## Abstract

An analytical expression for calculating the position of the reflection-peak wavelength of a chirped sampled fiber Bragg grating (C-SFBG) is obtained for what is believed to be the first time. Using Fourier theory, the chirped sampling function of the C-SFBG is expanded, and an equivalent local Bragg period is then obtained to derive the expression of the peak wavelength. The calculated results based on the expression are in excellent agreement with the numerical reflection spectra obtained by the conventional transfer-matrix method.

© 2008 Optical Society of America

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### Equations (9)

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(1)
$$b(i)={b}_{0}[1+(i-1){c}_{s}]\mathrm{,}$$
(2)
$$\mathrm{\Delta}n(z)=\mathrm{\Delta}{n}_{0}s(z)\mathrm{Re}[\mathrm{exp}(j\frac{2\pi}{{\mathrm{\Lambda}}_{0}}\left)\right]\mathrm{,}$$
(3)
$$s(z)=\mathit{rect}\left(\frac{z}{a}\right)*\sum _{m=-\infty}^{+\infty}\delta \{z-m{b}_{0}[1+\frac{(m-1)}{2}{c}_{s}\left]\right\}\mathrm{,}$$
(4)
$$s(z)=\sum _{k=-\infty}^{+\infty}{F}_{k}\mathrm{exp}(2\pi jkz/p)\mathrm{,}$$
(5)
$$\mathrm{\Delta}n(z)=\mathrm{\Delta}{n}_{0}\sum _{k=-\infty}^{+\infty}{F}_{k}\mathrm{exp}\left[2\pi j\right(\frac{1}{\mathrm{\Lambda}(k)}\left)z\right]\mathrm{,}$$
(6)
$$\mathrm{\Lambda}(k)=\frac{p{\mathrm{\Lambda}}_{0}}{p-k{\mathrm{\Lambda}}_{0}}\mathrm{.}$$
(7)
$${\lambda}_{\mathrm{max}}(k)=2{n}_{0}\mathrm{\Lambda}(k)\mathrm{.}$$
(8)
$${\lambda}_{\mathrm{max}}(k)=2{n}_{0}\xb7\frac{{b}_{0}{\mathrm{\Lambda}}_{0}[2+(N-1){c}_{s}]}{{b}_{0}[2+(N-1){c}_{s}]-2k{\mathrm{\Lambda}}_{0}}\mathrm{.}$$
(9)
$$\mathrm{\Delta}\lambda (k)=2{n}_{0}\left[\frac{k{\mathrm{\Lambda}}_{0}^{2}({b}_{0}-p)}{({b}_{0}-k{\mathrm{\Lambda}}_{0})(p-k{\mathrm{\Lambda}}_{0})}\right]\mathrm{,}$$