Extended Cauchy equations

In the off-resonance region, the right terms in Equations (6.6) and (6.7) can be expanded by a power series to the i term and then combined with Equation (6.9) to form the extended Cauchy equations for describing the wavelength-dependent refractive indices of anisotropic LCs as 

In Equation (6.10), A and are three Cauchy coefficients. By measming the refrac- 

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Although the extended Cauchy equation fits experimental data well [7], its physical origin is not clear. A better physical meaning can be obtained by the three-band model which takes fliree major electronic transition bands into consideration. 

To model the refractive indices of an LC mixture, we could expand Equation (6.12) into power series because in the visible and IR spectral regions, A By keeping up to the terms, the above extended Cauchy equation (6.10) is again derived. 

Table 6.1 Fitting parameters for the three-band model and the extended Cauchy equations. LC 5CB at...

Prove that the extended Cauchy equations (Equation (6.10)) derived from LC compounds can be applied to liquid crystal mixtures. Hint see Ref. 13. 

J. Li and S. T. Wu, Extended Cauchy equations for the refractive indices of liquid crystals, J. Appl. Phys. 

For most LC displays [14], the cell gap is controlled at aroimd 4 pm so that the required birefringence is smaller than 0.12. Thus Equation (6.13) can be used to describe the wave-length-dependent refractive indices. For infrared applications, high birefringence LC mixtures are required [15]. Under such circumstances, the three-coefficient extended Cauchy model (Equation (6.10)) should be used. 

Figure 6.9 Wavelength-dependent refractive index of NOA65 and the ordinary refractive index of E48, E44, and E7 at T= 20°C. The open squares, upward-triangles, fiUed circles, and downward-triangles are the measured refractive index of E48, E44, NOA65 and E7, respectively. The solid lines represent the fittings using the extended Cauchy model (Equation (6.10)). The fitting parameters are listed in Table 6.3.

The LEM, elaborated and extended by one of the authors [4], [5] for the general nonlinear operators of the polynomial type, results In qualitative representations for the solution of the B.-E. equation associated Cauchy problem of arbitrary Initial data. These representations form a basis for the present study. We... 

These are the Cauchy-Riemann relations. If, at any point in the liquid, y is taken along a contour of constant 0 then dip/dx is zero, showing that in the x direction at that point 0 is a constant. This argument can be extended to all points on a curve with 0 = constant. Therefore, each value of the constant in the equation 0( > y) = constant, represents a member of a family of curves that cut orthogonally each member of the family given by 0(x,y) = constant. 

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