## Abstract

Based on the off-axis theory, a model for describing the far field with the bipeak structure of a high-power laser diode is proposed. The computed results agree well with the measured far-field data of practical devices. A minimum overall error criterion for fitting the theoretical model with the measured data is also given. The results show that the overall error of this model is less than 5% for popular laser diodes. This model has a simple mathematical structure and can be easily used to design the beam-shaping system and to analyze the propagation properties when the laser beam passes through an optic system.

© 2004 Optical Society of America

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### Equations (11)

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(1)
$$E\left(x,y,z\right)=-\frac{\mathit{iz}}{\mathrm{\lambda}r}\frac{exp\left(\mathit{ikr}\right)}{r}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}{u}_{0}\left(x\prime ,y\prime \right)\times exp\left[-\frac{\mathit{ik}}{r}\left(\mathit{xx}\prime +\mathit{yy}\prime \right)\right]\mathrm{d}x\prime \mathrm{d}y\prime ,$$
(2)
$${u}_{\pm}\left(x\prime ,y\prime \right)={u}_{0\pm}exp\left(-p|x\prime |\right)exp\left(-{q}_{\pm}y{\prime}^{2}+{\mathit{ik}}_{\pm}y\prime \right),$$
(3)
$${E}_{\pm}\left(x,y,z\right)={A}_{\pm}\frac{z}{r}\frac{exp\left(\mathit{ikr}\right)}{r}\frac{{\mathrm{\Gamma}}^{2}}{{\mathrm{\Gamma}}^{2}+{x}^{2}}\times exp\left[-\frac{{\left(rsin{\mathrm{\theta}}_{\pm}-y\right)}^{2}}{\mathrm{\Omega}_{\pm}{}^{2}}\right],$$
(4)
$${A}_{\pm}=-{u}_{0\pm}\frac{2i}{\mathrm{\lambda}p}\sqrt{\frac{\mathrm{\pi}}{{q}_{\pm}}},$$
(5)
$${\mathrm{\Gamma}}^{2}=\frac{{p}^{2}}{{k}^{2}}{r}^{2},$$
(6)
$$\mathrm{\Omega}_{\pm}{}^{2}=\frac{4{q}_{\pm}}{{k}^{2}}{r}^{2},$$
(7)
$$sin{\mathrm{\theta}}_{\pm}={k}_{\pm}/k.$$
(8)
$$I\left(x,y,z\right)=|{A}_{-}{|}^{2}\frac{{z}^{2}}{{r}^{4}}{\left(\frac{{\mathrm{\Gamma}}^{2}}{{\mathrm{\Gamma}}^{2}+{x}^{2}}\right)}^{2}\left\{{\left|\frac{{A}_{+}}{{A}_{-}}\right|}^{2}\times exp\left[-\frac{2{\left(rsin{\mathrm{\theta}}_{+}-y\right)}^{2}}{\mathrm{\Omega}_{+}{}^{2}}\right]+exp\left[-\frac{2{\left(rsin{\mathrm{\theta}}_{-}-y\right)}^{2}}{\mathrm{\Omega}_{-}{}^{2}}\right]\right\}.$$
(9)
$$I\left(x,0,z\right)=B\frac{{z}^{2}}{{r}^{4}}{\left(\frac{{\mathrm{\Gamma}}^{2}}{{\mathrm{\Gamma}}^{2}+{x}^{2}}\right)}^{2},$$
(10)
$$I\left(0,y,z\right)=|{A}_{-}{|}^{2}\frac{{z}^{2}}{{r}^{4}}\left\{{\left|\frac{{A}_{+}}{{A}_{-}}\right|}^{2}exp\left[-\frac{2{\left(rsin{\mathrm{\theta}}_{+}-y\right)}^{2}}{\mathrm{\Omega}_{+}{}^{2}}\right]+exp\left[-\frac{2{\left(rsin{\mathrm{\theta}}_{-}-y\right)}^{2}}{\mathrm{\Omega}_{-}{}^{2}}\right]\right\}.$$
(11)
$$\mathrm{\epsilon}=\frac{|{S}_{m}-{S}_{t}|}{{S}_{m}}.$$