Download Electro-optics/ Acousto-optics

Chapter 9 – Electro-Optics
Gabriel Popescu
University of Illinois at Urbana‐Champaign
y
p g
Beckman Institute
Quantitative Light Imaging Laboratory
Quantitative
Light Imaging Laboratory
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
Electro‐Optics
Electro
Optics

1st order effect:
Pi  
w
1
0
DC
 ii jj rijk E j ( ) Ek (0)
11Oz
Di   0 ni Ei   0 ni n j rijk E j Ek ( DC )
2
2
2
Dx   0 n02   0 n04 r12 3 E y E Dc   0 n02 ne r13 3 E z E DC
6
 Dy   0 n02   0 n04 r213 E x E Dc  0
4
Dz   0 ne 2 E z
Chapter 9: Electro‐Optics
5
r53  0
kDP
2
'1
226
ELECTRO-OPTICS
By using the contracted indices (7.1-11), the equation of the index
ellipsoid in the presence of an electric field can be written
(:~ + rlkEk ) x
2
+ (:; + r2kEk) y2 + (:; + r3kEk) Z2 (7.2-3)
+2yzr4k E k + 2zxr5kEk
I
I
I
Table 7.2. Electro-optic: Coefficients in Contracted Notation for All Crystal
Symmetry Oasses
Q
Centrosymmetric (I, 2/m, mmm, 4/m, 4/mmm, 3, 3m 6/m,6/mmm, m3, m3m):
000
000
000
000
000
I
+ 2xyr6kEk = 0
000
where Ek (k = 1,2,3) is a component of the applied electric field and
summation over repeated indices k is assumed. Here 1,2,3 correspond to
the principal dielectric axes x, y, z, and nx ' ny, nz are the principal refractive indices. This new ellipsoid (7.2-3) reduces to the unperturbed ellipsoid
(7.1-1) when Ek = O. In general, the principal axes of the ellipsoid (7.2-3) do
not coincide with the unperturbed axes (x, y, z).
A new set of principal axes can always be found by a coordinate rotation,
which is known as the principal-axis transformation of a quadratic form.
The dimensions and orientation of the ellipsoid (7.2-3) are, of course,
dependent on the direction of the applied field as well as the 18 matrix
elements rlk. We have argued above that in crystals possessing an inversion
symmetry (centrosymmetric), rlk = O. The form, but not the magnitude, of
the tensor r1k Can be derived from symmetry considerations, which dictate
which of the 18 coefficients r1k are zero, as well as the relationships that exist
between the remaining coefficients. In Table 7.2 we give the form of the
electro-optic tensor for all the noncentrosymmetric crystal classes. The
electro-optic coefficients of some crystals are listed in Table 7.3.
Triclinic:
rij =
0
0
0
0
0
r63
'13
'23
'31
'32
'33
'41
'42
'43
'51
'52
'53
'61
'62
'63
2 (2 II X2)
0
0
0
2 (2" X3)
0
0
'13
0
0
'23
0
0
0
0
0
'41
0
'43
'41
'42
0
'52
0
'51
'52
0
0
'61
0
'63
0
0
'63
'12
'22
'32-
m (m J. X2)
Consider the specific example of a crystal of potassium dihydrogen phosphate (KH 2 P04 ), also known as KDP. The crystal has a fourfold axis of
symmetry, which by strict convention is taken as the z (optic) axis, as well as
two mutually orthogonal twofold axes of symmetry that lie in the plane
normal to z. These are designated as the x and y axes. The symmetry group
of this crystal is 42m. Using Table 7.2, we write the electro-optic tensor in
the form
0
0
0
0
r41
0
'12
'22
Monoclinic:
7.2.1. Example: 1be Electro-optic Effect in KH 1 P04
0
0
0
r41
0
0
'Il
'21
'33
m (m J. X3)
'21
0
0
'23
'21
'22
'31
0
'33
'31
0
'42
0
0
'32
0
'43
'51
0
'53
0
0
'53
0
'62
0
'61
'62
0
0
0
0
222
0
0
0
0
0
'41
0
0
'Il
'13
'II
0
'12
0
0
Orthorhombic:
(7.2-4)
0
0
'52
0
2mm
0
0
0
0
0
0
'51
'63
0
0
0
0
'42
0
0
'\3
'23
'33
0
0
0
227
"r
THE LINEAR ELECTRO-OPTIC EFFECT
229
Table 7.2. ( Continued).
Table 7.1- ( Continued).
Hexagonal:
Tetragonal:
'13
0
4
0
0
'13
0
0
0
'33
0
0
4
0
0
0
0
'41
'51
'51
-'41
0
0
0
0
0
'41
-'51
0
0
'51
'41
0
0
4mm
0
0
0
0
0
0
0
'51
0
0
'51
0
'13
-'13
0
'\3
'33
0
0
0
'63
(211
0
-'41
0
6
0
0
0
0
0
0
0
0
0
'13
'41
'51
0
'51
-'41
0
0
0
0
0
0
0
'51
'51
0
0
0
'33
0
0
'11
-'22
'22
0
0
0
0
0
0
0
0
0
0
-'II
0
0
0
0
0
0
'41
0
'63
-'22
'13
0
-'22
-'22
'22
622
0
0
0
0
0
0
0
'\3
'33
'41
0
0
0
0
0
6m2 (m .L XI)
-'11
0
0
0
0
0
0
0
0
'41
0
'\3
6
XI)
0
0
6mm
0
0
0
0
0
0
0
0
0
0
0
0
'41
0
0
42m
'\3
0
0
0
422
0
0
0
0
0
0
0
-'41
0
0
0
6m2 (m.l X2)
'II
0
0
0
0
0
0
-'11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-'11
0
Cubic:
Trigonal:
32
3
-'22
'22
0
'33
0
'41
'51
0
'41
0
0
0
0
'51
-'41
0
0
-'41
-'22
-'II
0
0
-'11
'II
-'II
0
3m
0
0
0
0
'51
-'22
(m .1
'13
'11
'\3
-'11
XI)
3m
(m .1 X2)
0
0
0
-'22
'13
'11
'22
'13
-'II
'33
0
0
'51
'51
0
0
-'11
0
'51
0
0
0
0
0
0
0
0
0
0
0
'13
'13
'33
0
0
0
0
0
0
'41
0
0
43m,23
0
0
0
0
0
'41
0
0
0
0
0
'41
0
0
0
0
0
0
432
0
0
0
0
0
0
0
0
0
0
0
0
°The symbol over each matrix is the conventional symmetry-group designation.
so that the only nonvanishing elements are '41 = '52 and '63' Using Eqs.
(7.2-3) and (7.2-4), we obtain the equation of the index ellipsoid in the
presence of a field E(Ex' Ey ' Ez ) as
where the constants involved in the first three terms do not depend on the
field and, since the crystal is uniaxial, are taken as nx = ny = no' n z = ne'
We thus find that the application of an electric field causes the appearance
of "mixed" terms in the equation of the index ellipsoid. These are the terms
with xv. xz. vz. This means that the major axes of the ellipsoid, with a field
228
ECE 460 – Optical Imaging
Electro‐Optics
Electro
Optics

