Jones Matrix Method

When light propagates through films in which the slow and fast axes as well as the refractive indices are a function of position z, the Jones matrix method can still be used as an approximation method as long as the the refractive indices do not change much in one wavelength. We divide the film into N slabs as shown in Figure 3.2. When the thickness Ah = hIN of the slabs is sufficiently small, then within each slab, the slow axis can be considered fixed. 

The Jones matrix method has the limitation that it only works well for normally incident and paraxial rays. It neglects the reflection and refraction from the interface between two optic media whose refractive indices are different. The extended Jones matrix method takes account of the reflection and refraction, but still neglect multiple reflection, and can be used to calculate the optical properties of media for obhquely incident hght [4—7]. 

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] ... 

We characterized the diffraction properties of the LC grating based on the Jones matrix method and Fraunhofer diffraction theory. By considering that the nematic LC layer consists of M plates with a thickness of dm = d/M, the grating vector is parallel to the x-direction, the substrates are parallel to the ay-plane, and that the substrate normal is parallel to z-direction, the Jones matrix of the LC grating is... 

In summary, using a two-step exposure method with a line-and-space photomask we fabricated an LC grating inner coated with PLCP photoalignment films. The diffraction efficiencies and the unique polarization conversion properties of the LC grating were well explained based on the simple binary model involving planar and TN alignments, the Jones matrix method, and diffraction theory. 

For arbitrary angle 4) or director axis angular and spatial distributions, and more complicated cell structure, the phase shift, and therefore the transmission of light through the cell and other accompanying polarization selective elements, is not amenable to simple analytical treatment. More sophisticated Jones matrix methods or nmnerical technique such as the finite difference time domain (FDTD) nmnerical methods discussed in the next chapter are needed to solve such a complex propagation problem. 

We now consider the application of the Jones matrix method to the simple problem of polarized light through a birefringent phase plate or retardation plate... 

Jones Matrix Method for Propagation Through a Nematic Liquid Crystal Cell... 

Instead of the 4X4 matrix method described in the preceding section, oblique incidence can also be treated using the so-called extended Jones matrix method " involving 2X2 matrices, if the effects of multiple reflections can be neglected. To avoid confusion, we employ the cgs units adopted by these authors. Maxwell equations become... 

In the extended Jones matrix method, the liquid crystal cell is divided into (usually several hundreds for accurate computation) layers, with the dielectric tensor of each layer differing from that of its adjacent layer. The input and output polarizers are considered as two separate layers characterized by dielectric tensors of the form of Equation (7.71). In each layer, there are four eigenwaves two transmitted and two reflected waves. At each interface, the boundary condition that applies is that the tangential components of the electric field are continuous. 

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