## Abstract

Building on the optimal-rotation-angle method, an algorithm for the design of inter-element coherent arrays of diffractive optical elements (DOEs) was developed. The algorithm is intended for fan-out DOE arrays where the individual elements fan-out to a sub-set of points chosen from a common set of points. By iteratively optimizing the array of elements as a whole the proposed algorithm ensures that the light from neighbouring elements is in-phase in all fan-out points that are common to neighbouring DOEs. This is important in applications where a laser beam scans the DOE array and the fan-out intensities constitute a read-out of information since the in-phase condition ensures a smooth transition in the read-out as the beam moves from one DOE to the next. Simulations show that the inter-element in-phase condition can be imposed at virtually no expense in terms of optical performance, as compared to independently designed DOEs.

©2004 Optical Society of America

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### Equations (13)

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(1)
$${U}_{\mathit{km}}={A}_{k}^{\mathit{inc}}\mathrm{exp}\left[j\left({\phi}_{k}^{\mathit{inc}}+{\phi}_{k}\right)\right]{A}_{\mathit{km}}\mathrm{exp}\left(j\phi \mathit{km}\right)$$
(2)
$${A}_{\mathit{km}}\mathrm{exp}\left(j{\phi}_{\mathit{km}}\right)=\frac{1}{4\pi}(-j{k}_{z}-\frac{{\mathit{L}}_{\mathit{km}}}{{r}_{\mathit{km}}^{c}}\{j{k}_{0}-\frac{1}{{r}_{\mathit{km}}^{c}}\})$$
(2)
$$\times \frac{4{A}_{\mathit{k}}^{\mathit{inc}}}{{r}_{\mathit{km}}^{c}}\mathrm{exp}\left(j{k}_{0}{r}_{\mathit{km}}^{c}\right)\frac{\mathrm{sin}\left({\tilde{k}}_{x}\frac{a}{2}\right)\mathrm{sin}\left({\tilde{k}}_{y}\frac{b}{2}\right)}{{\tilde{k}}_{x}{\tilde{k}}_{y}}$$
(3)
$${S}_{1}={\Sigma}_{m}{A}_{\mathit{km}}\mathrm{cos}{\Phi}_{\mathit{km}}$$
(3)
$${{S}_{2}=\Sigma}_{m}{A}_{\mathit{km}}\mathrm{sin}{\Phi}_{\mathit{km}}$$
(3)
$${\alpha}_{k}=\mathrm{arctan}({S}_{2}/{S}_{1})$$
(3)
$$\begin{array}{cc}\Delta {\phi}_{k}^{\mathit{opt}}={\alpha}_{k}& \mathrm{if}{\phantom{\rule{.2em}{0ex}}S}_{1}>0\\ \Delta {\phi}_{k}^{\mathit{opt}}={\alpha}_{k}+\pi & \mathrm{if}\phantom{\rule{.2em}{0ex}}{S}_{1}<0\\ \Delta {\phi}_{k}^{\mathit{opt}}=\pi /2& \mathrm{if}{\phantom{\rule{.2em}{0ex}}S}_{1}=0\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{S}_{2}>0\\ \Delta {\phi}_{k}^{\mathit{opt}}=-\pi /2& \mathrm{if}\phantom{\rule{.2em}{0ex}}{S}_{1}=0\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{S}_{2}<0\end{array}$$
(4)
$${S}_{1}={\Sigma}_{m}{w}_{m}{A}_{\mathit{km}}\mathrm{cos}{\Phi}_{\mathit{km}}$$
(4)
$${S}_{2}={\Sigma}_{m}{w}_{m}{A}_{\mathit{km}}\mathrm{sin}{\Phi}_{\mathit{km}}$$
(5)
$${w}_{m}^{\mathit{new}}={w}_{m}^{\mathit{old}}{\left(\frac{{I}_{m}^{\mathit{desired}}}{{I}_{m}}\right)}^{p}$$
(6)
$$1.\phantom{\rule{.2em}{0ex}}{w}_{n}^{\mathit{BD}}:={w}_{n}^{\mathit{BD}}+{b}_{c}\left({I}_{n}-\stackrel{\u0305}{I}\right)/\stackrel{\u0305}{I}$$
(6)
$$2.\phantom{\rule{.2em}{0ex}}{w}_{n}^{\mathit{BD}}:={w}_{n}^{\mathit{BD}}-min\left({w}_{n}^{\mathit{BD}}\right)$$
(7)
$$\mathrm{unif}.\mathrm{err}.=\frac{{({I}_{\mathit{mn}}\u2044{I}_{\mathit{mn}}^{\mathit{desired}})}^{\mathit{max}}-{({I}_{\mathit{mn}}\u2044{I}_{\mathit{mn}}^{\mathit{desired}})}^{\mathit{min}}}{{({I}_{\mathrm{mn}}\u2044{I}_{\mathrm{mn}}^{\mathit{desired}})}^{\mathit{max}}+{({I}_{\mathit{mn}}\u2044{I}_{\mathit{mn}}^{\mathit{desired}})}^{\mathit{min}}}$$