Ch 4-EM Waves in Anisotropic Media

7/28/2019 Ch 4-EM Waves in Anisotropic Media

1/51

Ch 4,

1


2/51

Ch 4,

The dielectric tensor

Plane wave propagation in anisotropic media

The index ellipsoid

Phase velocity, group velocity and energy

Crystal types

Propagation in uniaxial crystalsPropagation in biaxial crystals

Optical activity and Faraday rotation

Coupled mode analysis of wave propagation2


3/51

Ch 4,

relates the electric field to the electricdisplacement by

In anisotropic materials the polarization maynot be in the same direction as the drivingelectric field. Why?

= E= 0E+ P

3


4/51

Ch 4,

Charges in a material arethe source of polarization.They are bound to neighboringnuclei like masses on springs.

In an anisotropic materialthe stiffness of the springsis different depending on theorientation.

The wells of the electrostatic potential that thecharges sit in are not symmetric and therefore thematerial response (material polarization) is notnecessarily in the direction of the driving field.

4


5/51

Ch 4,

What every-day phenomena can have a responsein a direction different than the driving force?

5


6/51

Ch 4,

The susceptibility tensor relates thepolarization of the material to the drivingelectric field by

or

Thus the material permittivity in

is also a tensor

Px

Py

Pz

= 0

11 12 13

21 22 23

31 32 33

Ex

Ey

Ez

P = 0 E

= 0

I +

= E= 0

E+ P

6


7/51

Ch 4,

For brevity, we often express tensor equationssuch as

in the form

where summation over repeated indices isassumed, so that this is equivalent to

7

D = 0I+

E

Di = 0(ij + ij)Ej

Di =

j

0(ij + ij)Ej


8/51

Ch 4,

The dielectric tensor is Hermetian such that

In a lossless material all elements of are realso the tensor is symmetric!! ! ! ! ! and canbe described by 6 (rather than 9) elements

ij =

ji

11 12 13

12 22 23

13 23 33

ij = ji

8


9/51

Ch 4,

For a given propagation direction in a crystal (orother anisotropic material) the potential wellsfor the charges will be ellipsoidal, and so there

will be two directions for the driving field forwhich the material response is in the samedirection.

These two polarization states each have a

(different) phase velocity associated with them.Light polarized at either of these angles willpropagate through the material with thepolarization unchanged.

9


10/51

Ch 4,

From Maxwells equations in a dielectric (=J=0),using d/dti and ik we have

E+

B

t = 0

ik E+ i B = 0

H

D

t= J

ik H i D = J

k k E+ 2 E = 0

k H= Ek E= H

Which is the wave equation for a plane wave, most easily

analyzed in the principle coordinate system where is

diagonal (i.e. a coordinate system aligned to the crystal axes) 10


11/51

Ch 4,

k k E+ 2 E = 0

for which the determinant of the matrix must be zero forsolutions (other than k==0) to exist

2x k

2

y k2

zkxky kxkz

kykx2

y

k2

x

k2

z kykzkzkx kzky

2z k2

x k2

y

Ex

EyEz

= 0

or

2

xk2

yk2

z kxky kxkzkykx

2y k2

x k2

zkykz

kzkx kzky 2z k

2

x k2

y

= 0

This defines two surfaces in k-space called the normal shells.11


12/51

Ch 4,

In a given direction, going out from theorigin, a line intersect this surface attwo points, corresponding to themagnitude of the two k-vectors (andhence the two values of the phasevelocity) wave in this direction canhave.

The two directions where the surfaces

meet are called the optical axes.Waves propagating in these directionswill have only one possible phasevelocity.

kx

kz

12


13/51

Ch 4,

k k E+ 2 E = 0

2x k

2

y k2

zkxky kxkz

kykx2

y

k2

x

k2

z kykzkzkx kzky 2z k

2

x k2

y

Ex

EyEz

= 0

or

where

Ex

Ey

Ez

= E0

kx

k2

2

x

kyk22y

kzk22z

E2

0 E

2

x+E

2

y+E

2

z

13


14/51


15/51

Ch 4,

Gauss lawin a dielectric (=0) can be written as

implying the propagation direction is orthogonalto the displacement vector .

