8. Propagation in Nonlinear Media

8.1. Microscopic Description of Nonlinearity.8.1.1. Anharmonic Oscillator.

Use Lorentz model (electrons on a spring) but with nonlinear response, or anharmonic spring

b) Harmonic spring; Low intensityc) Anharmonic spring (overstretched); High intensity

22

02

2 3

2 3

NL

NL

f x td x t dx t e tx t

dt dt m mf a x t a x tm

E

At strong incident linearly polarized fields electron trajectories no longer straight lines

8.1.2. Lorentz Force as Source of Nonlinearity

At strong incident linearly polarized fields electron trajectories no longer straight linesMolecule oscillates over other degrees of freedomLorentz force becomes significante v B

Simple case where electric field polarized along x, magnetic field lies on y

,0,0xEE 0, ,0yHH

ˆ ˆ ˆ0

em

x y y

f e

e E zB x y exB z

E v B

22

02

22

02

22

02

( ) ( )

0

( ) ( )

tx y

y

d x t dx t ex t E z t B t

dt dt m

d y t dy ty t

dt dt

d z t dz t ez t x t B t

dt dt m

Equations of motion from E and H fields, no force acting in y direction. Solve for x and z displacements.Take Fourier transform, obtain following equations:

2 20

2 20 .

ry

y

eE ex i i z B

m me

z i i x Bm

2 2

2 2.

x

x

D eEx

D b m

b eEz

D b m

Equations obtained from method in 5.1Ignore magnetic field, we get linear response

2 20

.y

D i

eb i B

m

Express magnetic field in terms of electric field

xy

i

EB

ic

k E B 22

x xz eE eE

x me mc

Order of magnitude relation between x and z displacements

Solve for when electric field and associated irradiance where x and z are comparable

2

12

2

10 ,

x x z

mcE

eV

m

2

212

1

2

10

xI E

Wm

22

2 2 2

22

2 2

2 2

1.

x

x

x

x

ez E

m c D b

eE

m c D

DeEx

m D b

eE

m D

xe Eb Dmc Using we have:

Electron oscillates at on z axis, DC term is called optical rectification

2

22 1 cos 2 .

i t i tx x

x

z E e E e

E t

8.1.3. Dropping the Complex Analytic Signal Representation of Real Fields.

cosrU A tReal field Complex analytic signal

i tU Ae

22

2

2 2

1

2

1 cos 2

.

i t i tr

i t

U A e e

A t

U A e

2Re[ ] A cos(2 ) rU t U

Complex analytic signal does not capture DC term.

Whenever we deal with fields raised to powers higher than one, we use

1*

2rU U U

8.2. Second-Order Susceptibility 2

220 22

2d x t dx t E tx t a x t

dt dt m

Want solution of form where is the nonlinear perturbation to linear solution. 1 2x x x

1 1212

02

2 2222

02

2 21 1 2 2

2 2 22 0.

d x t dx t eE tx t

dt dt m

d x t dx tx t

dt dt

a x t a x t x t a x t

Simplify by neglecting and

2

2 2 2 120 2x t x t x t a x t

2 1 12

2

2 ,

D x a x x

E Eea

m D D

ⓥ

ⓥ

*1 1 1 1

*2 2 2 2

E E E

E E

Illuminating the nonlinear crystal with two monochromatic fields of different frequencies

22

2

2

2

1

' '1'.

' '

E Eex a

m D D D

E Eea d

m D D D

ⓥ

a b a b ⓥ

E

22 1 1

E

E E ⓥ

22 2 1

12 0 2 1 2 1

12 222 1

All possible frequencies:

22

2 1 1 2 2

221 1 1

1 1 2 1 21 1

*1 2 1 2

222 2 2

2 1 2 1 22 2

*1 2 1 2

1

21

21

ex a f f f f

m D

E E

f E ED D

E E

E E

f E ED D

E E

1 21

1 2

1 22

1 2

22

2 1 2

22211 1 2

1 221 1 1 2 1 21

*1 2

*1 2 1 2

22222 1 2

2 222 2 1 2 1 22

*1 2

1

. . . .

2 0

. .

