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 E and is related by a scalar quantity (the susceptibility) that is independent of the field direction. - 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 requires that 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 state of polarization as well as its direction of propagation. - For given direction of propagation in the medium, there exist, in general, two eigenwaves with 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 anisotropic medium 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 normal surface 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 the intersections of the direction of propagation and the normal surface. These two k values correspond to two different phase velocities of the waves propagating 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 only one 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 of energy flow (given by the Poynting vector ExH) is not, in general collinear with the direction 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 U e 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 two corresponding directions of D associated with two independent plane waves that propagate along an arbitrary direction s in a crystal: 1) find the intersection ellipse between a plane through the origin which is normal to the direction 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 transverse impermeability tensor with eigenvalues. There are two orthogonal eigenvectors,, corresponding to the two normal modes of propagation 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 andwavevectork. Each plane wave component satisfies the following Maxwell's eqn in momentum space. [Proof of ] Consider an infinitesimal change, and the corresponding changes : Using, weobtain 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, LiNbO 3 (lithium niobate), -Letkbe the wavevector and c be a unit vector in the direction of the c axis (the z axis) The polarizations for the displacement vectors,,aregivenby
- 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, is a mixture of two eigenmodes. In an uniaxial crystal, a mixture of the ordinary and extraordinary 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 linearly polarized 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 power is 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 enters the 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 is 188 o /cm - The EM theory of the optical activity: Born et al, Condon The optical activity represented a parameter where p= the induced dipole moment of the molecule; for a linear molecule,. Nonzero arises from an intrinsic helical structure in the molecule. (1) For plane-wave propagation in a homogeneous medium, the material equation for an optically active material is written as where = the dielectric tensor with 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 antisymmetric tensor [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 polarization with 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 electron involves 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 normal modes that have well-defined polarization states and phase velocities, and are obtained by diagonalizing 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, thenormal modes of propagation can always be found (see Sec. 4.2). - When the perturbation is small ( ), the wave propagation is described in terms of a 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 from the unperturbed case ( ). The dependence of is due to the presence of the dielectric perturbation - The wave eqn: with for the wave propagation along s.
for a homogeneous medium and plane waves. Ignoring the longitudinal component of E and assuming that whereweused - 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 on the polarization state ofthewaveisgiven.
- 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 complex number 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-mode formalism 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 the direction 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 for the 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-index matrix 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)