## Abstract

In this paper we present a model to investigate the thermal limitations of volume Bragg gratings (VBGs) used in lasers for spectral control. Also presented are the limiting optical powers, to which intracavity VBGs of different length could be subjected, before the laser operation rapidly deteriorates. The results revealed that the power limit of a VBG-locked laser is highly dependent on the length of the employed VBG. Furthermore, the power limit expressed in incident power related linearly to the radius of the laser beam irradiating the VBG.

© 2013 Optical Society of America

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### Equations (15)

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(1)
$$\left[\begin{array}{c}{E}_{12}^{+}\\ {E}_{12}^{-}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1+\frac{{n}_{1}}{{n}_{2}}& 1-\frac{{n}_{1}}{{n}_{2}}\\ 1-\frac{{n}_{1}}{{n}_{2}}& 1+\frac{{n}_{1}}{{n}_{2}}\end{array}\right]\left[\begin{array}{c}{E}_{11}^{+}\\ {E}_{11}^{-}\end{array}\right]={B}_{2,1}\left[\begin{array}{c}{E}_{11}^{+}\\ {E}_{11}^{-}\end{array}\right].$$
(2)
$$\left[\begin{array}{c}{E}_{11}^{+}\\ {E}_{11}^{-}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1+\frac{{n}_{2}}{{n}_{1}}& 1-\frac{{n}_{2}}{{n}_{1}}\\ 1-\frac{{n}_{2}}{{n}_{1}}& 1+\frac{{n}_{2}}{{n}_{1}}\end{array}\right]\left[\begin{array}{c}{E}_{12}^{+}\\ {E}_{12}^{-}\end{array}\right]={B}_{1,2}\left[\begin{array}{c}{E}_{12}^{+}\\ {E}_{12}^{-}\end{array}\right].$$
(3)
$$\left[\begin{array}{c}{E}_{22}^{+}\\ {E}_{22}^{-}\end{array}\right]=\left[\begin{array}{cc}\mathrm{exp}\{-i{k}_{2}d\}& 0\\ 0& \mathrm{exp}\{i{k}_{2}d\}\end{array}\right]\left[\begin{array}{c}{E}_{12}^{+}\\ {E}_{12}^{-}\end{array}\right]$$
(4)
$$\left[\begin{array}{c}{E}_{12}^{+}\\ {E}_{12}^{-}\end{array}\right]=\left[\begin{array}{cc}\mathrm{exp}\{i{k}_{2}d\}& 0\\ 0& \mathrm{exp}\{-i{k}_{2}d\}\end{array}\right]\left[\begin{array}{c}{E}_{22}^{+}\\ {E}_{22}^{-}\end{array}\right]={P}_{2}\left[\begin{array}{c}{E}_{22}^{+}\\ {E}_{22}^{-}\end{array}\right].$$
(5)
$$\left[\begin{array}{c}{E}_{11}^{+}\\ {E}_{11}^{-}\end{array}\right]={B}_{1,2}{P}_{2}{B}_{2,3}\left[\begin{array}{c}{E}_{23}^{+}\\ {E}_{23}^{-}\end{array}\right]=M\left[\begin{array}{c}{E}_{23}^{+}\\ {E}_{23}^{-}\end{array}\right].$$
(6)
$$M={B}_{0,1}{P}_{1}{B}_{1,2}{P}_{2}{B}_{2,3}{P}_{3}\dots {B}_{N-1,N}{P}_{N}{B}_{N,N+1}=\left[\begin{array}{cc}{m}_{11}& {m}_{12}\\ {m}_{21}& {m}_{22}\end{array}\right],$$
(7)
$$\left[\begin{array}{c}{E}_{0}^{+}\\ {E}_{0}^{-}\end{array}\right]=\left[\begin{array}{cc}{m}_{11}& {m}_{12}\\ {m}_{21}& {m}_{22}\end{array}\right]\left[\begin{array}{c}{E}_{N+1}^{+}\\ 0\end{array}\right]=\left[\begin{array}{cc}{m}_{11}& {E}_{N+1}^{+}\\ {m}_{21}& {E}_{N+1}^{+}\end{array}\right].$$
(8)
$$r=\frac{{E}_{0}^{-}}{{E}_{0}^{+}}=\frac{{m}_{21}}{{m}_{11}}.$$
(9)
$$\mathrm{DE}={|r|}^{2}.$$
(10)
$${\lambda}_{B}=2{n}_{0}\mathrm{\Lambda}.$$
(11)
$${n}_{1}=\frac{{\lambda}_{B}}{\pi L}{\mathrm{tanh}}^{-1}\left(\sqrt{{\mathrm{DE}}_{\mathrm{max}}}\right).$$
(12)
$$I(z)=\frac{(1+\mathrm{DE})}{n(0){|E(0)|}^{2}}n(z){|E(z)|}^{2},$$
(13)
$$-\nabla \xb7(k\xb7\nabla T)=Q.$$
(14)
$${C}_{p}(T)=9.36\times {10}^{-9}{T}^{3}\phantom{\rule{0ex}{0ex}}-1.54\times {10}^{-5}{T}^{2}\phantom{\rule{0ex}{0ex}}+8.22\times {10}^{-3}T\phantom{\rule{0ex}{0ex}}-0.637,$$
(15)
$$Q(x,y,z)={\alpha}_{\text{abs}}{I}_{0,\text{inc}}\text{\hspace{0.17em}}\mathrm{exp}\{-2\frac{{x}^{2}+{y}^{2}}{{\omega}^{2}}\}I(z).$$