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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
f NL  x  t  
d 2 x t 
dx  t 
eE  t 
2







x
t


0
dt 2
dt
m
m
2
3
f NL
 a2 x  t   a3 x  t   
m
b) Harmonic spring; Low intensity
c) Anharmonic spring (overstretched); High intensity
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 lines
Molecule oscillates over other degrees of freedom
Lorentz force ev  B becomes significant
Simple case where electric field polarized along x, magnetic field lies on y
E   Ex , 0, 0 
H   0, H y , 0 
f em  e  E  v  B 
 y zˆ
 y  xˆ  0  yˆ  exB
 e  Ex  zB
d 2 x t 
dx  t 
e  t 
2


x
t
Ex  z (t ) By (t ) 






0
2

dt
dt
m
d 2 y t 
dy  t 

 0 2 y  t   0

2
dt
dt
d 2 z t 
dz  t 
e
2






z
t
x (t ) By (t )


0
2
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:
eEr  
e
 i  z     By  
m
m
e
z   0 2   2  i   i  x     By   .
m
x   0 2   2  i   
x   
D  
eEx  

D 2    b 2  
m
b  
eEx  
z    2

.
D    b 2  
m
D    0 2   2  i
b    i
Express magnetic field in terms of electric field
Equations obtained from method in 5.1
Ignore magnetic field, we get linear response
e
 By   .
m
Order of magnitude relation between x and z displacements
z   eEx
eEx 


x   me 2 mc 2
k  E    i B  
By   
Ex
ic
Solve for when electric field and associated irradiance where x and z are comparable
1 2
2 mc 2
I
Ex
E x x    z   

2
e
 1012 V
m
,
 1021 W
m2
Using b   
e Ex
mc
 D   we have: e2

z     2 Ex 2   2
mc
D    b 2  

e2
  2 Ex 2   2
mc
D  
z     Ex  eit  Ex e  it 
2
 2 Ex 2 1  cos 2t  .
Electron oscillates at 2 on z axis, DC term is called optical rectification x    

D  
eEx
 2
m D    b 2  
eEx
1

.
m D  
8.1.3. Dropping the Complex Analytic Signal Representation of Real Fields.
Real field Complex analytic signal i t
U r  A cos t
U  Ae
1 2 it  it 2
Ur  A e  e 
2
 A2 1  cos 2t 
U  A2 e i 2  t .
Re[U ]  A 2 cos(2t )  U r
Complex analytic signal does not capture DC term.
Whenever we deal with fields raised to powers higher than one, we use
Ur 
1
U  U *
2
8.2. Second‐Order Susceptibility
d 2 x t 
dx  t 
2E t 
2
2







x
t
a
x
t




0
2
dt 2
dt
m
Want solution of form where x  x   x 
1
2
is the nonlinear perturbation to linear solution. Simplify by neglecting 2
 d 2 x 1  t 
dx 1  t 
eE  t  
2 1
x
t




0



2
dt
m 
 dt

x 2  t    x  2  t   0 2 x 2  t   a2  x 1  t  
2
D   x 2    a2 x 1   ⓥx 1  
d 2 x 2  t 
dx  2  t 


 0 2 x  2   t  
2
dt
dt
2
and 2
 a2  x 1  t    2a2 x 1  t  x  2  t   a2  x  2  t    0.
2
 e   E   E   
 a2   
ⓥ
,
 m   D   D   
Illuminating the nonlinear crystal with two monochromatic fields of different frequencies
E    E1   1   E1*   1  
 E2   2   E2*   2 
2
1  E   E   
e
x    a2  
ⓥ


m
D
D
D









 


 2

E  ' E    '
1
e
d '.
 a2  


m
D
D
'
D
'






     
 
   a  ⓥ   b      a  b 
2
E  
E  
2
All possible frequencies:
1
1
2

E   ⓥE  
22 2  1 21 2  1   0 2  1 21
2  1
22

2
1
e
x  2    a2  
 f 1   f  1   f 2   f  2  
 m  D  
 E12   21   E1     


1
  E1 E2   1  2    
f 1  
D 1  D   1  

*
  E1 E2    1  2  

2
 E2 2   22   E2 2     


1
  E1 E2   1  2   

f  2  
D  2  D    2  

  E1 E2*   1  2  

2
e
x  t    a2    g1  t   c.c.  g 2  t   c.c.
m
 2
E1
E12 ei 21t
E1 E2 e  1 2 
g1  t  



