nonlinear optics: brief background

IPT 544000-Selected Topics in Ultrafast Optics

Chen-Bin Huang 1/22

Second-Order Ultrafast Nonlinear Optics1

Foreword:

Nonlinear optical effects play a very important role in ultrafast optics. We will first

introduce the formalism of nonlinear optics and then focus on second order nonlinear

processes, including second harmonic generation (SHG), sum frequency generation

(SFG), difference frequency generation (DFG), and optical parametric amplification

(OPA). These processes are important for conversion of short pulses from

mode-locked lasers to new frequency ranges and, particularly for the case of second

harmonic generation, for pulse measurements.

Nonlinear optics: brief background

Maxwell's equations in vacuum are linear. Nonlinear effects arise in materials, since

the material response may be nonlinear in the applied field. The nonlinear material

response can couple back to the optical field, giving rise to nonlinear optics. For a

nonlinear medium, the electric flux density (sometimes referred as the electric

displacement field) is expressed as

PED 0 , (1)

where the polarization density is expressed in the time-domain (in scalar form) as

...)( 3)3(2)2()1(0 EEEP , (2)

where (k) denotes the k-th order susceptibility.

In general, we can have two input electric fields at different frequencies

.].)(~

)(~

[2

121

21 cceEeEE tjtj . (3)

1 Special acknowledgement to Prof. S.-D. Yang for his notes covering a majority of this topic.


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Insert this back to Eq. (2) and only look at the term quadratic with fields,

.}.

*)(~

)(~

),:0(*)(~

)(~

),:0(

*)(~

)(~

),:(2

)(~

)(~

),:(2

)(~

)(~

),:2(

)(~

)(~

),:2({4

2222)2(

1111)2(

)(212121

)2(

)(212121

)2(

222222

)2(

211111

)2(0)2(

21

21

2

1

cc

EEEE

eEE

eEE

eEE

eEEP

tj

tj

tj

tjNL

(4)

we can see that more frequency components are generated through the field

interaction with the nonlinear polarization.

Now in vectorial form, we may write the optical field and polarization as

.].)(~

)(~

[ˆ2

1 3

121

21

i

tji

tjii cceEeEu E (5a)

.].)(~

[ˆ2

1 3

1

)(21

)2( 21

i

tjiiNL ccePu P , (5b)

where

)(~

)(~

),:(2

)( 2

3

1,12121

)2(021

)2(

kkj

jijki EED

P

. (6)

and D is the degeneracy factor (D=2 for sum-frequency generation, while D=1 for

second harmonic generation).

Comments:

1. We have assumed that the process is off-resonance, so that the nonlinear

susceptibility can be taken as real.

2. We have assumed the medium is dispersionless, so the nonlinear

susceptibility is frequency independent.

3. Nonlinear polarization is having a fast response, and the nonlinear processes


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are treated as instantaneous.

4. When crystal symmetries are considered, many elements within the dielectric

tensor reduce to zero. For crystal showing inversion symmetry, second order

susceptibility is essentially zero.

5. It is common to use the engineering notation: )(2

2

1ijkijkd . For common

materials, dijk ranges from {10-13 to 10-11} m/V.

Forced wave equation

We now express the electric flux density of a nonlinear medium as

NLPED )(1 , (7)

where the EPEE 02

)1(0)1( n and NLP represent the linear and nonlinear

polarization densities, respectively.

Assumptions made for our medium:

1. Isotropic in its linear properties, so )1( is a scalar.

2. Homogeneous, so that the linear and nonlinear susceptibilities are not space

dependent.

3. Source free and nonmagnetic.

From our assumptions above, the wave propagation equation reduces to

2

2

02

2

)1(02

ttNL

PE

E . (8)

Now for a linearly polarized plane wave and the excited nonlinear polarization

propagating in the z-direction ( ),(ˆ tzeeE , ),(ˆ tzpp NLNL P with their own unit

vectors), Eq. (8) is simplified to:


Chen-Bin Huang 4/22

)ˆˆ(),(),(),(2

2

02

2

)1(02

2

petzpt

tzet

tzez NL

. (9)

The Fourier transform of Eq. (9) with respect to time is then (note here the

Fourier-transform relation of temporal derivatives)

)ˆˆ(),(),(),( 02

)1(02

2

2

pezPzEzEz NL , (10)

where )},({),( tzeFzE and )},({),( tzpFzP NLNL .

