## Abstract

The three-dimensional distribution of light intensity that is modulated by a pure phase-shifting apodizer is studied. Results show that the Strehl ratio can be altered by the proposed apodizer and by the waist width of incident Gaussian beams. By changing geometrical parameters of the proposed apodizer, we can increase the focal depth to several times that of the original system. The proposed apodizer can also be used to realize focal splitting and local minimum of intensity, which may be advantageous for constructing an optical trap. Furthermore, the local minimum of intensity number is tunable by changing the parameters of the apodizer.

© 2005 Optical Society of America

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### Equations (2)

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(1)
$$G(\mathrm{\rho},\hspace{0.17em}u)=2\sum _{j=1}^{3}\text{exp}(i{\mathrm{\varphi}}_{j}){\int}_{{r}_{j-1}}^{{r}_{j}}r{J}_{0}(\mathrm{\rho}r)\times \text{exp}\left[-\left(\frac{1}{{w}^{2}}+\frac{iu}{2}\right){r}^{2}\right]\text{d}r,$$
(2)
$$\begin{array}{l}S={[{G}_{a\ne b}(0,\hspace{0.17em}0)/{G}_{a=b=0}(0,\hspace{0.17em}0)]}^{2}\\ =\{4\hspace{0.17em}\text{exp}\left(-\frac{2}{{w}^{2}}{b}^{2}\right)+1+4\hspace{0.17em}\text{exp}\left(-\frac{2}{{w}^{2}}{a}^{2}\right)\\ +\hspace{0.17em}\text{exp}\left(-\frac{2}{{w}^{2}}\right)-4\hspace{0.17em}\text{exp}\left(-\frac{1}{{w}^{2}}{b}^{2}\right)+4\hspace{0.17em}\text{exp}\left(-\frac{1}{{w}^{2}}{a}^{2}\right)\\ -\hspace{0.17em}2\hspace{0.17em}\text{exp}\left(-\frac{1}{{w}^{2}}\right)-8\hspace{0.17em}\text{exp}\left(-\frac{{a}^{2}+{b}^{2}}{{w}^{2}}\right)\\ +\hspace{0.17em}4\hspace{0.17em}\text{exp}\left(-\frac{{b}^{2}+1}{{w}^{2}}\right)-4\hspace{0.17em}\text{exp}\left(-\frac{{a}^{2}+1}{{w}^{2}}\right){\}}^{2}\\ \times \hspace{0.17em}{\left[1-\text{exp}\left(-\frac{1}{{w}^{2}}\right)\right]}^{-4}.\end{array}$$