D  21  D 2 1  D  0  D 1  2 D 1  2  D 1  D 2 
2
Fourier
Transform
→
i   t
E1 E2*e  1 2 

 c.c.
D 1  2  D 1  D* 2 
i   t
E2
E2 2 e i 2  2 t
E1 E2 ei1 2 t
g2 t  



D  22  D 2 2  D  0  D 2  2 D 1  2  D 1  D 2 
2

E1 E2*e  1 2 
 c.c.
D 1  2  D 1  D* 2 
i   t
Equation 8.26 indicates that, as the result of the second-order nonlinear interaction,
the resulting field has components that oscillate at frequencies 22 (second harmonic
of 2), 21 (second harmonic of 1), 1 +2 (sum frequency), 1 -2 (difference
frequency) and 0 (optical rectification terms).
Nonlinear susceptibility from induced polarization
P
 2
 i   j    0 
 2
 i   j ; i ,  j  E  i  E   j  →
*
*
  2  i   j ; i ,  j   
P  2  i   j    Nex  2  i   j  , i, j  1, 2
P
 2
r    0  r  E r  E r 
 2
 r  E  r  E  r 
 P  r  .
 0 
0
x
2
 i  i 
a2 Ne3

 g  t   g 2  t   c.c.
 0m2  1
Importantly,   2 vanishes in centrosymmetric media
 2
Ne
P
2
  2  r     2  r 
 r    Ner
 Ne  r 
2
 P   r  .
Fulfilled simultaneously only if P  2  r   0
So second‐order nonlinear processes require noncentrosymmetric media
8.2.1. Second Harmonic Generation (SHG)
a2 Ne3
1
  21 ; 1 , 1  

 0 m 2 D  21  D 2 1 
a2 Ne3
1
.
  22 ; 2 , 2  

 0 m 2 D  22  D 2 2 
Nonlinear susceptibility a s function of linear chi: Linear:
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.
Ne 2
1
.
   

 0 m D  
1
Nonlinear:

 2
2
a2 0 2 m 1
1


 21 ; 1 , 1   2 3   21    1   .
N e
8.2.2. Optical Rectification (OR)

 2
a2 Ne3
1

 0; 1 , 1  
2
 0 m D  0  D 1  D  1 
a2 0 2 m 1
  0   1 1   1  1  .

2 3
N e
OR: a DC polarization is created in a chi(2) material.
8.2.3. Sum Frequency Generation (SFG)

 2
a2 Ne3
1

1  2 ; 1 , 2  
2
 0 m D 1  2  D 1  D 2 

a2 0 2 m 1
 1  2   1 1   1 2  .
2 3
N e
8.2.4. Difference Frequency Generation (DFG)

 2
a2 Ne3
1
1  2 ; 1 , 2  
 0 m 2 D 1  2  D 1  D  2 
a2 0 2 m 1

 1  2   1 1   1  2  .
2 3
N e
8.2.5. Optical Parametric Generation (OPG)
The time reverse process of SFG
1  2  3
2
  2
3
8.3. Third‐Order Susceptibility
Anharmonic oscillator

x  t    x  t   0 2 x  t   a3 x3  t   
eE  t 
.
m
Solve using perturbation theory, solution of form
eE  t  
 1
1
2 1




x
t
x
t
x
t











0
m


3
x  t    x  t   0 x  t   a3  x1  t   x 3  t    0.
 
 3
 3
2
 3

1
3
First term vanishes, approximate a3 x   x 

3
 
 a3 x  
1
3
3
→
3
3
3
1

x   t    x    t   0 2 x   t   a3  x   t   ,
e E   E    E 
1
x  t   x    
,
m D  
1
  1   E1*   1  
 E2   2   E2*   2  
 E3   3   E3*   3  .
1
For scalar fields perturbation displacement is: e

x  t    x  t   0 2 x  t   a3  
m
 3
 3
3
3

m , n , p 3
Em En E p e

 i m   n   p

D  m  D  n  D   p 
So, e 3 En e  int
 c.c.
x t    
m n 1 D n 
1
Induced polarization both for electromagnetic fields in terms of displacement and susceptibility
Pi 
Pi
3
 3
    Nex   
3
q
q
3
3
     
