Download Propagation in anisotropic media

11/29/2010
Propagation in Anisotropic Media
Gabriel Popescu
University of Illinois at Urbana‐Champaign Beckman Institute
Quantitative Light Imaging Laboratory
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
Introduction
 Isotropic media easy, same index & velocity in all directions
 Anisotropic media
1) always 2 “transverse” modes
2) in general, different velocities
 
3) k  E  0 in general
 What is different?
isotropic
anisotropic
 Both are charge on springs
 2 orthogonal modes
identical springs
unequal springs
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Introduction

E
Px   o ( 
'
11E x

'
12 E y

'
13 E z )
Py   o (  '21Ex   '22 E y   '23 Ez )
Pz   o (  '31Ex   '32 E y   '33 Ez )
_
displacement


x(t )  eE
 
P  E
 Always possible to diagonalize a 3x3 matrix
 Means a set of coordinates that is rotated with respect to crystal axis co‐ordinate system
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Introduction
 in that reference frame: Px   o 11Ex
Py   o  22 E y
Pz   o  33 Ez
 'ij
 Values different from !
 Axis set is called principal dielectric axes
 Can also discuss this in terms of dielectric (versus) susceptibility tensor (no new physics!)
Dx   '11Ex   '12 E y   '13 Ez
can again diagonalize
end up with same principal axes!
Dy   '21Ex   '22 E y   '23 Ez
Dz   '31Ex   '32 E y   '33 Ez
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Introduction

 
 of course have D   o E  P   ij   o (1   ij )
 ij
  ij   ji
Hermitian + symmetry
 if are real
 ij
Hermitian
 if are complex
  ij   ji*
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Propagation in Anisotropic Medium
 


 assume plane wave E (r , t )  Ee j (t  k r )

 

H ( r , t )  He j (t k r )
Taking F.T. in both time and space:



 

B 
  E  

k  E   H
t 







   D 

k  H   E
  H  J 

t 

0
  


 eliminating H  k  (k  E )   2  E  0
 x 0

 with    0  y
0

Propagation in Anisotropic Media
0
0
0 
 z 
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Propagation in Anisotropic Medium
zˆ  k y Ez  k z E y 


k z   k z Ex  k x Ez 
Ez  k x E y  k y Ex 
xˆ
yˆ
zˆ
  
k kE 
kx
ky
kz
k y Ez  k z E y k z Ex  k x E z k x E y  k y E x
 Look at component, for example
x̂
xˆ
 
k  E  kx
Ex

yˆ
ky
Ey

xˆ  k y k x E y  k y 2 E x  k z 2 E x  k z k x Ex 
  
k  k  E xˆ   2  x E x   2  x  k y 2  k z 2 E x




 k x k y E y  k x k z Ez  0
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Propagation in Anisotropic Medium
 Can write the whole thing in matrix form
 2  x  k y 2  k z 2
 E 
kxk y
kxkz

 x
k ykx
k y kz
 2  y  k x 2  k z 2

 Ey   0


kzkx
kz k y
 2  z  k x 2  k y 2   E z 

 For non‐trivial solution

(trivial solution is )
E 0
0
 2  x  k y 2  k z 2
kxk y
kxkz
k ykx
 2  y  k x 2  k z 2
k ykz
kz kx
kz k y
 2  z  k x 2  k y 2
 i.e. Propagation in Anisotropic Media
0
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Propagation in Anisotropic Medium

 3 dimensional surface in k
2
2
2
2
define  x  n1 ( nx )  y  n2 (n y )
 z  n32 (nz 2 )
e.g. Optic axis in k
g p
plane (see next slide)
(
)
x‐ky p
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
k surface

same speed
Optic axis
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Propagation in Anisotropic Medium
 “normal” surface
 Crossing points define “optic axis” (axes)
 in general, can have 2 “optic axes”
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Propagation in Anisotropic Medium

 For any given direction , there are two “orthogonal” fields, k
different phase velocities

(for the vector)
D

 Fields given by 
kx
 2

2
k



x

 E 
x


ky
 
 2
  Ey 
2
k




y
 

  Ez 
 2 kz2

 k    z 

 along optic axis, is degenerate 
k
1 phase velocity
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Classification of Materials
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Classification of Materials
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Classification of Materials
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Classification of Materials
 2 refractive indices no , ne  uniaxial ne  no positive uniaxial
no  ne negative uniaxial
 nx , n y , nz biaxial
convention i  nz  n y  nx
(may be totally unrelated to crystal axes!) Propagation in Anisotropic Media
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Uniaxial Crystals
nx  n y  no
no

2
c2
nz  ne
 k y2  kz 2
k ykx
kz kx
Propagation in Anisotropic Media
kxk y
no

2
c2
kxkz
 kx2  kz 2
kz k y
k ykz
ne

2
c2
0
 kx2  k y 2
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Uniaxial Crystals
 some mathematics
 k x 2  k y 2 k z 2  2  k x 2  k y 2  k z 2  2 

 2  2 
 2 0
 ne 2
no
c 
no 2
c 





ellipsoid of revolution

 along z‐axis k x , k y  0

sphere
 k 2  2  k 2  2 
  z 2  2  z 2  2   0
c  no
c 
 no
 Surfaces “touch” along z‐axis
 modes are “degenerate”
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Double Refraction at Boundaries
 Always have to get light into crystal to have an interaction

E ||
 for tangential fields to be continuous (e.g. )

 k || Preserved

 However not the same for all polarizations
k
 2 different
2 diff
waves excited
i d
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Double Refraction at Boundaries
 waves are not collinear!
 waves are not necessarily orthogonal!!
 
k1  k 2
( )
 ko sin  o  k1 sin 1  k2 sin  2 ‐ Snell’s Law
 because of isotropy in x‐y plane, 1 wave is always
b
fi
i
l
1
i l
ordinary wave
ko sin  o  no sin 1
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Biaxial Crystals
 nz  n y  n x
 optic axis is propagation direction for which there are two degenerate modes
kz
n(Ѳ) varies from nzto nx
ny lies between these limits
nz
Ѳ
ny
kx
 optic axis lie in x‐z plane where
sin 2  cos 2 
1

 2
nz 2
nx 2
ny
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Biaxial Crystals
 solve for Ѳ
1/2
2
2
nz  n y  nx 
 2
2 
nx  nz  n y 
VNC do not touch
surfaces intersect!
 tan  
beam with
divergence
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Biaxial Crystals
 In other planes, surfaces do not intersect
kz
ky
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Biaxial Crystals
 again have double refraction at an interface
 also new phenomenon Conical Refraction
 back to optic axis in x‐z plane
 at intersection “singular” point, energy can go anywhere along g
p
,
gy
g
y
g
a cone of angles
 called “conical diffraction”
 no unique surface normal (group velocity property)
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Aplications of Birefringent Crystals
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Optical Activity
 can undo rotation by reflecting off a mirror of going back through crystal
(different from Faraday rotation!) Propagation in Anisotropic Media
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Optical Activity
 Ρ‐specific rotary power (angular rotation per unit distance)
 “dextrarotary” ‐ right‐handed
((counter‐clockwise as seen by observer)
y
)
 “levarotary” ‐ left‐handed
(clockwise as seen by observer)
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Faraday Rotation
 also rotates plane of polarization
 but going back through medium does not reverse the rotation
 occurs in all materials, need strong magnetic fields
 what is going on??
 look at response of electros in matter
 Electron on springs with
k1
Equal force constants
k2
k3
B (parallel to z-axis)

 Also optical field present E
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