n02
n04 r63 EDc 0 


4
2
  ij   0   n0 r63 E Dc
n0
0 

2
0
0
n
e 

 ij '  W (  )  W (  ) 
2
 cos   sin    n0    cos 


 

2
 sin  cos     n0    sin 
sin  
cos  
b
i
 n0 2  

0
0



2
   ij  0  0
n0   0  ;   n0 4 r63 Ez ( DC )
4

2

0
0
n
e


biaxial crystal
Chapter 9: Electro‐Optics
3
ECE 460 – Optical Imaging
Modulators

Eg (tetra ) g KDP(
y
42m ) r41 , r52 , r63 only three nonzero elements
n x 2  n0 2  ;

1

1
nx  n0    n0 1  2  n0  n0 2  n0  n0
2 n0
2
n0
1 3
 n0  n0 r63 Ez ( DC )
2
1 
n y  n0 
2 r01
2
Chapter 9: Electro‐Optics
4
ECE 460 – Optical Imaging
Modulators
2
 2 3

(nx  n y ')d  
n0 r63 Ez ( DC )d

n0 0
V
    V 

T  ssin
2
2
Chapter 9: Electro‐Optics
0
2n03r63
Add
linearize
  V 
QW  T  sin  

4
2
V
 

2
5
ECE 460 – Optical Imaging
Modulators

Let V  Vm sin mt
 

T  sin    m sin mt  
4 2

2
1

 1
 1  cos    m sin mt    1  sin   m sin mt 
2
2
 2
linear
m  1  T 
Chapter 9: Electro‐Optics
1
1   m sin mt    m
2
6
ECE 460 – Optical Imaging
Quadratic (Kerr)
Quadratic (Kerr)
Pi
( )
Chapter 9: Electro‐Optics

