Understanding the propagation of light in continuous inhomogeneous materials is important to design optical structures and devices. To have accurately numerical calculations Berreman's 4×4 propagation matrix method is generally used, and layer approximation, i.e., the whole one-dimensional continuous inhomogeneous material is divided into many small homogeneous layers, is assumed. However, this layer approximation is only correct up to the second-order of the layer thickness. To efficiently solve Berreman's first-order differential equation, a simple fourth-order symplectic integrator is presented. The efficiency of the fourth-order symplectic integrator was studied for a cholesteric liquid crystal. Numerical results of reflectance spectra show that the fourth-order symplectic integrator is highly efficient in contrast to the extensively used fast 4×4 propagation matrix.
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