Comments:

1. The above derivation assumes the nonlinear polarization is weak, so we are

using a perturbation approach.

2. )1( is a constant in Eq. (9), but can be frequency-dependent in Eq. (10). We

see that the frequency domain formalism is good when dispersion is

considered.

3. The nonlinear polarization take a generalized form, we have not yet made

any assumption of its dependence on the optical field.

4. In Eq. (10), we can see that the nonlinear polarization can only drive the field

of the same frequency.

Frequency-domain formulation

We express the optical field and the nonlinear polarization as:

0

])([ ..),(2

ˆ2

1),( ccez

detze zktj

(11a)

0..),(

2ˆ

2

1),( ccez

dptzp tj

NLNL

, (11b)

where cnk )()()( )1(0 is the dispersive wave number.


Chen-Bin Huang 5/22

Inserting Eq. (11) and the dispersive wave number into Eq. (10), assuming pe ˆˆ ,

zjkNL ezz

zjkz

z)(

02

2

2

),(),()(2),(

. (12)

If the change in the electric field envelope due to the effect of nonlinear propagation is

small on the order of a few optical wavelengths, then we may invoke the slowly

varying envelope approximation (SVEA):

22

2

2 kz

kz

, (13)

which leads us to the result of the forced wave equation in the frequency domain

zjkNL ez

n

cjz

z)(0 ),(

)(2),(

. (14)

The nonlinear polarization shows up as a source term which modifies or excites the

complex spectral amplitude of the electric field.

(What is the implication of SVEA?)

Comments:

1. The dispersion of electric field is fully taken into account by the term

zjke )( ; while the dispersion for the nonlinear polarization is embedded

within ),( zNL .

2. Explicit separation of the zjke )( term helps to eliminate the )1(02

term in Eq. (10).

3. We have not yet made any assumption on the pulse duration, so Eq. (14) is

valid even for very short pulses (very broad spectrum).

4. ),( z and ),( zNL are spectral envelopes centered at 0 .

Time-domain formulation for the FWE

In the time domain formulation, we express the field and the nonlinear polarization in


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terms of slowly varying envelope functions multiplied by a carrier as:

eetzatze zktj ˆ}),(Re{),( )( 00 , (15a)

petzptzP tjNLNL ˆ}),(~Re{),( 0

. (15b)

The E-field base-band spectral envelop is related to the time-domain envelope

through

dtetzatzaFzA tj ~),()},({)~,( , (16)

where 0~ . Substituting Eq. (16) into Eq. (9) and assuming pe ˆˆ , we get

zjkNLNL

NL

tj

et

p

t

pjp

ez

A

z

AjkAkk

d

0

2

2

0200

2

2

020

2

2

22

~~

~

))((~ ~

. (17)

To simplify Eq. (17), we introduce the following assumptions:

1. We invoke SVEA in both space and time:

Akz

Ak

z

A 2002

2

2

and at

a

t

a 2002

2

2

(18)

2. We assume no backward traveling waves, so that 00 2)( kkk .

3. No group velocity dispersion: so that gv

kk

kk

~~)( 00

.

After the inverse Fourier transform to the left-hand side of Eq. (17) and all the above

assumptions, we have the resulting forced wave equation in the time domain:

zjkNL

g

etzpn

cj

t

tza

vz

tza0),(~

2

),(1),(

0

00

. (19)

Comments:

1. Without nonlinear polarization, the pulse propagates with a group velocity


Chen-Bin Huang 7/22

without distortion.

2. In the derivation of Eq. (19), higher order dispersions are neglected, therefore

this equation might not be valid for very short pulses.

Continuous-wave second harmonic generation

Basic formulations

Let e(z,t) consist of two CW waves at frequencies of 0 and 20:

..)()(2

1),( )2(

2)( 200 ccezEezEtze zktjzktj

, (20)

where E(z)and E(z)are complex amplitudes, and k= k(0), k=k(20) are the

propagation constants. The 2nd-order nonlinear polarization pNL(z,t) induced by e(z,t)

and crystal nonlinearity is:

),(2),( 20 tzedtzp effNL , (21)

where the effective nonlinear coefficient deff is determined by the field polarization

and material properties. Substituting Eq. (20) into Eq. (21), we derive two nonlinear

polarization temporal envelopes driving CW waves at 0 and 20, respectively:

zkkj

effNL ezEzEdtzp 2)()(2),(~

20, , (22a)

zkj

effNL ezEdtzp 22

02, )(),(~ . (22b)

Substituting these two equations into Eq. (19) gives rise to a system of coupled

equations:

zkjeff eEE

cn

dj

z

E )(2

0

(23a)

zkjeff eE

cn

dj

z

E )(2

2

02

, (23b)


Chen-Bin Huang 8/22

where

kkk 22 , (23c)

is the wave vector mismatch between the fundamental-harmonic (FH) and

second-harmonic (SH) waves, and n=n(0), n=n(20).

