Extended Jones matrix method

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. 

Let and be the incident field. The transmitted field is then given by 

With the inclusion of the transmission at the air-glass interfaces at the entrance and exit, the net optical transmission of the pixel becomes 

Finally, the transmission for parallel (p) and vertical (s) components of the transmission are given by 

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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]. 

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