 475 480

2

3

4

1

Signal wavelength (nm)

Dye-laser wavelength (nm)

FWM intensity (arb. units) Transmission

Fig. 5.26 Intensity of the ωa = 2ω1 −ω2 four-wave mixingsignal from the hollow PCF with the length of 8 cm versusthe wavelength of dye-laser radiation with λ1 ranging from630 to 665 nm and λ2 = 1064 nm: (1) the measured spectrumof the FWM signal, (2) fiber transmission for dye-laserradiation, (3) fiber transmission for the FWM signal, and(4) the spectral profile of the factor M

and using the above expression for M(Δαl, αa l, δβl)with δβL ≈ 0 to fit the frequency dependence ofthe FWM signal. Dots with error bars (line 1) inFig. 5.26 present the intensity of the anti-Stokes sig-nal from hollow PCFs measured as a function of thefrequency of the dye laser. Dashed lines 2 and 3 inthis figure display the transmission of the PCF fordye-laser radiation and the anti-Stokes signal, respec-tively. Solid line 4 presents the calculated spectralprofile of the factor M(ΔαL, αa L, 0). Experimen-tal frequency dependencies of the FWM signals, ascan be seen from the comparison of lines 1 and 4in Fig. 5.26, are fully controlled by the spectral con-tours of PCF passbands (lines 2 and 3), indicating thatphase-mismatch effects are much less significant forthe chosen PCF lengths than variations in radiationlosses.

The second series of experiments was intended todemonstrate the potential of waveguide CARS in a hol-low PCF for the sensing of Raman-active species. Forthis purpose, the frequency difference of the second-harmonic and dye-laser pump fields was scannedthrough the Raman resonance, ω1 −ω2 = 2πcΩ, withO−H stretching vibrations of water molecules, adsorbedon the inner PCF walls. The frequencies Ω of O−H

stretching vibrations of water molecules typically fallwithin a broad frequency band of 3200–3700 cm−1.The frequency dependence of the FWM signal fromthe PCF substantially deviates from the spectral pro-file of the factor M(ΔαL, αa L, 0) (cf. lines 1 and 4 inFig. 5.27), clearly indicating the contribution of Raman-active species to the FWM signal. To discriminatebetween the CARS signal related to water moleculesadsorbed on the PCF walls against the OH contamina-tion of the PCF cladding, we measured the spectrum ofthe CARS signal from a PCF heated above a burner.Heating by 30 K during 30 min reduced the amplitudeof the Raman resonance in the spectrum of the CARSsignal by a factor of about seven. The high level of theCARS signal was then recovered within several days.This spectrum of the CARS signal from the dry PCFwas subtracted from the CARS spectrum recorded at theoutput of the hollow PCF under normal conditions. Thedifference spectrum was normalized to the spectral pro-file of the factor M(ΔαL, αa L, 0). The result of thisnormalization is shown by line 5 in Fig. 5.27.

670

10

5

0

1.0

0.5

0.0650

452

655 660 665

448450 446 444 442

FWM intensity (arb. units) Transmission

Signal wavelength (nm)

Dye-laser wavelength (nm)

1

5 2

3

4

Fig. 5.27 Intensity of the ωa = 2ω1 −ω2 four-wave mixingsignal from the hollow PCF with the length of 8 cm versusthe wavelength of dye-laser radiation with λ1 = 532 nm andλ2 ranging from 645 to 670 nm: (1) the measured spectrumof the FWM signal, (2) fiber transmission for dye-laser ra-diation, (3) fiber transmission for the FWM signal, (4) thespectral profile of the factor M and (5) the spectrum of theFWM signal corrected for the factor M upon the subtrac-tion of the spectrum of the CARS signal from the heatedhollow PCF

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Notably, the contrast of the experimental wavelengthdependence of the FWM intensity (squares with er-ror bars) in Fig. 5.26 is higher than the contrast ofa similar dependence for the CARS signal in Fig. 5.27.This variation in the ratio of the maximum amplitudeof the nonlinear signal correlates well with the be-havior of transmission for dye-laser radiation and thenonlinear signal, shown by curves 2 and 3 in bothfigures. With the dye-laser radiation wavelength setaround 650 nm, both the pump and nonlinear signalwavelengths λ1 and λa in Fig. 5.26 are close to therespective maxima of PCF transmission. The CARSsignal, on the other hand, is detected away from themaximum transmission for the dye-laser radiation andthe nonlinear signal (Fig. 5.27). It is therefore impor-tant to normalize the measured CARS spectrum to thewavelength dependence of the M factor, taking into ac-count wavelength-dependent losses introduced by thePCF. This normalization procedure considerably im-proves the contrast of the CARS spectrum, as shownby curve 5 in Fig. 5.27.

Experiments presented above demonstrate the po-tential of waveguide CARS in PCFs to detect traceconcentrations of Raman-active species, suggesting PCFCARS as a convenient diagnostic technique. However,CARS signals detected in these experiments do not al-low the origin of Raman-active species in the fiber tobe reliably identified, as it is not always possible todiscriminate between the contributions to the CARSsignal provided by the hollow core and PCF walls.As an example of a more easily quantifiable Ramanmedium, permitting the CARS signal from the PCF coreto be separated from the signal from PCF walls, wechose gas-phase molecular nitrogen from atmospheric-pressure air filling the hollow core of a PCF. A two-colorRaman-resonant pump field used in these experimentsconsisted of 15 ns second-harmonic pulses of Nd:YAGlaser radiation with a wavelength of 532 nm (ω1) anddye-laser radiation (ω2) with a wavelength of 607 nm.The dye-laser frequency was chosen in such a way as tosatisfy the condition of Raman resonance ω1 −ω2 = Ω

with a Q-branch Raman-active transition of N2 at thecentral frequency Ω = 2331 cm−1. Coherently excitedQ-branch vibrations of N2 then scatter off the second-harmonic probe field, giving rise to a CARS signalat the frequency ωCARS = 2ω1 −ω2 (corresponding toa wavelength of 473 nm). The hollow PCF (shown ininset 1 to Fig. 5.28) was designed to simultaneously pro-vide high transmission for the air-guided modes of thesecond harmonic, dye-laser radiation, and the CARSsignal. With an appropriate fiber structure, as can be

2324 2338

1.0

0.5

0.02331

800

1

0.1

700600500400λ (nm)

Transmission (arb. units)

473 532 607nm

4

2

0

–2

2340λ (nm)

σ� (cm–1)

23302320

CARS intensity (arb. units)

Ω (cm–1)

1

2

3

Fig. 5.28 CARS spectrum of Q-branch Raman-active vi-brations of N2 molecules in the atmospheric-pressure airfilling the hollow core of the PCF. The insets show: (1) animage of the PCF cross section; (2) the transmission spec-trum of the hollow PCF designed to simultaneously supportair-guided modes of the pump, probe, and CARS signalfields (their wavelengths are shown by vertical lines) inthe hollow core of the fiber; and (5.3) the mismatch of thepropagation constants δβ = 2β1 −β2 −βa calculated for theωa = 2ω1 −ωa2 CARS process in fundamental air-guidedmodes of the hollow PCF with β1, β2, and βa being the prop-agation constants of the Nd:YAG-laser second-harmonic(ω1) and dye-laser (ω2) pump fields and the CARS signal(ωa) in the hollow PCF

seen from inset 2 to Fig. 5.28, PCF transmission peakscan be centered around the carrier wavelengths of theinput light fields and the CARS signal (shown by ver-tical lines in inset 2 to Fig. 5.28). Phase matching forCARS with the chosen set of wavelengths has beenconfirmed [5.168] (inset 3 in Fig. 5.28) by a numeri-cal analysis of PCF dispersion based on a modificationof the field-expansion technique developed by Poladianet al. [5.169].

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Nonlinear Optics 5.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources 53

The resonant CARS signal related to Q-branch vi-brations of N2 in these experiments can be reliablyseparated from the nonresonant part of the CARS sig-nal originating from the PCF walls. The spectra of theCARS signal at the output of the PCF Fig. 5.28 areidentical to the N2 Q-branch CARS spectrum of theatmospheric air [5.99] measured in the tight-focusingregime. In view of this finding, the CARS signal canbe completely attributed to the coherent Raman scatter-ing in the gas filling the fiber core with no noticeablecontribution from the nonlinearity of PCF walls.

Results presented here show that large-core-areahollow PCFs bridge the gap between standard, solid-cladding hollow fibers and hollow PCFs in terms of

effective guided-mode areas, allowing energy fluencescaling of phase-matched waveguide four-wave mix-ing of laser pulses. We used hollow PCFs with a corediameter of about 50 μm to demonstrate phase-matchedFWM for millijoule nanosecond laser pulses. IntenseCARS signal has been observed from stretching vibra-tions of water molecules inside the hollow fiber core,suggesting CARS in hollow PCFs as a convenient sens-ing technique for pollution monitoring and trace gasdetection. Hollow PCFs have been shown to offer muchpromise as fiber-optic probes for biomedical Raman ap-plications, suggesting the way to substantially reducethe background related to Raman scattering in the coreof standard biomedical fiber probes [5.170].

