## Abstract

Narrow linewidth transmission filters in lossy materials based phase-shifted fiber Bragg gratings have been investigated experimentally and analytically. A novel matrix technique has been developed in calculation of the transmission loss and linewidth. The elements of the matrix simply consist of the coefficients of the coupled mode equations. Simulation shows a small fiber loss could result in a significant transmission loss, which has not been explained properly yet to our knowledge. For phase-shifted gratings in erbium-doped fibers, the absorption could result in over
$20\text{\hspace{0.17em} dB}$ loss at transmission wavelengths. Such an approach can also be used to analyze cladding modes, radiation mode, and complex structure gratings.

© 2007 Optical Society of America

Full Article |

PDF Article
### Equations (10)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$\frac{\mathrm{d}{y}_{i}\left(z\right)}{\mathrm{d}z}={\displaystyle \sum _{j=1}^{N}{a}_{ij}\left(z\right){y}_{j}\left(z\right)}\text{\hspace{1em}}\left(i=1,2,3\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}N\right)\text{,}$$
(2)
$$\left[\begin{array}{c}\hfill {y}_{1}\left(L\right)\hfill \\ \hfill {y}_{2}\left(L\right)\hfill \\ \hfill \cdots \hfill \\ \hfill {y}_{N}\left(L\right)\hfill \end{array}\right]=M\left[\begin{array}{c}\hfill {y}_{1}\left(0\right)\hfill \\ \hfill {y}_{2}\left(0\right)\hfill \\ \hfill \cdots \hfill \\ \hfill {y}_{N}\left(0\right)\hfill \end{array}\right]\text{,}$$
(3)
$$M={\displaystyle \prod _{k=1}^{Q}\left[\begin{array}{cccc}\hfill 1+{a}_{11}\Delta z\hfill & \hfill {a}_{12}\Delta z\hfill & \cdots \hfill & \hfill {a}_{1N}\Delta z\hfill \\ \hfill {a}_{21}\Delta z\hfill & \hfill 1+{a}_{22}\Delta z\hfill & \hfill \cdots \hfill & \hfill {a}_{2N}\Delta z\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill {a}_{N1}\Delta z\hfill & \hfill {a}_{N2}\Delta z\hfill & \hfill \cdots \hfill & \hfill 1+{a}_{NN}\Delta z\hfill \end{array}\right]}\text{.}$$
(4)
$$\begin{array}{l}{y}_{1}\left(0\right)=1\hfill \\ {y}_{k}\left(L\right)=0\hfill \end{array}\text{\hspace{1em}}\left(k=2,3\text{,\hspace{0.17em}\u2026\hspace{0.17em},\hspace{0.17em}}N\right)\text{.}$$
(5)
$$\left[\begin{array}{c}\hfill {y}_{1}\left(L\right)\hfill \\ \hfill {y}_{2}\left(0\right)\hfill \\ \hfill \cdots \hfill \\ \hfill {y}_{N}\left(0\right)\hfill \end{array}\right]=-{\left[\begin{array}{ccccc}-1& {M}_{12}& {M}_{13}& \cdots & {M}_{1N}\\ 0& {M}_{22}& {M}_{23}& \cdots & {M}_{2N}\\ \text{\hspace{0.17em}}& \text{\hspace{0.17em}}& \cdots & \text{\hspace{0.17em}}& \text{\hspace{0.17em}}\\ 0& {M}_{N2}& {M}_{N3}& \cdots & {M}_{NN}\end{array}\right]}^{-1}\times \left[\begin{array}{c}\hfill {M}_{11}\hfill \\ \hfill {M}_{21}\hfill \\ \hfill \cdots \hfill \\ \hfill {M}_{N1}\hfill \end{array}\right]\text{.}$$
(6)
$$\frac{\mathrm{d}A\left(z\right)}{\mathrm{d}z}=j\kappa \left(z\right){e}^{\left(\alpha -j2\delta \right)z}B\left(z\right)\text{,}$$
(7)
$$\frac{\mathrm{d}B\left(z\right)}{\mathrm{d}z}=\pm j\kappa \left(z\right){e}^{\left(-\alpha +j2\delta \right)z}A\left(z\right)\text{,}$$
(8)
$$M={\displaystyle \prod _{k=1}^{Q}\left[\begin{array}{cc}1& j\kappa \left({z}_{i}\right){e}^{\left(\alpha -j2\delta \right){z}_{k}}\Delta z\\ \pm j\kappa \left({z}_{i}\right){e}^{\left(-\alpha +j2\delta \right){z}_{k}}\Delta z& 1\end{array}\right]}\text{,}$$
(9)
$$\left[\begin{array}{c}t\\ r\end{array}\right]={\left[\begin{array}{cc}-1& {M}_{12}\\ 0& {M}_{22}\end{array}\right]}^{-1}\left[\begin{array}{c}{M}_{11}\\ {M}_{21}\end{array}\right]\text{,}$$
(10)
$$\text{exp}\left[-\text{6}{\left(\frac{z-L/2}{L/2}\right)}^{2}\right]$$