Polarizing Optical Elements

Such a crystal is double refracting since the two values of the refractive index will cause the two polarizations to propagate along different paths. Separation of the two polarizations occurs at the interface where the horizontally polarized light is totally internally reflected. The vertically polarized light, on the other hand, is transmitted across the interface where the second half of the polarizing cube returns this polarized beam to its 

Also relying on light reflection to polarize light are Brewster angle windows based on the effect discussed in section 1.6. This type of window is also used in most laser cavities 

Since the polarizers discussed above involve light reflection combined with the real part of the refractive index tensor, they can be used effectively over a broad spectral range about a central wavelength. Calcite Glan-Thompson polarizers, for example, operate successfully over the entire visible spectrum. When fabricated of crystalline quartz, these polarizers can be used to polarize ultraviolet light as well as visible light. 

A second class of polarizers uses dichroism to produce linearly polarized light. Techniques to produce dichroic polarizing sheets were pioneered by Land [52], These are made by dissolving a strongly dichroic, small molecule into an amorphous, transparent polymer. The dichroic molecules are then oriented by a uniaxial stretching of the polymer matrix. Since this is accomplished below the polymers glass transition temperature, this ori- 

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Generally, polarimetry requires the establishment of the type, quality, azimuth and ellipticity of the polarization. In addition, polarization of some type of irradiated or dispersed light [irradiated by natural and artificial objects or some of light-source] is difficult and it can contain simultaneously linear, elliptic, circular or partial polarized components [94]. The whole polarimetric analysis of such objects needs to investigate. At the same rime, appheation of the standard polarization optical elements, for the investigation of the polarization state of each section, it is necessary to take each complete set of different gradient elements, and for this reason it is difficult and labor-intensive to carry out such investigations. 

Accurate experimental determinations require that not only the handedness of the produced light, but its exact degree of polarization, are known. The theoretical performance of an undulator may be in practice be degraded by magnetic defects, and the optical beam can be further depolarized by reflections along the beamline. Again, the dephasing on optical elements can in principle be... 

Ellipsometric techniques in which amplitude ratios and phase shifts for reflected light are directly measured as opposed to the previous technique in which the phase shift is indirectly obtained. This is difficult to do over large wavelength regions because of requirements on optical elements such as polarizers and retarders. 

We may represent a beam of arbitrary polarization, including partially polarized light, by a column vector, the Stokes vector, the four elements of which are the Stokes parameters. In general, the state of polarization of a beam is changed on interaction with an optical element (e.g., polarizer, retarder, reflector, scatterer). Thus, it is possible to represent such optical elements by a 4 X 4 matrix (Mueller, 1948). The Mueller matrix describes the relation between incident and transmitted Stokes vectors by incident is meant before interaction with the optical element, and by transmitted is meant after interaction. As an example, consider the Mueller matrix for an ideal linear polarizer. Such a polarizer transmits, without change of amplitude, only electric field components parallel to a particular axis called the transmission axis. Electric field components in other directions are completely removed from the transmitted beam by some means which we need not explicitly consider. The relation between incident field components (E, E i) and field components ( l, E () transmitted by the polarizer is... 

The usefulness of the Mueller formulation becomes apparent when we realize that Mueller matrices give us a simple means of determining the state of polarization of a beam transmitted by an optical element for an arbitrarily polarized incident beam. Moreover, if a series of optical elements is interposed in a beam, the combined effect of all these elements may be determined by merely multif ying their associated Mueller matrices. As an example, let us consider how a circular polarizer can be constructed by superposing a linear polarizer and a hnear retarder. The beam is first incident on a linear polarizer with horizontal transmission axis ( = 0°), the Mueller matrix for which is obtained from (2.87) ... 

Thus, if unpolarized light or, indeed, light of arbitrary polarization is incident on the optical system described by the Mueller matrix (2.92), the transmitted light will be 100% right-circularly polarized. Note that matrix multiplication is not commutative the order of elements in a train must be properly taken into account. Further details about Mueller matrices and experimental means for realizing polarizers, retarders, and other optical elements are found in the excellent book by Shurcliff (1962). 

The simplest, and probably most obvious, way to measure scattering matrix elements is with a conventional nephelometer (Fig. 13.5) and various optical elements fore and aft of the scattering medium. Recall that we introduced Stokes parameters in Section 2.11 by way of a series of conceptual measurements of differences between irradiances with different polarizers in the beam. Although we did not specify the origin of the beam, it could be light scattered in any direction. Combinations of scattering matrix elements can therefore be extracted from these kinds of measurements. There are now, however, two beams—incident and scattered—and many possible pairs of optical elements these are discussed below. 

The incident beam, which we may take to be unpolarized, encounters three optical elements, each of which is represented by a matrix we recall from Section 2.11 that matrices for polarizers and retarders depend on their orientation. The Stokes parameters of the light emerging successively from the polarizer, the modulator, and the scattering sample are (from right to left)... 

Most optical experiments consist of a cascade of optical elements, and each will be represented by a Jones or Mueller matrix. Such a series is shown in Figure 2.2. The polarization vector generated by a train of n elements is... 

Figure 2.2 Propagation of polarized light through a cascade of optical elements.

Once a Jones or Mueller matrix of an optical element is obtained for one orthonormal basis set (ep e2, for example), the corresponding matrices for the element relative to other basis sets can be obtained using standard rotation transformation rules. The action of rotating an optical element through an angle 0 and onto a new basis set ej, e2 is pictured in Figure 2.3. In the nonrotated frame, the exiting polarization vector is ... 

It is of interest to consider the response of circularly polarized light to optically active materials. The action of an optical element represented by a Jones matrix formulated using the basis set for linearly polarized light is... 

The majority of Jones matrices for transmission polarizing elements have been presented in this chapter. As mentioned earlier, it is generally more convenient to use Mueller matrices and not Jones matrices when analyzing cascades of optical elements making up a particular experiment. For the purpose of such calculations, a list of Jones and Mueller matrices for most of the elements encountered in practice can be found compiled in Appendix I. 

The measurement of the polarization properties of light can be automated and improved by introducing a modulation of the polarization. Here a regular, time-dependent variation is introduced onto the optical properties of certain devices within either (or both) the PSG or PSA sections of the instrument. The modulation can be one of two types rotation of an optical element with fixed optical properties, or the modulation of the optical properties (retardation, for example) of an element with a fixed orientation. These are referred to as rotary modulators or field effect modulators, respectively. The latter name reflects the use of external fields (stress, electric or magnetic) to impart the modulation in these devices. In any case, a periodic oscillation is introduced into the signals that are measured that can effectively isolate specific optical properties in the sample. 

A rotary polarization modulator simply consists of an optical element that rotates uniformly at a frequency Q about the transmission axis of light. In practice, retardation plates and polarizers are used. In either case, the Mueller matrix of such a device is found by simply replacing the angle 6 by Q.t in the equations listed in Appendix I. Typical PSGs based on rotary modulators and the associated Stokes vectors, Sp G, that are produced are listed in table 8.2. 

The following list of Jones and Mueller matrices has been compiled for most optical elements encountered in optical instruments where polarization effects must be taken into account. In writing these matrices, the following notation has been used a - 2nn d/X a" = 2nn"d/ k, where n = n -iti" is the isotropic refractive index, d is the sample thickness ... 

Also, the heterogeneity of the ruby power density cross section in the sample interaction volume caused a large shot to shot variation in the non-linear signal intensity. In the RIKES experiments, all optical elements were placed outside the crossed polarizers only the sample cell windows remained. This arrangement prevented strain birefringence from interfering with the RIKES spectrum. 

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