## Abstract

We calculated and measured the difference between focal positions of radially and azimuthally polarized beams after passing through a uniaxial crystal. Calculations were carried out on the basis of the ray optics and the vector diffraction theory. The results of the calculations were in good agreement with those of the experiment. In addition, we discussed the polarization selection in a hemispherical laser cavity that was used for the generation of a radially polarized beam by use of the birefringence of a *c*-cut $\mathrm{Nd}:\mathrm{Y}\mathrm{V}{\mathrm{O}}_{4}$ laser crystal [Opt. Lett. **31**, 2151 (2006)
]. The stability range of the laser cavity length for the generation of a radially polarized beam was also in good agreement with the differences mentioned above.

© 2008 Optical Society of America

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### Equations (5)

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(1)
$${E}_{r}(r,z)=A{\int}_{0}^{\alpha}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{1\u22152}\left(\theta \right)\mathrm{sin}\left(2\theta \right){l}_{0}\left(\theta \right){J}_{1}\left(kr\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)\mathrm{exp}\left(ikz\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)\mathrm{d}\theta ,$$
(2)
$${E}_{a}(r,z)=2A{\int}_{0}^{\alpha}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{1\u22152}\left(\theta \right)\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta {l}_{0}\left(\theta \right){J}_{1}\left(kr\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)\mathrm{exp}\left(ikz\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)\mathrm{d}\theta ,$$
(3)
$${l}_{0}\left(\theta \right)=\mathrm{exp}\left(ik\Phi \left(\theta \right)\right),$$
(4)
$${\Phi}_{o}\left(\theta \right)=({n}_{o}\overline{ab}+\overline{bc})-\overline{ad}=D\{\frac{{n}_{o}}{\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{o}}+\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta (\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta -\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\theta}_{o})-\frac{1}{\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta}\}=D({n}_{o}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{o}-\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta ),$$
(5)
$${\Phi}_{e}\left(\theta \right)=D({(\frac{{\mathrm{cos}}^{2}\phantom{\rule{0.2em}{0ex}}{\theta}_{e}}{{n}_{o}^{2}}+\frac{{\mathrm{sin}}^{2}\phantom{\rule{0.2em}{0ex}}{\theta}_{e}}{{n}_{e}^{2}})}^{-1\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{e}-\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\theta ).$$