But the Poynting vectorwhich gives the direction of energy flow isorthogonal to the electric field , not thedisplacement vector. Thus energy does not flowin the direction of the waves propagation if thepolarization of the wave ( ) is not an eigen-

vector of the materials dielectric tensor!

ik D = 0

D =

=

E H

k

D

E

E

15


16/51

Ch 4,

Image 8.26 from Hecht16


17/51

Ch 4,

For any direction of propagation there exists twopolarization directions in which the displacementvector is parallel to the transverse component of

the electric field. These are the eigenstates ofthe material.

Requiring D and E be parallel in the matrixequation

is equivalent to finding the eigenvalues andeigenvectors of satisfying

D = E

i ui = ui

_

17


18/51

Ch 4,

Index of refraction as a function of polarization angle

The polarization directions that have a max and minindex of refraction form the major and minor axes ofan ellipse defining n() the index fora wave with theelectric displacement vector at an angle of in thetransverse plane.

k

n1

n2

18


19/51

Ch 4,

The sum of all index ellipses plotted in threedimensions is the index ellipsoid, defined by

x2

n2x

+

y2

n2y

+

z2

n2z

= 1

19


20/51

Ch 4,

Find the angle of the direction of the opticalaxis for Lithium Niobate

20


21/51

Ch 4,

Find the angle of the direction of the opticalaxis for Topaz

21


22/51

Ch 4,

The index ellipsoid is determined by the threeprinciple indices of refraction nx, ny and nz.

A crystal with nx

ny

nz will have ____optical axes

A crystal with nx=nynz will have ____optical axes

A crystal with nx=ny=nz will have _____optical axes

22


23/51

Ch 4,

The geometry of a crystal determines the formof the dielectric tensor

23


24/51

Ch 4,


Isotropic

= 0n2

0 0

0 n2

0

0 0 n2

24


25/51

Ch 4,


Unaxial

= 0

n2

00 0

0 n2

00

0 0 n2e

25


26/51

Ch 4,


Biaxial

= 0

n2x

0 0

0 n2

y0

0 0 n2z

26


27/51

Ch 4,

By convention nx


28/51

Ch 4,

By convention nonx=ny and nenz

The optical axis in a uniaxial crystal is always in

the z-direction

A negative uniaxial crystal is one in which neno.

28


29/51

Ch 4,

A wave is propagating in the [1,1,1] direction inMica, what are the two principle indices ofrefraction and in which direction are the

eigenpolarizations?

29


30/51

Ch 4,

A wave is propagating in the [1,1,1] direction inMica, what are the two principle indices ofrefraction and in which polarization directions do

these correspond to?plane of polarization is given by r.k=0 so x+y+z=0.The intersection of this plane and the index ellipsoid, given by

is,

Find extremes of r2=x2+y2+z2 subject to the preceding two constraints to finddirections of transverse eigenpolarizations. Plug directions back into r=(x2+y2

+z2)1/2 to find index for each polarization

x2

n2x

+y2

n2y

+z2

n2z

= 1

k r = 0

x2

n2x

+y2

n2y

+ (x + y)2

n2z

= 1

30


31/51

Ch 4,31


32/51

Ch 4,

The index of refraction of the ordinary ray isalways equal to the index seen when the raypropagates along the optical axis

The extra-ordinary ray sees an index ofrefraction that is a function of the raypropagation angle from the optical axis. In auniaxial crystal

1n2e()

= cos2

n2o

+ sin2

n2e

32


33/51

Ch 4,

kisini knsinn

o

i

noene

Snells law requires the tangential component ofthe k-vector be continuos across a boundary

kisini knsinn

n

i

n

33


34/51

Ch 4,

The normal surface is a surfaceof constant in k-space. Groupvelocity vg=k is perpendicular

to the normal surfaces, but in abiaxial crystal the normalsurfaces are singular along theoptical axis.

