## Abstract

The optimum phase defocus grating for wavefront curvature sensing is proposed. It features an equidistantly quantized, binary-phase-step defocus grating with a phase-step height of *π*. The diffractive efficiency of the phase defocus grating is theoretically computed. The optical transfer function is obtained. The optimum phase defocus grating is fabricated. The high diffractive efficiencies of the $\pm 1$ diffraction orders are verified experimentally, the average values of which are 38.08% and 40.36%, respectively.

© 2007 Optical Society of America

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### Equations (15)

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(1)
$$t\left(\xi \right)={e}^{i2\pi (\left[\xi \right]+1-\xi )}=\sum _{m=-\infty}^{+\infty}{a}_{m}{e}^{i2\pi m\xi},$$
(2)
$${a}_{m}={\int}_{0}^{1}t\left(\xi \right){e}^{-i2\pi m\xi}\mathrm{d}\xi ={e}^{i\pi (1-m)}\mathrm{sinc}(1+m),$$
(3)
$${\eta}_{m}={\mid {a}_{m}\mid}^{2}={\mathrm{sinc}}^{2}(1+m).$$
(4)
$$t\left(\xi \right)={e}^{i2\pi {Q}_{k}}.$$
(5)
$${a}_{m}=\sum _{k=1}^{L}({\xi}_{k}-{\xi}_{k-1})\mathrm{sinc}\left[m({\xi}_{k}-{\xi}_{k-1})\right]{e}^{i\pi [2{Q}_{k}+m({\xi}_{k}+{\xi}_{k-1})]}.$$
(6)
$${\eta}_{m}=\frac{1}{{L}^{2}}\phantom{\rule{0.2em}{0ex}}{\mathrm{sinc}}^{2}\left(\frac{m}{L}\right){\mid \sum _{k=1}^{L}{e}^{-\frac{i\pi}{L}\left[2k(1+m)\right]}\mid}^{2},$$
(7)
$$4k\u2215L=2n,\phantom{\rule{1em}{0ex}}n\u220a\mathrm{Integer},\phantom{\rule{1em}{0ex}}k=1,\dots ,L-1,L.$$
(8)
$$t\left(\xi \right)=\mathrm{exp}\left(i\pi S\right)rect\left(\frac{\xi}{w}\right)*\sum _{n=0}^{N-1}\delta [\xi -(n+\frac{w}{2})]+rect\left(\frac{\xi}{1-w}\right)*\sum _{n=0}^{N-1}\delta [\xi -(n+\frac{1+w}{2})],$$
(9)
$${a}_{0}={\int}_{0}^{w}{e}^{iS\pi}\mathrm{d}\xi +{\int}_{w}^{1}\mathrm{d}\xi ={e}^{iS\pi}w+1-w.$$
(10)
$${\eta}_{0}={\mid {e}^{iS\pi}w+1-w\mid}^{2}={\mid \mathrm{cos}\left(S\pi \right)w+1-w+i\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\left(S\pi \right)\mid}^{2}.$$
(11)
$${a}_{m}={\int}_{0}^{w}{e}^{iS\pi}{e}^{-i2\pi m\xi}\mathrm{d}\xi +{\int}_{w}^{1}{e}^{-i2\pi m\xi}\mathrm{d}\xi ={e}^{-im\pi (1+w)}\{(1-w)\mathrm{sinc}\left[m(1-w)\right]+{e}^{i\pi (m+S)}w\phantom{\rule{0.2em}{0ex}}\mathrm{sinc}\left(mw\right)\}.$$
(12)
$${\eta}_{m}={\mid {a}_{m}\mid}^{2}=\frac{4\phantom{\rule{0.2em}{0ex}}{\mathrm{sin}}^{2}[\left(\pi S\right)\u22152]{\mathrm{sin}}^{2}\left(\pi mw\right)}{{\pi}^{2}{m}^{2}}.$$
(13)
$$t\left(\xi \right)=-2i\sum _{m=-\infty}^{+\infty}{e}^{-i\pi m}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\left(\pi \frac{m}{2}\right)\mathrm{sinc}\left(\frac{m}{2}\right){e}^{i2\pi m\xi}.$$
(14)
$$t(x,y)=-2i\sum _{m=-\infty}^{+\infty}{e}^{-i\pi m}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\left(\pi \frac{m}{2}\right)\mathrm{sinc}\left(\frac{m}{2}\right){e}^{2\pi im({x}^{2}+{y}^{2})\u22152\lambda f}.$$
(15)
$$t(x-{x}_{0},y)=-2i{e}^{\pi im{x}_{0}^{2}\u2215\lambda f}\sum _{m=-\infty}^{+\infty}{e}^{-i\pi m}\mathrm{sin}\left(\pi \frac{m}{2}\right)\mathrm{sinc}\left(\frac{m}{2}\right){e}^{2\pi im({x}^{2}+{y}^{2})\u22152\lambda f}{e}^{-2\pi imx{x}_{0}\u2215\lambda f}.$$