nonlinear optics - institut optique

NONLINEAR OPTICS

Ch. 1 INTRODUCTION TO NONLINEAR OPTICS •  Nonlinear regime - Order of magnitude

• Origin of the nonlinearities

-  Induced Dipole and Polarization

-  Description of the classical anharmonic oscillator

model

-  Linear polarization and susceptibility

-  Nonlinear polarization and susceptibility

•  Nonlinear interactions : a brief description

2 N. Dubreuil - NONLINEAR OPTICS

Induced dipole and macroscopic polarization

A simplistic description of the interaction between a wave and an atom :

- Simplistic model for an atom =

Nucleus (charge +)

Electronic cloud (charge - )

- An applied static electric field acts on the electrons trajectories

E

Change in position of the center of mass of the electrons

€

! F = q

! E + ! v ×

! B ( )

∆x

Lorentz Force

ü  Microscopic scale



- With an applied EM fiels = temporal deformation of the electronic cloud

Abrev. : EM field means Electromagnetic Field

E

•  time variation of the position of the center of mass : ∆x(time)

•  the electronic cloud is linked with the nucleus, presence of a restoring

force (linear function of the displacement of the electrons)

•  vibrating dipole ( induced dipole) :

time p

time

Induced Dipole

€

! p = q ×Δ

! x (t)

In a linear regime : linear relation between the displacement of the electrons and the restoring force

Induced dipole frequency ω = applied EM field frequency ω

The magnetic part of the Lorentz force is negligible



ü  Macroscopic scale

- LINEAR MEDIUM case made of N atoms (N identical dipoles)

= « macroscopic »

source term

E time

p time

Induced dipole

P time

Polarisation

Average

€

E(t) = Acos(ωt + kz)

P(t)∝ χ (1)Acos(ωt + kz)

Case of a z propagative EM wave with freq. ω

Source term @ ω

€

! P (t) = N ×

! p = ε0χ

(1)! E (t)

Def. : LINEAR susceptibility of the medium

The polarization vector P(t) acts as a source term in the wave equation


Classical anharmonic oscillator model

•  Classical anharmonic oscillator - Description

Induced dipole (microscopic quantity) :

Nucleus (fixed)

Electron Electronic cloud

Polarization (MACROscopic quantity) : x(z,t) ??

•  Equation of motion

E time

p time

Induced Dipole



- Two applied EM wave with frequencies ω1 and ω2 :

E time

p time

Induced Dipole

LINEAR CASE : « LINEAR » response of the vibrating dipole

Vibrating dipoles with frequencies = ω1 and ω2

A material with a collection of N dipoles in a linear

vibrating regime = LINEAR MEDIUM



- Two applied EM wave with frequencies ω1 and ω2 :

E time

p time

NONLINEAR CASE : « NONLINEAR » dependence of the restoring force with the displacement of the center of mass

Induced Dipole

Vibrating dipoles with frequencies = ω1 and ω2 + harmonics

A material with a collection of N dipoles in a nonlinear

vibrating regime = NONLINEAR MEDIUM



•  Equation of motion

Solution : perturbation method, taking into account

Driven Coulomb force Restoring force Damping term

⇤20x � �x2 � ⇥x3

x(1) � x(2) � x(3)


Linear polarization

Harmonic oscillator : equation of motion

Driven solution

Induced dipole

Linear polarizability (microscopic quantity)

Macroscopic Polarization Linear susceptibility


Linear polarization

Harmonic oscillator : equation of motion

Dispersion Absorption (or amplification)

Lorentzian line shape

Linear Susceptibility

�(1)(⇥) = ��(⇥) + ı��(⇥)


2nd Order Nonlinear Polarization

Anharmonic oscillator : equation of motion

Driven solution



Anharmonic oscillator : equation of motion

Nonlinear Polarization

2nd order Nonlinear Susceptibility

Optical RECTIFICATION

(induces static electric field)

2nd Harmonic Generation

creation of an electric field with

a 2ω frequency component



Anharmonic oscillator : equation of motion Conclusion & Comments

•  The macroscopic polarization induced inside the material is then given by the sum :

With : (Complex amplitudes)

•  Phase mismatching between the polarization component @ 2ω and the free propagative wave @ 2ω :

wavevector related to P(2ω) ≠ wavevector of E(2ω) 2k(ω) ≠ k(2ω)

•  Strong enhancement of the nonlinear susceptibility is expected once ω or 2ω (or both) is close to a material transition (@ω0)


3rd Order Nonlinear Polarization

Anharmonic oscillator : equation of motion Case of a centro-symmetric material

Driven solution

Linear and Nonlinear Polarization

3rd Harmonic generation

Optical Kerr Effect

x(z, t) = �x(1)(z, t) + �2x(2)(z, t) + �3x(3)(z, t) x(2) = 0

⇤(3)(⌅1,⌅2,⌅3) =�N�e4

⇥0m3D(⌅1 + ⌅2 + ⌅3)D(⌅1)D(⌅2)D(⌅3)


3rd Order Nonlinear Polarization

Conclusion & Comments •  The macroscopic polarization induced inside the material is then given by the sum :

With : (Complex amplitudes)

•  Phase mismatching between the polarization component @ 3ω and the free propagative wave @ 3ω :

3k(ω) ≠ k(3ω)

•  Strong enhancement of the nonlinear susceptibility is expected once ω or 3ω (or both) is close to a material transition (@ω0)

⇤(3)(⌅1,⌅2,⌅3) =�N�e4

⇥0m3D(⌅1 + ⌅2 + ⌅3)D(⌅1)D(⌅2)D(⌅3)