q
0
j , k ,l 1 m , n , p 3
 ijkl 3 q ; m , n ,  p  E j m  Ek n  El  p 
  0 d   ijkl 3 q ; m , n ,  p  E j m  Ek n  El  p ,
jkl
Where d is the degeneracy factor
3
General expression for   
a3 Ne*
1
,
 ijkl q ; m , n ,  p  

d  0 m3 Di q  D j m  Dk n  Dl  p 
 3
Da    0 a 2   2  i
Express   3 in terms of  1
a3 m 03
1
1
1
1

 ijkl q ; m , n ,  p  







 p   ,







i
q
j
m
k
n
l
3 4 
dN e
 3
a
1
Ne 2
1

  
 0 m Da  
8.3.1. Third Harmonic Generation (THG) signal contained by terms that oscillate at 3ω
The THG susceptibility for one component of the electric field (scalar case) is

 3
a3 Ne 4
1

 3;  ,  ,   
d  0 m3 D  3   D    3


3
a3 m 03 1
1


 3 4   3       ,
N e
D    0 2   2  i
8.3.2. Two‐Photon Absorption (TPA) and Intensity‐Dependent Refractive Index
If
1   , 2   , 3   ,

 3
a3 Ne4
1

;  ,  ,   
d  0 m3 D 2    D    2

2
2
a3m 03  1
1

 3 4        
N e

 3



2
2
2
a3m 03 1
   3 4      R1      I1    i 2  R1    I1  
N e
  R     i  I
3
3
  ,
Define effective refractive index 1
 3
    
    3    E   
 n0 2  1  3 3   E  
2
2
 n 2  1,
n  n0  n2 I
Find expression for n0  2n2 I  1   ( )
2
3   3
n2   
4n0 2 0 c

3
  R3    i  I3   

4n0  0 c 
2
 n2 '    in2 ''   .
• 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 ; 1 , 2 , 3  
   1  2  3
a3 Ne
1

d  0 m3 D   D 1  D 2  D 3 
k  k1  k 2  k 3
k‐vector conservation, phase matching condition
a) Generic four‐wave mixing process. b) Momentum conservation.
PNL  6 0    A1 A2 A3*e 
3
i k1  k 2  k 3 r
.
8.3.4. Phase Conjugation via Degenerate (all  are the same) Four‐Wave Mixing
Phase conjugation via degenerated four wave mixing: field E4 emerges as the phase conjugate of E3, i.e. E4=E3*.
8.3.5. Stimulated Raman Scattering (SRS).
8.3.5.1. Spontaneous Raman Scattering
Population of the excited vibrational levels obeys the Maxwell‐Boltzmann distribution
E
1  k BT
PE  e ,
z
Spontaneous Raman scattering: ‐a) Stokes shift
‐b) anti‐Stokes shift
P  E1 
P  E0 
Thus, the Stokes component is typically orders of magnitude stronger !
e
 h10
k BT
1
p   E.
the molecular optical polarizability, 
 t   0 

xv
xv  t   Av eivt  Av*eivt ,
P t   N P
xv  t  .
xv  0


 N  0 
xv


xv  t   E  t  .

xv  0
P  t   PL  t   PNL  t 
PL  t   N 0  AL e iLt  c.c.
PNL  t   N
  * iL v t
 i L v t


 c.c. .
.
.
A
A
e
c
c
A
A
e
v
L
v L

xv 0
Harmonic vibration occurs spontaneously due to Brownian motion
8.3.5.2. Stimulated Raman Scattering
To derive expression for SRS susceptibility, solve equation of motion for vibrational mode of resonant frequency and damping , d 2 xv  t 
dxv  t 
F t 
2

 v xv  t  
,
dt 2
dt
m
1
p t  E t  t
2
1
  E 2 t 
t
2
 2
1
d
 0 
xv  E  t 

2
dxv 0 
F t  
W
dW  Fdxv .

t
dW
dxv
1 d
2 dxv
E 2 t  .
t
0
1 d
F   
E   ⓥE  
2 dxv 0
E  t   AL e  iLt  AS e  iS t  c.c.
E    AL    L   AS    S   AL*   L   AS *   S 



