Chap. 4. Electromagnetic Propagation in Anisotropic Media- Optical properties depend on the direction of propagation and the polarization of the light.

- Crystals such as calcite, quartz, KDP, and liquid crystals.

- Double refraction (birefringence), optical rotation, polarization effects, electro-optical effects.

- Prism polarizers, sheet polarizers, birefringent filters

4.1. The Dielectric Tensor of An Anisotropic Medium

- In an isotropic medium, the induced polarization P is always parallel to the electric field Eand is related by a scalar quantity (the susceptibility) that is independent of the fielddirection.

- In anisotropic media,

Along the principal axes of the crystal (vanishing off-diagonal elements),

- In terms of the dielectric permittivity tensor , where

- For a homogeneous, nonabsorbing, and magnetically isotropic medium,the energy density of the stored electric field in the anisotropic medium

⋅

Differentiating the above eqn,

.

Using the Poynting theorem, the net power flow into a unit vol in a lossless medium is

∇⋅× ⋅⋅ → ∇⋅× ⋅

The first term must be equal to (the Poynting vector corresponds to the energy flux).

→ (symmetric)

* For a lossless medium, : the conservation of electromagnetic field energy requiresthat the dielectric tensor be "Hermitian".

4.2. Plane Wave Propagation in Anisotropic Media

- In an anisotropic medium such as a crystal, the phase velocity of light dependes on its stateof polarization as well as its direction of propagation.

- For given direction of propagation in the medium, there exist, in general, two eigenwaveswith well-defined eigen-phase velocities and polarization directions.

[Question] Find two eigen-polarizations and the corresponding eigen-indices of refraction.

- Consider a monochromatic plane wave of angular frequency propagating in the anisotropicmedium with an electric field ⋅ and a magnetic field ⋅ where

k is the wavevector with s is a unit vector along the propagation,and n is the refraction index to be determined.

Maxwell's eqn: × ×→ × ×

- In the principal coordinate system,

Then, the wave eqn is given by

For nontrival solutions,

representing

a 3-dim. surface of k space (momentum space). This surface is known as the normalsurface and consists of two shells, which, in general, have 4 points in common

(see Fig. 4.1)

→ The two lines going through the origin and these points are known as the optic axes.

- Given a direction of propagation, there are in general two k values which are theintersections of the direction of propagation and the normal surface.These two k values correspond to two different phase velocities of the wavespropagating along the chosen direction.The two phase velocities always correspond to two mutually orthogonal polarizations.

- The direction of the electric field vector associated with these propagation:

- For propagation in the direction of the optic axes, there is only one value of k and thus onlyone phase velocity. There are two independent directions of polarization.

- In terms of the direction cosines of the wavevector, using for the plane wave,(Fresnel's equation of wave normals)

and

For each direction of polarization , two solutions for (quadratic eqn in ).

[Homework] Problem 4.2 (Fresnel equation)

- Consider for the linearly polarized eigenwaves associated with and

- ∇⋅ → are orthogonal to s. ⋅

⇒ : a orthogonal triad used as a coordinate system.

- The resultant Maxwell's eqn: ×

×

* D and H are both perpendicular to the direction of propagation s, thus the direction ofenergy flow (given by the Poynting vector ExH) is not, in general collinear with thedirection of propagation s.

- Using the identity × × ⋅⋅,

× ×

⋅

Since ⋅ and

⋅ ⋅

- Orthogonal relations: ⋅ ⋅ ⋅ ⋅ ⋅

In general are not orthogonal !!

- The orthogonality of the eigenmodes of propagation: ⋅×

4.2.1 Orthogonal Properties of the Eigenmodes

- Orthogonality and Lorentz reciprocity theorem: ⋅× ⋅×

⋅× ⋅

⋅× ⋅

Using the identity ⋅× ⋅×,

×⋅×

×⋅×

Since this equation must hold for any arbitrary direction of propagation s with ≠,it is satisfied only when both sides vanish → ⋅× ⋅×

4.3 The Index Ellipsoid

- The surfaces of constant energy density Ue in D space:

- Replacing and defining → ,

: the equation of a general ellipsoid with major axes parallel to

the x, y, z directions whose respective lengths are .

- The index ellipsoid is used mainly to find the two indices of refraction and the twocorresponding directions of D associated with two independent plane waves that propagatealong an arbitrary direction s in a crystal:

1) find the intersection ellipse between a plane through the origin which is normal to thedirection of propagation s and the index ellipsoid.

