## Abstract

In a recent publication [J. Opt. Soc. Am. A **13**, 1013 (1996)] a modified formulation of a rigorous Fourier-expansion eigenmode method was presented in TM polarization, which produces the correct effective refractive indices of form-birefringent stratified media in the lowest-order approximation. In this communication it is shown that a further refinement is consistent, for any angle of incidence, with the extraordinary nature of the TM-polarized wave in the quasi-static limit.

© 1997 Optical Society of America

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### Equations (19)

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(1)
$${\epsilon}_{m}=\frac{1}{d}{\int}_{0}^{d}{\u220a}_{r}(x)exp(i2\pi \mathit{mx}/d)\mathrm{d}x,$$
(2)
$${\xi}_{m}=\frac{1}{d}{\int}_{0}^{d}\frac{1}{{\u220a}_{r}(x)}exp(i2\pi \mathit{mx}/d)\mathrm{d}x$$
(3)
$$({k}^{2}\mathbf{E}-\mathbf{AIA})\mathbf{E}={\gamma}^{2}\mathbf{E},$$
(4)
$$\mathbf{E}({k}^{2}\mathbf{I}-\mathbf{AXA})\mathit{H}={\gamma}^{2}\mathit{H},$$
(5)
$$\mathbf{E}({k}^{2}\mathbf{I}-{\mathbf{AE}}^{-1}\mathbf{A})\mathit{H}={\gamma}^{2}\mathit{H},$$
(6)
$${\mathbf{X}}^{-1}({k}^{2}\mathbf{I}-\mathbf{AXA})\mathit{H}={\gamma}^{2}\mathit{H},$$
(7)
$${\mathbf{X}}^{-1}({k}^{2}\mathbf{I}-{\mathbf{AE}}^{-1}\mathbf{A})\mathit{H}={\gamma}^{2}\mathit{H},$$
(8)
$${\gamma}^{2}={k}^{2}({\epsilon}_{0}-{n}_{\mathrm{in}}^{2}{sin}^{2}{\theta}_{\mathrm{in}})$$
(9)
$${\gamma}_{\mathrm{I}}^{2}={k}^{2}{\epsilon}_{0}(1-{\xi}_{0}{n}_{\mathrm{in}}^{2}{sin}^{2}{\theta}_{\mathrm{in}})$$
(10)
$${\gamma}_{\mathrm{II}}^{2}={k}^{2}({\epsilon}_{0}-{n}_{\mathrm{in}}^{2}{sin}^{2}{\theta}_{\mathrm{in}})$$
(11)
$${\gamma}_{\mathrm{III}}^{2}={k}^{2}({\xi}_{0}^{-1}-{n}_{\mathrm{in}}^{2}{sin}^{2}{\theta}_{\mathrm{in}})$$
(12)
$${\gamma}_{\mathrm{IV}}^{2}={k}^{2}{\xi}_{0}^{-1}(1-{\epsilon}_{0}^{-1}{n}_{\mathrm{in}}^{2}{sin}^{2}{\theta}_{\mathrm{in}})$$
(13)
$${\overline{n}}_{\mathrm{TE}}={\epsilon}_{0}^{1/2},$$
(14)
$${\overline{n}}_{\mathrm{TM}}={\xi}_{0}^{-1/2},$$
(15)
$${\overline{n}}_{\mathrm{TE}}=[{n}_{1}^{2}f+{n}_{2}^{2}(1-f){]}^{1/2},$$
(16)
$${\overline{n}}_{\mathrm{TM}}={n}_{1}{n}_{2}[{n}_{2}^{2}f+{n}_{1}^{2}(1-f){]}^{-1/2},$$
(17)
$$\frac{{\alpha}^{2}}{{\overline{n}}_{\mathrm{TE}}^{2}}+\frac{{\gamma}^{2}}{{\overline{n}}_{\mathrm{TE}}^{2}}={k}^{2},$$
(18)
$$\frac{{\alpha}^{2}}{{\overline{n}}_{\mathrm{TM}}^{2}}+\frac{{\gamma}_{\mathrm{III}}^{2}}{{\overline{n}}_{\mathrm{TM}}^{2}}={k}^{2},$$
(19)
$$\frac{{\alpha}^{2}}{{\overline{n}}_{\mathrm{TE}}^{2}}+\frac{{\gamma}_{\mathrm{IV}}^{2}}{{\overline{n}}_{\mathrm{TM}}^{2}}={k}^{2}.$$