2 0

i ti t

i t

i ti t

i t

ex t a g t c c g t c c

m

EE e E E eg t

D D D D DD D

E E ec c

D D D

EE e E E eg t

D D D D DD D

E E e

D

*

2 1 2

. .c cD D

FourierTransform

Equation 8.26 indicates that, as the result of the second-order nonlinear interaction, the resulting field has components that oscillate at frequencies 22 (second harmonic of 2), 21 (second harmonic of 1), 1 +2 (sum frequency), 1 -2 (difference frequency) and 0 (optical rectification terms).

Importantly, 2 vanishes in centrosymmetric media 2 2 r r

22

0

2

0

.

P r r E r E r

r E r E r

P r

Fulfilled simultaneously only if

2

2 .

Ne

Ne

P r r

r

P r

2 0P r

So second-order nonlinear processes require noncentrosymmetric media

2 2 * *0

2 2

; ,

, , 1, 2

i j i j i j i j

i j i j

P E E

P Nex i j

Nonlinear susceptibility from induced polarization

2 2

0

32

1 220

; ,

. .

i j i j i i

Nex

a Neg t g t c c

m

→

8.2.1. Second Harmonic Generation (SHG)

32

1 1 1 2 20 1 1

32

2 2 2 2 20 2 2

12 ; ,

2

12 ; , .

2

a Ne

m D D

a Ne

m D D

Nonlinear susceptibility a s function of linear chi:

2

1

0

1.

Ne

m D

a) SHG: pumping the chi(2) material at omega yields both the fundamental frequency (omega) and its second harmonic (2omega). b) Description in terms of virtual energy levels.

2 22 1 12 0

1 1 1 1 12 32 ; , 2 .

a m

N e

Linear:

Nonlinear:

8.2.2. Optical Rectification (OR)

OR: a DC polarization is created in a chi(2) material.

32 2

1 1 20 1 1

21 1 12 0

1 12 3

10; ,

0

0 .

a Ne

m D D D

a m

N e

8.2.3. Sum Frequency Generation (SFG)

32 2

1 2 1 2 20 1 2 1 2

21 1 12 0

1 2 1 22 3

1; ,

.

a Ne

m D D D

a m

N e

8.2.4. Difference Frequency Generation (DFG)

32 2

1 2 1 2 20 1 2 1 2

21 1 12 0

1 2 1 22 3

1; ,

.

a Ne

m D D D

a m

N e

8.2.5. Optical Parametric Generation (OPG)

The time reverse process of SFG 2

31 2 3 2

8.3. Third-Order Susceptibility

2 30 3 .

eE tx t x t x t a x t

m

Anharmonic oscillator

Solve using perturbation theory, solution of form 𝑥 (𝑡 )=𝑥 (1 ) (𝑡 )+𝑥 ( 3) (𝑡 )

1 1 120

33 3 3 1 320 3 0.

eE tx t x t x t

m

x t x t x t a x t x t

First term vanishes, approximate 3 31 3 1

3 3a x x a x 3

3 3 3 120 3 ,x t x t x t a x t →

*1 1 1 1

*2 2 2 2

*3 3 3 3 .

E E E

E E

E E

1 1 ,Ee

x t xm D

3

1

1

. .ni t

n

n n

E eex t c c

m D

So,

3 3

3 3 20 3

, , 3

m n pi

m n p

m n p m n p

E E E eex t x t x t a

m D D D

For scalar fields perturbation displacement is:

3 3

3 33 3

0, , 1 , , 3

30

; , ,

; , , ,

i q q

i q ijkl q m n p j m k n l pj k l m n p

ijkl q m n p j m k n l pjkl

P Nex

P E E E

d E E E

Induced polarization both for electromagnetic fields in terms of displacement and susceptibility

