## Abstract

The developed iteration algorithm for simulation of lasers with an open resonator was employed in the study of transverse mode formation. The simulations of an axially symmetrical resonator rely on an analytical description of radiation diffraction from a narrow ring. Reflection of an incident wave with a specified amplitude-phase distribution from the mirror is calculated by the Green-function method. The model also includes an active medium homogeneous along the resonator axis that is represented by the formula for saturating gain. The calculations were performed for two types of lasers: with on-axis and off-axis gain maximum. In the first type of laser one can obtain either a principal mode or “multimode” generation. The latter means quasi-stationary generation with regular or chaotic oscillations. In the second type of laser high order single-mode generation is possible. Experimental results obtained on a fast axial flow 4 kW ${\mathrm{CO}}_{2}$ laser are also presented. They are in good agreement with the calculations.

© 2012 Optical Society of America

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### Equations (6)

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(1)
$$\mathrm{DEL}(L,\theta ,{r}_{0})\approx 2\pi {r}_{0}ik\frac{{e}^{ikL}}{L}{e}^{ik\frac{{r}_{0}^{2}}{2L}}{J}_{0}(k{r}_{0}\theta ),$$
(2)
$$E(L,\theta )\approx {\int}_{0}^{{r}_{m}}{E}_{0}({r}_{0})\mathrm{DEL}(L,\theta ,{r}_{0})\mathrm{exp}(-ik\frac{{r}_{0}^{2}}{R})\mathrm{d}{r}_{0}\phantom{\rule{0ex}{0ex}}=2\pi \frac{{e}^{ikL}}{L}{\int}_{0}^{{r}_{m}}{r}_{0}{E}_{0}({r}_{0})\phantom{\rule{0ex}{0ex}}\mathrm{exp}\left[ik(\frac{{r}_{0}^{2}}{2L}-\frac{{r}_{0}^{2}}{R})\right]{J}_{0}(k{r}_{0}\theta )\mathrm{d}{r}_{0}\mathrm{.}$$
(3)
$$\alpha (r)=\frac{{\alpha}_{0}(r)}{1+I(r)/{I}_{\text{sat}}(r)},$$
(4)
$${E}_{2}^{n}(L,\theta )=2\pi \frac{{e}^{ikL}}{L}{\int}_{0}^{{r}_{m}}{r}_{0}\mathrm{exp}\left(ik\frac{{r}_{0}^{2}}{L}(1-\frac{L}{{R}_{1}})\right)\xb7\mathrm{exp}\left[\frac{{\alpha}_{0}(r)\xb7L}{1+{[{E}_{1}^{n-1}({r}_{0})/{E}_{\text{sat}}(r)]}^{2}}\right]\phantom{\rule{0ex}{0ex}}\times ({E}_{1}^{n-1}({r}_{0})+{E}_{\mathrm{sn}}({r}_{0}))\xb7{k}_{m}\xb7{J}_{0}(k{r}_{0}\theta )\xb7\mathrm{d}{r}_{0}\mathrm{.}$$
(5)
$${E}_{1}^{n-1}\Rightarrow {E}_{2}^{n}[{E}_{1}^{n-1},{R}_{1},{k}_{m}]\Rightarrow {E}_{1}^{n}[{E}_{2}^{n},{R}_{2}]\Rightarrow {E}_{1}^{n}\mathrm{.}$$
(6)
$${g}_{1}\xb7{g}_{2}=\frac{1+\mathrm{cos}\text{\hspace{0.17em}}\theta}{2};\phantom{\rule[-0.0ex]{2em}{0.0ex}}\theta =2\pi \frac{K}{N};\phantom{\rule[-0.0ex]{2em}{0.0ex}}0\le K\le N/2,$$