P    N  0 
xv    E  
xv 0


F  
1
xv   
m v 2   2  i
1 d

2m dxv
0
N  d
PNL   

2m  dxv
AL* AS    L  ⓥ   S 
v 2   2  i
 N 0 E     N

xv   E  
xv 0
 PL    PNL   .
2
 AL* AS    L  ⓥ   S 
 AL   L   AL*   L  

2
2
v    i
0
  0 d   R    AL* AS    L  S    AL   L   AL*   L  
N  d
PNL  t  

2m  dxv
Induced
polarization for
SRS



 i t
*
* 
 AL AS AL e S  AL e L S

2
2


 i L  S 





0
v
L
S
2
i 2  t
nonlinear contribution
N  dx
S 
 t  

12m 0  dxv
 L  v   S
2

1
  2
2
v  L  S   i L  S 
0
→
N  dx
S 
  t   i

12m 0  dxv
2

1
 

   L  S 
0
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 
N  d
  AS   t   i

12m 0  dxv
 
S
2

1

  L  S 
0
t 
*
Strong attenuation
a) Raman susceptibility at Stokes frequency; indicates amplification. b) Raman susceptibility at anti‐Stokes frequency; indicates absorption.
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
 3
 CARS
 A  21  2 ; 1 , 1 , 2 
 3
 CSRS
S  22  1; 2 , 2 , 1 
CARS is a powerful method currently used in microscopy we will hear about it during student presentations.
8.4. Solving the Nonlinear Wave Equation.
8.4.1. Nonlinear Helmholtz Equation
B  r, t 
  E  r, t   
t
D  r, t 
  H  r, t  
 j  r, t 
t
  D  r, t   
  B  r, t   0.
 2 D  r, t 
    E  r, t   0
 0,
t 2
follow the standard procedure of eliminating B and H from the
equations
 0
j  0.
D  r, t    0 E  r, t   P  r, t 
  0 E  r, t   PL  r, t   PNL  r, t 
  0 r E  r, t   PNL  r, t 
    E  r, t       E    2 E
  2 E.
2
 2 PNL  r, t 
n 2  E  r, t 
 E  r, t   2
 0
,
c
dt 2
t 2
2
 2E  r,     2 E  r,    
1
0
 0 2 PNL  r,   ,
Nonlinear wave equation after approximation,
  D    E
term negligible
Propagation of the Sum Frequency Field
SFG nonlinear polarization has the form
 2
PNL
 r; 3  1  2 ; 1 , 2    0   2 3 ; 1 , 2   E1  r, t  E2  r, t 
E1  r, t   A1  r  e i1t  1z   c.c.
E2  r, t   A2  r  e
E3  r, t   A3  r  e
1  n 1  1 c
 i  2 t   2 z 
 i 3t  3 z 
 c.c.
 c.c.
 E3  r, t   n 3 
2
E3
2
x

E3
y
32
c2
E3  r, t    032 0    E1  r, t  E2  r, t 
2
0
d 2   z   i3t  3 z  
 E3  r, t   2  A3 e

dt
 d 2  z
  i3t  3 z 
dA3
2
  2 A3  2i 3
 3 A3  e
.
dz
 dz

2
d 2 A3  t 
dA3  z 
i  1   2  3  z
 2
2
i
A
A
e

2







.
3
0 3 0
1 2
2
dt
dz
Simplifying approximation, A1 and A2 , do not change with z (do not deplete).
d 2 A3  t 
dA3  t 
ikz
i
Be

2



3
dt 2
dt
B   32    A1 A2
2
k  1   2   3 ,
amplitude is slowly varying
d 2 A3  z 
dA3  t 
 3
.
dz 2
dz
2
i  ikz
dA3  t  
e dz
23
L
iB
A3  L  
eikz dz

23 0
The intensity of SFG field is
I3  z  
iB  eikL  1 



23  ik 
kL 
iB i2kL  2sin
2

e
k
23




kL 
iBL i2kL  sin
2

e
23
 kL

2 


2
1
A3  z 
2


     0 n
B 2 L2

sinc 2 kL ,
2
2
83
Net output power can vanish at kL
2
  , 2 ,...

iB i2kL
e sinc kL .

