## Abstract

The role of the excitation of guided waves propagating in a corrugated dielectric waveguide is discussed in view of the resonance anomalies in reflectivity. Narrow-wavelength filtering properties that are due to these sharp anomalies have been a topic of interest for some time, but a proper understanding of device performances requires an analysis of tolerances with respect to the incident-beam collimation and to waveguide losses. Such an analysis is proposed in this paper, and the conclusion is that the incident-beam divergence plays a critical role in reducing the maximum reflectivity for narrow-band filters.

© 2001 Optical Society of America

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### Equations (16)

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(1)
$${n}_{c}sin{\theta}_{i}=\frac{{\beta}_{g}}{{k}_{0}}+m\frac{\mathrm{\lambda}}{d},$$
(2)
$${r}_{m}={r}_{0,m}\frac{(\alpha -{\alpha}_{m}^{r,z})}{(\alpha -\alpha {}^{p})},$$
(3)
$${t}_{m}={t}_{0,m}\frac{(\alpha -{\alpha}_{m}^{t,z})}{(\alpha -\alpha {}^{p})},$$
(4)
$$\frac{1}{{\beta}_{0}^{r}}\sum _{m\in U}({\beta}_{m}^{r}|{r}_{m}{|}^{2}+{\beta}_{m}^{t}|{t}_{m}{|}^{2})=1,$$
(5)
$$({\beta}_{m}^{r,t}{)}^{2}={n}_{c,s}^{2}-{\alpha}_{m}^{2},$$
(6)
$$sin{\theta}_{i}=\frac{1}{{n}_{c}}{\alpha}_{0}^{t,z}.$$
(7)
$$min({t}_{0})\propto \frac{\mathrm{Im}({\alpha}_{m}^{t,z})}{\mathrm{Im}({\alpha}^{p})}.$$
(8)
$${E}_{z}^{i}(x,y)={\int}_{-\infty}^{+\infty}p(\delta )exp[i(\alpha x-{\beta}^{r}y)]\mathrm{d}\delta ,$$
(9)
$$p(\delta )=exp\left[-{\left(\frac{\delta -{\delta}_{0}}{\mathrm{\Delta}}\right)}^{2}\right].$$
(10)
$${E}_{z}^{r}(x,y)={\int}_{-\infty}^{+\infty}r(\delta )p(\delta )exp[i(\alpha x+{\beta}^{r}y)]\mathrm{d}\delta ,$$
(11)
$${\eta}_{0}=\frac{{\int}_{-\infty}^{+\infty}|r(\delta )p(\delta ){|}^{2}{\beta}^{r}(\delta )\mathrm{d}\delta}{{\int}_{-\infty}^{+\infty}|p(\delta ){|}^{2}{\beta}^{r}(\delta )\mathrm{d}\delta}.$$
(12)
$${I}^{i,r}\propto {\int}_{-\infty}^{+\infty}{P}_{y}^{i,r}(x,y){|}_{y=\infty}\mathrm{d}x.$$
(13)
$${P}_{y}(x,y)\propto {E}_{z}{\overline{H}}_{x}\propto {E}_{z}\frac{\partial {\overline{E}}_{z}}{\mathrm{d}y},$$
(14)
$${P}_{y}^{i}(x,y)\propto {\int}_{-\infty}^{+\infty}\int \mathrm{d}{\delta}^{\prime}\mathrm{d}{\delta}^{\u2033}p({\delta}^{\prime})\overline{p}(\delta {}^{\u2033}){\beta}^{\u2033}\times exp[i({\alpha}^{\prime}-{\alpha}^{\u2033})x-i({\beta}^{\prime}-{\beta}^{\u2033})x],$$
(15)
$${P}_{y}^{r}(x,y)\propto {\int}_{-\infty}^{+\infty}\int \mathrm{d}{\delta}^{\prime}\mathrm{d}\delta {}^{\u2033}r({\delta}^{\prime})\overline{r}(\delta {}^{\u2033})p({\delta}^{\prime})\overline{p}(\delta {}^{\u2033}){\beta}^{\u2033}\times exp[i({\alpha}^{\prime}-{\alpha}^{\u2033})x-i({\beta}^{\prime}-{\beta}^{\u2033})x].$$
(16)
$${\int}_{-\infty}^{+\infty}exp[i({\alpha}^{\prime}-{\alpha}^{\u2033})x]\mathrm{d}x=2\pi \delta ({\alpha}^{\prime}-{\alpha}^{\u2033}),$$