Polarizer Jones matrix

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... 

Circular birefringence will induce a differential retardation in the phase of the orthogonal states of circularly polarized light. Circular dichroism, on the other hand, results in anisotropic attenuation of left- and right-circularly polarized light. The Jones matrix of circularly dichroic materials is normally written as ... 

The presence of a thin film, or a stack of films, at an interface will affect the polarization properties of reflected and transmitted light. The analysis for isotropic materials is simplified by the fact that the Jones matrix will not contain off-diagonal elements. This means that the reflection and transmission of p and s polarized light can be treated separately and these subscripts can be dropped in the description of the components of the electric vector. 

In the general case, when s-polarized light is converted into p-polarized light and/or vice versa, the standard SE approach is not adequate, because the off-diagonal elements of the reflection matrix r in the Jones matrix formalism are nonzero [114]. Generalized SE must be applied, for instance, to wurtzite-structure ZnO thin films, for which the c-axis is not parallel to the sample normal, i.e., (1120) ZnO thin films on (1102) sapphire [43,71]. Choosing a Cartesian coordinate system relative to the incident (Aj) and reflected plane waves ( > ), as shown in Fig. 3.4, the change of polarization upon reflection can be described by [117,120]... 

In Section V we used the system transfer matrix to study the effect of an optical system on the parameters of a Gaussian beam. A similar formalism exists for studying the polarization evolution of a Jones vector as a beam traverses a polarization-transforming system. In this case the system transfer matrix is called a Jones matrix. The simplest Jones matrix is the matrix that describes the polarization vector reflected from an ideal mirror. In order to satisfy the boundary conditions of vanishing tangential E, we need... 

It is important to note that the preferred orientation of the grid for nonnormal incidence is with the grid in the plane of incidence of the Gaussian beam (Erickson, 1987). This corresponds to i) = 90°. For this orientation of the grid, nonidealities due to cross-polarized components are minimized. The field transmitted by the polarizer is described by the complement of the Jones matrix of the polarizer, namely, which... 

It is important to emphasize that this modeP explains the polarization dependence of the writing and reading beams, and it provides a full picture of what one should expect in terms of the diffraction efficiency for the various experimental conditions. Note, however, that the formation of the SRG, which contributes with a phase difference to be input in the Jones matrix formalism, was assumed a priori. The model is not, therefore, aimed at explaining the origin of the mass transport, unlike the case of the models in references 9, 30-33, 36, and 37. 

The coherency matrix method permits the so-called dominant type of deterministic polarization transformation i.e., the corresponding deterministic part, Mueller-Jones matrix, of the initial Mueller matrix. A Jones matrix J is a 2x2 complex valued matrix containing generally eight independent parameters from the real and imaginary parts for each the four matrix elements, or seven parameters if the absolute (isotropic) phase which is not of interest for polarizations is excluded. Every Jones matrix can be transformed into an equivalent Mueller matrix but the converse assertion is not necessarily true. Between Jones J and Mueller Mj matrices that describe deterministic objects there exist a one-to-one correspondence ... 

The Mueller-Jones matrix provides a complete description of the anisotropy properties of an object [9,10]. However, the information in the matrix is in implicit form. The history of the problem of analysis of the Jones and Mueller-Jones matrix goes back to the derivation of three equivalence theorems by Hurwitz and Jones [17]. According to the first theorem, an optical system (object) composed of any number of retardation plates (that is an object with linear phase anisotropy) and rotators (circular phase anisotropy) is optically equivalent to a system containing only two elements a retardation plate, and a rotator. The second theorem is analogous to the first and but is concerned with partial polarizers (linear amplitude anisotropy) and rotators. The third theorem claims that an optical system composed of any number of partial polarizers, retardation plates, and rotators is optically equivalent to a system containing only four elements two retardation plates, a partial polarizer, and rotator. 

Another approach to the analysis of Jones and Mueller-Jones matrix exploits the polar decomposition theorem [18]. This approach was first suggested in [19] and was explored in [20,21]. The polar decomposition of a Jones matrix J can be represented as ... 