2
23
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
k  0
 3  1   2
n 3 
n 1  1


 n 2  2 , 3  1  2
c
c
c
3
n 3  
n 1  1  n 2  2
.
1  2
Normal dispersion
Abnormal dispersion.
Normal dispersion curves for a positive uniaxial crystal.
Negative  ne  no 
Positive ne  no
Type I
ne 3  3  no 1  1  no 2  2
no 3  3  ne 1  1  no 2  2
Type II
ne 3  3  ne 1  1  no 2  2
no 3  3  no 1  1  ne 2  2
Table 8-1.
cos 2  sin 2 


2
2
n  
n0
ne 2
1
phase matching can be achieved by angle tuning, that is, selecting the angle  that ensures k
Type I
no 3  3  n 1 ,  1  n 2 ,  2
Type II
no 3  3  no 1  1  n 2 ,  2
Type II phase matching by angle tuning in a positive uniaxial crystal: o‐ordinary wave, e‐extraordinary wave, c‐optical axis.
0
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
 
Pi  ;  , 0    0  ijk
  ;  , 0  E j    Ek  0  .
2
Kerr effect
 3
Pi  ;  , 0, 0    0  ijkl
;  , 0, 0  E j   Ek   El   ,
Due to the electro‐optic effect, an optical field can suffer voltage‐dependent polarization and phase changes.
Pockles effect
Pi 
2
 ;  , 0   
rijk 
 0 2  ijk
1
0
 ii jj rijk E j   Ek  0  .
  ijk
 ii jj  ni 2 n j 2
rijk  rjik
Change in the rank of the tensor, from 3 to 2
r11k  r1k
r22 k  r2 k
r33k  r3k
r12 k  r21k  r6 k
r13k  r31k  r5 k
r23k  r32 k  r4 k .
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
32  27 to 3  6  18 .
Electro‐Optic Effect in Uniaxial Crystals
Use KDP (KH2PO4, or potassium dihydrogen phosphate) as a
specific example of uniaxial crystal
0

0
0
r 
 r41
0

0
0
0
0
0
r41
0
0

0
0

0
0

r63 
The KDP refractive index tensor is (in the normal coordinate
system of interest)
 no

n 0
0

assume that the voltage is applied only along z, such that only r63 is relevant.
0
no
0
0

0
ne 
D   0 n2E  P
Di   0 ni 2 Ei   0 ni 2 n j 2 rijk E j   Ek  0 
  ij E j   ,
electric displacement can be expressed for each component as
(i=x, j=y, k=z)
Dx   0 no 2 Ex     0 no 4 r63 E y   Ez  0 
Dy   0 no 2 E y     0 no 4 r63 Ex   Ez  0 
Dz   0 ne 2 Ez   .

 no 4 r63 Ez  0  0 
no 2


no 2
0 .
   0  no 4 r63 Ez  0 
2

0
0
n
e 

Electro‐optic effect in a KDP crystal. For E(0) parallel to z, the normal axes rotate by 45 degrees around the z‐axis.
express  ij in a reference system rotated about the z-axis by an arbitrary angle, say 
 '    R     R  
 cos 
 0 
 sin 
 sin    no 2  D   cos 


cos     D no 2    sin 
 no 2  D sin 2
 0 
  D cos 2
 D cos 2 
,
2
no  D sin 2 
 no 2  D
0
0 


 '  0  0
no 2  D 0  .
2
 0
0
n
e


sin  

cos  
D  no 4 r63 Ez  0 
Electro‐Optic Modulators (EOMs)
defining the normal refractive indices
nij '   ij '
 n 'x

n'   0
 0

0
n 'y
0
0 

0 
n 'z 
phase retardation
   n 'x  n ' y 

2
0

c
L
no 3r63 Ez  0  L,
n 'x  no 2  D
1D
 no 
2 no
1
 no  no 3 r63 Ez  0 
2
1
n ' y  no  no 3 r63 Ez  0 
2
n 'z  ne .
Ez  0  
V
.
d
half-wave voltage
V 
0
 
d

.
3
2no r63 L
 V   
V
.
V
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.
KDP longitudinal modulator reads
E ' V   Ee
ino k0 L
 Ee
ino k0 L
e
e
1
k0 no3 r63V
2
i V 
2
.
(x’,y’) is rotated by 45° with respect to (x,y)
 Ex ' 
  V  
0
E
i
'
sin


R
W
V
R






45
45
0








 
x

2 
1

 Ey ' 
2
 i2

I
E
'
'

x
x
0 
e
W0 V   
 i 

2   V 
 0 e2 
sin




,
2


 2

2


2
2
.
R  45   

2
2


2 
 2
voltage is modulated sinusoidally, at frequencies,  , much lower than the optical
frequency,
V  t   V0 sin t
 t   
V0
 sin t.
V
phase and intensity modulations
1 V
x '  t    0 sin t
2 V
1 V