When the FH pump is non-depleted (i.e. E(z)E(0)=E), Eqs. (23a-b) are

decoupled:

zkjeEjzE

z)(2

2 )(

, and cn

deff

2

0 , (24)

direct integration of Eq. (24) from z=0 to z=L results in:

222 2

sinc)(kL

j

eLk

LEjLE . (25)

(1) When k=0:

Perfect phase matching, we have LEjLE 22 )( , SH intensity I2(L)=

2220 )(

2

1LEcn = 222

IL , and the CW SHG efficiency is formulated as:

ILI

ISHG

222 , 22

30

2202

2

nnc

deff . (26)

Since Eq. (26) is a monotonically increasing function of L, energy is always

converted from FH wave into SH wave. We also note that under perfect phase

match condition, nn 2 .

(2) k0:

Phase mismatch exists, Eq. (26) will be degraded by a sinc2 factor: SHG

2

sinc22 LkL

2sin 2 Lk

. SHG efficiency becomes a periodic function

of L, the direction of energy conversion will be reversed for every coherence


Chen-Bin Huang 9/22

length lc /k.

Phase matching using birefringence

Here we consider phase matching using optical birefringent crystals. Due to different

crystal lattices, some crystals have indices of refraction that depend on the e-field

polarization. For a uniaxial crystal, there exists one direction (called the optical axis),

where a plane wave only sees the same index of refraction no matter how it is

polarized. Light propagating other than on the optical axis partially sees a constant

index (ordinary ray, O-ray) and partially see an angle-dependent index (extraordinary

ray, or E-ray).

Mathematically, birefringence can be incorporated into the linear polarization as:

2

2

2

0

00

00

00

e

o

o

ij

n

n

n

, (27)

where on is the index seen by the ordinary ray, while en is the index seem by the

extraordinary ray.

For the O-ray, the propagation constant is c

nk o .

For the E-ray, the propagation constant is c

nk e )( . The angular dependent


Chen-Bin Huang 10/22

indices generate an index ellipsoid expressed as:

2

2

2

2

2

sincos

)(

1

eoe nnn

. (28)

Now we focus on phase matching in the case where oe nn (positive uniaxial

crystals), using the following figure:

Type I matching ( oee ):

For perfect phase matching, we require that ),()2( 00 eo nn . This requires

phase matching by angle tuning of the crystal, from Eq. (28), we have:

]

)(

1

)(

1[

])(

1

)2(

1[

sin

02

02

02

02

2

oe

oop

nn

nn

. (29)

Type II matching ( oeo ):

The phase matching condition in this case is simply:

),()()2(2 000 eoo nnn . (30)

Due to birefringence, the Poynting vector of e-ray is typically non-parallel to the

wave-vector. We will discuss briefly in class over different focusing and walk-off

effects in the SHG power.



Ultrashort pulse second harmonic generation

Basic formulation

We denote the input field and the second harmonic fields as:

..),(2

1

2

1),(

0

])([ ccdezEtze zktj

, (31a)

..),(2

1

2

1),(

0

])([22

2 ccdezEtze zktj

. (31b)

The Fourier transform of Eq. (21) is:

),(),(2),( 0 zEzEdzP effNL . (32)

If the FH pump is non-depleted, )(),0(),( EEzE , we only need to

analyze the evolution of SH signal driven by nonlinear polarization centered at 20.

We can evaluate the positive frequency components first by inserting Eq. (31a) into

Eq. (32):

0

)()(02, )()(

2),( zkkj

effNL eEEd

dzP

. (33)

Substituting it into Eq. (14), we get the forced wave equation for SH signal:

0

),(2 )()(

2),( zkjeEE

djzE

z

, (34)

where is defined in Eq. (24). The wave vector mismatch is defined as:

)()()(),( 2 kkkk , (35)

which is a function of both (SH band) and ’(FH band) in general.

