Simple Analytical Solution for Light Incident Parallel to the Helical Axis

3 Simple Analytical Solution for Light Incident Parallel to the Helical Axis 

Our task is to find the spectrum of eigenmodes propagating along the helical axis of a cholesteric liquid crystal and discuss some consequences of that. It is very rare and even unique case when, despite chirality and anisotropy of a medium, there is an analytical solution found many years ago by De Vries [6]. Here, we follow a rather simple and very elegant analytic solution of this problem given by Kats [7]. 

We again consider an electromagnetic wave propagating parallel to the helical axis of an infinite cholesteric medium (k II qo II z) in the geometry corresponding to Fig. 12.3. Therefore non-zero components of the electric field and Ey depend only on z. The Helmholtz wave equation 

Using Ex z,t) = Fx z)expi(ot,Ey z,t) = Fy z)expi(at we exclude the time dependence and, on account of = co E2d /c, obtain the set of equations for the field amphtudes, which contains all optical properties of the cholesteric  

These equations become simpler if one introduces circular field components E+ =Fx + iFy and E =Fx iFy (12.11) 

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