5.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

5.9.1 Wavelength-Tunable Sources andProgress in Nonlinear Spectroscopy

The progress in wavelength-tunable light sourcesthrough the past decades has been giving a powerfulmomentum to the development of nonlinear laser spec-troscopy. Nonlinear Raman spectroscopy, in particular,has benefited tremendously more than 30 years ago fromthe application of tunable laser sources, as optical para-metric oscillators [5.171] and dye lasers [5.172] weredemonstrated to greatly simplify measurements basedon coherent anti-Stokes Raman scattering (CARS), mak-ing this technique much more informative, efficient, andconvenient. Broadband laser sources later contributedto the technical and conceptual progress in nonlinearRaman spectroscopy [5.99, 106–110], allowing single-shot CARS measurements. In the era of femtosecondlasers, several parallel trends have been observed inthe development of laser sources for nonlinear Ramanspectroscopy [5.113]. One of these tendencies was toadapt broadband femtosecond pulses for spectroscopicpurposes [5.113, 173, 174] and to use different spatialphase-matching geometries to simultaneously generatecoherent Stokes and anti-Stokes, as well as degeneratefour-wave mixing signals [5.113, 127, 128]. The rapidprogress in nonlinear materials, on the other hand, re-sulted in the renaissance of optical parametric oscillatorsand amplifiers (OPOs and OPAs) for nonlinear spec-troscopy [5.175]. Chirped pulses were used [5.176,177]to probe broad spectral regions and large ranges ofdelay times, suggesting efficient single-shot nonlinearspectroscopic approaches [5.178–180].

In this section, we focus on the potential of photonic-crystal fibers [5.36,37] as novel efficient sources for non-linear spectroscopy. PCFs are unique waveguide struc-tures allowing dispersion [5.40] and spatial field [5.181]profiles to be engineered by modifying the designof the fiber structure. Nonlinear-optical PCF compo-nents and novel PCF-based light sources have beenintensely used through the past few years in frequencymetrology [5.74,77], biomedical optics [5.81], ultrafastphotonics [5.78, 79], and photochemistry [5.182].

Coherent nonlinear spectroscopy and microscopyopen up a vast area for applications of PCF lightsources and frequency shifters. Efficient frequency con-version and supercontinuum generation in PCFs havebeen shown to enhance the capabilities of chirped-pulseCARS [5.183] and coherent inverse Raman spec-troscopy [5.184]. Cross-correlation frequency-resolvedoptically gated CARS (XFROG CARS) has recentlybeen demonstrated [5.80] using specially designedPCF frequency converters for ultrashort laser pulses.Novel light sources based on frequency shifting inPCFs provide a useful tool for the measurementof second-order optical nonlinearities in organic ma-terials [5.185] and offer interesting new options inCARS microscopy [5.186]. Efficient spectral broad-ening of ultrashort pulses in PCFs with carefullyengineered dispersion profiles [5.42–44] makes thesefibers ideal light sources for pump–supercontinuumprobe time- and frequency-resolved nonlinear-opticalmeasurements [5.187].

Below in this section, we demonstrate applica-tions of PCF light sources for chirped-pulse CARS

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and nonlinear absorption spectroscopy. We will showthat PCFs can provide efficient nonlinear-optical trans-formations of femtosecond Cr:forsterite laser pulses,delivering linearly chirped frequency-shifted broadbandlight pulses with central wavelengths ranging from400 to 900 nm. These pulses were cross-correlated inour experiments with the femtosecond second-harmonicoutput of the Cr:forsterite laser in toluene solution,used as a test object, in boxcars geometry to measureCARS spectra of toluene molecules (XFROG CARS).The blue-shifted chirped-pulse output of a photonic-crystal fiber with a spectrum stretching from 530 to680 nm is shown to be ideally suited for the nonlinearabsorption spectroscopy of one- and two-exciton bandsof thiacarbocyanine J aggregates in a polymer film ex-cited by femtosecond second-harmonic pulses of theCr:forsterite laser.

5.9.2 Photonic-Crystal FiberFrequency Shifters

Spectroscopic measurements were performed withmulticomponent-glass PCFs [5.51,188], fabricated withthe use of the standard PCF technology [5.36, 37]. InPCFs used in our experiments (Fig. 5.29a), the solid fibercore is surrounded by a single ring of thin-wall capillar-ies whose outer diameters are equal to the diameter of thePCF core. The outer part of the microstructure claddingconsists of 11 rings of capillaries with outer diametersapproximately three times larger than the diameter of thePCF core and a high air-filling fraction. Dispersion andnonlinearity of the PCFs were managed by scaling thegeometric sizes of the PCF structure. Technologically,this was realized by using the same preform to fabri-cate PCFs with the same type of the structure, but withdifferent magnifying factors. This procedure allowed toscale the sizes of PCF structure without changing itsgeometry.

The laser system used in experiments consistedof a Cr4+:forsterite master oscillator, a stretcher, anoptical isolator, a regenerative amplifier, and a compres-sor [5.189]. The master oscillator, pumped with a fiberytterbium laser, generated 30–60 fs light pulses of radi-ation with a wavelength of 1.23–1.25 μm at a repetitionrate of 120 MHz. These pulses were then transmittedthrough a stretcher and an isolator, to be amplified ina Nd:YLF-laser-pumped amplifier and recompressed tothe 200 fs pulse duration with the maximum laser pulseenergy up to 20 μJ at 1 kHz. The energy of laser pulsesused in experiments presented in this paper ranged from0.5 up to 200 nJ.

400 900

1

0.1

0.01

500 600 700 800

a)

b)

Wavelength (nm)

PCF output (arb. units)

6 5 4

3

2

1

Fig. 5.29 (a) An SEM image of soft-glass photonic-crystalfibers. (b) Intensity spectra of the dispersion-managed blue-shifted output of soft-glass PCFs. The wavelength offset δ

is 50 nm (1), 110 nm (2), 150 nm (3), 190 nm (4), 220 nm(5), and 300 nm (6).

For the PCFs used in experiments, the central wave-length of Cr:forsterite laser pulses falls within the rangeof anomalous dispersion. Such pulses can thereforeform solitons in the fiber. High-order dispersion in-duces wave-matching resonances between solitons anddispersive waves [5.21, 22], giving rise to intense blue-shifted emission, observed as prominent features in thePCF output spectra (Fig. 5.29b). The central wavelengthof this blue-shifted emission correlates well with thewavelength where dispersive waves guided in the fun-damental mode of the PCF are phase-matched withthe soliton produced by Cr:forsterite laser pulses. Thephase matching between the soliton produced by theinput laser pulse and the dispersive wave, which de-fines the frequency of the dominant peaks in output

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spectra of PCFs, is controlled by the dispersion of thefiber. The central frequency of the blue-shifted signalin the output of PCFs can thus be tuned by modifyingthe dispersion profile of the fiber. The GVD profiles ofwaveguide modes were modified in our experiments byscaling the geometric sizes of the fiber without chang-ing the type of the structure shown in Fig. 5.29a. Thecore diameter of PCFs fabricated for these experimentswas varied from 0.9 to 3.8 μm. Figure 5.29b shows thespectra of the blue-shifted output of the fiber, tunedby changing the offset δ = λ0 −λz between the centralwavelength of the input laser field, λ0, and the zero-GVD wavelength λz. With the PCF length remainingunchanged (10 cm), larger blue shifts are achieved byincreasing the offset δ (cf. curves 1–6 in Fig. 5.29b).The power of the input laser field needs to be in-creased for larger δ in these experiments to keep theamplitude of the blue-shifted signal constant. In theseexperiments, dispersion-managed soft-glass PCFs areshown to serve as frequency shifters of femtosecondCr:forsterite laser pulses, providing an anti-Stokes out-put tunable across the range of wavelengths from 400 to900 nm.

Experiments presented in this section show thatthe structural dispersion and nonlinearity managementof multicomponent-glass photonic-crystal fibers allowsa wavelength-tunable frequency shifting and white-lightspectral transformation of femtosecond Cr:forsteritelaser pulses. We have explored the ways toward op-timizing non-silica PCFs for frequency shifting andwhite-light spectral superbroadening of femtosecondCr:forsterite laser pulses and identified important ad-vantages of multicomponent-glass PCFs over silicamicrostructure fibers for the spectral transformationof laser pulses in the 1.2–1.3 μm spectral range. Bycoupling 200 fs pulses of 1.24 μm Cr:forsterite laserradiation into different types of multicomponent-glassPCFs, where the zero-GVD wavelength is tuned byscaling the sizes of the fiber structure, we have demon-strated spectrally tailored supercontinuum generationand frequency upshifting providing a blue-shifted out-put tunable across the range of wavelengths from 400 to900 nm.

5.9.3 Coherent Anti-Stokes RamanScattering Spectroscopywith PCF Sources

In this section, we show that PCF frequency shifterscan serve as convenient sources of chirped wavelength-tunable pulses for CARS spectroscopy. In CARS

Raman medium

ObjectiveCr: forsteriteLaser LBO

Delay line

Lens

PCFMS fiber

FilterObjective

SpectrographCARSsignal

Lens

Fig. 5.30 Diagram of femtosecond CARS spectroscopywith the use of ultrashort pulses frequency-upconvertedand chirped in a photonic-crystal fiber. The inset showsthe cross-sectional view of the central part of the photonic-crystal fiber

experiments, sub-microjoule Cr:forsterite-laser pulseswith an initial duration of about 90 fs are launched intothe central core of the PCF (Fig. 5.30), resulting in theefficient generation of a blue-shifted signal (Fig. 5.31),with a central wavelength dictated by phase matching fordispersive-wave emission and controlled by fiber disper-sion. The wavelength of the blue-shifted signal can befinely tuned by changing the intensity of the pump pulse(Fig. 5.31) due to the nonlinear change in the refractiveindex of the fiber core and the spectral broadening of thepump pulse.

