## Abstract

By introducing the rate equations and light intensity propagating equations into the fast Fourier transform-based calculation, the optical gain served as the connection between the light field and light intensity, its influence over mode pattern was studied. Thermal lens effect was also investigated by means of finite element analysis. The analysis was applied to a slab laser with a hybrid cavity. A similar experimental study was also carried out in the laboratory. ${\mathrm{TEM}}_{00}$ mode with sidelobes along the unstable direction was observed both in the calculation and experiment. As predicted in the analysis, the homogeneity of the pump light improved the beam quality. Numerical and experimental results of pump threshold and slope efficiency were also presented.

© 2013 Optical Society of America

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### Equations (11)

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(1)
$${\xi}_{A}({\nu}_{x},{\nu}_{y})=\iint {U}_{A}(x,y,{z}_{A})\mathrm{exp}[j(2\pi {\nu}_{x}x+2\pi {\nu}_{y}y)]\mathrm{d}{\nu}_{x}\mathrm{d}{\nu}_{y},$$
(2)
$${\xi}_{B}({\nu}_{x},{\nu}_{y})={\xi}_{A}({\nu}_{x},{\nu}_{y})\mathrm{exp}[-jkL\sqrt{1-{\lambda}^{2}{\nu}_{x}^{2}-{\lambda}^{2}{\nu}_{y}^{2}}],$$
(3)
$${U}_{B}(x,y,{z}_{B})=\iint {\xi}_{B}({\nu}_{x},{\nu}_{y})\mathrm{exp}[-j(2\pi {\nu}_{x}x+2\pi {\nu}_{y}y)]\mathrm{d}{\nu}_{x}\mathrm{d}{\nu}_{y}.$$
(4)
$${I}_{B}(x,y,{z}_{B})=\frac{{|{U}_{B}(x,y,{z}_{B})|}^{2}}{\iint {|{U}_{B}(x,y,{z}_{B})|}^{2}\mathrm{d}x\mathrm{d}y}\iint {I}_{A}(x,y,{z}_{A})\mathrm{d}x\mathrm{d}y.$$
(5)
$$\frac{d{n}_{2}}{dt}={W}_{p}{n}_{0}-\frac{{n}_{2}\sigma ({I}^{+}+{I}^{-})}{h{\nu}_{\text{laser}}}-\frac{{n}_{2}}{\tau},$$
(6)
$${n}_{2}+{n}_{0}={n}_{\text{tot}},$$
(7)
$$\frac{d{I}^{\pm}}{dz}={n}_{2}\sigma {I}^{\pm}-{\epsilon}_{\text{loss}}{I}^{\pm},$$
(8)
$${g}^{\pm}({x}_{i},{y}_{j},{z}_{k})=\frac{\mathrm{ln}[{I}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})/{I}^{\pm}({x}_{i},{y}_{j},{z}_{k})]}{\mathrm{\Delta}z}.$$
(9)
$${U}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})={U}^{\pm}({x}_{i},{y}_{j},{z}_{k})\sqrt{\mathrm{exp}[\mathrm{\Delta}z\xb7g({x}_{i},{y}_{j},{z}_{k})]}.$$
(10)
$${I}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})=\frac{{|{U}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})|}^{2}}{\iint {|{U}^{\pm}(x,y,{z}_{k\pm 1})|}^{2}\mathrm{d}x\mathrm{d}y}\iint {I}^{\pm}(x,y,{z}_{k})\mathrm{d}x\mathrm{d}y,$$
(11)
$${U}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})={U}^{\pm}({x}_{i},{y}_{j},{z}_{k})\sqrt{{I}^{\pm}({x}_{i},{y}_{j},{z}_{k\pm 1})/{I}^{\pm}({x}_{i},{y}_{j},{z}_{k})}.$$