2) the two axes of the intersection of ellipse are equal in length to .These axes are parallel to the directions of of the two allowed solutions.(See Fig. 4.2)

- Define the impermeability tensor (inverse dielectric tensor)

then, → × ×

- In a new coordinate system with one axis in the direction of propagation of wave,

→

since ⋅

- Ignoring and defining a transverse impermeability ,then the wave equation becomes

The polarization vectors of the normal modes are eigenvectors of the transverseimpermeability tensor with eigenvalues .There are two orthogonal eigenvectors, , corresponding to the two normal modes ofpropagation with refractive indices .

4.4 Phase Velocity, Group Velocity, and Energy Velocity

- Phase velocity , the group velocity ∇, and

the velocity of energy flow where S is the Poynting vector.

In an anisotropic medium,

- A wave packet can be viewed as a linear superposition of many monochromatic plane waves,each with a definite frequency and wave vector k. Each plane wave component satisfiesthe following Maxwell's eqn in momentum space.

× ×

[Proof of ]Consider an infinitesimal change , and the corresponding changes :

×× × ×

Using ⋅× ⋅× ⋅×, we obtain

⋅×⋅× ⋅⋅ ⋅×⋅× ⋅⋅

Subtracting the above two equations,

⋅× ⋅⋅ ⋅×⋅ ×→

where the symmetry properties of the tensors used: ⋅ ⋅⋅ ⋅

Finally, we have ⋅× ⋅⋅

According to the definition of the energy flow and the Poynting vector,

⋅ ⋅ ∇⋅ ⋅, thus for an arbitrary

4.5 Classification of Anisotropic Media

- In uniaxial optical materials (two of principal indices are equal),

where

The normal surface consists of a sphere and an ellipsoid of revolution.

The z axis is the only optic axis -> uniaxial

The ordinary and extraordinary indices :

positive if , negative if

(See Table 4.1 in p. 83 for optical symmetry in crystals)

4.6 Light Propagation in Uniaxial Crystals

- In uniaxial crystals such as quartz, calcite, LiNbO3 (lithium niobate),

- Let k be the wavevector and c be a unit vector in the direction of the c axis (the z axis)The polarizations for the displacement vectors, , are given by

× ×

×

×

- The index of refraction for varies from for ∘to for

∘

Substituting

into

→

The direction of polarization for the extraordinary field: Eq. (4.2.9),

4.7 Double Refraction at a Boundary

- For a plane wave incident on an anisotropic medium, the refracted wave, in general, isa mixture of two eigenmodes. In an uniaxial crystal, a mixture of the ordinary andextraordinary waves.

- For the refracted waves, the kinematic condition :

The values of are not, in general, constant; rather, they vary with the directions of

4.8 Light Propagation in Biaxial Crystals

- Set ; two factors in the secular eqn Eq. (4.2.8)

4.9 Optical Activity

- Through certain optical materials, there exists a rotation of the polarization plane of linearlypolarized light: first observed in quartz.

- The amount of rotation is proportional to the path length of light in the medium:conventionally, the rotary power is given in degrees per centimeter, the specific rotary poweris defined as the amount of rotation per unit length.

- The sense of rotation:1) dextro-rotary (right-handed) if the sense of the rotation of the polarization plane is

counterclockwise as viewed by the observer facing the approaching light beam.2) levo-rotary (left-handed) if clockwise.

- quartz, cinnabar, sodium chlorate, turpentine, sugar, tellurium, selenium, silver thiogallate(See Table 4.3)

- In 1825, Fresnel recognized that the optical activity arises from "circular" double refraction.

and

where are unit Jones vectors for the circular polarizations.

- For a linearly polarized beam of amplitude that is polarized along the x-axis and entersthe medium z = 0, it is represented by the sum of two waves with amplitudes .

At distance z,

→

where

The specific rotary power: ; right-handed if

(ex) the specific rotary power of quartz at Å is 188o/cm → ×

- The EM theory of the optical activity: Born et al, Condon

The optical activity represented a parameter where p= the induced dipolemoment of the molecule; for a linear molecule, . Nonzero arises from an intrinsichelical structure in the molecule.

(1) For plane-wave propagation in a homogeneous medium, the material equation for anoptically active material is written as × where = the dielectric tensorwith no optical activity, G = the gyration vector parallel to the direction of propagation.

(2) The vector product × is always represented by the product of an antisymmetrictensor [G] with E.