Cross-correlation frequency-resolved optical gat-ing (XFROG) [5.190, 191] was used to characterizethe blue-shifted output of the PCF. An XFROG sig-nal was generated by mixing the blue-shifted signalfrom the fiber Ea with the 620 nm 90 fs second-harmonic output of the Cr:forsterite laser ESH in a BBOcrystal. A two-dimensional XFROG spectrogram,S(ω, τ) ∝ | ∫ ∞

−∞ Ea(t) ESH(t − τ) exp(−iωt) dt|2, wasthen plotted by measuring the XFROG signal as a func-tion of the delay time τ between the second-harmonicpulses and the blue-shifted output of the PCF andspectrally dispersing the XFROG signal. The XFROGspectrogram shown in inset 1 to Fig. 5.31 visualizes

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600 900

1.0

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0.0700 800


λ (nm)

1.2

0.8

0.4

0.0

t (fs)2000–2000 0

40

20

0

–20

Intensity (arb. unit) Phase (rad)

340

330

320

310–10 0–5 5 10 15

τ/100 (fs)

2

3

λ (nm)

340

330

320

310

λ (nm)

1.0

0.8

0.6

0.4

0.2

0.0

τ/100 (fs)–10 0 10

1

Fig. 5.31 The spectrum of the blue-shifted output ofa photonic-crystal fiber pumped by 1.24 μm 90 fsCr:forsterite-laser pulses with an input energy of (solid line)170 nJ, (dashed line) 220 nJ, and (dash–dotted line) 270 nJ.The insets show: (1) the intensity of the sum-frequency sig-nal generated in a BBO crystal by the second-harmonicpulse from the Cr:forsterite laser and the blue-shifted PCFoutput as a function of the wavelength and the delay timeτ between the second-harmonic and anti-Stokes pulses,(2) theoretical fit of the XFROG trace, and (3) the pulseenvelope and the phase of the anti-Stokes pulse providingthe best fit

the temporal envelope, the spectrum, and the chirp ofthe blue-shifted PCF output. A reasonable fit of theexperimental XFROG trace was achieved (inset 2 inFig. 5.31) with a blue-shifted pulse having a durationof about 1 ps and a linear positive chirp correspond-ing to the phase ϕ(t) = αt2 with α = 110 ps−2. Thespectrum and the phase ϕ of the blue-shifted pulse re-constructed with this procedure are shown in inset 3 inFig. 5.31.

1000

100

10

1

0.1

592

588

584

580

576

1.00.80.60.40.20.0

950 1000 1050 1100

588

584

580

576

–7 –4 –1 2 5

–1–4 2

I2

I1

ω

τ

˜|�(3)

|2

a)

Ω

λCARS (nm)b)

c)

(ω1 – ω2) /2πc (cm–1)

d)

τ /100 (fs)

τ /100 (fs)

1ω1

ω2

2

2

1

Fig. 5.32 (a) A diagram of femtosecond CARS spec-troscopy with chirped pulses is shown on the left. The firstpulse (frequency ω1) is transform limited. The second pulse(frequency ω2) is linearly chirped. The linear chirp of thesecond pulse maps the delay time between the pulses on thefrequency, allowing the frequency difference ω1 −ω2 to bescanned through Raman resonances Ω by tuning the de-lay time τ . (b) The intensity of the CARS signal generatedin a noncoplanar boxcars geometry from toluene solutionas a function of the wavelength and the delay time τ be-tween the second-harmonic and the blue-shifted PCF outputpulses used as a biharmonic pump. (c) A model of the Ra-man spectrum, including a doublet of Lorentzian lines anda frequency-independent nonresonant background used inthe theoretical fit. (d) Theoretical fit of the XFROG CARSspectrogram

The linear chirp defines a simple linear mapping be-tween the instantaneous frequency of the blue-shiftedPCF output and the delay time τ , allowing spectral mea-surements to be performed by varying the delay timebetween the pump pulses. Experiments were performedwith 90 fs second-harmonic pulses of the Cr:forsteritelaser (at the frequency ω1) and the linearly chirpedpulses from the PCF (the frequency ω2) as a bihar-monic pump for the CARS spectroscopy of toluenesolution. The frequency difference ω1 −ω2 was scanned

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through the frequencies of Raman-active modes oftoluene molecules by tuning the delay time between thepump pulses (Fig. 5.32a). The second-harmonic pulsealso served as a probe in our CARS scheme, generatingthe CARS signal at the frequency ωCARS = 2ω1 −ω2through the scattering from Raman-active vibrations co-herently excited by the pump fields. The light beamswith frequencies ω1 and ω2 were focused into a cellwith toluene solution at a small angle with respectto each other (Fig. 5.30). The CARS signal generatedin the area of beam interaction in this non-coplanarboxcars geometry had a form of a sharply directedlight beam with a low, phase-matching-controlled an-gular divergence spatially separated (Fig. 5.30) fromthe pump beams. Figure 5.32b presents the map ofCARS spectra from the toluene solution measuredfor different delay times τ between the biharmonicpump pulses. This procedure of measurements, infact, implements the XFROG technique [5.190, 191].However, while FROG-based techniques [5.192] areusually employed to characterize ultrashort pulses, ourgoal here is to probe Raman-active modes of toluenemolecules, used as a test object, by means of CARSspectroscopy.

In the case of a positively chirped pulse from the PCF(insets 2 and 3 in Fig. 5.31), small delay times τ corre-spond to the excitation of low-frequency Raman-activemodes (τ ≈ −200 fs in Fig. 5.32b). In particular, the1004 cm−1 Raman mode of toluene is well resolved inthe presented XFROG CARS spectrogram. This mode isexcited with the second harmonic of Cr:forsterite-laserradiation and the spectral slice around the wavelength ofλ2 ≈ 661 nm picked with an appropriate τ (Fig. 5.32a)from the positively chirped blue-shifted PCF output,giving rise to a CARS signal with the wavelengthλCARS ≈ 584 nm. Raman modes with higher frequen-cies are probed at larger delay times (τ ≈ 100–200 fs inFig. 5.32b).

As a simple model of the Raman spectrum oftoluene molecules in the studied spectral range, we em-ploy a doublet of Lorentzian lines interfering with thecoherent nonresonant background (Fig. 5.32c). The fre-quencies of the peaks in these spectra are used as fittingparameters. The ratio of the peak value of the resonantpart of the CARS susceptibility, χ(3), to the nonreso-nant susceptibility, χ(3)nr, was estimated by measuringthe intensities of the CARS signal on and off the Ramanresonances, yielding |χ(3)nr|/|χ(3)| ≈ 0.05. The best fit(Fig. 5.32d) is achieved with the Raman peaks centeredat 1004 and 1102 cm−1, which agrees well with earlierCARS studies of toluene [5.99].

Although the slope of the XFROG CARS trace andpositions of Raman peaks can be adequately describedwith the use of this simple model, some of the spectro-scopic features of the experimental XFROG CARS tracedeviate from the theoretical fit. These deviations mayoriginate from variations in the coherent background asa function of the frequency. For a quantitative spectro-scopic analysis, these inhomogeneities in the frequencydependence of the coherent background should be care-fully measured and included in the fit. On the otherhand, the nonresonant contribution to CARS spectralprofiles can be efficiently suppressed [5.99, 106] by us-ing three input pulses in the CARS arrangement insteadof two and by introducing the delay time between thethird pulse (probe) and the two-color pump, tuned toa Raman resonance under study.

5.9.4 Pump-Probe NonlinearAbsorption Spectroscopyusing Chirped Frequency-ShiftedLight Pulsesfrom a Photonic-Crystal Fiber

Time-resolved nonlinear-optical spectroscopy of mo-lecular dynamics and fast excitation transfer processestypically involves specifically designed sequences ofpump and probe pulses with a variable delay timeand a smoothly tunable frequency of the probe field.In a laboratory experiment, such pulse sequences canbe generated by femtosecond optical parametric ampli-fiers (OPAs). Femtosecond OPAs, however, inevitablyincrease the cost of laser experiments and make thelaser system more complicated, unwieldy, and diffi-cult to align. An interesting alternative strategy ofpump–probe spectroscopy employs a pump field in theform of a broadband radiation (supercontinuum) witha precisely characterized chirp. A time–frequency map,defined by the chirp of such a supercontinuum pulse,allows time- and frequency-resolved measurements tobe performed by tuning the delay time between thetransform-limited pump pulse and chirped supercontin-uum pulse [5.176,193]. The supercontinuum probe pulsefor pump–probe experiments is most often generated byfocusing amplified femtosecond pulses into a silica orsapphire plate. The chirp of the probe pulse in such an ar-rangement is dictated by the regime of nonlinear-opticalspectral transformation and dispersion of the nonlinearmaterial, leaving little space for the phase tailoring ofthe probe field.

In this section, we demonstrate the potential ofPCFs as compact and cost-efficient fiber-optic sources

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of probe pulses with a tunable frequency and tailoredphase for a time-resolved nonlinear spectroscopy ofmolecular aggregates. We will show that PCFs canprovide efficient nonlinear-optical transformations offemtosecond Cr:forsterite laser pulses, delivering lin-early chirped frequency-shifted broadband light pulses,optimized for pump–probe nonlinear absorption spec-troscopy of molecular aggregates. The blue-shiftedoutput of a photonic-crystal fiber with a spectrumstretching from 530 to 680 nm will be used to probeone- and two-exciton bands of thiacarbocyanine J ag-gregates in a polymer film excited by femtosecondsecond-harmonic pulses of the Cr:forsterite laser.

Molecular aggregates are interesting and practi-cally significant objects encountered in many physical,chemical and biological systems [5.194]. Interactionsbetween the molecules forming aggregates give rise tocollective electronic states, which can be delocalizedover large chains of molecules, modifying the optical re-sponse of the system [5.195]. Specific types of molecularaggregation, known as J and H aggregation, give riseto a pronounced spectral shift and dramatic narrowingof absorption bands, indicating the cooperative natureof the optical response of molecular aggregates [5.196].In natural systems, molecular aggregates are involvedin the processes and functions of vital importance asthey play the key role in light harvesting and primarycharge separation in photosynthesis [5.197]. On opti-cal resonances, the nonlinear susceptibility of molecularaggregates displays a collective enhancement [5.195],scaling as N2 with the number of particles N formingan aggregate. Dramatic enhancement of optical nonlin-earity and the available pathways for ultrafast relaxationof excited states [5.194] suggest a variety of interestingapplications of molecular aggregates, such as terahertzdemultiplexing of optical signals [5.198], spectral sensi-tization in optical data storage and photography [5.199],energy transfer from light-harvesting antennas and com-plexes in artificial photosynthesis [5.197], mode lockingin laser cavities, and creation of novel devices for ultra-fast photonics.

