## Abstract

The concept and the generation of polarized pseudonondiffracting
beams (PNDB’s) from polarization-selective diffractive phase
elements (DPE’s) are presented for what we believe is the first
time in a monochromatic illuminating system. The polarized PNDB’s
behave as segmented almost constant axial-intensity distributions with
individual different polarization states in different segments. The
pure polarization state in each segment can be arbitrarily
preset. The design of polarization-selective DPE’s is achieved
with the use of the conjugate-gradient method. The simulation
results show that the DPE’s we designed can successfully implement the
desired polarization modulation. Furthermore the PNDB’s
characteristics of axial-intensity uniformity and beamlike shape are
well achieved with the DPE’s we designed.

© 1999 Optical Society of America

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### Equations (5)

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(1)
$${U}_{2}\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)=\frac{2\mathrm{\pi}}{i\mathrm{\lambda}{z}_{\mathrm{\beta}}}exp\left(i2\mathrm{\pi}{z}_{\mathrm{\beta}}/\mathrm{\lambda}\right)\times {\int}_{0}^{{R}_{1m}}{\mathrm{\rho}}_{1}\left({r}_{1},\mathrm{\alpha}\right)exp\left[i\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\left({n}_{\mathrm{\alpha}}-1\right)h\left({r}_{1}\right)\right]\times exp\left(i\mathrm{\pi}r_{1}{}^{2}/\mathrm{\lambda}{z}_{\mathrm{\beta}}\right){r}_{1}\mathrm{d}{r}_{1}.$$
(2)
$$E=\sum _{\mathrm{\alpha}=1}^{2}\sum _{\mathrm{\beta}=1}^{{N}_{z}}W\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right){\left[{\tilde{\mathrm{\rho}}}_{2}\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)-|{U}_{2}\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)|\right]}^{2},$$
(3)
$$\frac{\partial E\left[y\left(\mathrm{\xi}\right)\right]}{\partial y\left(\mathrm{\xi}\right)}=-2I\mathrm{m}\left[\sum _{\mathrm{\alpha}=1}^{2}\sum _{\mathrm{\beta}=1}^{{N}_{z}^{e\left(o\right)}}W\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)\times \left(\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\left({n}_{\mathrm{\alpha}}-1\right){h}_{m}cos\left[y\left(\mathrm{\xi}\right)\right]{\mathrm{\rho}}_{1}\left(\mathrm{\xi},\mathrm{\alpha}\right)\times exp\left\{i\frac{2\mathrm{\pi}}{\mathrm{\lambda}}\left({n}_{\mathrm{\alpha}}-1\right){h}_{m}sin\left[y\left(\mathrm{\xi}\right)\right]\right\}\times \left[1-\frac{{\tilde{\mathrm{\rho}}}_{2}\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)}{|{U}_{2}\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)|}\right]\times G\left({z}_{\mathrm{\beta}},\mathrm{\xi}\right){U}_{2}*\left(\mathrm{\alpha},{z}_{\mathrm{\beta}}\right)\right)\right],$$
(4)
$$G\left({z}_{\mathrm{\beta}},\mathrm{\xi}\right)=\frac{2\mathrm{\pi}}{i\mathrm{\lambda}{z}_{\mathrm{\beta}}}exp\left(i2\mathrm{\pi}{z}_{\mathrm{\beta}}/\mathrm{\lambda}\right)exp\left(i\mathrm{\pi}{\mathrm{\xi}}^{2}/\mathrm{\lambda}{z}_{\mathrm{\beta}}\right)\mathrm{\xi}.$$
(5)
$${\mathrm{PCT}}_{e\left(o\right)}=\frac{{\displaystyle \sum _{\mathrm{\beta}=1}^{N\begin{array}{l}e\left(o\right)\\ z\end{array}}}|{U}_{2}\left[o\left(e\right),{z}_{\mathrm{\beta}}\right]{|}^{2}}{{\displaystyle \sum _{\mathrm{\beta}=1}^{N\begin{array}{l}e\left(o\right)\\ z\end{array}}}|{U}_{2}\left[e\left(o\right),{z}_{\mathrm{\beta}}\right]{|}^{2}},$$