→ or ′

(3) Solve for the eigenwave equations of propagation × ×′ with ′.(4) The resulting Fresnel eqn for the eigen-indices of refraction using G = Gs:

Let

be the roots of the Fresnel equation with G = 0:

For propagating along the optic axes, , then

±

Since G is small, ±

↔

(5) Find the polarization states of the eigenmodes represented by the Jones vectors.(See the texbook pp. 99-101)

4.10 Faraday Rotation

- For light propagating along the magnetic field, the rotation of the plane of the polarizationwith distance:

where = the magneto-gyration of coefficient.- The specific rotation (rotation per unit length): with V = Verdet constant

- The material relation × since the induce dipole moment of the electroninvolves a term that is proportional to ×.

4.11 Coupled-Mode Analysis of Wave Propagation in Anisotropic Media

- The propagation of em radiation in anisotropic media can be described in terms of normalmodes that have well-defined polarization states and phase velocities, and are obtained bydiagonalizing the transverse impermeability tensor .→ Any wave propagation in an anisotropic medium can be decomposed into a linear

combination of these normal modes with constant amplitudes.

- Let = the vectors representing the polarization direction of E of the normal modes and = the corresponding wave numbers.

A general wave propagation is written as

with = constants, = the distance along the propagation direction ( ⋅)

- Under an external (or internal) "perturbation" such as stress, magnetic field, electric field,or even the presence of optical activity, are NOT the eigenvectors of propagation.

′ (the unperturbed part and the change in dielectric permittivity)Once is known, the normal modes of propagation can always be found (see Sec. 4.2).

- When the perturbation is small (≪ ), the wave propagation is described in terms ofa linear combination of the unperturbed normal modes.

Due to the perturbation, the mode amplitudes are no longer constants since

are not, in general, normal modes.

→

If are given, the field E is uniquely specified because , are known fromthe unperturbed case ( ).

The dependence of is due to the presence of the dielectric perturbation

- The wave eqn: ∇× ∇× with ∇→ for the wave propagationalong s.

⋅ for a homogeneous medium and plane waves.

Ignoring the longitudinal component of E and assuming that⋅ ⋅ ⋅ ⋅

where we used

- Assuming that is slowly varying function of ( ≪ in ),

≪ .

Then,

in the coordinate

→ The coupled-mode eqns can be uniquely solved provided that an initial condition onthe polarization state of the wave is given.

- Example: the propagation of em radiation in an optically active and birefringent medium.

1) For small perturbation from the optical activity,

the matrix elements is given as

2) Let = the amplitudes at ; at a general point

Then,

where are the refractive indices of modes (i.e., ) and

*The perturbation causes energy exchange between modes 1 and 2.

*With no optical activity (G=0), →

3) Let = the initial polarization and = the polarization state at in the complexnumber representation.

*The polarization state is a periodic function of with period .

*For ,

where

=rotary power

*Initial linear polarization , the polarization at , → the plane of the polarization is rotated by an angle of .

- Derivation of normal modes in the presence of the perturbation within the coupled-modeformalism using the Jones vector.

≡

→

The "wave" matrix

In the presence of optical activity with the dielectric perturbation ,

In terms of normal modes written as , we have (eigenvalue eq)

the secular eqn:

or

the roots (the refractive indices):

±

the corresponding wave numbers:

±

the eigenvectors (polarizations):

4.12 Equation of Motion for the Polarization State

- Equation of motion for the displacement field vector D, which is perpendicular to thedirection of propagation:

wave eqn ∇× ∇×

with ∇→

then, × ×

→

where = the 2x2 transverse impermeability.Defining a 2x2 matrix N such that , and multiplying the above eq by

,

→

*The matrix N is called the refractive-index matrix and reduced to the refractive index forthe case of an isotropic medium.

- In the presence of a perturbation,

For small perturbation (≪ ),

Once N is known,

→

can be solved uniquely.

*The normal modes of propagation can be found by diagonalizing the refractive-indexmatrix in the presence of perturbation.

- In an optically active medium, let represent the effect of the optical activity:

The refractive-index matrix N can be written as

A normal mode of propagation, by definition, must have a well-defined polarization state

and a well-defined wave number: →

→

where .

The secular eqn for n;

The roots are

±

The corresponding Jones vectors for the polarization state of the normal modes are

±

±

Problem Set #1: 4.9, Hermitian dielectric tensor (p. 116 in textbook)#2: 4.10, Displacement eigenmodes (p. 118 in textbook)