Transcript Chapter 6 - Physics & Astronomy

Light
and
Matter
The tensor nature of susceptibility
Tim Freegarde
School of Physics & Astronomy
University of Southampton
Birefringence
• asymmetry in crystal structure
causes polarization dependent
refractive index
• ray splits into orthogonally polarized
components, which follow different
paths through crystal
• note that polarization axes are not
related to plane of incidence
2
Anisotropic media
• difference in refractive index (birefringence)
or absorption coefficient (dichroism)
depending upon polarization
P   0 E   0   1E
2
c 

2
1          i

2 

E
P
2
• recall that
where
• difference in  or

implies difference in

1
• susceptibility  cannot be a simple scalar
3
Linear dichroism
• fields normal to the conducting wires are
transmitted
• fields parallel to the conducting wires are
attenuated
• current flow is not parallel to field
I
• susceptibility not a simple scalar
WIRE GRID POLARIZER
4
Birefringence – mechanical model
• springs attach electron cloud to fixed ion
z
• different spring constants for x, y, z axes
• polarization easiest along axis of
weakest spring
• polarization therefore not parallel to field
• susceptibility not a simple scalar
y
x
5
Susceptibility tensor
• the Jones matrix can map
 a , a    a  , a 
 x y   x y 
• similarly, a tensor can describe the
susceptibility of anisotropic media
• e.g.
Px   0 11Ex  12 E y  13 Ez 
 ax   a11 a12  ax 
 
 

 a  a
a 
a
y
   21 22  y 
JONES MATRIX
 11
P   0   21
  31
12 13 
 22  23  E
 32  33 
SUSCEPTIBILITY TENSOR
6
Diagonizing the susceptibility tensor
• if the polarization axes are aligned with
the principal axes of birefringent crystals,
rays propagate as single beams
• the susceptibility tensor is then diagonal
• a matrix may be diagonalized if symmetrical:
 ij   ji
 11
P   0  0
 0
0
 22
0
0 
0  E
 33 
DIAGONAL
SUSCEPTIBILITY TENSOR
• the optical activity tensor is not symmetrical;
it cannot be diagonalized to reveal principal
axes
7
The Fresnel ellipsoid
• surface mapped out by electric field vector
for a given energy density
E z
E D  U
• symmetry axes x’,y’,z’ are principal axes
• semi-axes are
1
 xx
,
1
 y y 
,
1
 z z 
E y
FRESNEL
ELLIPSOID
E x
8
The Fresnel ellipsoid
• allows fast and slow axes to be determined:
• allows fast and slow axes to be determined:
• establish ray direction through Poynting
vector
E z
S
• electric field must lie in normal plane
• fast and slow axes are axes of elliptical
cross-section
• axis lengths are
1
 fast
,
1
 slow
E y
FRESNEL
ELLIPSOID
E x
9
The optic axis
• if the cross-section is circular, the refractive
index is independent of polarization
E z
• the Poynting vector then defines an optic axis

 yy, the single optic axis lies

 yy, there are two, inclined,
• if xx 
along zˆ
• if xx 
optic axes
S
(uniaxial crystals)
(biaxial crystals)
E y
FRESNEL
ELLIPSOID
E x
10
Uniaxial crystals
• the single optic axis lies along
zˆ
optic axis E
z
• one polarization is inevitably perpendicular
to the optic axis
(ordinary polarization)
S
• the second polarization will be orthogonal
to both the ordinary polarization and the
Poynting vector (extraordinary polarization)
• positive uniaxial:
• negative uniaxial:
e  o
o  e
E y
FRESNEL
ELLIPSOID
E x
11
Poynting vector walk-off
E S  0
• wavevector vector obeys D k  0
• Poynting vector obeys
D
E
• in anisotropic media, E and D are not
necessarily parallel
• the Poynting vector and wavevector may
therefore diverge
S
Poynting
vector
k
wavevector
Fresnel
ellipsoid
12
How light interacts with matter
• atoms are polarized by applied fields
D   0E  P
  0 1   E
V x 
P  0 E
m02 2
x
2
E
• Lorentz model: harmonically bound classical particles
dV
mx  
 m02 x
dx
x
13
Quantum description of atomic polarization
energy
2 e i0t
0
• harmonic
• oscillator
two-level atom

