linear optical properties of dielectrics introduction to crystal optics introduction to nonlinear...
TRANSCRIPT
• Linear optical properties of dielectrics
• Introduction to crystal optics
• Introduction to nonlinear optics
• Relationship between nonlinear optics and electro-optics
Bernard Kippelen
Maxwell's Equations and the Constitutive Equations
Light beams are represented by electromagnetic waves propagating in space. An electromagnetic wave is described by two vector fields: the
electric field E(r, t) and the magnetic field H(r, t).
In free space (i.e. in vacuum or air) they satisfy a set of coupled partial differential equations known as Maxwell's equations.
0
0
0
0
t
t
EH
HE
E
H MKS
0 and 0 are called the free
space electric permittivity and the free space magnetic permeability, respectively, and satisfy the condition c2 = (1 / (0 0)), where c is the
speed of light.
0
t
t
DH j
BE
D
B
D(r,t) the electric displacement field, and B(r,t) the magnetic induction field
In a dielectric medium: two more field vectors
and j are the electric charge density (density of free conduction carriers) and the current density vector in the medium, respectively. For a transparent dielectric and j =0.
0
0
D E P E
B H
Constitutive equations:
is the electric permittivity (also called dielectric function) and P = P (r, t) is the polarization vector of the medium.MKS
Linear, Nondispersive, Homogeneous, and Isotropic Dielectric Media
Linear: the vector field P(r, t) is linearly related to the vector field E(r, t).
Nondispersive: its response is instantaneous, meaning that the polarization at time t depends only on the electric field at that same time t and not by prior values of E
Homogeneous: the response of the material to an electric field is independent of r.
Isotropic: if the relation between E and P is independent of the direction of the field vector E.
0( ) ( ) (MKS)
( ) ( ) (CGS)
, t , t
, t , t
P r E r
P r E r
0 (1 ) (MKS)
= 1 + 4 (CGS)
is called the optical susceptibility
Wave equation
From Maxwell’s equations and by using the identity: EEE 2)()(
2 22
2 20
n
c t
EE
0
1 (MKS)
1 4 (CGS)
rn
n
Solutions of Wave Equation
i( i( i(( ) = ( e e et) t) t)1,t cos t) Re c .c . E c .c .
2 kr kr krE r kr
Real number
Complex number
Complex conjugate
Time
Space
Wavelength = 2/k
Period T = 2/
nk
v c
Dispersion relationship
In real materials: polarization induced by an electric field is not instantaneous
t
(t) (t ) ( )dP E
Which can be rewritten in the frequency domain
( ) ( ) ( )P E
• the susceptibility is a complex number: has a real and imaginary part (absorption)
• the optical properties are frequency dependent
Lorentz oscillator model
Refractive Index
Absorption
Photon Energy
Nucleus
Electron
Displacement around equilibrium position due to Coulomb force exerted by electric field
Optics of Anisotropic Media
Optical properties (refractive index depend on the orientation of electric field vector E with respect to optical axis of material
Need to define tensors to describe relationships between field vectors
ij
X 11 X 12 Y 13 Z
Y 21 X 22 Y 23 Z i jj
Z 31 X 32 Y 33 Z
P E E E
P E E E P E
P E E E
X
Y
Z
11 12 13
21 22 23
31 32 33
Uniaxial crystals – Index ellipsoid
2X11
222 Y
233 Z
n 0 00 0
0 0 0 n 0
0 0 0 0 n
nX = nY ordinary index
nZ = extraordinary index
Refractive index for arbitrary direction of propagation can be derived from the index ellipsoid
2 2
2 2 2o e
1 cos sin
n ( ) n n
Introduction to Nonlinear Optics( 2 ) ( 3 )
LP(E) E E E EEE ...
Linear term Nonlinear corrections
Example of second-order effect: second harmonic generation (Franken 1961):
Symmetry restriction for second-order processes
n
n
i tn
n
i tn
n
( ,t) ( )e c .c
( ,t) ( )e c .c
E r E
P r P
Several electric fields are present
( 2 )i n m n m n m j n k m
jk
P( ) D ( ; , ) E ( ) E ( )ijk
2X
2( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) YX XXX XYY XZZ XYZ XXZ XXY
2( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) Z
Y YXX YYY YZZ YYZ YXZ YXY
( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) Y ZZ ZXX ZYY ZZZ ZYZ ZXZ ZXY
X Z
X Y
E
EPEP
2 E EP
2 E E
2 E E
Nonlinear polarization
Tensorial relationship between field and polarization
Second-order nonlinear susceptibility tensor
)2(
36)2(
35)2(
34)2(
33)2(
32)2(
31
)2(26
)2(25
)2(24
)2(23
)2(22
)2(21
)2(16
)2(15
)2(14
)2(13
)2(12
)2(11
)2(~
( 2 )15
( 2 )( 2 )15
( 2 ) ( 2 ) ( 2 )31 31 33
0 0 0 0 0
0 0 0 0 0
0 0 0
18 independent tensor elements but can be reduced by invoking group theory
Example: tensor for poled electro-optic polymers
Contracted notation for last two indices: xx = 1; yy = 2, zz = 3; zy or yz = 4; zx or xz = 5; xy or yx = 6
Introduction to Electro-opticsJohn Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a dc or low
frequency electric field
3 3 20 0 0 0
1 1n(E ) n( E 0 ) n r E n s E ...
2 2
In this formalism, the effect of the applied electric field was to deform the index ellipsoid
2 2 22 2 2 2
1 2 3 4
2 25 6
1 1 1 1X Y Z 2 YZ
n n n n
1 12 XZ 2 XY 1
n n
3
ij 0 j2i j 1
1r E
n
Index ellipsoid equationCorrections to the coefficients
Electro-optic tensor
21
22 11 12 13
21 22 230 X2
3 31 32 330Y
41 42 430 Z2
51 52 534
61 62 632
5
26
1
n
1
r r rn
r r r1E
r r rnE
r r r1E
r r rn
r r r1
n
1
n
Relationship with second-order susceptibility tensor:
( 2 )ij ji4
8r
n
Simplification of the tensor due to group theory
Example of tensor for electro-optic polymers
13
13
33
13
13
0 0 r
0 0 r
0 0 rr
0 r 0
r 0 0
0 0 0
Application of Electro-Optic Properties
Light
Applied voltage changes refractive index
Electro-Optic Properties of Organics
A D
P E E E ( ) ( ) ( ) ...1 2 2 3 3
If the molecules are randomly oriented inversion symmetry
nonlinear susceptibilities
hyperpolarizabilities
a
b
c
d
e
D e p h a s i n g
T r a p p i n g
T r a n s p o r t
S p a c e
The The Photorefractive EffectEffect
Convert an intensity distribution into a refractive index distribution