Jones vectors

If we are interested only in the polarization state of the wave, it is convenient to use the normalized Jones vector which satisfies the condition 

Fundamentals of Liquid Crystal Devices, Second Edition. Deng-Ke Yang and Shin-Tson Wu. 2015 John Wiley Sons, Ltd. Published 2015 by John Wiley Sons, Ltd. 

Jones vectors for right- and left-handed circularly polarized light are respectively. The Jones vector of various polarization states is listed in Table 2.1. 

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The discretized adiabatic procedure, and its analog with STIRAP, is but one possibility for achieving broadband response of an optical device. An alternative, which we discuss, relies on the analogy between the Jones vector description of an optical beam and the two-state time-dependent Schrodinger equation (TDSE). This equation has two commonly used solutions. One is rapid adiabatic passage (RAP), produced by swept detuning (a chirp), and the other is Rabi oscillations, specifically a pi pulse. The RAP has theoretical connection with STIRAP, but the pi pulses have no such connections. We describe application of a procedure that has been used to extend the traditional pi pulses to broadband excitation. This can accomplish the present goal of PAP, under complementary conditions. 

This is the differential equation for propagation alterations of a Jones vector [10], having two complex-valued components and y, passing through nonreflecting dielectric media. 

This is the basic equation for propagation of the Jones vector components through material in which there are no reflections and only slowly varying optical properties. Like Eq. (5.7), the replacement of independent variables, t, makes this an analog of the two-state TDSE [11, 12], cf. Eq. (5.A.2). 

The Jones vector is not directly observable. Most experiments use square law detectors that measure the intensity [1],... 

The subscripts l and r refer to left- and right-circularly polarized light, respectively. Any electric field, A, can be expressed in terms of either of these two sets. The Jones vectors for the two representations are related by the transformation ... 

Although the Stokes vector, with its greater number of components, appears to be a more cumbersome representation of the electric vector, it is often more convenient to use than the Jones vector. This is because its components are observable quantities. For monochromatic, perfectly polarized light, the four components of the Stokes vector are not linearly independent, but related according to... 

The connection between the Stokes and Jones vectors, given by equation (1.59) can be used to relate the sixteen-component Mueller matrix to the four-component Jones matrix. Combining equations (2.1), (2.2), and (1.59), we have, using a notation similar to that developed in Azzam and Bashara [5],... 

Since the Maxwell equations involve the components of the Jones vector, it is normally easier to derive the Jones matrix, J, for complex, anisotropic media. Once J is obtained, it is generally convenient to transform it to the Mueller matrix representation for the purpose of analyzing the quantities measured in specific optical trains. This is because the components of the Stokes vector are observable, whereas the Jones vector components are not. Since it is the intensity of light that is normally required, only the first element of Sn,... 

Isotropic materials are characterized by a scalar refractive index, n = n + in", as defined in equation (1.17). The real and imaginary parts of the refractive index induce phase shifts and attenuation of the electric vector, respectively. This is seen by examining the Jones vector, Aj, of light exiting an element of isotropic material with thickness d. If the element is surrounded by a medium of refractive index, nQ, then... 

This boundary condition reflects the fact that the Jones vector will by unchanged by passage through an element of zero thickness. In reference 7, Jones was able to show that the solution to (2.44) is ... 

This problem was treated in section 1.6 of Chapter 1, where the Fresnel coefficients for reflected and refracted light were calculated and presented in equations (1.74) to (1.77). The problem being treated is pictured in Figure 1.4, and it is convenient to represent the electric vector as a Jones vector having orthogonal components that are either parallel... 

Thus, if light is linearly polarized along x, along y, and at 45° from both x and y, the Jones vectors are, respectively,... 

We next calculate the null setting of an ellipsometer from the reflection matrix in an anisotropic sample. The Jones vector for the reflected light is given by Eq. (2.15.44) for an anisotropic sample the off-diagonal elements of reflection matrix R are nonzero. 

In order to proceed beyond a qualitative description of how a PTR operates, it is convenient to use a mathematical description of coherent polarization states, which are a good approximation to the output of solid-state near-millimeter sources. The Jones vector formalism is well known (Hecht and Zajac, 1979, pp. 268-270 LeSurf, 1990) and well suited to the present purpose. Any transverse polarization vector can be represented by an equation of the form E = -I- E y), where H and V are... 

The Jones vectors of a horizontally polarized Gaussian beam E and a vertically polarized Gaussian beam Ey of field strength Eq at the beam waist may be represented as... 

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... 

A grid polarizer is the next object we need to consider. First we will define the Jones vectors for linear polarization at + 45° with respect to the y axis. These cases correspond to the situation shown in Fig. 8a. The required Jones vectors are... 

An interesting result gives the investigation of holographic recording in the azo-dye material with Weigert s effect in the general case of linear polarization of the object waves. For the simple theoretical calculations the Jones vector-matrix method of was used (Jones, 1941 Kakichashvili, 1974). 

We consider interference of mutually coherent polarized light in uniaxial anisotropic media. As illustrated in Fig. 1, the xz -plane is the incident plane and the z -axis is taken normal to the film plane. Assuming that the two recording beams are plane waves and that the amplitude of their incident angles is small, the electric field of interference light is described using the Jones vector as... 

Qx i-lntioZlX). Therefore the Jones vector Eyo.Eyo) of outgoing light will be... 

In the principal frame, Equation (3.6) is valid. The Jones vector of the incident light in the principal frame is related to the Jones vector , in the lab frame by... 

For layer j, the angle of the slow axis with respect to the x axis is Pj and the phase retardation is Tj = l7t[ne z=j h)-no z=jish)]ishlX. In the lab frame, the Jones vector of the incident... 

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