## Abstract

Pseudorandom encoding is a method of statistically approximating desired complex values with those values that are achievable with a given spatial light modulator. Originally developed for phase-only modulators, pseudorandom encoding is extended to modulators for which amplitude is a function of phase. This is accomplished by transforming the phase statistics to compensate for the amplitude coupling. Example encoding formulas are derived, evaluated, and compared with a noncompensating pseudorandom-encoding algorithm. Compensating algorithms encode a smaller area of the complex plane and can produce more noise than is possible for arbitrary pseudorandom algorithms. However, the encoding formulas have greatly simplified numerical implementations.

© 1998 Optical Society of America

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### Equations (32)

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(1)
$$\u3008\mathbf{a}\u3009=\int \mathbf{a}p(\mathbf{a})\mathrm{d}\mathbf{a}$$
(2)
$${I}_{c}({f}_{x})={\left|\sum _{i}{\mathbf{A}}_{\mathit{ci}}\right|}^{2}={\left|\mathcal{F}\left\{\sum _{i}{\mathbf{a}}_{\mathit{ci}}\right\}\right|}^{2},$$
(3)
$$\u3008I({f}_{x})\u3009={I}_{c}({f}_{x})+\sum _{i}(\u3008|{\mathbf{A}}_{i}{|}^{2}\u3009-|{\mathbf{A}}_{\mathit{ci}}{|}^{2}),$$
(4)
$$\u3008\mathbf{a}\u3009=\int a(\psi )p(\psi )exp(j\psi )\mathrm{d}\psi \equiv {a}_{0}exp(j{\psi}_{0}),$$
(5)
$$\gamma =\underset{i}{max}({\mathbf{a}}_{\mathit{ci}})$$
(6)
$$\eta =\frac{1}{N}\sum _{i=1}^{N}|{\mathbf{a}}_{\mathit{ci}}{|}^{2}.$$
(7)
$$p(\psi ;\u3008\psi \u3009,\nu )=\frac{1}{\nu}\mathrm{rect}\left(\frac{\psi -\u3008\psi \u3009}{\nu}\right),$$
(8)
$$\u3008\mathbf{a}\u3009=\mathrm{sinc}(\nu /2\pi )exp(j\u3008\psi \u3009)\equiv {a}_{0}exp(j{\psi}_{0}).$$
(9)
$${a}_{0}=|\mathrm{sinc}(\nu /2\pi )|,$$
(10)
$${\psi}_{0}=\left\{\begin{array}{ll}\u3008\psi \u3009,& \mathrm{sinc}(\nu /2\pi )>0\\ \u3008\psi \u3009+\pi ,& \mathrm{sinc}(\nu /2\pi )<0\end{array}\right.,$$
(11)
$$\u3008\mathbf{a}\u3009=\mathrm{sinc}(\nu /2\pi )exp(j\u3008\psi \u3009),\hspace{1em}0\u2a7d\nu \u2a7d2\pi .$$
(12)
$$\nu =2\pi {\mathrm{sinc}}^{-1}({a}_{c}).$$
(13)
$$\psi =\u3008\psi \u3009+\nu [\mathrm{ran}(s)-1/2].$$
(14)
$$\u3008\psi {\u3009}_{\mathrm{eff}}=\int \psi {p}_{\mathrm{eff}}(\psi )\mathrm{d}\psi ,$$
(15)
$$P(\psi )={\int}_{-\infty}^{\psi}p(\varphi )\mathrm{d}\varphi .$$
(16)
$$\psi ={P}^{-1}(s).$$
(17)
$$a(\psi )=m\psi +b,\hspace{1em}\hspace{1em}\psi \in [-2\pi ,2\pi ],$$
(18)
$${p}_{\mathrm{eff}}(\psi )\propto \mathrm{rect}[(\psi -{\psi}_{0})/\nu ].$$
(19)
$$p(\psi )=\frac{1}{\psi +b/m}\times {\left[ln\left(\frac{{\psi}_{0}+\nu /2+b/m}{{\psi}_{0}-\nu /2+b/m}\right)\right]}^{-1}\mathrm{rect}\left(\frac{\psi -{\psi}_{0}}{\nu}\right).$$
(20)
$$\psi =\frac{({\psi}_{0}+\nu /2+b/m{)}^{s}}{({\psi}_{0}-\nu /2+b/m{)}^{s-1}}-b/m.$$
(21)
$$\u3008\mathbf{a}\u3009=m\nu {\left[ln\left(\frac{{\psi}_{0}+\nu /2+b/m}{{\psi}_{0}-\nu /2+b/m}\right)\right]}^{-1}\mathrm{sinc}\left(\frac{\nu}{2\pi}\right)exp(j{\psi}_{0}).$$
(22)
$${p}_{\mathrm{eff}}(\psi )\propto \delta (\psi -{\psi}_{0}+\nu /2)+\delta (\psi -{\psi}_{0}-\nu /2),$$
(23)
$$p(\psi )=\frac{a({\psi}_{0}+\nu /2)\delta (\psi -{\psi}_{0}+\nu /2)+a({\psi}_{0}-\nu /2)\delta (\psi -{\psi}_{0}-\nu /2)}{a({\psi}_{0}-\nu /2)+a({\psi}_{0}+\nu /2)}$$
(24)
$${\mathbf{a}}_{c}=\u3008\mathbf{a}\u3009=\frac{2a({\psi}_{0}+\nu /2)a({\psi}_{0}-\nu /2)}{a({\psi}_{0}+\nu /2)+a({\psi}_{0}-\nu /2)}cos(\nu /2)exp(j{\psi}_{0}).$$
(25)
$$\psi =\left\{\begin{array}{l}{\psi}_{0}-\nu /2\hspace{1em}\hspace{1em}\mathrm{if}\hspace{0.5em}s\u2a7d\frac{a({\psi}_{0}+\nu /2)}{a({\psi}_{0}-\nu /2)+a({\psi}_{0}+\nu /2)}\\ {\psi}_{0}+\nu /2\hspace{1em}\hspace{1em}\mathrm{otherwise}\end{array}\right..$$
(26)
$$\u3008\mathbf{a}\u3009=d\u3008\mathbf{a}{\u3009}_{\mathbf{u}}+(1-d)\u3008\mathbf{a}{\u3009}_{\mathbf{l}},$$
(27)
$$\u3008\mathbf{a}\u3009=[{\mathit{da}}_{u}+(1-d){a}_{l}]exp(j{\psi}_{0}),$$
(28)
$$p(\psi ;d)={\mathit{dp}}_{u}(\psi ;{\psi}_{0},{\nu}_{u})+(1-d){p}_{l}(\psi ;{\psi}_{0},{\nu}_{l}),$$
(29)
$$\u3008\mathbf{a}\u3009=d{\mathbf{a}}_{1}+(1-d){\mathbf{a}}_{2},$$
(30)
$$\u220a=\int {a}^{2}(\psi )p(\psi )\mathrm{d}\psi -|{\mathbf{a}}_{c}{|}^{2}.$$
(31)
$$\u220a=d(1-d)[a_{1}{}^{2}+a_{2}{}^{2}-2{a}_{1}{a}_{2}cos({\psi}_{1}-{\psi}_{2})]\hspace{1em}\mathrm{subject}\hspace{0.5em}\mathrm{to}\hspace{0.5em}{\mathbf{a}}_{c}=d{\mathbf{a}}_{1}+(1-d){\mathbf{a}}_{2},$$
(32)
$$\u220a=|{\mathbf{a}}_{c}-{\mathbf{a}}_{1}||{\mathbf{a}}_{c}-{\mathbf{a}}_{2}|.$$