Comment:

1. Eqs. (34-35) mean that every SH frequency component is driven by

infinitely many combinations of FH field components at frequencies ' and



', i.e. viewed as a result of the sum-frequency generations by the entire

FH spectrum instead of by a single frequency component at /2.

2. The nonlinear polarization driving the SH fields takes the form of the

auto-convolution of the FH fields. Therefore, if for transform-limited FH

pulses, we can expect the SH spectral envelope to be slightly broader than

that of the FH field.

General solution of ultrafast SHG

By integrating Eq. (34) (assuming no SH at the input) from – 2L to 2

L , we obtain

the general output SH spectrum:

0 2 2

)',(sinc)()(

2),(

LkLEE

djLE

(36)

If the GVD effect is negligible, k(,’) can be reduced to a function of only:

)(2

2)(2)2(),(

0

02

2002

00

kk

kkkkk

, (37)

where

)(

1

)2(

1

00

gg vv. (38)

Phase mismatch depends on both phase velocity mismatch and the inverse group

velocity mismatch (GVM) between FH and SH pulses. This will allow the terms

within the bracket of Eq. (36) being pulled out from the integration, leading to a

transfer function relation:

),(),( 22 LHPjLE NL , (39)

where



)()( EEPNL (40)

and

2]2[sinc),( 02 sTLLH , (41)

represent the nonlinear polarization and PM spectra for SH signal, respectively. The

GVM walk-off Ts within the nonlinear crystal of length L, and the frequency detuning

of H2(L,) due to carrier wave vector mismatch (k 0) are defined as:

LTs , k . (42)

Comments:

1. Not all of the SH frequencies can be driven by the nonlinear polarization

even under phase-matching condition. This is true since phase matching does

not guarantee in GVM. Thus in the real world, the SH spectrum can by

substantially narrower than the FH spectrum.

2. We have neglected higher-order dispersion terms, which mean the FH pulse

broadening during propagation within the crystal is neglected.

Ultrafast SHG for thin and thick crystals

Since only spectral envelopes are important for short pulses, we focus on the

baseband representations of Eqs. (39-41):



),(~

)(~

),(~

2 LHPjLA NL , (43)

where )2,(),(~

022 LELA , )()(~

0 EA , ))(~~

()(~ AAPNL ,

]2/)(sinc[),(~ sTLLH .

We will only examine the waveform and efficiency of pulsed SHG by two phase

matched extreme cases:

(1) Thin crystal (long pulses):

With a broad PM bandwidth: LLHLH )0,(~

),(~ , Eq. (43) is approximated by

LPjLA NL )(~

),(~

2 , whose temporal representation is:

)()},(~

{),( 22

12 tLajLAtLa F , (44a)

)()( 2222 tILtI . (44b)

The output SH pulse energy U2= dttLIAeff ),(2 =effeff tA

UL

222

, where Aeff is the

effective beam cross-sectional area, teff dttIdttI )()( 22

is the effective

FH pulse duration. For FH pulses with average power P , repetition trep, the

pulsed SHG efficiency is formulated as:

,22

222

, peakeff

rep

effthinSHG ILP

t

t

A

L

U

U

. (45)

(2) Thick crystal (short pulses):

For narrow PM bandwidth: Eq. (43) is approximated by: ),(~

2 LA

),(~

)0(~ LHPj NL , whose temporal representation is:



s

NL

T

tPjtLa

)0(

~),(2 (46a)

s

NL

T

tPstLI

2

2

2

)0(~

),( , (46b)

where 222

00 2cnds eff , and the rectangular function in Eq. (46b) is defined as:

(x)=

otherwise ,0 ;

2

1 for ,1 x . The output SH pulse energy

U2= dttLIAeff ),(2 = sNL

eff TP

sA2

2)0(

=

22 )( dtta

sLAeff

. If the FH pulse has

no nonlinear temporal phase, I(t)= 2/)(20 tacn , U2= 2

2

U

A

L

eff

. The

pulsed SHG efficiency becomes:

s

effpeak

effthickSHG T

tIL

A

UL

U

U

,

222

2, . (47)

Comments:

1. By Eq. (44), the SH field is simply the square of its FH input, as long as the

PM bandwidth is much broader than the FH spectrum.

