Berreman matrix

In the medium the wave vector is k = nk . From Equations (3.146) and (3.147) we have eigenvalues of the Berreman matrix... 

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... 

Berreman used a 4x4 matrix multiplication method. Assuming the incident and reflected wavevectors to be in the xz plane, z being along the helical axis of the cholesteric, the dependence on the y coordinate may be ignored altogether. Writing exp [i( ut—etc., it is easily verified... 

To calculate the propagation of light through the twisted nematic structure, we start with the 4x4 matrix linear differential equation approach of Berreman > and Teitler and Henvis. As we were not interested in the interference effects and reflections internal to the twisted nematic layer are negligible, we were able to reduce the calculations to a 2x2 matrix linear differential equation. Further reduction in the amount of computation was achieved by solving this equation in terms of the natural variable X of the twisted nematic structure, rather than in terms of distance. 

For stratified optical media (whose refractive indices are only a function of the coordinate normal to the film), Berreman introduced a 4 x 4 matrix method (now known as the Berreman 4x4 method) [14—18], in which the electric field and magnetic field (the sum of the fields of the light beam propagating in forward and backward directions) are considered. When the film is divided into slabs, the reflection at the interface between the slabs is taken into account. The Berreman 4x4 method works well for both normal and obliquely incident fight. Consider an optical film, such as a cholesteric liquid crystal in the planar texture (twisted nematic), whose dielectric tensor is only a function of the coordinate z, which is perpendicular to the film ... 

D. W. Berreman, Ultrafast 4x4 matrix optics with averaged interference Mnges, SLD International Symposium, Seminar and Exhibition, 101-104, Seattle, USA 16-21. May (1993). 

Ais the Berreman s matrix, which for normal incidence of light reduces to... 

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 equation (11) may be solved either numerically (by making use of an algorithm much simpler than the one required by the full 4x4 Berreman s matrix), or using a standard perturbative treatment. In both cases, use is made of the interaction matrix formalism, in which V plays the role of the perturbative term. 

The transmitted light intensity was calculated by the Berreman 4x4 matrix method [22]. The director tensor is determined by the director tilt angle 7 and the director twist angle (f), which are expressed as... 

The plots in Figure 17.3 were created using the Berreman 4x4 matrix method, simulating the optical response of a multilayer. Variations in the anisotropy of the refractive index, the pitch of the layers, the number of layers, and the average refractive index are shown. The changes shown by the Berreman simulation match the intuitive understanding of ordinaiy Bragg reflectors. 

Another analytical method which had been extensively applied to liquid crystal structures was developed by Berreman [35]. It is based on a 4x4 matrix representation of optical elements [36], where the incident and transmitted light are described by a 4x1 column vector consisting of components of both the electric and magnetic field associated with the electromagnetic wave. 

Berreman DW (1972) Optics in stratified and anisotropic media 4x4-matrix formulation. Journal of the Optical Society of America 62 502-510. 

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