## Abstract

Generation of vectorial optical fields with complex spatial distribution in the cross section is of great interest in areas where exotic optical fields are desired, including particle manipulation, optical nanofabrication, beam shaping and optical imaging. In this work, a vectorial optical field generator capable of creating arbitrarily complex beam cross section is designed, built and tested. Based on two reflective phase-only liquid crystal spatial light modulators, this generator is capable of controlling all the parameters of the spatial distributions of an optical field, including the phase, amplitude and polarization (ellipticity and orientation) on a pixel-by-pixel basis. Various optical fields containing phase, amplitude and/or polarization modulations are successfully generated and tested using Stokes parameter measurement to demonstrate the capability and versatility of this optical field generator.

© 2013 OSA

Full Article |

PDF Article
### Equations (11)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${E}_{d}\left(x,y\right)={A}_{d}\left(x,y\right){e}^{j{\varphi}_{d}\left(x,y\right)}\left(\begin{array}{c}{E}_{xd}\left(x,y\right)\\ {E}_{yd}\left(x,y\right){e}^{j{\delta}_{d}\left(x,y\right)}\end{array}\right),$$
(2)
$$\begin{array}{c}{M}_{PR}={M}_{QWP\_135\xb0}{M}_{refl}{M}_{SLM}{M}_{QWP\_45\xb0}\\ =\frac{1}{4}\left(\begin{array}{cc}1-j& -1-j\\ -1-j& 1-j\end{array}\right)\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}{e}^{i\delta \left(x,y\right)}& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1-j& 1+j\\ 1+j& 1-j\end{array}\right)\\ ={e}^{j\frac{\delta \left(x,y\right)}{2}}\left(\begin{array}{cc}-\mathrm{sin}\left(\frac{\delta \left(x,y\right)}{2}\right)& -\mathrm{cos}\left(\frac{\delta \left(x,y\right)}{2}\right)\\ \mathrm{cos}\left(\frac{\delta \left(x,y\right)}{2}\right)& -\mathrm{sin}\left(\frac{\delta \left(x,y\right)}{2}\right)\end{array}\right)\\ ={e}^{j\left(\frac{\delta \left(x,y\right)}{2}\right)}R\left(\frac{3\pi}{2}-\frac{\delta \left(x,y\right)}{2}\right),\end{array}$$
(3)
$${J}_{\text{1}}\left(x,y\right)={e}^{j{\varphi}_{1}\left(x,y\right)}{E}_{\text{0}}\left(x,y\right)\left(\begin{array}{c}1\\ 0\end{array}\right),$$
(4)
$${J}_{\text{2}}\left(x,y\right)={e}^{j\left({\varphi}_{1}\left(x,y\right)\text{+}\frac{{\varphi}_{2}\left(x,y\right)}{2}\text{+}\pi \right)}\mathrm{sin}\left(\frac{{\varphi}_{2}\left(x,y\right)}{2}\right){E}_{\text{0}}\left(x,y\right)\left(\begin{array}{c}1\\ 0\end{array}\right),$$
(5)
$${\varphi}_{2}\left(x,y\right)=2{\mathrm{sin}}^{-1}\left({A}_{d}\left(x,y\right)\right).$$
(6)
$$\begin{array}{c}{J}_{3}\left(x,y\right)={E}_{0}\left(x,y\right){e}^{j\left({\varphi}_{1}\left(x,y\right)+\frac{{\varphi}_{2}\left(x,y\right)}{2}+\frac{{\varphi}_{3}\left(x,y\right)}{2}+\pi \right)}\\ \cdot \mathrm{sin}\left(\frac{{\varphi}_{2}\left(x,y\right)}{2}\right)\left(\begin{array}{c}\mathrm{cos}\left(\frac{{\varphi}_{3}\left(x,y\right)}{2}+\frac{\pi}{2}\right)\\ \mathrm{sin}\left(\frac{{\varphi}_{3}\left(x,y\right)}{2}+\frac{\pi}{2}\right)\end{array}\right).\end{array}$$
(7)
$${\varphi}_{3}\left(x,y\right)=2{\mathrm{tan}}^{-1}\left(\frac{\left|{E}_{yd}\left(x,y\right)\right|}{\left|{E}_{xd}\left(x,y\right)\right|}\right)-\pi .$$
(8)
$$\begin{array}{c}{J}_{4}\left(x,y\right)={E}_{0}\left(x,y\right){e}^{j\left({\varphi}_{1}\left(x,y\right)+\frac{{\varphi}_{2}\left(x,y\right)}{2}+\frac{{\varphi}_{3}\left(x,y\right)}{2}+\pi \right)}\\ \cdot \mathrm{sin}\left(\frac{{\varphi}_{2}\left(x,y\right)}{2}\right)\left(\begin{array}{c}\mathrm{cos}\left(\frac{{\varphi}_{3}\left(x,y\right)}{2}+\frac{\pi}{2}\right){e}^{j{\varphi}_{4}\left(x,y\right)}\\ \mathrm{sin}\left(\frac{{\varphi}_{3}\left(x,y\right)}{2}+\frac{\pi}{2}\right)\end{array}\right),\end{array}$$
(9)
$${\varphi}_{\text{4}}\left(x,y\right)=-{\delta}_{d}\left(x,y\right),$$
(10)
$${\varphi}_{output}\left(x,y\right)={\varphi}_{\text{1}}\left(x,y\right)+\frac{{\varphi}_{\text{2}}\left(x,y\right)}{2}+\frac{{\varphi}_{\text{3}}\left(x,y\right)}{2}+\pi .$$
(11)
$${\varphi}_{\text{1}}\left(x,y\right)={\varphi}_{d}\left(x,y\right)+{\varphi}_{c}\left(x,y\right),$$