## Abstract

We demonstrate theoretically and experimentally that two orthogonal polarizations of light see an unequal index contrast in a UV-induced fiber Bragg grating. We have found that in a Bragg grating the phase-velocity mismatch is a function of detuning, and hence we can introduce an effective birefringence. The description of the polarization dynamics that we developed was confirmed by a set of experiments with a fiber grating. We show theoretically that by adjusting the value of the unequal index contrast relative to the average birefringence it should be possible to eliminate the phase-velocity mismatch between polarizations at a given frequency.

© 2002 Optical Society of America

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### Equations (21)

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(1)
$${n}_{x}(z)=[{n}_{0}+\delta {n}_{x}(z)]+\delta {n}_{x}(z)cos(2{k}_{0}z),$$
(2)
$${n}_{y}(z)=[{n}_{0}+\delta {n}_{y}(z)]+\delta {n}_{y}(z)cos(2{k}_{0}z),$$
(3)
$${n}_{x}(z)={\overline{n}}_{x}+\delta {n}_{x}(z)cos(2{k}_{0}z),$$
(4)
$${n}_{y}(z)={\overline{n}}_{y}+\delta {n}_{y}(z)cos(2{k}_{0}z),$$
(5)
$${n}_{b}\equiv {\overline{n}}_{y}-{\overline{n}}_{x},$$
(6)
$$\delta {n}_{b}\equiv {\overline{\delta n}}_{y}-{\overline{\delta n}}_{x}.$$
(7)
$$M\equiv \frac{1}{2}\frac{\delta {n}_{b}}{{n}_{b}}.$$
(8)
$$\delta {n}_{i}(z)=\left\{\begin{array}{ll}{\overline{\delta n}}_{i}exp[-(ln2)(z-3{z}_{h\omega}{)}^{2}/z_{h\omega}{}^{2}]& \hspace{1em}0z3{z}_{h\omega}\\ {\overline{\delta n}}_{i}& \hspace{1em}3{z}_{h\omega}z(L-3{z}_{h\omega}),\\ {\overline{\delta n}}_{i}exp\{-(ln2)[z-(L-3{z}_{h\omega}){]}^{2}/z_{h\omega}{}^{2}\}& \hspace{1em}(L-3{z}_{h\omega})zL\end{array}\right.$$
(9)
$${n}_{i}(z)={\overline{n}}_{i}+\delta {N}_{i}(z)+\delta {n}_{i}(z)cos(2{k}_{0}z),$$
(10)
$$\delta {N}_{i}(z)=\left\{\begin{array}{ll}-{\overline{\delta N}}_{i}\frac{\pi (2.6{z}_{h\omega}-z)}{0.4{z}_{h\omega}}& \hspace{1em}0<z<2.6{z}_{h\omega},\\ {\overline{\delta N}}_{i}sin\left[\frac{\pi (z-2.6{z}_{h\omega})}{0.4{z}_{h\omega}}\right]& \hspace{1em}2.6{z}_{h\omega}z3{z}_{h\omega},\\ 0& \hspace{1em}3{z}_{h\omega}z(L-3{z}_{h\omega}),\\ {\overline{\delta N}}_{i}sin\left\{\frac{\pi [z-(L-3{z}_{h\omega})]}{0.4{z}_{h\omega}}\right\}& \hspace{1em}(L-3{z}_{h\omega})z(L-2.6{z}_{h\omega}),\\ -{\overline{\delta N}}_{i}\frac{\pi [z-(L-2.6{z}_{h\omega})]}{0.4{z}_{h\omega}}& \hspace{1em}(L-2.6{z}_{h\omega})zL,\end{array}\right.$$
(11)
$$0=\pm i\frac{\partial {A}_{i\pm}(z;\omega )}{\partial z}+\left[{\sigma}_{i}(z)+\frac{{\overline{n}}_{i}}{c}(\omega -{\omega}_{0i})\right]{A}_{i\pm}(z;\omega )+{\kappa}_{i}(z){A}_{i\mp}(z;\omega ),$$
(12)
$${\kappa}_{i}(z)=\frac{1}{2}\frac{\delta {n}_{i}(z)}{{\overline{n}}_{i}}{k}_{0},\hspace{1em}\hspace{1em}{\sigma}_{i}(z)=\frac{\delta {N}_{i}(z)}{{\overline{n}}_{i}}{k}_{0},$$
(13)
$${n}_{i}(z)={\overline{n}}_{i}+({\overline{\delta n}}_{i})cos(2{k}_{0}z),$$
(14)
$${f}_{i}=\frac{{\overline{n}}_{x}}{c{\overline{\kappa}}_{x}}({\omega}_{i}-{\omega}_{0x}),\hspace{1em}\hspace{1em}{\delta}_{i}=\frac{{k}_{i}-{k}_{0}}{{\overline{\kappa}}_{x}}.$$
(15)
$${f}_{x}({\delta}_{x})=\pm \sqrt{\delta _{x}{}^{2}+1},$$
(16)
$$\nu =2\frac{{n}_{b}}{{\overline{\delta n}}_{x}},$$
(17)
$${f}_{y}({\delta}_{y})=\frac{{\overline{n}}_{x}}{{\overline{n}}_{y}}\left(-\nu \pm {\left\{\delta _{y}{}^{2}+{\left[\frac{{\overline{n}}_{x}}{{\overline{n}}_{y}}(1+M\nu )\right]}^{2}\right\}}^{1/2}\right)\sim -\nu \pm [\delta _{y}{}^{2}+(1+M\nu {)}^{2}{]}^{1/2},$$
(18)
$${F}_{b}(f)=\frac{{\delta}_{y}(f)-{\delta}_{x}(f)}{\nu}=\pm \frac{1}{\nu}\{[(f+\nu {)}^{2}-(1+M\nu {)}^{2}{]}^{1/2}-\sqrt{{f}^{2}-1}\},$$
(19)
$${f}_{\mathrm{no}-\mathrm{bi}}=\frac{\nu}{2}({M}^{2}-1)+M.$$
(20)
$${{\omega}_{x}}^{\prime}=\frac{c}{{\overline{n}}_{x}}\frac{\sqrt{{f}^{2}-1}}{f},$$
(21)
$${{\omega}_{y}}^{\prime}=\frac{c}{{\overline{n}}_{y}}\frac{[(f+\nu {)}^{2}+(1+M\nu {)}^{2}{]}^{1/2}}{f+\nu}.$$