Collective electronic eigenstates in aggregates ofstrongly coupled molecules are grouped into excitonicbands [5.194–196]. In the ground state of this bandstructure, all N molecules of the aggregate reside in theground state. In the lowest excited band, the moleculescoupled into an aggregate share one excitation. Theeigenstates of this first excited band (one-excitons, orFrenkel excitons) are represented by linear combina-tions of basis states corresponding to one excited andN −1 ground-state molecules. Eigenstates that can be

reached from the ground state via two optical transi-tions, making the molecules of an aggregate share twoexcitations, form a two-exciton band. The third-ordersusceptibility χ(3), responsible for four-wave mixing(FWM) processes, can involve only one- and two-excitons. Although three-exciton states can be reached

800

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1.0

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0.0700600500


λ (nm)

Optical density (arb. units)

3

21

Two-exciton band

One-exciton band

fs pump

Chirped probe

Ground state

Fig. 5.33 (a) Absorption spectrum of the thiacarbocya-nine film (1), spectrum of the second-harmonic outputof the Cr:forsterite laser (2), and spectrum of the blue-shifted PCF output (3). (b) Diagram of the pump–probenonlinear-absorption spectroscopy of exciton bands in mo-lecular aggregates using a femtosecond (fs) pump pulse anda broadband chirped probe field

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770700630560

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– 0.02

– 0.04

550 600 650 700 750

0.03

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0.00

– 0.01

– 0.02

– 0.03550 600 650 700 750

a) Differential absorption (arb. units)

b)

c)

λ (nm)

by three transitions, they do not contribute to the third-order polarization as they do not have a transition dipolethat would couple them to the ground state. Time-resolved nonlinear spectroscopic methods based on χ(3)

processes, such as spectroscopy of nonlinear absorp-tion, degenerate four-wave mixing, and third-harmonicgeneration, have proven to be convenient and infor-

Fig. 5.34a–c Differential spectra of nonlinear absorptionmeasured by the pump–probe technique for thiacarbocya-nine J aggregates in a polymer film with a delay timebetween the probe and pump pulses equal to (a) 100 fs,(b) 500 fs, and (c) 1100 fs

mative techniques for the characterization of one- andtwo-exciton states of aggregates, providing the dataon the strength of dipole–dipole interaction betweenmolecules in aggregates, as well as on the disorderand typical relaxation and exciton annihilation times inaggregates [5.194].

Experiments [5.200] were performed with thin-filmsamples of J aggregates of thiacarbocyanine dye. Filmsamples of J aggregates were prepared by spin-coatingthe solution of thiacarbocyanine dye on a thin sub-strate. The concentration of thiacarbocyanine dye was5×10−3 mol/l. The spinning speeds ranged from 1000up to 3000 revolutions per minute. A mixture of ace-tonitrile, dichloroethane, and chloroform with a volumeratio of 2:2:1 was used as a solvent. The thickness of thedye layer applied to a substrate was estimated as 30 nm,and the total thickness of the sample was about 1 μm.

Absorption spectra of J-aggregate films (curve 1 inFig. 5.33a) display two pronounced peaks. The broaderpeak centered at 595 nm corresponds to thiacarbocya-nine monomers, while the narrower peak at 660 nmrepresents the excitonic absorption of J aggregates.To provide a rough estimate of the delocalizationlength Nd of excitons in aggregates, we apply theformula [5.201] NW

d ≈ (3π2|J |3π2|J |W)1/2 −1, whichexpresses Nd through the half-width at half-maximumW of the aggregate peak in the absorption spectrumand the energy J of dipole–dipole interaction betweennearest-neighbor molecules in the aggregate (J < 0for J aggregates). With the J parameter estimated asJ ≈ 900 cm−1 from the aggregation-induced red shiftof the absorption peak in Fig. 5.33a, we find Nd ≈ 6.Experimental results presented below in this papershow that, with such a delocalization length of excitonsin molecular aggregates, spectra of nonlinear absorp-tion display well-pronounced nonoverlapping featuresindicating [5.194–196] bleaching through transitions be-tween the ground and one-exciton states and inducedabsorption via transitions between one- and two-excitonstates of molecular aggregates.

The laser system used in experiments [5.200] wasbased on the Cr4+:forsterite laser source with re-generative amplification, as described in Sect. 5.9.2.A 1 mm-thick BBO crystal was used to generate thesecond harmonic of amplified Cr:forsterite laser radi-

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ation. The spectrum of the second-harmonic output ofthe Cr:forsterite laser was centered at 618 nm (curve 2in Fig. 5.33a). Second-harmonic pulses with a pulsewidth of about 120 fs and the energy ranging from10 to 80 nJ were used as a pump field in our ex-periments on the nonlinear spectroscopy of molecularaggregates.

Frequency-tunable upconversion of fundamental-wavelength Cr:forsterite-laser pulses was performedthrough the nonlinear-optical spectral transformationof these pulses in soft-glass PCFs with the cross-section structure shown in Fig. 5.29a. The propertiesof such fibers and the methods of frequency conver-sion of Cr:forsterite laser pulses in these PCFs havebeen discussed in Sect. 5.9.2 Intensity spectrum of thefrequency-shifted output of the PCF best suited asa probe field for time-resolved nonlinear-absorptionspectroscopy of J aggregates is presented by curve 3 inFig. 5.33a. At the level of 20% of its maximum, the in-tensity spectrum of the blue-shifted PCF output stretchesfrom 530 to 680 nm. Dispersion of the PCF frequencyshifter used in our experiments provided a linear chirpof the output pulse Sect. 5.9.3 with the pulse chirp ratecontrolled by the fiber length.

The spectrum of 120 fs second-harmonic pulses ofthe Cr:forsterite laser partially overlaps the absorptionspectrum of molecular aggregates (Fig. 5.33a). Thesepulses were used in our experiments to excite the aggre-gates through the transitions from the ground state to theone-exciton band (Fig. 5.33b). The spectra of absorptionmodified by the pump field were measured by chirpedblue-shifted pulses delivered by the PCF (Fig. 5.33b).Figures 5.34a–c present the results of experimentalmeasurements upon the subtraction of absorption spec-tra measured in the absence of the pump pulse andnormalization to the spectrum of the probe field. Non-linear absorption spectra shown in Figs. 5.34a–c displaywell-pronounced minima at 665 nm and blue-shiftedpeaks at 640 nm. Such features are typical of nonlin-ear absorption spectra of J aggregates measured bythe pump–probe technique (see [5.194] for a review).

The negative feature is indicative of bleaching throughpump-induced transitions between the ground state andthe one-exciton band, while the blue-shifted peak origi-nates from induced absorption due to transitions betweenone- and two-exciton bands of molecular aggregates(Fig. 5.34a–c).

For highly ordered aggregates, the spectrum ofnonlinear absorption is dominated by transitions be-tween the ground state and lowest one- and two-excitonstates [5.195, 196]. The exciton delocalization lengthcan be then estimated from the spectral shift Δ ofthe induced-absorption peak relative to the bleach-ing minimum using the following formula [5.202]:NΔ

d ≈ (3π2|J |/Δ)1/2 −1. With the spectral shift es-timated as Δ ≈ 470 cm−1, we find that NΔ

d ≈ 6, inperfect agreement with the value of NW

d obtained fromaggregate absorption spectra.

The amplitudes of both positive and negativefeatures in nonlinear absorption spectra decay ona sub-picosecond time scale with increasing delay timebetween the pump and probe pulses (Fig. 5.34a–c),indicating a sub-picosecond relaxation rate of the one-exciton state of molecular aggregates in our experiments.This finding suggests, in agreement with earlier stud-ies of ultrafast excitation energy transfer processes inmolecular aggregates, that the relaxation dynamics ofaggregates in our experimental conditions is mainly con-trolled by the quenching of excited states of aggregatesthrough exciton–exciton annihilation [5.194].

We have thus shown that photonic-crystal fibers witha specially designed dispersion offer the ways to cre-ate efficient sources of ultrashort pulses for coherentnonlinear spectroscopy. These fibers provide a high effi-ciency of frequency upconversion of femtosecond laserpulses, permitting the generation of sub-picosecondlinearly chirped anti-Stokes pulses ideally suited forfemtosecond coherent anti-Stokes Raman scatteringspectroscopy. Experimental studies demonstrate thatPCFs can deliver linearly chirped frequency-shiftedbroadband light pulses, optimized for pump–probe non-linear absorption spectroscopy.

5.10 Surface Nonlinear Optics, Spectroscopy, and Imaging

In this section, we will dwell upon the potentialof nonlinear-optical methods for the investigation ofsurfaces and interfaces. The ability of second-ordernonlinear-optical processes, such as SHG and SFG(Sect. 5.3), to probe surfaces and interfaces is most

clearly seen in the case of a centrosymmetric mater-ial. In this situation, the electric-dipole SHG and SFGresponse from the bulk of the material vanishes. Ata surface or an interface, the bulk symmetry is bro-ken, and electric-dipole second-order nonlinear-optical

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Nonlinear Optics 5.10 Surface Nonlinear Optics, Spectroscopy, and Imaging 61

effects are allowed. Such surface-specific SHG andSFG processes enable a highly sensitive nondestruc-tive local optical diagnostics of surfaces and interfaces(Fig. 5.35a–b). Illuminating and physically insightfuldiscussion of this technique can be found in the classicaltexts on nonlinear optics [5.9, 203].