Where d is the degeneracy factor

General expression for 3

*

3 33

0

1; , , ,ijkl q m n p

i q j m k n l p

a Ne

d m D D D D

2 2

0a aD i

Express in terms of 3 1

3

3 1 1 1 13 03 4

; , , ,ijkl q m n p i q j m k n l p

a m

dN e

21

0

1a

a

Ne

m D

8.3.1. Third Harmonic Generation (THG) signal contained by terms that oscillate at 3ω

43 3

330

3 31 13 03 4

13 ; , ,

3

3 ,

a Ne

d m D D

a m

N e

The THG susceptibility for one component of the electric field (scalar case) is

2 20D i

8.3.2. Two-Photon Absorption (TPA) and Intensity-Dependent Refractive Index

1 , 2 , 3 ,

43 3

23 20

3 2 21 13 03 4

1; , ,

a Ne

d m D D

a m

N e

If

3 2 2 23 1 1 1 1 13 0

3 4

3 3

2

,

R I R I

R I

a mi

N e

i

0 2n n n I

Define effective refractive index

21 3

2320

2

3

1 3

1,

E

n E

n

20 22 1 ( )n n I

Find expression for

3

2 20 0

3 3

20 0

2 2

3

4

3

4

' '' .

R I

nn c

in c

n in

• A plane wave undergoes an intensity-dependent loss of factor e^-alpha.

• b) Energy level diagram for single photon absorption (left) and two-photon absorption (right).

8.3.3. Four Wave Mixing

3 3

1 2 3 30 1 2 3

1; , ,

a Ne

d m D D D D

1 2 3 1 2 3 k k k k

k-vector conservation, phase matching condition

a) Generic four-wave mixing process. b) Momentum conservation.

1 2 33 *0 1 2 36 .i

NLP A A A e k k k r

Phase conjugation via degenerated four wave mixing: field E4 emerges as the phase conjugate of E3, i.e. E4=E3*.

8.3.4. Phase Conjugation via Degenerate (all are the same) Four-Wave Mixing

Population of the excited vibrational levels obeys the Maxwell-Boltzmann distribution

1,B

E

k TP E ez

Spontaneous Raman scattering: -a) Stokes shift-b) anti-Stokes shift

8.3.5. Stimulated Raman Scattering (SRS).8.3.5.1. Spontaneous Raman Scattering

10

1

0

1B

h

k TP Ee

P E

Thus, the Stokes component is typically orders of magnitude stronger !

.p E the molecular optical polarizability,

0

0

.v

vv x

t x tx

0

0

.v

vv x

P t N P

N x t E tx

* ,v vi t i tv v vx t A e A e

Harmonic vibration occurs spontaneously due to Brownian motion

0

*

0

. .

. . . . .

L

L v L v

L NL

i tL L

i t i tNL v L v L

v

P t P t P t

P t N A e c c

P t N A A e c c A A e c cx

8.3.5.2. Stimulated Raman Scattering

22

2,v v

v v

d x t dx t F tx t

dt dt m

To derive expression for SRS susceptibility, solve equation of motion for vibrational mode of resonant frequency and damping

.vdW Fdx

2

20

0

1

21

2

1

2

t

t

v tv

W p t E t

E t

dx E t

dx

2

0

0

1.

2

1

2

v

tv

v

dWF t

dx

dE t

dx

dF E E

dx

ⓥ

* *

. .SL i ti tL S

L L S S L L S S

E t A e A e c c

E A A A A

2 2

*

2 2

0

1

1

2

vv

L S L S

v v

Fx

m i

A Ad

m dx i

ⓥ

0

0

0

0

.

vv

vv

L NL

P N x Ex

N E N x Ex

P P

Induced polarization for SRS

2*

*2 2

0

* *0

2L S L S

NL L L L Lv v

RL S L S L L L L

A AN dP A A

m dx i

d A A A A

ⓥ

nonlinear contribution

2 2* *

220

2

L SS i ti tL S L L

NLv v L S L S

A A A e A eN dP t

m dx i

2

220 0

1

12S

v v L S L S

N dxt

m dx i

L v S

2

0 0

1

12S

v L S

N dxt i

m dx

→

By resonantly enhancing the vibration mode the Stokes component can be amplified significantly; in practice, this amplification can be many orders of magnitude higher than for the spontaneous Raman.

susceptibility associated with the anti-Stokes component

L AS v L S

2

0 0

*

1

12AS

v L S

S

N dt i

m dx

t

a) Raman susceptibility at Stokes frequency; indicates amplification. b) Raman susceptibility at anti-Stokes frequency; indicates absorption.

Strong attenuation

8.3.6. Coherent Anti-Stokes Raman Scattering (CARS) and Coherent Stokes Raman Scattering (CSRS)

• Coherent Anti-Stokes Raman Scattering (CARS) and Coherent Stokes Raman Scattering (CSRS) are also established methods for amplifying Raman scattering.

