Chap. 5. Jones Calculus and Its Application to Birefringent Optical Systems

Transcription

1 Chap. 5. Jones Calculus and Its Application to Birefringent Optical Systems - The overall optical transmission through many optical components such as polarizers, EO modulators, filters, retardation plates. - R. C. Jones, 11940: The Jones calculus, 2x2 matrix method in which the state of polarization is represented by a two-component vector Jones Matrix Formulation - The light propagation in a birefringent crystal consists of a linear combination of two eigenwaves whose phase velocities and the directions of polarization are well-defined. - In an uniaxial crystal, the eigenwaves are the ordinary and extraordinary waves. The directions of polarization for these waves are mutually orthogonal and are called the "slow" and the "fast" axes of the crystal. - Retardation plates are normally cut in such a way that the c-axis lies in the plane of the plate surfaces. Thus, the propagation direction of normally incident light is perpendicular to the c-axis. - In formulating the Jones matrix method, no reflection is assumed from both surfaces of the plate and the light is totally transmitted the plate surfaces. *In practice, the Fresnel reflections at the plate surfaces not only decrease the transmitted

2 intensity but also affect the fine structure of the spectral transmittance because of multi-reflection interfence. - The Jones vector where are two complex numbers.

3 - Decopose the light into a linear combination of the "fast" and "slow" eigenwaves of the crystal. with the rotation matrix -Let be the refractive indices for the "slow" and "fast" components: The polarization state of the emerging beam in the crystal "sf" coordinate system is given by with l the thickness of the plate. The plate retardation ; a measure of the relative change in phase, not the absolute change. *Since, the absolute change in phase caused by the plate is a few 100 times greater than the phase retardation; the mean absolute phase change.

4 -Intermsof and, - The Jones vector of the polarization state of the emerging beam in the crystal "xy" coordinate system: - The transformation due to the retardation plate: with *The phase factor can be neglected if interference effects are not important. - A retardation plate is characterized by its phase retardation and its azimuthal angle, and is represented by the product of three matrices: with (unitary)

5 - The Jones matrix of an ideal homogeneous linear polarizer orientated with its transmission axis parallel to the laboratory x-axis: with = the absolute phase accumulated due to the finite optical thickness of the polarizer. The Jones matrix of a polarizer rotated by an angle : Example: A Half-Wave Retardation Plate - A half-wave plate:, the thickness The azimuthal angle of the plate is take as, and the incident beam as vertically polarized:. Then, the Jones matrix for the half-wave plate is given by The Jones matrix for the emerging beam:, horizontally polarized. *The effect of the half-wave plate is to rotate the polarization by. *For a general azimuthal angle, the half-wave plate rotates the polarization by

6 *For a right-hand circularly polarized a left-hand circularly polarized

7 Example: A Quarter-Wave Retardation Plate - A quarter-wave plate:, the thickness

8 The azimuthal angle of the plate is take as, and the incident beam as vertically polarized:. Then, the Jones matrix for the quarter-wave plate is given by The Jones matrix for the emerging beam: polarized., left-hand circularly *For a horizontally polarized beam, a right-hand circularly polarized beam is coming out Intensity Transmission - Consider a light beam after it passes through a polarizer: The electric field vector, then the intensity If emerging from the polarizer, transmissivity

9 Example: A Birefringent Plate between Parallel Polarizers - The plate is oriented so that the "slow" and "fast" axes are at with respect to the polarizer, and the birefringence, the plate thickness = d. The phase retardation, the Jones matrix For an unpolarized incident beam passing through the first polarizer, The transmitted beam: The transmitted beam id vertically polarized with the intensity given by Example: A Birefringent Plate between Crossed Polarizers - The transmitted beam:.

10 The transmitted beam id vertically polarized with the intensity given by 5.3. Polarization Interference Filter - Spectral filters: bandwidth, tuning capability - A series of wave plates and polarizers: matrix multiplication The Chebyshev's identity with [Reading Assignment] Sec & 5.3.2, Solc filters

11 5.4. Light Propagation in Twisted Anisotropic Media - The propagation of em radiation through a "slowly twisting" anisotropic medium: - N identical twisted plates, each plate is a wave plate with a phase retardation and an azimuthal angle; twisting is linear and the azimuthal angle is Let l be the total thickness of the twisted medium, be the total twist angle. and Each plate has a phase retardation, the plates are oriented at azimuthal angles, with The overall Jones matrix for N plates: where Then,

12 Using the Chebyshev's identity and taking, with The polarization of the light exiting from the twisted medium Adiabatic Following - In some cases, the phase retardation is much larger than the twist angle : (ex) l =25um,,, then for = 0.5um. -For, If the incident light is linearly polarized along either the slow or fast axis at the entrance plane, the light will remain linearly polarized along the local slow or fast axis; The polarization vector follows the rotation of the local axes: adiabatic following - The operation of the Jones matrix: the phase retardation matrix and the polarization rotation.

13 - TN LCD under parallel polarizers; and polarizer along the x-axis. Therotationmatrix after passing through the entrance The polarization state The y-component will be blocked by the exit polarizer, and the transmissivity is then