The χ(2) signal from a surface or an interface is, how-ever, not entirely background-free, as the second-order

ω 1

ω 2

ω 1

ωCARS

ν 2

ν 1

a) Reflectedsecondharmonic

Incidentbeam

Medium 1

Medium 2

Surface /Interface

Transmittedsecondharmonic

c)

Lens

Filter

Sample

Dichroicmirror

LensFilter

Forward CARSdetection

Epl-CARSdetection

Objective

Objective

ω 1 ω 2

CARS

CARS

b)

ωSFω 2

ω 1

ν 2

ν 1

ω 1

ω 2 ωSF

ω 2ω

2ω

nonlinear optical processes are not strictly forbid-den even in a centrosymmetric medium. Beyond theelectric-dipole approximation, the second-order nonlin-ear signal, as can be seen from (5.6–5.8) and (5.14),can be generated through the electric-quadrupole andmagnetic-dipole effects. It is generally very difficult, of-ten impossible, to completely distinguish between thesurface and bulk contributions to the nonlinear signal.Luckily enough, the electric-quadrupole and magnetic-dipole components in the nonlinear-optical responseare typically ka times less significant than the dipole-allowed part [5.9], with k = 2π/λ and a being a typicalsize of an atom or a unit cell in a crystal. The ra-tio of the surface dipole-allowed susceptibility χ

(2)s to

the bulk susceptibility γ(2)b can be then estimated as

|χ(2)s |/|γ (2)

b | ∼ d/(ka), where d is the thickness of thesurface layer. In reflection SHG (Fig. 5.35a), the bulkcontribution is typically generated in a subsurface layerwith a thickness of d ∼ λ/(2π). The ratio of the sur-face part of the total reflected SHG signal to the bulkcontributions in this case is on the order of d2/a2,which can be easily made much larger than unity. Thisratio can be substantially enhanced on frequency res-onances or with an appropriate choice of polarizationarrangement.

A combination of the high spatial and temporal res-olution with a spectral selectivity makes χ(2) techniquesa powerful tool for time-resolved species-selective stud-ies of surfaces and buried interfaces (see, e.g., [5.204] fora comprehensive review of recent results), detection andsize and shape analysis of adsorbed species, nanopar-ticles, and clusters on surfaces [5.205, 206], as well asimaging and microscopy of biological species [5.207].The sensitivity and selectivity of the χ(2) techniqueare enhanced when the frequency of one of the laserfields (ω1 in the inset to Fig. 5.35b) is tuned to a reso-nance with a frequency of one of the vibrations typicalof species on a surface or an interface (Fig. 5.35b).This method of surface analysis is referred to assum-frequency surface vibrational spectroscopy. Thecapabilities of this technique have been impressively

Fig. 5.35 Nonlinear optics, spectroscopy, and imaging ofsurfaces and buried structures. (a) Nonlinear-optical prob-ing of surfaces and interfaces using second-harmonicgeneration. (b) Surface vibrational spectroscopy using sum-frequency generation. Diagram of vibrational transitionsprobed by SFG is shown on the right. (c) Nonlinear mi-croscopy based on coherent anti-Stokes Raman scattering.Diagram of Raman-active transitions selectively addressedthrough CARS is shown on the left

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demonstrated for vapor–liquid and liquid–solid inter-faces [5.205].

In sum-frequency surface vibrational spectroscopy[5.207] an infrared laser pulse E1 with a frequency ω1overlaps on the surface of a sample with the secondlaser pulse E2, which typically has a frequency ω2 inthe visible, to induce a second-order polarization at thesum frequency ωSF = ω1 + ω2:

P(2)SF (ωSF) =�χ(2) (ωSF; ω1, ω2) : E1E2 . (5.226)

The intensity of the optical signal at the sum frequencyis given by

ISF ∝∣∣∣χ(2)

eff

∣∣∣2 I1 I2 . (5.227)

Here, I1 and I2 are the intensities of the laser beams and

χ(2)eff = (�LSF · eSF

) ·�χ(2)s : (�L2 · e2

) (�L1 · e1)

, (5.228)

where �L1, �L2 and �L3 are the Fresnel factors at thefrequencies ω1, ω2, and ωSF, respectively, e1, e2, andeSF are the unit polarization vectors of the laser andsum-frequency fields, and the surface quadratic suscep-tibility �χ(2)

s is written as

�χ(2)s =�χ(2)nr +

∑q

�aq

ω1 −ωq + iΓq, (5.229)

with �χ(2)nr being the nonresonant quadratic susceptibil-ity and�aq , ωq , and Γq being the strength, the frequency,and the damping constant for the q-th vibrational mode,respectively.

When the infrared field is scanned over the frequencyof the q-th vibrational mode, the SFG signal is reso-nantly enhance, and its spectrum gives the spectrum ofthe vibrational mode. Tunable dye lasers [5.204], opti-cal parametric oscillators and amplifiers [5.205], or PCFfrequency shifters [5.185,208] are employed as sourcesof frequency tunable radiation, allowing selective prob-ing of vibrational (as well as electronic and excitonic)transitions in molecules and molecular aggregates.

Through the past few years, nonlinear-optical meth-ods of surface spectroscopy have been extensivelyinvolved in the rapid growth of nonlinear microscopictechniques based on χ(2) and χ(3) processes. In particu-lar, SHG and THG processes have proven to be conve-nient techniques for high-resolution three-dimensionalmicroscopy of biological objects [5.207,209], as well aslaser-produced plasmas and micro-explosions [5.210].In early experimental demonstrations of SHG mi-croscopy, grain structures and defects on the surface

of thin films were visualized using SHG in transmis-sion [5.211] and surface monolayers were imaged byreflection-geometry SHG microscopy [5.212]. In recentyears, progress in laser technologies and the advent ofnew-generation imaging and scanning systems made itpossible to extend nonlinear microscopy techniques tothree-dimensional structures, buried objects, and biolog-ical tissues [5.207,209]. In this modification of nonlinearmicroscopy, the nonlinear signal is generated in thefocal region of the laser beam in the bulk of a sam-ple, originating from optical micro-inhomogeneities,which break the point-group symmetry of the mediumor change phase-matching conditions. In two transversedimensions, the high spatial resolution of nonlinear mi-croscopy techniques is controlled by the nonlinear natureof the process, tightly confining the area where the sig-nal is generated to the focal region. Resolution in thedirection of probing is achieved either due to symmetrybreaking, similar to nonlinear-optical surface-imagingtechniques, or through phase-matching modification.

In CARS microscopy [5.207,209,213,214], the non-linear signal is resonantly enhanced when a frequencydifference between two laser fields is tuned to a Raman-active mode of molecules under study, as described inSect. 5.4.8 (see also Fig. 5.35c). This makes microscopyalso species-selective as Raman resonances serve asfingerprints of a certain type of molecules or molecu-lar aggregates. Forward CARS and backward CARS(also called epi-CARS) geometries have been devel-oped (Fig. 5.35c). In the forward-CARS microscopy,the mismatch |Δk| of the wave vectors involved inwave mixing (Sect. 5.7) is typically much smaller than2π/λCARS, where λCARS is the wavelength of the CARSsignal. For epi-CARS, |Δk| is larger than 2π/λCARS.The intensity of the epi-CARS signal can thus be com-parable with the forward-CARS signal intensity onlyfor a thin sample with a sample thickness d meetingthe inequality |Δk|d < π. CARS microscopy has beenone of the most successful recent developments in thefield of nonlinear-optical microscopy. In addition to thehigh spatial resolution, this technique has a numberof other important advantages over, for example, mi-croscopy based on spontaneous Raman scattering. Inparticular, in CARS microscopy, laser beams with mod-erate intensities can be used, which reduces the risk ofdamaging biological tissues. The anti-Stokes signal inCARS microscopy is spectrally separated from the laserbeams and from fluorescence, as the anti-Stokes wave-length is shorter than the wavelengths of the laser beams.In transparent materials, CARS microscopy can be usedto image tiny buried objects inside the sample in three

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Nonlinear Optics 5.11 High-Order Harmonic generation 63

dimensions. Due to the coherent nature of CARS, thecapabilities of CARS microscopy can be enhanced by

means of coherence control, as recently demonstratedby Dudovichet al. [5.117].

5.11 High-Order Harmonic generation

5.11.1 Attosecond Metrology –Historical Background

The invention of the laser in 1960 opened many newfields of research. One of them is nonlinear optics trig-gered by Franken [5.215] with the first demonstrationof frequency doubling in a crystal (1961) and pioneeredby Bloombergen and coworkers. Third-harmonic gen-eration in gases was observed for the first time by Newand Ward in 1967 [5.216], fifth and higher-order har-monic generation a few years later by Reintjes and others[5.217]. The main objectives of this research were toincrease the conversion efficiency, to cover a large (con-tinuous) spectral range with, for example, frequencymixing processes and to reach very short wavelengths.The most natural route for the latter goal was to use fun-damental fields with the shortest possible wavelength, toproduce short-wavelength radiation through a low-ordernonlinear-optical process.

Another research area, which started in the late six-ties, is the study of atoms in strong laser fields. Theobjective of this fundamental research was simply to un-derstand the behavior of atoms and molecules exposedto intense electromagnetic fields. This field evolved inparallel with the development of pulsed lasers towardsincreasing peak powers, increasing repetition rates anddecreasing pulse durations. The character of the laser–atom interaction also evolved from being essentiallyperturbative for laser intensities below 1013 W/cm2 tostrongly nonperturbative for higher intensities. for manyyears, this regime of nonlinear optics was studied only bylooking at ionization processes. The number and chargeof the produced ions, as well as the energy and angularemission of the electrons, were experimentally detectedand compared with theoretical predictions.