• These techniques involve two laser frequencies for excitation

31 2 1 1 22 ; , ,CARS A

32 1 2 2 12 ; , ,CSRS S

CARS is a powerful method currently used in microscopy we will hear about it during student presentations.

8.4.1. Nonlinear Helmholtz Equation

,,

,, ,

,

, 0.

tt

tt

t tt

t

t

B rE r

D rH r j r

D r

B r

follow the standard procedure of eliminating B and H from the equations

0

0.j

2

0 2

,, 0,

tt

t

D r

E r

0

0

0

, , ,

, , ,

, ,

L NL

r NL

t t t

t t t

t t

D r E r P r

E r P r P r

E r P r

2

2

,

.

t

E r E E

E

8.4. Solving the Nonlinear Wave Equation.

2 222

02 2 2

, ,, ,NLt tnt

c dt t

E r P r

E r Nonlinear wave equation after approximation,

term negligible D E

2 2 20

0

1, , , ,NL

E r E r P r

Propagation of the Sum Frequency Field

2 23 1 2 1 2 0 3 1 2 1 2; ; , ; , , ,NLP E t E t r r r

SFG nonlinear polarization has the form

1 1

2 2

3 3

1 1

2 2

3 3

, . .

, . .

, . .

i t z

i t z

i t z

E t A e c c

E t A e c c

E t A e c c

r r

r r

r r

2

22 2 233 3 3 0 3 0 1 22

, , , ,E t n E t E t E tc

r r r r

11 1n c

3 3

3 3

22

3 32

223

3 3 3 32

,

2 .

i t zz

i t zz

dE t A e

dt

dAdA i A e

dz dz

r

3 3 0E Ex y

1 2 3

223 3 2

3 0 3 0 1 222 .i zd A t dA zi A A e

dt dz

23 3

32

223 1 2

1 2 3

2

,

i kzd A t dA ti Be

dt dt

B A A

k

Simplifying approximation, 1A and 2A , do not change with z (do not deplete).

223 3

32.

d A z dA t

dz dz

amplitude is slowly varying

332

i kzidA t e dz

33 0

3

2

3

2

3

2

3

2

1

2

2sin 22

sin 22

2

sinc .22

Li kz

i kL

i kL

i kL

i kL

iBA L e dz

iB e

i k

kLiBe

k

kLiBLe

kL

iB kLe

2

3 3

2 22

23

1

2

sinc ,28

I z A z

B L kL

0n

The intensity of SFG field is

Net output power can vanish at , 2 ,...2kL

a) The SFG output field has a phase that depends on the position where the conversion took place.b) The overall SFG intensity oscillates with respect to <eq>

Parametric Processes: Phase Matching

3 1 2

1 13 23 2 ,

nn n

c c c

0k

3 1 2

1 1 2 23

1 2

.n n

n

Normal dispersion Abnormal dispersion.

Normal dispersion curves for a positive uniaxial crystal.

Negative e on n Positive e on n

Type I 3 3 1 1 2 2e o on n n 3 3 1 1 2 2o e on n n

Type II 3 3 1 1 2 2e e on n n 3 3 1 1 2 2o o en n n

Table 8-1.

2 2

2 2 20

1 cos sin

en n n

phase matching can be achieved by angle tuning, that is, selecting the angle that ensures 0k

Type II phase matching by angle tuning in a positive uniaxial crystal: o-ordinary wave, e-extraordinary wave, c-optical axis.

3 3 1 1 2 2, ,on n n

3 3 1 1 2 2,o on n n

Type I

Type II

Electro-Optic Effect

The electro-optic effect is the charge in optical properties of a material due to an applied electric field that oscillates at much lower frequencies than the optical frequency.

Electro-Optic Tensor

a) Linear interaction with a birefringent crystal. b) Electro-optic (nonlinear) interaction. p is the induced dipole, E(omega) is the optical field, and E(0) is the static field. These sketches should be interpreted as 3D representations.

The induced polarization for the Pockels effect

20; ,0 ; ,0 0 .i ijk j kP E E

Due to the electro-optic effect, an optical field can suffer voltage-dependent polarization and phase changes.

