• 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