At the end of the 1980s, it was realized that look-ing at the emitted photons would bring complementaryinformation on the physical processes taking place. In-deed, efficient photon emission in the extreme ultraviolet(XUV) range, in the form of high-order harmonics ofthe fundamental laser field, was observed for the firsttime in 1987, in Saclay [5.218] (33rd harmonic ofa Nd:YAG laser) and in Chicago [5.219] (17th harmonicof a KrF laser). The harmonic spectra were character-ized by a decrease in efficiency for the first harmonics,

followed by a broad plateau of nearly constant con-version efficiency, ending up by an abrupt cutoff. Theexistence of such a plateau was clearly a nonpertur-bative signature of the laser-atom interaction. Most ofthe early work concentrated on the extension of theplateau, i. e. the generation of harmonics of higherfrequency and shorter wavelength going progressivelyfrom ≈ 20 nm at the end of the 1980s to ≈ 7 nm bythe middle of the 1990s [5.220–224]. It was soon re-alized that, in contrast to the ideas promoted in thenonlinear-optics community, the shortest wavelengthswere obtained with long-wavelength lasers. Today, har-monic spectra produced with short and intense laserpulses extend to the water window (below the carbonK-edge at 4.4 nm) [5.225,226].

A breakthrough in the theoretical understanding ofhigh-order harmonic generation processes was initi-ated by Krause and coworkers [5.227] who showedthat the cutoff position in the harmonic spectrum fol-lows the universal law Ip +3Up, where Ip is theionization potential, whereas Up = e2 E2/4mω2, is theponderomotive potential, i. e. the mean kinetic en-ergy acquired by an electron oscillating in the laserfield. Here e is the electron charge, m is its mass,and E and ω are the laser electric field and its fre-quency, respectively. An explanation of this universalfact in the framework of a simple semiclassical the-ory was found shortly afterwards [5.228, 229], andconfirmed by quantum-mechanical calculations [5.230,231].

Progress in experimental techniques and theoret-ical understanding stimulated numerous studies ofharmonic generation. The influence of the laser po-larization was investigated in great detail [5.232–239](see also [5.240–242] for the theory). The nonlinearconversion process was optimized with respect to thelaser parameters [5.243, 244], and to the generatingmedium [5.245–250]. Finally, the spatial [5.251–256]and temporal [5.257–263] properties of the radiationwere characterized and optimized. The specificationsof the harmonic emission (ultrashort pulse duration,high brightness, good coherence) make it a uniquesource of XUV radiation, used in a growing number ofapplications ranging from atomic [5.264, 265] and mo-lecular [5.266–268] spectroscopy to solid-state [5.269–

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271] and plasma [5.272–274] physics. Finally, it has re-cently been demonstrated that the low-order harmonicsare intense enough to induce nonlinear optical processesin the XUV range [5.275–277].

Almost immediately after the first observation of theharmonic plateau, it was realized that, if the harmonicswere emitted in phase, i. e. phase-locked, the tempo-ral structure of the radiation emitted from the mediumwould consist of a “train” of attosecond pulses sep-arated by half the laser period [5.278, 279]. There isa clear analogy here with mode-locked lasers, where ax-

Photons (arb. units)

Energy (eV)50

10

8

6

4

2

060 70 80

GasjetLens

Laserpulse ...

H31

H37 H43

H49

H53

ω3ω

5ω

ial modes oscillating in a laser cavity are locked in phase,leading to the production of trains of short pulses. At-tosecond pulses have remained, however, a theoreticalprediction until recently [5.280–282]. A first possibleindication of harmonic radiation containing an attosec-ond sub-structure was in a high-order autocorrelationof the driving laser pulse [5.283]. This was followedby a beautiful experiment showing evidence for phase-locking between five consecutive harmonics generatedin argon, thus indicating that trains of 250 as pulses wereformed [5.284]. In a series of experiments performed inVienna, single pulses of duration of a few hundred at-toseconds were demonstrated by using ultrashort (5 fs)laser pulses and spectrally filtering a few harmonics inthe cutoff spectral region [5.285,286]. These experimen-tal results are the beginning of a new field of researchattophysics, where processes in atoms and moleculescan be studied at an unprecedented time scale.

The purpose of this section is to present to the nonex-pert reader a simple description of high-order-harmonicgeneration, and its application to attosecond metrology.In Sect. 5.11.2, we describe the most important aspectsof high-order-harmonic generation processes. We be-gin with a short description of the experimental set upneeded to obtain high-order harmonics. Then we discussthe microscopic and macroscopic physics underlyingthe generation of high-order harmonics. We focus onthe physics of importance for the generation of attosec-ond pulse trains and single attosecond pulses. We referthe reader to several review articles for a more com-plete overview of this research topic [5.245, 287, 288].In Sect. 5.12, the emerging field of attosecond scienceis reviewed. The different measurement techniques aredescribed and the first application of attosecond pulsesis presented.

5.11.2 High-Order-Harmonic Generationin Gases

Experimental MethodGenerating high-order harmonics is experimentally sim-ple. A typical set up is shown in Fig. 5.36. A laser pulse

Fig. 5.36 Schematic representation of a typical experimen-tal setup for high-order-harmonic generation. An intenseshort pulse laser is focused into a vacuum chamber contain-ing a gas medium. Harmonics are emitted along the laserpropagation axis. A photograph of a gas target is shownat the bottom, on the left. A typical spectrum obtained inneon is shown on the right. It shows a plateau ending witha sharp cutoff

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Harmonic conversion efficiency

Harmonic order30 300

10–6

10–7

10–8

10–9

10–10

10–11

10–12

100

Intensity

Order

Ionisation energy Ip

Ar

Ne

He

Dephasing limited

OxygenK-edge

CarbonK-edgeAbsorption limited

2He4.003

10Ne20.18

18Ar

39.95

36Kr

83.80

54Xe131.3

Fig. 5.37 Illustration of the influence of the gas species on the generation efficiency (left). The result presented at theright has been obtained with 5 fs 800 nm laser pulses (after [5.302])

is focused into a vacuum chamber containing a rathersmall gas target with an atomic pressure of at leasta few mbar. The harmonics (only of odd order, ow-ing to inversion symmetry) are emitted along the laserpropagation axis. Many types of lasers have been used,excimers, Nd:YAG, Nd:glass, Ti:sapphire, dye lasers,etc. [5.218,219,222,223,289,290]. In addition, the sec-ond harmonic of these lasers [5.291–293] as well asthe radiation from sum- or difference-frequency mix-ing processes [5.294, 295] for example to get into themid-infrared range [5.296], have also been employed.Typical energies are between a fraction to a few tensof mJ, typical pulse durations are from a few femtosec-onds to a few tens of picoseconds. In the last five years,the favorite tool has become the Ti:sapphire laser, pro-viding very short pulse lengths, high laser intensities athigh repetition rates. The shortest laser pulses used to-day to study atoms in strong laser fields are about 5 fs(two cycles) long [5.285]. The advantage of using shortpulses is that atoms get exposed to higher laser intensitybefore they ionize, leading to higher-order harmonics.The different parameters of the laser pulses, such as thepolarization, the focusing characteristics, the spatial andtemporal profiles are often varied and optimized in theexperiments. Recent studies do not simply vary a givenparameter, but attempt to shape a laser pulse (by varying,for example, its phase [5.244], its degree of elliptic-ity [5.297–299], or its spatial properties [5.300,301]) totailor the harmonics for different applications.

The gas medium is provided by a gas jet, hollowfiber [5.246–248] or a (small) gas cell [5.249,250,303].Figure 5.36 shows a photograph of such a gas tar-get containing argon atoms irradiated by an intense

laser pulse. Rare gases are the favored species, for ob-vious technical reasons. In addition, some work hasbeen done with alkali atoms [5.296], ions [5.220, 304],molecules [5.305–308] and clusters [5.309–311]. Pho-tons are separated in energy and detected by an XUVspectrometer, including a grating, sometimes a re-focusing mirror, and a detector (electron multiplier,microchannel plates, etc.). A typical experimental spec-trum is shown in Fig. 5.36. This result has been obtainedin a gas of neon using a 100 fs 800 nm Ti:sapphirelaser [5.312]. It shows odd harmonic peaks up to the53rd order, with a rapid decrease beyond the 49th har-monic, characteristic of the cutoff region. The spectralrange of the harmonic emission as well as the conver-sion efficiency depends strongly on the gas medium.As schematically illustrated in Fig. 5.37 (top), the effi-ciency is highest in the heavy atoms Ar, Xe, Kr, butthe highest photon energies are obtained in He andNe [5.222, 245, 302]. Figure 5.37 (bottom) presents

Laser AtomHarmonicemissiom

Coherentsuperposition

MediumLaser

Fig. 5.38 Illustration of the two aspects of high-order-harmonic generation. (top) Harmonic emission froma single atom (bottom). Phase matching in the nonlinearmedium

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a comparison between the generation efficiency for threerare gases Ar, Ne, He, when ultrashort laser pulses (ofduration 5 fs) are used for the excitation [5.245,302].

As illustrated in Fig. 5.38, two conditions are neces-sary to observe efficient harmonic emission. First, eachindividual atom must generate light at these frequencies,requiring a highly nonlinear response to the radiationfield. Second, the harmonic field results from the co-herent superposition of all the emitting atoms in themedium. Harmonic generation will be efficient only ifphase matching is achieved, requiring the generated fieldto be in phase with the nonlinear polarization creating itover the medium’s length. We discuss these two aspectsin more details in the following sections.