Nonlinear interactions

ü  A short review :

- LINEAR MEDIUM case made of N atomes (N identical dipoles)

= « macroscopic »

source term

E time

p time

Induced dipole

P time

Polarisation

Average

€



Case of a z propagative EM wave with freq. ω

Source term @ ω

€

! P (t) = N ×

! p = ε0χ

(1)! E (t)

Def. : LINEAR susceptibility of the medium

The polarization vector P(t) acts as a source term in the wave equation



ü  A short review : - Case of a NONLINEAR MEDIUM : collection of N identical atomes The nonlinear response of the medium can be expressed as (we have assumed that the medium have an instantaneous response - Case of a lossless and a dispersionless medium ) :

€

P(t) = ε0χ(1)E(t) +ε0χ

(2)E(t)E(t) +ε0χ(3)E(t)E(t)E(t)

= ε0χ(1)E(t) +ε0χ

(2)E 2(t) +ε0χ(3)E 3(t) +!

Considering a z propagative EM wave @ ω

€


Linear response

€


Source term @ ω

1st Order

2nd Order + 3rd Order Nonlinear Response

€

P(t)∝ χ (2)A2 cos2(ωt + kz)2

∝ χ (2)A2 cos(2ωt +2kz) + χ (2)A2

2nd Order

Source term @ 2ω !!!



ü  A short review :

- Case of a NONLINEAR MEDIUM : collection of N identical atomes

The nonlinear response of the medium can be expressed as :

€

P(t) = ε0χ(1)E(t) +ε0χ

(2)E(t)E(t) +ε0χ(3)E(t)E(t)E(t)

= ε0χ(1)E(t) +ε0χ

(2)E 2(t) +ε0χ(3)E 3(t) +!


€


Linear response

€


Source term @ ω

1st Order

2nd Order + 3rd Order Nonlinear Response

€

PNL (t)∝ χ (3)A3 cos3(ωt + kz)

∝ χ (3)A3 cos(3ωt + 3kz) + χ (3)A2Acos(ωt + kz) +!

3rd Order

Source term @ 3ω Third harmonic generation

Source term @ ω Optical Kerr effect


Example : Optical Kerr Effect

ü  Variation of the refractive index

€

P(t) = ε0χ(1)E(t) +ε0χ

(3)E 3(t)


€


I ∝ A2

Kerr effect induces a variation of the refractive index directly proportional to the wave intensity.

€

P(t) = PL (t) +PNL (t)∝ χ (1)Acos(ωt + kz) + χ (3)A2Acos(ωt + kz)

€

P(t)∝ χ (1) + χ (3)A2( )Acos(ωt + kz) NONLINEAR Regime

€

PL (t)∝ χ (1) Acos(ωt + kz) LINEAR Regime

Wave intensity

Directly related to the refractive index

Case of a lossless medium



⇒  2nd order nonlinearities

•  optical rectification : induces a static electric field •  2nd harmonic generation : generate a EM wave @ 2ω •  Sum and Difference frequency generation : generate a EM wave @ ω1± ω2, fluorescence, amplification and parametric oscillation

⇒  3rd order nonlinearities •  Optical Kerr effect : nonlinear refractive index change •  Solitons •  Self-phase modulation •  Four-wave mixing : nonlinear interaction of 4 degenerates or non-degenerates waves •  Raman scattering •  Brillouin scattering


E

time p

time

Ind

uc

ed

dip

oe

€

E1(t) = A1 cos(ω1t + k1z)E2(t) = A2 cos(ω2t + k2z)E3(t) = A3 cos(ω3t + k3z)

Nonlinear interaction of 3 waves with 3 different frequencies

€

PNL (t)∝ χ (3)E1(t)E2(t)E3(t)

Frequency components of the nonlinear polarization :

ω1+ ω2+ ω3 ; ω1+ ω2 - ω3 ; ω1 - ω2+ ω3 ; ω1 - ω2 - ω3 ;

ω3

ω2

ω1

ω3 ω2 ω1

Example : Four-wave mixing


Example : Source term @ ω4 = ω1+ ω2 - ω3

€

PNL (t)∝ χ (3)A3 cos (ω1 +ω2 −ω3) t + (k1 + k2 − k3)z( )

ω3

ω2

ω1


z

z

Assumption : low dispersive medium / The refractive indices @ ω1, ω2, ω3, ω4 are equals ⇒ the polarization @ ω4 radiates in phase with a z propagative

field @ ω4 = « phase matching condition »

€

E4 (t) = A4 cos(ω4 t + k4z)

Once the phase matching condition is fulfilled, the in-phase array of

dipoles contributes to efficiently generate a propagative field @ ω4 €

k4 = k1 + k2 − k3

€

k4 = k1 = k2 = k3


ω3

ω2

ω1

€

PNL (t)∝ χ (3)A3 cos (ω1 +ω2 −ω3) t + (k1 + k2 − k3)z( )

€

E4 (t) = A4 cos(ω4 t + k4z)

Assumption : dispersive medium / The refractive indices @ ω1, ω2, ω3, ω4 are no more equals ⇒ the radiating polarization @ ω4 is no more in phase

with the z propagative field @ ω4 = « NO phase matching condition »

The EM field @ ω4 can not be efficiently generated because of a strong phase mismatch between the source term and

the propagative wave @ ω4


€

k1 + k2 − k3 ≠ k4

Example : Source term @ ω4 = ω1+ ω2 - ω3

z

z

nonlinear optics - institut optique

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