Chap. 2. Polarization of Optical Waves

Transcription

1 Chap. 2. Polarization of Optical Waves 2.1 Polarization States - Direction of the Electric Field Vector : r E = E xˆ + E yˆ E x x y ( ω t kz + ϕ ), E = E ( ωt kz + ϕ ) = E cos 0 x cos x y 0 y - Role : Reflectance at Boundary between Two Materials Refractive Index of Anisotropic Materials Rotation of Polarization in Optically Active Materials Absorption by Certain Materials Scattering from Matter - Classification : Elliptically Polarized Light Linearly Polarized Light Circularly Polarized Light y

2 Linear Polarization Circular & Elliptical Polarization

3 E E 2 x 2 0x E + E 2 y 2 0y ExEy 2cos( ϕ ) = sin E E 0x 0y 2 ( ϕ)

4 2.2. Jones Vector

5 2.3. Partially Polarized & Unpolarized Light : Stokes parameters with Stokes parameters: The degree of polarization: and Poincare sphere:

6 Chap. 3. EM Propagation in Anisotropic Media 3.1 Maxwell Equations and Dielectric Tensor with constitutive equations (material equations) - In anisotropic media such as LCs, calcite, quartz, LiNbO 3, (real and symmetric for nonmagnetic and transparent materials) Principal dielectric constants:

7 - : freq. dependent In the regime of optical waves, Hz. In the low freq. regime (< 10 9 Hz), molecular reorientation of LCs. 3.2 Plane Waves in Homogeneous Media and Normal Surface - Monochromatic plane wave: where

8 In the principal coordinate system, where For nonvanishing solutions to exist, the secular equation (the determinant of the matrix = 0), At given frequency, the secular eqn. gives 3-dim. surface in space (the normal surface consisting of two shells). Given a direction of propagation, in general, two k values correspond to two different phase velocities of the waves propagating along the chosen direction. - Fresnel's eqn. of wave normals: "eigenvalues of index of refraction" & "direction of polarization" In terms of the direction cosines of the wavevector for the plane wave of.

9 and where = the unit vector in the direction of propagation. - For each direction of polarization (a set of ), two solutions for can be obtained. = the electric field vectors and = the displacement vectors of the linearly polarized normal modes associated with and.:. Since, the three vectors { } form an orthogonal triad. - The direction of energy flow, the Poynting vector is, in general, not collinear with the direction of propagation s.

10 - Orthogonality of normal modes (eigenmodes): Using, and Eliminating H, Since and, - Orthogonal relations In general, are not orthogonal to each other.

11 - The orthogonality relation of the eigenmodes of propagation: Lorentz reciprocal theorem: or * Along an arbitrary direction of propagation s, there exist two independent "plane-wave", linearly polarized propagation modes. These modes have phase velocities, where are the two solutions of Fresnel's eqn k in the xy Plane:

12 where is the angle of the wavevector measured from x axis k in the yz Plane: k in the zx Plane: Classification of Media - The normal surface of propagation is uniquely determined by the principal indices of refraction. Biaxial: Uniaxial: two of three indices equal:

13 In uniaxial materials, Power Flow in Anisotropic Media - The Poynting power flow is not, in general, not collinear with the direction of propagation s. Given a direction of propagation,. The magnetic field - The power orthogonality theorem: the total power flow along the direction of propagation is a "sum" of the "individual" mode power.

14 3.3. Light Propagation in Uniaxial Media - The normal surface of propagation in uniaxial materials: where - O wave (ordinary) : and E wave (extraord) : where = the angle between the direction of propagation and the optic axis. Using, the electric field E of the O wave can be calculated:

15 O wave: E wave: where - The unit vector where is the angle of propagation in the spherical coordinate. - O wave: O Wave:

16 - If a polarized light inside the "uniaxial" medium is generated that is to propagate along a direction s, D is given as a "linear" combination of these two normal modes: As the light propagates inside the medium. a phase retardation between these two components is built up as a result of the difference in their phase velocities: a "new polarization" state Propagation Perpendicular to the c axis: Propagation in the xz Plane: - where is measured from the c axis. Eigen refractive index of the ordinary wave & Eigen refactive index of the extraord. wave The directions of the polarization: O wave- and E Wave- The phase difference for a distance d, Propagation along the c axis

17 3.4. Double Refraction at a Boundary - k i = the wavevector of the incident wave (See Fig. 3.3) k 1 and k 2 = the wavevectors of the refracted waves - In an uniaxial material, one shell of the normal surface is a "sphere". The corresponding wavevector k is a "constant" for all directions of propagation. Snell's law: with The other shell of the normal surface is an "ellipsoid" of revolution Anisotropic Absorption and Polarizers - Tourmaline, tin oxide crystals, Polaroid sheets (a thin layer of needle-like crystals) where = the extinction coefficients. - O-type polarizers: and E-type polarizers: and - An ideal O-type polarizers: and

18 Extinction Ratio and Real Sheet Polarizers - T 1,2 = transmission parallel (perpendicular) to the transmission axis Extinction ratio = T 2 / T 1 - For a pair of parallel (crossed) polarizers, the transmission for "unpolarized" light Field of View of Crossed Polarizers - Properties of polarizers at "oblique" incidence: the leakage T 2 = 0 for ideal polarizers - For most polarizers (O-type), assume that no extraordinary wave is transmitted and only the ordinary wave is transmitted. Let c 1, c 2 = unit vectors representing the c axes of the two polarizers (crossed). the polarization state through the 1st polarizer the polarization state through the 1st polarizer * In general, o 1 and o 2 are "not orthogonal" although c 1 and c 2 are orthogonal. (o 1 and o 2 are orthogonal only at normal incidence)

19 - The leakage of unpolarized light through a pair of crossed polarizers: Using, with The worst leakage occurs at large angles 3.6. Optical Activity and Faraday Rotation - "Optically active": rotary power that rotate the plane of polarization of a beam of light traversing through an optical material (1811, quartz crystal). - dextro(d) and levo(l): right-handed and left-handed *"right-handed": the sense of the plane of polarization is "counterclockwise" as viewed by an observing facing the "approaching" light beam. with R and L are unit vectors for circular polarizations. - The specific rotary power : "right-handed" for

20 3.7. Light Propagation in Biaxial Media - Let k y = 0 (see Fig. 3.8): Method of Index Ellipsoid - Impermeability tensor : In the principal cooridinate, - The index of ellipsoid:

21 - Wave equation: - For the wave eqn. becomes where Since the third component of D = 0. Ignore. The wave equation is written as where Perturbation Approach - Let : "uniaxial" + small perturbation

22 where * = the measure of the deviation from the uniaxial symmetry. - Assuming that the eigenvector of the biaxial crystal can be expressed in terms of two base vectors of D 1 and D 2 for the uniaxial case,. Eqn for the normal modes in the biaxial crystal