Kerr effect

30; ,0,0 ; ,0,0 ,i ijkl j k lP E E E

Pockles effect

2

0

1; ,0 0 .i ii jj ijk j kP r E E

20

2 2

ijkijkijk

ii jj i j

rn n

ijk jikr r

Change in the rank of the tensor, from 3 to 2

11 1

22 2

33 3

12 21 6

13 31 5

23 32 4 .

k k

k k

k k

k k k

k k k

k k k

r r

r r

r r

r r r

r r r

r r r

Electro-optic tensor can be represented by a 3x6 matrix. This tensor contraction, allowed by the permutation symmetry, reduces the number of independent elements from 23 27 to 3 6 18 .

Electro-Optic Effect in Uniaxial Crystals

41

41

63

0 0 0

0 0 0

0 0 0

0 0

0 0

0 0

rr

r

r

Use KDP (KH2PO4, or potassium dihydrogen phosphate) as a specific example of uniaxial crystal

The KDP refractive index tensor is (in the normal coordinate

system of interest)

0 0

0 0

0 0

o

o

e

n

n n

n

assume that the voltage is applied only along z, such that only 63r is relevant.

20 n D E P

Electro-optic effect in a KDP crystal. For E(0) parallel to z, the normal axes rotate by 45 degrees around the z-axis.

2 2 20 0 0

,

i i i i j ijk j k

ij j

D n E n n r E E

E

electric displacement can be expressed for each component as (i=x, j=y, k=z)

2 40 0 63

2 40 0 63

20

0

0

.

x o x o y z

y o y o x z

z e z

D n E n r E E

D n E n r E E

D n E

2 463

4 20 63

2

0 0

0 0 .

0 0

o o z

o z o

e

n n r E

n r E n

n

express ij in a reference system rotated about the z-axis by an arbitrary angle, say

2

0 2

2

0 2

'

cos sin cos sin

sin cos sin cos

sin 2 cos 2,

cos 2 sin 2

o

o

o

o

R R

n D

D n

n D D

D n D

463 0o zD n r E

2

20

2

0 0

' 0 0 .

0 0

o

o

e

n D

n D

n

Electro-Optic Modulators (EOMs)

defining the normal refractive indices

' 'ij ijn

2

363

363

' 0 0

' 0 ' 0

0 0 '

'

1

2

10

21

' 02

' .

x

y

z

x o

oo

o o z

y o o z

z e

n

n n

n

n n D

Dn

n

n n r E

n n n r E

n n

phase retardation

363

0

' '

20 ,

x y

o z

n n Lc

n r E L

0 .z

VE

d

half-wave voltage 03

63

.2 o

dV

n r L

.V

VV

a) Longitudinal modulators (electrodes are made of transparent material), d=L. b) Transverse modulator, d<L.

a) Electro-optic (EO) crystal operating as phase modulator: incident polarization parallel to the new normal axis. b) Amplitude modulation: input polarization parallel to original (i.e. when V=0) normal axis; P is a polarizer with its axis parallel to x.

30 63

0

0

1

2

2

'

.

oo

o

k n r Vin k L

i Vin k L

E V Ee e

Ee e

KDP longitudinal modulator reads

0

2

0

2

' 045 45

' 1

0

0

2 2

2 245 .2 2

2 2

x

y

i

i

ER W V R

E

eW V

e

R

(x’,y’) is rotated by 45° with respect to (x,y)

2

2

' sin 2

' '

sin ,2

x

x x

VE i

I E

V

0

0

sin

sin .

V t V t

Vt t

V

voltage is modulated sinusoidally, at frequencies, , much lower than the optical

frequency,

0

2 0

1' sin

2

1' sin sin .

2

x

x

Vt t

V

VI t t

V

phase and intensity modulations

0

0

0

1' 1 cos sin

2 2

11 sin sin

2

1sin .

2 2

x

VI t t

V

Vt

V

Vt

V

''

1.

2

xd tt

dt

V tV

phase modulators also modulate the frequency of the light

Liquid crystal spatial light modulator: a) phase mode; b) amplitude mode.

Acousto-Optic Effect

The acousto-optic effect is the change in optical properties of a material due to the presence of an acoustic wave

Elasto-Optic Tensor

Light wave (wavevector k, frequency omega, speed c) interacts with a travelling grating induced by a sound wave (wavevector script-k, frequency Omega, speed v).