2. Comparing Eq. (45) with Eq. (26), enhancement in ultrashort pulse SHG is

much more efficient than CW source by the ratio of repetition time to the

pulse duration.

3. By Eq. (46), GVM walkoff can seriously stretch the SH field to a

rectangular function of width Ts (regardless of the shape of FH pulse), if

the PM bandwidth is too narrow. In this case, FH pulse only determines the

SH pulse energy.

4. Comparing Eq. (47) with Eq. (45), narrowband SHG efficiency scales

linearly with crystal length L (instead of L2), and is reduced by a factor of



teff /Ts.

5. When SHG pulse duration is not important, high SHG efficiency can be

achieved using the thick crystal approach.

We will also discuss briefly quasi-phase matching for both CW and pulsed fields

in class.

Quasi Phase Matching (QPM)

We have derived under the assumption of perfectly phase matching, the second

harmonic intensity grows as a quadratic relation to the crystal length. On the other

hand, for non-perfectly phase matched case, the second harmonic intensity oscillates.

Perfect phase matching is difficult to achieve in practice due to fabrication tolerances,

in the class, we will briefly introduce the technique of quasi-phase matching for SHG

yield enhancement. The basic idea is to implement periodic spatial modulation to the

nonlinear coefficients, so that in the sense of turning )(zdd effeff , phase mismatch

may be canceled via the spatial modulation periodicity.



Three-wave interactions

Basic formulations

Thus far we restricted our discussions to SHG, in which the frequencies of the

interacting waves satisfy 111 2 . The generalized frequency conversion

process in second order nonlinearity involves three interacting three waves satisfying

the relation 321 . In our notation, the highest frequency term is denoted by 3.

To obtain higher conversion efficiency, phase matching or momentum conservation

also needs to be satisfied: 321 kkk .

We now express the total optical field as three individual frequency components

3

1

)( ..),(2

1),(

i

zktji ccetzatze ii . (48)

Using the formula for nonlinear polarization as expressed in Eq. (21), we show the

terms driving at frequencies of 1, 2, and 3 as

.}.

{),(])([*

13])([*

23

])([210

132231

213

cceaaeaa

eaadtzpzkktjzkktj

zkktjeffNL

. (49)

Insert this expression into the time-domain nonlinear wave equation (Eq. (19)), we

obtain basic equations for three-wave parametric interactions:

kzj

g

kzj

g

kzj

g

eaajt

a

vz

a

eaajt

a

vz

a

eaajt

a

vz

a

2133

3,

3

*132

2

2,

2

*231

1

1,

1

1

1

1

, (50a)

with



cn

d

i

effii 2

and )( 213 kkkk . (50b)

For CW fields, the time derivatives can be omitted, and this leads to the

Manley-Rowe relation, a manifestation of conservation of energy:

z

I

z

I

z

I

3

3

2

2

1

1

111

. (51)

Coming back to pulsed fields, we explicitly show the phase mismatch in the

frequency domain by rewriting iii . By assuming the crystal initially

satisfies 213 we obtain the relation

)(),,( 22

21

1

13

3

3321

kkk

k . (52)

Now with energy conservation and igi

i

vk

,

1

in mind, we may further express the

phase mismatch as

12,1,

32,3,

)11

()11

( gggg vvvv

k . (53)

Comments:

1. We can see that phase mismatch depends on the frequency variation of both

inputs.

2. For perfect phase matching, the lower frequency variation is tracking the

variation of the higher frequency field.

Sum frequency generation (SFG)

For sum-frequency generations, the two lower frequency inputs fields are used to

generate an up-converted signal field ( 321 ). Refer to Eq. (50), here we

assume the inputs are non-depleted, and phase-matched.

First we focus on the case where group-velocity walk-off=0, and the

up-converted signal can be expressed as



)()(1

2,2

1,13

3

3,

3

ggg v

zta

v

ztaj

t

a

vz

a

. (54)

And the intensity (in the retarded time-frame moving along with group velocity

3,'3 gvztt ) at the output end is

2'32

'31

3213

0

223'

33 )()(2

),( LtItInnnc

dtLI eff

. (55)

This result is very similar to the quasi-CW limit as we discussed for SHG.

