## Abstract

A new method for encoding high-efficiency radially symmetric computer-generated holograms is formulated. An iterative approach is used to determine fringe widths and phase values for phase-only holographic optical elements. Computer-simulation results indicate that this approach can be used to increase the efficiency of low-*f*-number elements significantly. The experimental results for an *f*/1 element with a diffraction efficiency of 90% are presented.

© 1993 Optical Society of America

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### Equations (20)

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(1)
$$N\le \frac{{T}_{\text{min}}}{\delta},$$
(2)
$${T}_{\text{min}}=\lambda {[1+{(2F/\#)}^{2}]}^{1/2}.$$
(3)
$$\eta ={\left[\frac{N\hspace{0.17em}\text{sin}(\pi /N)}{\pi}\right]}^{2}.$$
(4)
$$U({P}_{0})=\frac{1}{j\lambda}\iint U({P}_{1})\frac{\text{exp}(j\kappa {r}_{01})}{{r}_{01}}\text{cos}(\widehat{n},\overline{{r}_{01}})\text{d}s,$$
(5)
$$U({P}_{1})=H({P}_{1})A({r}_{21}).$$
(6)
$$U({P}_{0})=\frac{{z}_{01}}{j\lambda}{\int}_{0}^{R}\rho H(\rho ){\int}_{0}^{2\pi}A(\rho ,\theta )\frac{\text{exp}(j\kappa {r}_{01})}{{{r}_{01}}^{2}}\text{d}\theta \text{d}\rho ,$$
(7)
$${U}_{mn}=\frac{{z}_{01}}{j\lambda}\sum _{k=0}^{K}k\mathrm{\Delta}\rho {H}_{k}\sum _{L=0}^{2\pi /\mathrm{\Delta}\theta}{A}_{kL}\frac{\text{exp}(j\kappa {r}_{kLmn})}{{{r}_{kLmn}}^{2}}\mathrm{\Delta}\theta \mathrm{\Delta}\rho .$$
(8)
$${U}_{mn}=\sum _{k}{H}_{k}{C}_{kmn},$$
(9)
$${C}_{kmn}=\frac{{z}_{01}\mathrm{\Delta}\theta {(\mathrm{\Delta}\rho )}^{2}}{j\lambda}k\sum _{L}{A}_{kL}\frac{\text{exp}(j\kappa {r}_{kLmn})}{{{r}_{kLmn}}^{2}}.$$
(10)
$${\varphi}_{k}\in \left\{0,\frac{2\pi}{N},2\frac{2\pi}{N},3\frac{2\pi}{N},\dots ,\left(N-1\frac{2\pi}{N}\right)\right\}.$$
(11)
$${H}_{k}=\text{exp}(j{\varphi}_{k}).$$
(12)
$${U}_{mn}=\sum _{k=0}^{K}\text{exp}(j{\varphi}_{k}){C}_{kmn}.$$
(13)
$${U}_{mn}=\sum _{p=0}^{P}\text{exp}(j{\varphi}_{p}){S}_{pmn},$$
(14)
$${S}_{pmn}=\sum _{k={k}_{s}(p)}^{{k}_{s}(p+1)-1}{C}_{kmn}.$$
(16)
$$\eta =\frac{{\displaystyle \sum _{m=1}^{M}}{\displaystyle \sum _{n=1}^{N}}{P}_{mn}}{{P}_{\text{inc}}},$$
(17)
$$e=\sum _{q=1}^{Q}{a}_{q}(1-{\eta}_{q})$$
(18)
$${{U}_{mn}}^{\prime}={U}_{mn}+[\text{exp}(j\varphi )-\text{exp}(j{\varphi}_{p-1})]\mathrm{\Delta}{S}_{pmn},$$
(19)
$$\mathrm{\Delta}{S}_{pmn}={S}_{pmn}-{{S}_{pmn}}^{\prime}$$
(20)
$${{U}_{mn}}^{\u2033}={U}_{mn}+[\text{exp}(j{{\varphi}_{p}}^{\prime})-\text{exp}(j{\varphi}_{p})]{{S}_{pmn}}^{\prime},$$