2

,

k

v

the medium acts as a (phase) grating, which is capable of diffracting the light

induced polarization in the normal coordinate system

3

3 2 20

, , 1

.i i j ijkl kl jj k l

P n n P S E

1.

2k l

kll k

u uS

x x

strain tensor

Kerr electro-optic effect, induced polarization

3

3 2 20

1

0 0 ,i i j ijkl k l jijkl

P n n S E E E

0 0 .ijkl kl ijkl k lP S S E E

, , , , 1, 2,3; , 1, 2,...,6

11, 11; 1, 1

22, 22; 2, 2

33, 33; 3, 3

12, 12; 6, 6

13, 13; 5, 5

23, 23; 4, 4.

ijkl mnP P i j k l m n

ij kl m n

ij kl m n

ij kl m n

ij kl m n

ij kl m n

ij kl m n

electro-optic tensor, due to symmetry, the elasto-optic tensor can also be used using the following (Voigt) contraction

Photoelastic effect: the sound wave induces strain, which in turn modulates the refractive index. a) Longitudinal sound wave; b) Transverse (shear) sound wave. Lambda is the sound wavelength, with Omega the frequency and v the propagation speed.

Acousto-Optic Effect in Isotropic Media

Let us investigate in more detail the acousto-optic effect in isotropic media

elasto-optic tensor for isotropic media

11 12 12

12 11 12

12 12 11

11 12

11 12

11 12

' ' ' 0 0 0

' ' ' 0 0 0

' ' ' 0 0 0

10 0 0 ' ' 0 0

21

0 0 0 0 ' ' 02

10 0 0 0 0 ' '

2

mn

p p p

p p p

p p p

p pp

p p

p p

3 40 3 13 63 53

3 40 3 63 23 43

3 40 3 53 43 33 .

x x y z

y x y z

z x y z

P n S p E p E p E

P n S p E p E p E

P n S p E p E p E

33 3klS S S expressions for the polarization

43 53 63 0p p p

3 40 3 13

3 40 3 23

3 40 3 33 .

x x

y y

z z

P n S p E

P n S p E

P n S p E

320

,

n P

D E

E

Dielectric displacement

212

2 20 3 12

211

1 ' 0 0

0 1 ' 0 .

0 0 1 '

n p

n S n p

n p

dielectric tensor

medium becomes uniaxial, i.e. xx yy zz

3

1

21

2

z

i t kz

uS

z

ikAe

plane acoustic wave of amplitude A

.i t kzzu Ae

a) Brogg regime; b) Raman-Nath regime.

Bragg Diffraction Regime

wave equation for the total field (incident plus diffracted) reads

3222

02 2 2

31

1 2

, ,,

, ,

, , , ,

i t

E t P tnE t

c t t

P t BAe E t

E t E t E t

k r

r rr

r r

r r r

2 2 22 0 2 1

2

, , , ,

, .i

E n E F E

F BAe

k r

r r r r

r

Fourier transform

1 1, , .i te E t E r r

2 2 21 0 1, , 0E n E r r

32

1

1 1

, ,

,

i

V

E F e d

A

qrq r r

q

k k

1 1A

2 1

2 1 .

k k

Bragg condition

21

1

1

2

2

.

k

k

The Brogg condition: a) k1, k2 and script-k are, respectively, the incident, diffracted and acoustic wavevectors. b) Triangle that illustrates geometrically k2=k1+script-k.

1

1

2 sin

sin 2

k

n

Raman-Nath Diffraction Regime

a) Bragg diffraction (see solution for k2).

b) Raman-Nath diffraction when sound wave is curved (multiple solutions for k2).

scattering potential in the (x,z) coordinates 2

0

02 2, ,i x

i z zF x z BAe e

diffracted field is the Fourier transform of F

2

2 00

, , .2

xx z z

qE q q q

q is the scattering wavevector, 2 1 q k k .

2

2 12 1 0

0

.2

x xz z

k kk k

if the sound beam curvature satisfies

2

2 10

0

,2

x xk km

Where m=0,1,2,…, the diffracted beam has multiple solutions

2 1 0 0 .z zk k m