## Abstract

Recently, electromagnetic materials with negative permittivity and permeability have been given much attention. Propagation of electromagnetic waves in double negative (DNG) media and also in photonic crystals has been studied analytically and experimentally. The materials’ optical parameters are complex and frequency dependent to account for both dispersion and absorption. Here we have studied theoretically the dispersion as well the transmission and reflection of the visible light on a one-dimensional heterostructure combining anisotropic DNG and isotropic double positive (DPS) materials. Here our center of attention is the study of the non-Bragg band gaps, which are not based on interference, in a one-dimensional photonic crystal composed of alternating layers of DNG and DPS materials. We find that this type of photonic crystal in the visible wavelength range exhibits negative refraction in a wider frequency range than does a single DNG material.

© 2009 Optical Society of America

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### Equations (11)

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(1)
$$\mathrm{cos}\left(Kd\right)-\mathrm{cos}\left({k}_{1y}{d}_{1}\right)\mathrm{cos}\left({k}_{2y}{d}_{2}\right)+\frac{1}{2}(\frac{{p}_{2}{k}_{1y}}{{p}_{1}{k}_{2y}}+\frac{{p}_{1}{k}_{2y}}{{p}_{2}{k}_{1y}})\mathrm{sin}\left({k}_{1y}{d}_{1}\right)\mathrm{sin}\left({k}_{2y}{d}_{2}\right)=0.$$
(3)
$${p}_{2}{k}_{1k}={p}_{2}{k}_{2y},$$
(4)
$${k}_{1y}{d}_{1}=-{k}_{2y}{d}_{2},$$
(5)
$$\u03f5=\left(\begin{array}{ccc}{\u03f5}_{ii}& 0& 0\\ 0& {\u03f5}_{jj}& 0\\ 0& 0& {\u03f5}_{kk}\end{array}\right)\phantom{\rule{1em}{0ex}}\mu =\left(\begin{array}{ccc}{\mu}_{ii}& 0& 0\\ 0& {\mu}_{jj}& 0\\ 0& 0& {\mu}_{kk}\end{array}\right).$$
(6)
$$\u03f5\left(\omega \right)=1+\frac{{4.8412}^{2}}{{2.1101}^{2}-{\omega}^{2}},$$
(7)
$$\mu \left(\omega \right)=1+\frac{{4.2295}^{2}}{{3.6698}^{2}-{\omega}^{2}},$$
(8)
$${R}_{\mathit{TE}}=\frac{{r}_{ij}+{r}_{jk}{e}^{2i\phi}}{1+{r}_{ij}{r}_{jk}{e}^{2i\phi}},$$
(9)
$${r}_{ij}=\frac{{k}_{zi}-{k}_{zj}}{{k}_{zi}+{k}_{zj}},$$
(10)
$${R}_{\mathit{TM}}=\frac{{r}_{ij}+{r}_{jk}{e}^{2i\phi}}{1+{r}_{ij}{r}_{jk}{e}^{2i\phi}},$$
(11)
$${r}_{ij}=\frac{{k}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left({\theta}_{i}\right)-{k}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left({\theta}_{j}\right)}{{k}_{i}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left({\theta}_{j}\right)+{k}_{j}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left({\theta}_{i}\right)}.$$