5.11.3 Microscopic Physics

Electrons in an atom in the presence of a time-dependentradiation field oscillate. This is described by a dipolemoment d(t) = 〈Φ(t)|er|Φ(t)〉, where |Φ(t)〉 is the time-dependent electronic wavefunction, the solution of theSchrödinger equation. When the radiation field is weak,there is mainly one oscillation frequency, that of thefield. In a strong radiation field, the oscillatory motionbecomes distorted and the dipole moment now includesa series of higher-order frequencies, odd harmonics ofthe fundamental one. The harmonic emission from a sin-gle atom can thus be calculated by taking the Fouriertransform of the dipole moment. Theorists often use asingle-active-electron approximation, assuming that theinteraction with the field involves essentially one activeelectron, to describe the response of the atom to a stronglaser field. A number of methods have been proposedto solve this problem and it is beyond the scope of thispaper to review all of these approaches. The most re-alistic approach is probably the numerical solution ofthe time-dependent Schrödinger equation, pioneered byKulander at the end of the 1980s [5.313]. Many impor-tant results, such as the determination of the cutoff lawfor high-order harmonics in 1992 [5.227] or proposalsfor single attosecond pulse generation using few-cyclelaser pulses at the end of the 1990s [5.245,314,315] wereobtained directly from numerical calculations. The semi-classical strong field approximation, originating froma seminal paper of Keldysh in 1964 [5.316], applied byLewenstein in the 1990s to high-order-harmonic gener-ation [5.230, 231] allows to explore a larger parameterspace as well as to gain intuitive insight. This modeland a related one approximating the atomic potential bya δ(r) potential [5.240] have been extensively used tointerpret experimental results.

r

V(r)

5ω ω

r

V(r)

Fig. 5.39 Schematic picture of different regimes for har-monic generation. The multiphoton, perturbative regimeis illustrated on the top, while the semiclassical model ispresented on the bottom

Many insights in the physical understanding of theinteraction between atoms and strong laser fields havebeen provided by a simple semiclassical model, pro-posed first by Van der Linden, van der Heuvell, andMuller in the context of above-threshold ionizationand extended by Corkum and others [5.228, 229] tomultiple ionization and high-order-harmonic genera-tion. According to this model, illustrated in Fig. 5.39(right), the electron tunnels through the Coulomb en-ergy barrier modified by the presence of the relativelyslowly varying linearly polarized electric field of thelaser. It then undergoes (classical) oscillations in thefield, during which the influence of the Coulomb forcefrom the nucleus is practically negligible. If the elec-tron comes back to the vicinity of the nucleus, it may berescattered one or several times by the nucleus, thusacquiring a high kinetic energy, and in some cases,kicking out a second or third electron. It may alsorecombine back to the ground state, thus producinga photon with energy Ip, the ionization potential, plusthe kinetic energy acquired during the oscillatory mo-tion. We also show in Fig. 5.39 (left) for comparison themore traditional harmonic-generation process based on

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multiphoton absorption in the (barely perturbed) atomicpotential.

An intuitive understanding of some of the propertiesof harmonic generation can be gained by elementaryclassical calculations of the electron motion outside thebinding potential. Assuming the electron to have zerovelocity immediately after it has tunneled through thepotential barrier at time t = t0, and the laser field to besimply described by E = E0 sin(ωt), we obtain:

v = −v0 cos(ωt)+v0 cos(ωt0) , (5.230)

x = −v0

ωsin(ωt)

+ v0

ωsin(ωt0)+ (t − t0)v0 cos(ωt0) , (5.231)

where v0 = qE0/mω. Depending on the time at whichthe electron is released into the continuum (t0), it will fol-low different trajectories, as illustrated in Fig. 5.40. Onlythose electrons released between T/4 and T/2 (where T

τ2

τ1

Distance from the nucleus (arb. units)

Time0

Return energy (Up)

Release time (optical cycle)0.2 0.5

3.173

2

1

0

1.0

0.5

0.00.40.3

Time in the continuum (optical cycle)

T/4 T/2

Electric field

Time in the continuum

Return energy

Fig. 5.40 (Top) Electron trajectories in the continuum cor-responding to different release times. The laser electric fieldis represented in dotted line. (Bottom) Kinetic energy (solidline) and time in the continuum as a function of the releasetime

is the laser period) are of interest for harmonic gener-ation. When the laser field changes its sign, they comeback towards the core (at x = 0) with a certain kineticenergy. This energy, which determines the emitted har-monic order, is proportional to the square of the slopeof the trajectory as it crosses the time axis [open cir-cles in Fig. 5.40 (top)]. Except for the trajectory startingat approximately 0.3T , giving the highest kinetic en-ergy (cutoff), there are two (main) trajectories leadingthe same kinetic energy. This is illustrated in Fig. 5.40(bottom), showing the kinetic energy when the elec-tron returns to the core (solid line), as well as the timespent in the continuum (dashed line), as a function ofrelease time. As shown in the figure, for each energy,and hence for each harmonic order, there are mainly two

Dipole strength (arb. units)

Intensity (1014 W/cm2)1 5

10–10

10–12

10–14

10–16

10–18

0

–20

–40

–60

–80

–1002 3 4

Dipole phase (rad)

Intensity (1014 W/cm2)

α (10–14 cm2/W)–20 70

5

4.5

4

3.5

3

2.5

2

1.5–10 0 10 20 30 40 50 60

Fig. 5.41 (Top) Single atom response within the strong-field approximation. Intensity (dark line) and phase (lightline) of the 35-th harmonic in neon as a function of thelaser intensity. (Bottom) Quantum path analysis of the sameharmonic

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trajectories, a short and a long one, contributing to theharmonic emission. The periodicity (for a pulse severalcycles long) of the process implies that the light emis-sion is not continuous but at discrete (odd-harmonic)frequencies.

The influence of the complex electron dynamicsinherent to the harmonic-generation process is clearlyvisible on the intensity dependence of the harmoniccomponents of the quantum-mechanical dipole moment.Figure 5.41 shows for example the 35-th harmonicgenerated in neon, calculated within the strong field ap-

Coherence time (fs)

Harmonic order17 31

60

50

40

30

20

1019 21 23 25 27 29

Phase-lockedgenerating beams

Detectora)

b)

c)

Fig. 5.42a–c Spatially-resolved temporal coherence mea-surements of high-order harmonics. The principle of themeasurement is shown in (a). Two images obtained on the13-th harmonic in xenon obtained for two different time de-lays (0 and 25 fs) between the two pulses are shown in (b).A summary of the measured coherence times in the twospatial regions for high harmonics in xenon is presentedin (c)

proximation. The intensity and phase are representedrespectively in solid and dashed line as a function ofthe laser intensity. The sharp intensity and phase varia-tion at low intensity corresponds to the cutoff region.In the plateau, oscillations are clearly visible in thevariation of both the intensity and the phase. They origi-nate from interference effects between the contributionsfrom the multiple trajectories. This fascinating conclu-sion stimulated the development of analysis techniquesto extract the contributing electron trajectories (or ratherthe relevant quantum paths) from quantum-mechanicalapproaches [5.317, 318]. The result of such an analysisfor the 35-th harmonic in neon is shown at the bottom inFig. 5.41. The contributions to the dipole phase varyinglinearly with the intensity φdip(I) = α j I , correspondingto a quantum path ( j), are identified as vertical lines inFig. 5.41. In this case, the dominant trajectory is the “sec-ond” one, with a coefficient α2 = 24×10−14 cm2 W. Atlow intensity, in the cutoff region, there is only one quan-tum path. Note that here, the predictions of the strongfield approximation differ somewhat from those of thetime-dependent Schrödinger equation [5.318].

The microscopic physics (the quantum orbits) be-hind high-order-harmonic generation was shown ina series of experiments investigating the temporal co-herence of the harmonics [5.319–321]. We present inFig. 5.42 the principle as well as some results obtainedin xenon [5.320]. Two phase-locked spatially sepa-rated harmonic sources are created by splitting the laserinto two replicas in a Michelson interferometer and byslightly misaligning one of the arms. A variable timedelay can be introduced between the two pulses. Thegenerated harmonics are separated by a grating and in-terfere in the far field. The variation of the contrast ofthe fringes as a function of time delay gives the coher-ence time. Figure 5.42 (bottom) shows images obtainedfor the 13-th harmonic in xenon. (For experimental rea-sons, the images are not symmetrical.) These imagespresent two spatial regions with different coherencetimes. The central region exhibits a long coherence time,whereas the outer region a much shorter one. Figure 5.42(right) summarizes the measurements on the harmonicsgenerated in xenon.

These results can be interpreted in a simple wayby recalling that the harmonic field is a sum of thecontributions from each quantum path j

Eq(r, t) =∑

j

A j (r, t)e−[iqωt+α j I (r,t)] (5.232)

where A j is an envelope function, representing thestrength of each path. The temporal variation of the laser

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intensity I(t) will produce a chirp in the emitted field,and consequently a spectral broadening (or a reducedcoherence time). The radial variation of the laser inten-sity I(r) will affect the curvature of the phase front of theharmonics, and therefore their divergence. The contribu-tion from the quantum paths with a long excursion timein the continuum (Fig. 5.40), resulting in a large α j , willhave a short coherence time and a pronounced curvaturein the far field, whereas that from the quantum paths witha short excursion time will have a longer coherence timeand be more collimated [5.322]. Similar evidence forthe existence of (at least) two quantum paths has beenobtained by analyzing the harmonic spectra in combina-tion with measuring the harmonic pulse duration [5.321].Contributions of different quantum paths can be selectedmacroscopically, either by spatial or spectral filtering.