The surrounding region isconical, thus a wave along theoptical axis will spread outconically

34


35/51

Ch 4,

Quartz and other materials witha helical molecular structureexhibit Optical Activity, a

rotation of the plane ofpolarization when passingthrough the crystal.

35


36/51

Ch 4,

Optically active materials can be thought of asbirefringent, having a different index ofrefraction for right-circular polarization and for

left-circular polarization.

The specific rotary power describes how muchrotation there is per unit length

The source of this optical activity is the induceddipole moment of the molecule formed bychanging magnetic flux through the helicalmolecule

=

(nl nr)

36


37/51

Ch 4,

The displacement vector thus has an additional contribution in adirection perpendicular to the electric field

where ! ! ! ! ! ! ! is called the gyration vector and

so we can define an effective dielectric tensor

so that

allowing the wave equation to be solved for eigenpolarizations ofpropagation and two corresponding indices of refraction thatdepend on [G]

D = E+ i0G E

= + i0[G] = E

37

G = (gijkikj/k

2

0)k

G E=

i j kGx Gy GzEx Ey Ez

=

0 Gz GyGz 0 GxGy Gx 0

ExEyEz

= [G] E


38/51

Ch 4,

A form of optical activity induced by an externalmagnetic field.

where V is called the Verdet constant. The senseof rotation depends on the direction ofpropagation relative to the direction of themagnetic field

= V B

38


39/51

Ch 4,

The motion of the charges in a material driven by the electricfield feel a Lorentz force! ! ! ! ! resulting in an induceddipole with a term proportional to!! ! ! ! such that

where =-Vn00/.

For Faraday rotation the imaginary term is proportional to B,

while for optical activity it is proportional to k.

Thus a wave double passing (in opposite directions) a materialwill net zero polarization rotation from optical activity, but twicethe one-way rotation due to Faraday rotation.

qv BB E

D = E+ i0B E

39


40/51

Ch 4,

Given an isotropic material that is perturbed suchthat

where is

find the eigenpolarizations for the electric field and

associated indices of refraction. If a linearlypolarized plane wave were to propagate along thedirection of G through a thickness d of suchmaterial, what would the output wave look like?

= E+ i E

40

= o

0 Gz Gy

Gz 0 Gx

Gy Gx 0


41/51

Ch 4,41


42/51

Ch 4,

When the eigenstates of an unperturbed systemare known, a small perturbation can be treatedas a coupling between these states.

Rather than rediagonalizing the matrix for onecould solve the wave equation with theperturbed matrix +

E=

A1ei(k1zt)

e1 +

A2ei(k2zt)

e2

E=

A1(z)ei(k1zt)

e1 +

A2(z)ei(k2zt)

e2

orE= A3e

i(k3zt)e3 +A4ei(k4zt)e4

with

with +42


43/51

Ch 4,

Given an isotropic material that is perturbed suchthat

where is

find the coupled mode expression for the electric

field for a linearly polarized plane wavepropagating along the direction of G.

=

0 Gz Gy

Gz 0 Gx

Gy Gx 0

= E+ i E

43


44/51

Ch 4,44

The 1-dimensional wave equationin isotropic media


45/51

Ch 4,45

This is from wave equation solutionsto the unperturbed material


46/51

Ch 4,46

Envelope varies slowlycompared to optical frequency

These coupled differentialequations reduce to coupledequations that can be solvedusing linear algebra when weuse phasors to represent the

fields


47/51

Ch 4,47


48/51

Ch 4,

Consider the same problem where

Find the normal modes of propagation by finding

the eigenvectors of the dielectric tensor

48

=0

n2

iG 0

i G n2 0

0 0 n2


49/51

Ch 4,

Consider the same problem where

Find the normal modes of propagation by finding

the eigenvectors of the dielectric tensor=n2, n2-G,n2+G

u=[0,0,1], [i,1,0], [-i,1,0]49

=

n2

i0G 0

i0

G n2 0

0 0 n2


50/51


51/51

Ch 4,

Yariv & Yeh Optical Waves in Crystals chapter 4

Hecht Optics section 8.4

Fowles Modern Optics section 6.7

51

Documents

Ch 4-EM Waves in Anisotropic Media