## Abstract

There are three recognized low-loss configurations for waveguide laser resonators in which the waveguide is either closed at each end by a plane mirror (dual case I design) or one of the plane mirrors is replaced by a curved mirror at some distance from the guide exit. Some time ago, a variant of the latter design was proposed by exploiting the self-imaging properties of multimode waveguides. The resonator was predicted to produce a ${\mathrm{TEM}}_{00}$-like output with very low round-trip loss and excellent mode discrimination even though the curved mirror was placed much nearer to the guide exit (making the resonator more compact) than was conventional for achieving those results. In the present work, we show that the desirable features of the above design can be achieved even in a dual case I configuration by using end mirrors with suitable reflectivity profiles. Since there is no free space region between the waveguide and the mirrors, the resonator will have the additional advantages of being compact and portable. Furthermore, the absence of curved mirrors will also facilitate its realization in semiconductor integrated optics technology.

© 2009 Optical Society of America

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### Equations (16)

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(1)
$${E}_{m}(x,a)=\frac{1}{\sqrt{a}}\{\begin{array}{ll}\mathrm{cos}(\pi mx/2a)& ,\text{if \hspace{0.17em} \hspace{0.17em}}m\text{\hspace{0.17em} is odd}\\ \mathrm{sin}(\pi mx/2a)& ,\text{if \hspace{0.17em} \hspace{0.17em}}m\text{\hspace{0.17em} is even}\end{array}\mathrm{.}$$
(2)
$${\int}_{-a}^{a}{E}_{m}(x,a){E}_{n}(x,a)\mathrm{d}x={\delta}_{mn}\mathrm{.}$$
(3)
$${k}_{m}=\frac{2\pi}{\lambda}(1-\frac{1}{2}(\frac{m\lambda}{4a}{)}^{2})$$
(4)
$$F(x,0)=\sum _{m}{a}_{m}{E}_{m}(x,a)\mathrm{.}$$
(5)
$$F(x,z)=\sum _{m}{a}_{m}{E}_{m}(x,a)\mathrm{exp}(-i{\beta}_{m}z)\mathrm{.}$$
(6)
$${F}_{s}(x,L)={F}_{s}(x,0)\mathrm{exp}(-i\pi /4),\phantom{\rule[-0.0ex]{2em}{0.0ex}}{F}_{a}(x,2L)={F}_{a}(x,0),\phantom{\rule[-0.0ex]{2em}{0.0ex}}{F}_{g}(x,8L)={F}_{g}(x,0)\mathrm{.}$$
(7)
$${F}_{a}(x,L)=\frac{1}{2}[{F}_{a}(x-a,0)+{F}_{a}(x+a,0)]\mathrm{.}$$
(8)
$${F}_{s}(x,L/2)={F}_{s}(x-a/2,0)+{F}_{s}(x+a/2,0)\mathrm{.}$$
(9)
$${F}_{0}(x,0)=\sum _{m}{a}_{m}{E}_{m}(x,a)\mathrm{.}$$
(10)
$${F}_{1}(x,a)=\sum _{m,n,p}{a}_{m}\mathrm{exp}(-i{\beta}_{m}L/2){X}_{mn}^{(2)}\phantom{\rule{0ex}{0ex}}\mathrm{exp}(-i{\beta}_{n}L/2){X}_{np}^{(1)}{E}_{p}(x,a),$$
(11)
$$\sum _{m,n}{X}_{nq}^{(1)}\mathrm{exp}(-i{\beta}_{n}L/2){X}_{mn}^{(2)}\mathrm{exp}(-i{\beta}_{m}L/2){a}_{m}=\sigma {a}_{q}\mathrm{.}$$
(12)
$${X}_{mn}^{(1)}=\left[{r}_{2}\right({\int}_{-a}^{-b}\mathrm{d}x+{\int}_{b}^{a}\mathrm{d}x)+{r}_{1}{\int}_{-b}^{b}\mathrm{d}x]{E}_{m}(x,a){E}_{n}(x,a)={r}_{2}{\delta}_{mn}+({r}_{1}-{r}_{2}){I}_{mn}(b),$$
(13)
$${I}_{mn}(b)={\int}_{-b}^{b}{E}_{m}(x,a){E}_{n}(x,a)\mathrm{d}x,$$
(14)
$${X}_{mn}^{(2)}=\left[{r}_{1}\right({\int}_{-a}^{-c}\mathrm{d}x+{\int}_{c}^{a}\mathrm{d}x)+{r}_{2}{\int}_{-c}^{c}\mathrm{d}x]{E}_{m}(x,a){E}_{n}(x,a)={r}_{1}{\delta}_{mn}-({r}_{1}-{r}_{2}){I}_{mn}(c)\mathrm{.}$$
(15)
$${I}_{mn}(b)=\{\begin{array}{ll}{\gamma}_{m-n}(b/a)-{\gamma}_{m+n}(b/a)& ,\text{if \hspace{0.17em} \hspace{0.17em}}m\text{\hspace{0.17em} and \hspace{0.17em} \hspace{0.17em}}n\text{\hspace{0.17em} are both even}\\ {\gamma}_{m-n}(b/a)+{\gamma}_{m+n}(b/a)& ,\text{if \hspace{0.17em} \hspace{0.17em}}m\text{\hspace{0.17em} and \hspace{0.17em}}n\text{\hspace{0.17em} are both odd}\\ 0& ,\text{otherwise}\end{array}$$
(16)
$${\gamma}_{j}(x)=\{\begin{array}{ll}\frac{\mathrm{sin}(\pi jx/2)}{\pi j/2}& ,\text{for \hspace{0.17em} \hspace{0.17em}}j\ne 0\\ x& ,\text{otherwise}\end{array}\mathrm{.}$$