The Jones Vector

As demonstrated by the example of section 1.2.1, interaction of the electric field with anisotropic materials can cause its orthogonal components to have dissimilar phases and amplitudes. These properties of the electric vector describe the state of polarization of the electric vector. Since the electric vector lies in the plane perpendicular to the axis of propagation, a convenient description for this purpose is the two-component vector  

This describes the x and y components of the electric vector of light propagating along the z axis and through an isotropic material of refractive index n. Evidently, the light has had a prior interaction with an anisotropic material and these two components have differing amplitudes and phases. For example, in section 1.2.1, a sample with a uniaxial dielectric tensor was observed to induce a phase difference, 8X - 8y = (2k/X) (n, - n2) d, where 

As the electric vector propagates, it will rotate in space and time according to the difference in the phases 8 and 5. To illustrate the form of the electric vector, it is con- 

As an example, consider the case where each component has the same amplitude (Ax = Ay = A), but with a phase difference, 8X - 8y = 7t/2. Then the components of A are 

If the x and y components have the same phase (8-8=0), the light will be 

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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 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... 

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 ... 

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... 

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... 

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. 

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... 

Now we consider a TN display whose geometry is shown in Figure 3.3 [11,12]. The TN liquid crystal is sandwiched between two polarizers. The x axis of the lab frame is chosen parallel to the liquid crystal director at the entrance plane. The angles of the entrance and exit polarizers are a, and a , respectively. The Jones vector of the incident light is... 

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. 

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