5.11.4 Macroscopic Physics

We now turn to the second aspect of harmonic gener-ation, namely the response of the whole medium. Toachieve phase matching, i. e. to ensure efficient conver-sion, the wave vector difference (traditionally called thephase mismatch) between the harmonic wave and thepolarization must be minimized, so that the phase differ-ence varies as little as possible over the medium’s length.For an incident Gaussian beam, this phase difference onthe propagation axis (z) is given by

δφq = φq −φpolq

≈ kqz −qk1z +q arctan(2z/b)−φdip . (5.233)

In this equation, the first two terms characterize thedifference in dispersion at the fundamental and q-th har-monic frequencies, mainly due to the effect of the freeelectrons in the medium. The third one is a geometricalterm originating from the Gouy phase shift of a Gaus-sian beam with confocal parameter b across the focus.The fourth term is the dipole phase described above.According to the previous discussion, it is more phys-ically correct to consider separately the contributionsfrom the different quantum paths, before making a (co-herent) sum. The phase difference to minimize dependson the trajectory and is given by

δφq, j = φq −φpolq

≈ kqz −qk1z +q arctan(2z/b)−α j I(r, z, t) .

(5.234)

Figure 5.43 illustrates the variation of δφq along thepropagation axis (z). Most contributions are monotonic.Focusing as well as the free electron dispersion (not

δΦq (rad)

z (mm)–10 10

120

100

80

60

40

20

0

–20

–40–5 50

r (μm)

z (mm)–10 10

6040200

–20–40–60

10.80.60.40.20

–5 0 5

L coh (mm)

Fig. 5.43 Phase difference between the harmonic wave andthe polarization for the 35-th harmonic generated in neon.In dashed line, we show the term coming from the geo-metrical phase shift of a Gaussian beam and in dotted line,the term coming from the dipole phase. The laser intensityis 6× 1014 W/cm2. The confocal parameter is chosen to beb = 1 cm

shown in Fig. 5.43) lead to a contribution to δφq in-creasing with z, while the (normal) dispersion leads toa term decreasing with z, usually quite small. In con-trast, the contribution of the dipole phase, here duemainly to the second trajectory, with a transition to thecutoff for |z| ≥ 7.5 mm, first increases, then decreaseswith z. In the particular case shown in Fig. 5.43, phasematching on the propagation axis is best realized forz ≥ 7 m, requiring the medium to be located after thefocus. In general, the situation can be rather compli-cated. Methods have been developed to visualize where(in the nonlinear medium) and when (during the laserpulse) phase matching was best realized [5.317], usingin particular three-dimensional maps representing thelocal coherence length, defined as

Lq = π

(∂δφq

∂z

)−1

. (5.235)

An example of such a map for the situation correspond-ing to Fig. 5.43 (top) is shown at the bottom of the same

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figure. The white areas indicate where phase-matchingis best realized. The two parts of the figure correspondto the transition between cutoff and plateau.

The brief analysis presented above can easily be gen-eralized to include propagation in waveguides [5.246–248], which modifies somewhat the phase-matchingconditions, since the geometrical and dipole phase ef-fects are much reduced (owing to a constant intensityin the waveguide). Absorption in the nonlinear mediumstarts to play a role for high conversion efficiencies andlong medium lengths and/or high pressures. The op-timization of phase matching of high-order-harmonicgeneration has stimulated a great deal of efforts dur-ing the last five years. The so-called absorption limit,where the limitation on the conversion efficiency is dueto absorption, and not to the coherence length, has beenreached in different wavelength ranges [5.249,250,302].Recently, the use of extremely long focusing geometries(which minimize both the effect of the dipole phase andthat of the Gouy phase shift) has led to conversion ef-ficiencies as high as a few times 10−5 and energies inthe μJ range [5.303,323]. Phase matching of high-order-harmonic generation is by no means a solved issue, sinceit is a complicated three-dimensional problem, involv-ing a number of parameters (laser focusing, pressure,medium length and geometry, laser intensity). In add-ition, the problem is quite different for the low-orderharmonics with energy around a few tens of eV, wheremost of the work has been done, and for the high-order harmonics at 100 eV or more. An interesting idea,similar to quasi-phase-matching has recently been in-vestigated [5.324]. A modulated waveguide, used forthe generation of high-order harmonics, induces a pe-riodic variation of the degree of ionization leading toenhanced conversion efficiency, especially for the highharmonics.

Finally, a conclusion of importance for the gen-eration of attosecond pulse trains is that the differentquantum paths contributing to harmonic generation dis-cussed in the previous section are not phase matchedin the same conditions (5.234). The axial variation ofthe laser intensity I(z) leads to different phase match-ing conditions for the contributions from the differentquantum paths. In other words, depending on the ge-ometry, ionization, pressure conditions, phase matchingwill enhance one of the contributing trajectories to thedetriment of the others. This conclusion is extremelyimportant for the generation of attosecond pulse trains.As illustrated in Fig. 5.44 (left), the electron trajectoriescontributing to harmonic generation in the single-atomresponse lead to bursts of light at different times during

|eq | 2

0 Time (2π/ω0)0.5 1

|Eq | 2

0 Time (2π/ω0)0.5 1

eq

Esinω0t

τ1 τ2+

τ1

τ2

τ1

300 as

Fig. 5.44 Illustration of the temporal structure of a fewharmonics in the single atom response (top, middle). Insome conditions, phase matching selects only one burst oflight per half cycle, leading to phase locking (bottom)

the laser half-cycle. Phase locking between consec-utive harmonics is not realized. In some conditions,however, phase matching results in efficient generationof only one of these contributions [Fig. 5.44 (right)][5.325, 326], leading to a train of attosecond pulses.Another possibility, that makes use of the differentspatial properties of the contributing trajectories, is toselect the shortest trajectory by spatially filtering theharmonic beam with an aperture. Finally, a spectralfilter can be used to select only the harmonics in thecutoff region, where the electron dynamics is much sim-pler, with only one electron trajectory in a single-atomresponse.

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[5.330]. With this tool it should be possible to preciselycontrol the motion of energetic electron wave packetsaround atoms on attosecond timescales just as the motionof nuclear wave packets in molecules can be controlledwithin a few femtoseconds. The single sub-femtosecond

electron bunches and (XUV/X-ray) photon bursts thatarise from the recently gained ability to control elec-tron wave packets will enable the scientific communityto excite and probe atomic dynamics on atomics timescales.

References

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5.3 S. I. Vavilov: Microstructure of Light (USSR Acad. Sci.,Moscow 1950)

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5.9 Y. R. Shen: The Principles of Nonlinear Optics (Wiley,New York 1984)

5.10 D. Cotter, P. N. Butcher: The Elements of NonlinearOptics (Cambridge University Press, Cambridge 1990)

5.11 J. F. Reintjes: Nonlinear Optical Parametric Processesin Liquids and Gases (Academic, Orlando 1984)

5.12 Th. G. Brown, K. Creath, H. Kogelnik, M. A. Kriss,J. Schmit, M. J. Weber: The Optics Encyclopedia, ed.by Th. G. Brown, K. Creath, H. Kogelnik, M. A. Kriss,J. Schmit, M. J. Weber (Wiley-VCH, Weinheim 2004)Chap. Nonlinear Optics, p. 1617

5.13 R. W. Terhune, M. Nisenoff, C. M. Savage: Mixing ofdispersion and focusing on the production of opticalharmonics, Phys. Rev. Lett. 8, 21 (1962)

5.14 J. A. Giordmaine: Mixing of light beams in crystals,Phys. Rev. Lett. 8, 19 (1962)

5.15 N. Bloembergen, J. Ducuing, P. S. Pershan: Interac-tions between light waves in a nonlinear dielectric,Phys. Rev. 1217, 1918 (1962)

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5.17 R. B. Miles, S. E. Harris: Optical third-harmonic gen-eration in alkali metal vapors, IEEE J. QuantumElectron. 9, 470 (1973)

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5.20 T. Brabec, F. Krausz: Intense few-cycle laser fields:Frontiers of nonlinear optics, Rev. Mod. Phys. 72, 545(2000)

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5.22 N. Akhmediev, M. Karlsson: Cherenkov radiationemitted by solitons in optical fibers, Phys. Rev. A51, 2602–2607 (1995)

5.23 G. P. Agrawal: Nonlinear Fiber Optics (Academic,Boston 1989)

5.24 N. I. Koroteev, A. M. Zheltikov: Chirp control in third-harmonic generation due to cross-phase modulatio,Appl. Phys. B 67, 53–57 (1998)

5.25 S. P. Le Blanc, R. Sauerbrey: Spectral, temporal, andspatial characteristics of plasma-induced spectralblue shifting and its application to femtosecondpulse measurement, J. Opt. Soc. Am. B 13, 72 (1996)

5.26 M. M. T. Loy, Y. R. Shen: Theoretical interpretationof small-scale filaments of light originating frommoving focal spots, Phys. Rev. A 3, 2099 (1971)

5.27 S. A. Akhmanov, A. P. Sukhorukov, R. V. Khokhlov:Self-focusing and diffraction of light in a nonlinearmedium, Sov. Phys. Usp. 93, 19 (1967)

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5.29 G. Fibich, A. L. Gaeta: Critical power for self-focusingin bulk media and in hollow waveguides, Opt. Lett.25, 335 (2000)

5.30 G. Fibich, F. Merle: Self-focusing on bounded do-mains, Physica D 155, 132 (2001)

5.31 K. D. Moll, A. L. Gaeta, G. Fibich: Self-similar opticalwave collapse: observation of the townes profile,Phys. Rev. Lett. 90, 203902 (2003)

5.32 S. O. Konorov, A. M. Zheltikov, A. P. Tarasevitch,Ping Zhou, D. von der Linde: Self-channeling of sub-gigawatt femtosecond laser pulses in a ground-statewaveguide induced in the hollow core of a photoniccrystal fiber, Opt. Lett. 29, 1521 (2004)

5.33 R. A. Fisher: Optical Phase Conjugation (Academic,New York 1983)

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