1
0
 ii jj sijk  E j ( ) Ek ( DC ) Ee ( DC )
7
ECE 460 – Optical Imaging
Applications of EO
Applications of EO



longitudinal
g
transverse
For LiNiO3 : 
2
0
n0 L 
0
Chapter 9: Electro‐Optics
modulators
1 3
nx  n0  n0 r13 E
2
1
n y  n0  n03r13 E
2
1
nz  ne  ne3r33 E
2
 3
n0 r13V
v
Ed
V 
v
3
n0 r13

Phase mod(indep. of polariz.)
8
Chapter 9 – Acousto-optics
Gabriel Popescu
University of Illinois at Urbana‐Champaign
y
p g
Beckman Institute
Quantitative Light Imaging Laboratory
Quantitative
Light Imaging Laboratory
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
Acousto‐optics
Acousto
optics
Pi   0  n j ni pijkl Skl E j
2
2
jkl
12  6
13  5
23  4
x
• ac wave:
z
 S13  S5
 cos(t  kz )
U ( z , t )  xA
Chapter 9: Acousto‐optics
10
Chapter 1: Introduction
11
Acousto‐optics
Acousto
optics
k2
K

k1 
K
k2
k1
2nk0 sin   K
k0  2 / 
K  2 / 
sin  B 


; Bragg angle
ECE 460 – Optical Imaging
Acousto‐optics
Acousto
optics
  small
ll ( k  k0    0 )
  Δk  Δv
Doppler
Shift
 Kvs

Quantum
mechanics
Chapter 9: Acousto‐optics
k' k 
  '    
  
conservation of momentum
conservation of energy
13
ECE 460 – Optical Imaging
Anisotropic media
Anisotropic media
k ' k  
k'
'

n-different


- negative
k
k 'sin  ' k sin    ;   '
2 n '
2
2
sin  '
sin  
; 

0
0

n
0
  wavelength of sound
sin  ' 
 sin 
n ' n '
Chapter 9: Acousto‐optics
2 n
14
ECE 460 – Optical Imaging
Anisotropic
Ex:
C
k
'

k - (e)
k ' - scattered in prop. Plane – (o)
k ' (o)
0
ne
 sin  ' 
 sin 
n '  n0
0




 0

 '    
n0  ne
2
2


 '  , 
   0
2
Chapter 9: Acousto‐optics
2
n0  ne
0
n0  ne

0
n0  ne
15
ECE 460 – Optical Imaging
Small angle Scattering
Small angle Scattering
I scatt
 sin 2 (  L )
I inc
k0 ( n1n2 )3/2 
ei p ijke S ke e j

4 cos1 cos 2
 vs
Kin. Energy/ V = ½ Wtotal
 2
1
2
2 1
3
  vs  | u |   vs [ | U |]
2
2
 U



S
U
2
1


 I ac   vs 3 S
 z

2
1
I ac   vs
2
Chapter 9: Acousto‐optics
U
t
2
16
ECE 460 – Optical Imaging
Small angle Scattering
Small angle Scattering
S
2II ac
2
k0 ( n1n2 )3/2
2 I ac
 
P
3
 vs
 vs 3
4 cos1 cos 2
6
Small 
 cos 1
2
n p
M
 vs 3

p
table
k0 (n1n2 )3/2

4
Chapter 9: Acousto‐optics
 vs 3 M 2 I ac

6
3
n
 vs

MI ac  
20
17
ECE 460 – Optical Imaging
Small angle Scattering
Small angle Scattering
Detuning:
g
sin  B 

2k
sin  2  sin 1 
  k1 sin 1  k2 sin  2

k
; 2   B  
( Bragg )  2k sin   
 ( )  k1 cos11  k2 cos 2 2
1   2  
 ( )  k ((cos1  cos 2 ) 
 2k sin  B  
2
2



I scat
2



1

2
2
2

sin

L
1
;



s






 2  


2
I inc

2
1






2


    
2

Chapter 9: Acousto‐optics
18
ECE 460 – Optical Imaging
Finite Beams
Finite Beams
A’


B’
B

2

 
; 
 nw0
L
size of acoustic beam
     ;     1  ;  
2
2L
 f s 
f 
2nvs cos  20
4vs cos 
1

- Full   vs 
0
 nw0
 w0
2
1
; or (   )  f  2nvs cos 
W0
0
L
! Not overlap with undiffracted order
Chapter 9: Acousto‐optics
19
ECE 460 – Optical Imaging
N spots
N spots
N


 B 

 nW0  W0

f  N
2nvs cos 20
2vs


2nvs
0 f
f ; cond    B
2
0
0

f
2
n


f 

L 2nvs
f0
0 L
Chapter 9: Acousto‐optics
20