0
1
• weak electric field

expi
i0t 2
 r, t   a1 1  b 2 exp
0  
1
2
• electron density depends upon relative
phase of superposition components
14
Quantum description of atomic polarization
energy
2 e i0t
0
• harmonic
• oscillator
two-level atom

0
1 x/a
0
• weak electric field
1
2
x/a0

 r, t   1 
exp  it 2
0  
• electron density depends upon relative
phase of superposition components
15
Optical nonlinearity
V x 
• potential is anharmonic for large displacements
m 2
V x  
x  bx 3  cx 4  
2
2
0
m02 2
x
2
• polarization consequently varies nonlinearly with field


P   0  1 E   2  E 2   3 E 3  
• in quantum description, • uneven level spacing
• distortion of eigenfunctions
• higher terms in perturbation
x
16
Optical nonlinearity
V x 
• potential is anharmonic for large displacements
m 2
V x  
x  bx 3  cx 4  
2
2
0
• polarization consequently varies nonlinearly with field


P   0  1 E   2  E 2   3 E 3  
• in quantum description, • uneven level spacing
• distortion of eigenfunctions
• higher terms in perturbation
x
dV
dx
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Electro-optic effect
• exploit the nonlinear susceptibility


P   0  1 E   2  E 2   3 E 3  
• nonlinearity mixes static and oscillatory fields


P   0  1 2 E0  2  3E02  3  E
• susceptibility at
 hence controlled by E0
• Pockels effect; Kerr effect
E0
x
dV
dx
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Second harmonic generation
• again exploit the nonlinear susceptibility


P   0  1 E   2  E 2   3 E 3  
• distortion introduces overtones (harmonics)


P   0  1 E   2  E2  
where
E cos t 
2
 E 2 1  cos 2t  2
x
dV
dx
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Nonlinear tensor susceptibilities
• nonlinear contributions to the polarization depend upon products of
electric field components
e.g.
E1E2  E1,i E2, j i x,y,z; j x,y,z 
• each product corresponds to a different susceptibility coefficient
• the induced polarization has three components (i =x,y,z):


1
2 
Pi   0    ij E j    ijk E j Ek   
j ,k  x , y , z
 j  x, y , z

• terms in the susceptibility expansion are therefore tensors of increasing rank
20
Nonlinear tensor susceptibilities
• the susceptibility depends upon the frequencies of the field and polarization components
e.g. if
E  E1   E2  ,
1
1


3:2 E2
P



:

E


3 1
1
 0 3
  2  3:1,1 E1 E1   2  3:2,2 E2 E2
1
  2  3:1,2 E1 E2   2  3:2,1 E2 E1

• any susceptibility
 2  3:1,2   0 unless 3  1  2
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Symmetry in susceptibility
• the susceptibility tensor may be invariant under certain symmetry operations
e.g. • rotation
• reflection
• inversion
x, y, z   x, y, z
• the symmetries of the susceptibility must include – but are not limited to – those
of the crystal point group
• optically active materials fall outside the point group description (nonlocality)
• materials showing inversion symmetry have identically zero terms of even rank
22
Properties of susceptibility
• depends upon frequency
• dispersion and absorption in material response
• depends upon field orientation
• anisotropy in crystal and molecular structure
• tensor nature of susceptibility
• depends upon field strength
• anharmonicity of binding potential hence nonlinearity
• series expansion of susceptibility
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Pockels (linear electro-optic) effect
• nonlinearity mixes static and oscillatory fields

P   0 χ 1E  χ 2  :,0 E0E
 χ 2  :0,  E E0  

• applying intrinsic permutation symmetry,
ε  1  χ  2χ :,0 E0
 3χ 3 : ,0,0 E02  
1
2 
• in non-centrosymmetric materials,
χ 2  dominates
E0
x
dV
dx
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Kerr (quadratic electro-optic) effect
• nonlinearity mixes static and oscillatory fields

P   0 χ 1E  χ 2  :,0 E0E
 χ 2  :0,  E E0  

• applying intrinsic permutation symmetry,
ε   1  χ  2χ :,0 E0
 3χ 3 : ,0,0 E02  
1
2 
• in centrosymmetric materials,
χ 2   0
E0
x
dV
dx
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Pockels cell
• voltage applied to crystal controls
birefringence and hence retardance
• mounted between crossed linear
polarizers
• longitudinal and transverse
geometries for modulation field
• allows fast intensity modulation
and beam switching
polarizer
polarizer
modulation
voltage
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