## Abstract

Modulation-averaging reflectors have recently been proposed as a means for improving the link margin in self-seeded wavelength-division multiplexing in passive optical networks. In this work, we describe simple methods for determining key parameters of such structures and use them to predict their averaging efficiency. We characterize several reflectors built by arraying fiber-Bragg gratings along a segment of an optical fiber and show very good agreement between experiments and theoretical models.

© 2013 Optical Society of America

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### Equations (11)

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(1)
$$\mathrm{\Gamma}=\frac{\gamma}{{\gamma}_{0}}\xb7\frac{{P}_{0}}{P},$$
(2)
$${P}_{F0}={t}_{1}{P}_{F1}+{r}_{1}{P}_{R0},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{P}_{R1}={r}_{1}{P}_{F1}+{t}_{1}{P}_{R0}.$$
(3)
$$\left[\begin{array}{c}{P}_{Fk}\\ {P}_{Rk}\end{array}\right]=\mathbf{A}(k)\left[\begin{array}{c}{P}_{F,k-1}\\ {P}_{R,k-1}\end{array}\right].$$
(4)
$$\mathbf{A}(k)=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{{t}_{k}}& -\frac{{r}_{k}}{{t}_{k}}\\ \frac{{r}_{k}}{{t}_{k}}& \frac{{t}_{k}^{2}-{r}_{k}^{2}}{{t}_{k}}\end{array}\right].$$
(5)
$$\mathbf{B}=\prod _{k=1}^{n}\mathbf{A}(k)={\mathbf{A}}^{n}.$$
(6)
$${\mathbf{A}}^{n}={S}_{n-1}(x)\mathbf{A}-{S}_{n-2}(x)\mathbf{I},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}x={a}_{11}+{a}_{22},$$
(7)
$${S}_{n-1}(\mathrm{cosh}\text{\hspace{0.17em}}\theta )=\frac{\mathrm{sinh}\text{\hspace{0.17em}}n\theta}{\mathrm{sinh}\text{\hspace{0.17em}}\theta}.$$
(8)
$$\frac{x}{2}=\mathrm{cosh}\text{\hspace{0.17em}}\theta ,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}x=\frac{1+{t}^{2}-{r}^{2}}{t}.$$
(9)
$$\mathbf{B}={\mathbf{A}}^{n}=\frac{\mathrm{sinh}\text{\hspace{0.17em}}n\theta}{\mathrm{sinh}\text{\hspace{0.17em}}\theta}\left[\begin{array}{cc}{a}_{11}-\frac{\mathrm{sinh}(n-1)\theta}{\mathrm{sinh}\text{\hspace{0.17em}}n\theta}& {a}_{12}\\ {a}_{21}& {a}_{22}-\frac{\mathrm{sinh}(n-1)\theta}{\mathrm{sinh}\text{\hspace{0.17em}}n\theta}\end{array}\right].$$
(10)
$$\mathbf{C}=\left[\begin{array}{cc}\frac{1}{{t}_{C}}& 0\\ 0& {t}_{C}\end{array}\right],\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{\mathbf{C}}^{\prime}=\left[\begin{array}{cc}\frac{1}{{t}_{C}^{\prime}}& 0\\ 0& {t}_{C}^{\prime}\end{array}\right],\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{H}=\left[\begin{array}{c}\frac{{P}_{T}}{{t}_{0}}\\ \frac{{r}_{0}}{{t}_{0}}{P}_{T}\end{array}\right].$$
(11)
$$R(n)={t}_{C}^{2}\frac{{b}_{21}+{({t}_{C}^{\prime})}^{2}{r}_{0}{b}_{22}}{{b}_{11}+{({t}_{C}^{\prime})}^{2}{r}_{0}{b}_{12}},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}T(n)=\frac{{t}_{C}{t}_{0}{t}_{C}^{\prime}}{{b}_{11}+{({t}_{C}^{\prime})}^{2}{r}_{0}{b}_{12}}.$$