I x '  t   sin 2   0 sin t  .
 2 V

phase modulators also modulate the frequency of the light
dx '  t 
dt
1

V  t  .
2 V
 't  


V
1
I x '  t   1  cos    0 sin t  
2
V
2

 V0

1
 1  sin   sin t  
2
 V

1  V0
 
sin t.
2 2 V
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
the medium acts as a (phase) grating, which is capable of
diffracting the light
2

k

 ,
v
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).
induced polarization in the normal coordinate system
Pi
 3
3
    0 
j , k ,l 1
ni 2 n j 2 Pijkl Skl E j  .
electro-optic tensor, due to symmetry, the elasto-optic tensor can
also be used using the following (Voigt) contraction
strain tensor
1  u u
Skl   k  l
2  xl xk

.

Kerr electro-optic effect, induced polarization
Pi
 3
3
    0  ni 2 n j 2 Sijkl Ek  0  El  0  E j  ,
ijkl 1
Pijkl Skl  Sijkl Ek  0  El  0  .
Pijkl  Pmn , i, j , k , l  1, 2,3; m, n  1, 2,..., 6
ij  11, kl  11; m  1, n  1
ij  22, kl  22; m  2, n  2
ij  33, kl  33; m  3, n  3
ij  12, kl  12; m  6, n  6
ij  13, kl  13; m  5, n  5
ij  23, kl  23; m  4, n  4.
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
pmn
 p11 '

 p12 '
 p12 '


 0

 0


 0

p12 '
p11 '
p12 '
p12 '
0
0
0
0
p12 '
p11 '
0
0
0
0
1
 p11 ' p12 '
2
0
0
0
0
1
 p11 ' p12 '
2
0
0
0
0




0


0



0


1
 p11 ' p12 ' 
2

0
0
S kl  S33  S3
expressions for the polarization
Px 3     0 n 4 S3  p13 Ex    p63 E y    p53 Ez   
Dielectric displacement
D      0 n 2 E     P  3   
    0 n 4 S3  p63 Ex    p23 E y    p43 Ez  
3
Pz      0 n 4 S3  p53 Ex    p43 E y    p33 Ez    .
  E   ,
p43  p53  p63  0
dielectric tensor
Py
3
Px 3     0 n 4 S3 p13 Ex  
Py 3     0 n 4 S3 p23 E y  
Pz 3     0 n 4 S3 p33 Ez   .
1  n 2 p12 '

0
0


 0 n 2 S3 
0
1  n 2 p12 '
0
.
2


0
0
1
n
p11 ' 

medium becomes uniaxial, i.e.  xx   yy   zz
1 u z
S3 
2 z
1
 ikAe  i t  kz 
2
plane acoustic wave of amplitude A
uz  Aei t kz  .
a) Brogg regime; b) Raman‐Nath regime.
Bragg Diffraction Regime
wave equation for the total field (incident plus diffracted) reads
 2 P  3  r , t 
n 2 E  r, t 
 E  r, t   2
 0
c
t 2
t 2
3
 i t k r 
P    r, t    BAe 
E1  r, t 
2
E  r, t   E1  r, t   E2  r, t  ,
E2  q,     F  r,   eiqr d 3r
V
 A1        q   
    1       k  k1    ,
Fourier transform
 2 E2  r,    n 2  0 2 E2  r,    F  r,   E1  r,    
F  r,     2 BAeik r .
A1       1 
k 2  k1  
2  1  .
 2 E1  r,    n 2  0 2 E1  r,    0
eit E1  r, t   E1  r,     .
Bragg condition
k2 

2
1  
2
1
 k1.
2k1 sin   
  sin 1  2n 

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.

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
F  x, z ,     2 BAe
i 0 x 2
i 0 z
2z
e
diffracted field is the Fourier transform of F


qx 2  
E2  q x , q z ,      q z    0 
 .
2 0  


q is the scattering wavevector, q  k 2  k 1 .
k  k 
k2 z  k1z   0  2 x 1x
2 0
2
.
if the sound beam curvature satisfies
 k2 x  k1x 
2 0
2
 m 0 ,
Where m=0,1,2,…, the diffracted beam has multiple solutions
k2 z  k1z   0  m 0 .