Now we take group velocity walk-off 0 , and the signal field is expressed as

)()(),(

2321313

'33 ztaztaj

z

tza

(56a)

)11

(,, jgig

ij vv . (56b)

The general solution takes the form of

)()(),( 23'320 13

'313

'33 ztaztdzajtLa

L . (57)

In order to obtain better insight, let’s assume one of the inputs is a very short

pulse (a flat-top delta function), )()(1 tta , and the solution for the signal is

13

'312

213

'3

13

3'33

)(

2

1sq),(

ta

L

tjtLa . (58)

Comments:

1. The up-converted signal is a windowed, temporally scaled version of )(2 ta .

2. This scheme is useful for pulse measurement, where the cross-correlation of

the two lower frequencies yields up-converted signal that tracks the

group-velocity difference of the two inputs.

Now we discuss the case where group-velocity walk-off between the two inputs

is large ( 212 L , where 2 is the pulse duration of )(2 ta ). In this limit, it is

interesting to find that the output signal takes the form of



13

'312

213

3'33

)(),(

ta

jtLa . (59)

Comments:

1. The up-converted signal is independent of short input )(1 ta !

2. The up-converted signal is a true scaled version of )(2 ta in the time-domain,

and be stretched, compressed, and even time-reversed.

Cross-correlation measurements of intensity profiles of (a) input signal field and (b,c)

output fields in sum frequency generation experiment using 10-mm LBO crystal. The

duration of the reference pulse is 130 fs (not shown). Inverse group velocities are 5289

fs/mm for the input o-wave, 5435 fs/mm for the input e-wave, and 5472 fs/mm for the

sum frequency wave. (b) a1 is an o-wave; a2 is an e-wave. (c) a1 is an e-wave; a2 is an

o-wave.

Difference frequency generation (DFG)

Similar to how we treated for SFG, but now the signal is changed to a2. We quickly

obtain the expression for our signal intensity when inputs are non-depleted and

phase-matched:

2'23

'21

3213

0

222'

22 )()(2

),( LtItInnnc

dtLI eff

. (60)

We now turn to the case when group velocity walk-off 0 , we have



)()(*),( 23'230 12

'212

'22 ztaztdzajtLa

L , (61)

where 2,'2 gvztt . Again we assume input a1 is very short and can be

approximated as a delta function:

12

'213

312

'2

12

2'22

)(

2

1sq),(

ta

L

tjtLa . (62)

Again we see the signal is a scaled version of the longer input field.

Electric field profiles of 13.7 μm pulses generated by short pulse difference frequency

generation in GaSe, using ∼15 fs input pulses at 780 nm. (a) Input: a single pulse. (b)

Input: a pulse pair separated by 60 fs. The output field profiles are measured by

electro-optic sampling (see chapter 10). The group velocity mismatches are estimated to

be |η13| ≈ 160 fs/mm; |η12| ≈ 850 fs/mm.

Optical parametric amplification (OPA)

So far we have assumed the input field amplitudes are fixed during the nonlinear

interactions. In this part, we consider the case where the amplitudes of the two lower

frequency fields a1, a2 can vary, while a3 is strong and remains non-depleted. This

leads to the phenomenon called optical parametric amplification.

Since a3 can be viewed as constant, we are left with two coupled equations

between a1 and a2. It is easier to see the effect of OPA using CW fields:

*0*

0

* 2

1

32

31

2

1

a

a

aj

aj

a

a

z

, (63)

where we have assume that 0k and 2 is real. The solutions to Eq. (63) take the



form of ]exp[~11 zaa and ]exp[~

22 zaa . So the linear differential equation set is

reduced to

0** 2

1

32

31

a

a

aj

aj

. (64)

To obtain non-trivial solution, we require that 0det , and this gives

321 a . (65)

We can see that indeed is real, so amplification with exponential growth is

permitted.

We now focus on the case when initially there is only one lower frequency field

at the input ( 0)0(2 a ). With this boundary condition, the solutions yield

)sinh()0(*)(

)cosh()0()(

13

3

1

22

11

zaa

ajza

zaza

, (66)

where we note interestingly the indicated the phase sensitivity requirement between

the pump (a3) and the signal field (a1). We can now express the field intensities as

)(sinh)0()(

)(cosh)0()(

21

1

22

211

zIzI

zIzI

, (67)

and we see that the difference in photon numbers at and remains fixed throughout the

process, consistent with the Manley-Rowe relation.

nonlinear optics: brief background

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