Labarthet, F. L., Buffeteau, T., Sourisseau, C. (1999). Azopolymer holographic diffraction gratings time dependent analyses of the diffraction efficiency, birefringence, and surface modulation induced two linearly polarized interfering beams. /. Phys. Chem. B, Vol. 103, No. 32, (August 1999), pp. 6690-6699, ISSN 1089-5647 Lien, A. (1990). Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence. Appl. Phys. Lett, Vol. 57, No. 26, (December 1990), p>p. 2767-2769, ISSN 0003-6951... 

Key performance parameters for all types of optical isolators are insertion loss and isolation, and in addition, for the polarization-independent optical isolators, polarization-dependent loss (PDL) and polarization mode dispersion (PMD) are important parameters as well. Performances (insertion loss and isolation) of an optical isolator can be estimated by using the Jones matrix. For example, insertion loss and isolation of a polarization-dependent isolator can be expressed as... 

Use the Jones matrix method to numerically calculate the transmittance of a 90° twisted nematic display in the field-off state as a function of the retardation u = 2Anh/A. The polarizers are parallel to each other and are also parallel to the liquid crystal director at the entrance plane. Compare your result with Figure 3.4. 

Use the Jones matrix method and the Berreman matrix method separately to calculate the transmittance pattern in the following two cases as a function the polar and azimuthal angles 0 and tp of the incident light. A uniaxial bireffingent film is sandwiched between two crossed polarizers. The transmission axis of the polarizer at the entrance plane is along the x axis. (1) an a plate has the retardation And = A and its slow axis makes the... 

In a reflective cell shown in Figure 9.1, the incident light traverses the linear polarizer, A/4 film, LC layer, and is reflected back by the embedded mirror in the inner side of the rear substrate. In the voltage-off state, the normalized reflectance can be obtained by the Jones matrix method [12] ... 

The Jones representation shows its advantages when we consider the transmission of light through optical elements such as polarizers, A/4 plates, or beamsplitters. These elements can be described by 2 x 2 matrices, which are compiled for some elements in Table 2.1. The polarization state of the transmitted light is then obtained by multiplication of the Jones vector of the incident wave by the Jones matrix of the optical element. 

V/e are concerned with the polarization properties of the transmitted light in the quasi-adiabatic limit. As a consequence, it is possible to assume that, in a first approximation, the backward-propagating waves are negligible since they give rise only to second or higher order effects. By explicitly considering the waves propagating in the forward direction, the transformed Berreman s matrix may be reduced to a 2x2 Jones matrix... 

The Jones Matrix M 6) for any polarizing element, turned by an angle 0 against its original position Af(0) is obtained by the product... 

For example a linear horizontal polarizer turned by an angle 6 is described by the Jones matrix... 

Matrix representations are commonly used to provide a mathematical description of optical measurements in a compact and convenient manner. The Jones and the Mueller matrix representations are often used to describe the change in polarization state of the light upon interaction with a sample. In these representations, Jones and Stokes vectors represent the different polarization states of light, and matrices are used to represent elements which change those states, as optical elements or the sample being analyzed. Figure 2.14 illustrates a simple ellipsometer and shows the Jones matrix representations for each component in a general case. 

Fig. 2.14 Schematic diagram of a rotating analyzCT ellipsometer (RAE) and the Jones matrix representation for each comptment. The blue lines repiestait the trajectory of the electric field vector of the propagating light. The difference in the representations of the analyzer and polarizer is due to the angles a and ra being measured in opposite directions...

We can now combine these two eigenwaves to make any linear state of polarization that we require. If the polarization of the light were to bisect the x and y axes by 45°, we could represent this asV =Vy and use the two states combined into a single Jones matrix... 

In a similar fashion, we can tailor the thickness to give a quarter wave retardation of r=71/2. Such a wave plate is useful for converting to and from circularly polarized light. For a quarter wave plate with its fast axis aligned to the y axis (y/=0), the Jones matrix will be... 

A polarizer oriented at an angle y/ from the y axis can be written as a Jones matrix... 

In the Jones matrix representation an ideal linear polarizer, neglecting the absolute phase accumulated as a result of the finite optical thickness, can be given as ... 

---