## Abstract

It is generally accepted that diffractive elements designed for
multiwavelength operation require deep surface-relief profiles. We
show, however, that thin diffractive elements can be designed to
operate with more than one wavelength. A novel, to our knowledge,
optimization technique is introduced for this purpose. The maximum
phase delay is limited to only a few multiples of 2π, and the element
can implement different functions for different
wavelengths. Examples with fan-out gratings are
discussed.

© 1999 Optical Society of America

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### Equations (8)

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(1)
$${c}_{m}=\frac{1}{\mathrm{\Lambda}}{\int}_{0}^{\mathrm{\Lambda}}exp\left[i\mathrm{\Phi}\left(x\prime \right)\right]exp\left(-i2\mathrm{\pi}\mathit{mx}\prime /\mathrm{\Lambda}\right)\mathrm{d}x\prime ={\int}_{0}^{1}exp\left[i\mathrm{\Phi}\left(x\right)\right]exp\left(-i2\mathrm{\pi}\mathit{mx}\right)\mathrm{d}x,$$
(2)
$${c}_{m}=\sum _{k=1}^{N}exp\left(i{\mathrm{\varphi}}_{k}\right)exp\left[-i\mathrm{\pi}m\left({x}_{k}+{x}_{k-1}\right)\right]\times \left({x}_{k}-{x}_{k-1}\right)\mathrm{sinc}\left[m\left({x}_{k}-{x}_{k-1}\right)\right]=\sum _{k=1}^{N}exp\left(i{\mathrm{\varphi}}_{k}\right){g}_{m}\left({x}_{k},{x}_{k-1}\right),$$
(3)
$${c}_{m}\prime ={c}_{m}+\left[exp\left(i{\mathrm{\varphi}}_{q}\prime \right)-exp\left(i{\mathrm{\varphi}}_{q}\right)\right]g\left({x}_{q},{x}_{q-1}\right).$$
(4)
$${c}_{m}\prime ={c}_{m}+\left[{g}_{m}\left({x}_{q}\prime ,{x}_{q-1}\right)-{g}_{m}\left({x}_{q},{x}_{q-1}\right)\right]exp\left(i{\mathrm{\varphi}}_{q}\right)+\left[{g}_{m}\left({x}_{q+1},{x}_{q}\prime \right)-{g}_{m}\left({x}_{q+1},{x}_{q}\right)\right]exp\left(i{\mathrm{\varphi}}_{q+1}\right).$$
(5)
$$\mathrm{\gamma}=\frac{{\mathrm{\lambda}}_{D}}{\mathrm{\lambda}}\frac{n\left(\mathrm{\lambda}\right)-1}{n\left({\mathrm{\lambda}}_{D}\right)-1}.$$
(6)
$${\mathrm{\eta}}_{p}=\sum _{m\in {M}_{p}}{I}_{m},$$
(7)
$${\mathrm{\sigma}}_{p}={\left.\frac{max\left({I}_{m}\right)-min\left({I}_{m}\right)}{max\left({I}_{m}\right)+min\left({I}_{m}\right)}\right|}_{m\in {M}_{p},}$$
(8)
$$\mathrm{\mu}=\sum _{p=1}^{L}{w}_{p}\frac{{\displaystyle \sum _{m\in {M}_{p}}}{\left({I}_{m}-I_{m}{}^{0}\right)}^{2}}{{\displaystyle \sum _{m\in {M}_{p}}}{I}_{m}},$$