## Abstract

Iterative algorithms based on Fourier transform are used for the design of diffractive optical elements (DOEs), which produce a given intensity distribution, usually at the far field. For the near field, these algorithms can also be used by changing the Fourier transform for the Fresnel transform. However, when the distance between the DOE and the observation plane is short, the results obtained with this modification are not always valid. In the present work, we develop a technique for obtaining the desired intensity distribution in the near field using two DOEs in tandem. We have designed an algorithm based on the standard Gerchberg–Saxton algorithm to determine the modulation of the two DOEs. The best results are obtained when the first DOE modulates the amplitude and the second DOE modulates the phase.

© 2011 Optical Society of America

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### Equations (15)

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(1)
$${U}_{2}(x,y;z)={A}_{\text{target}}(x,y){e}^{j\phi (x,y{)}_{\text{random}}}.$$
(2)
$${U}_{3}(x,y)={A}_{3}(x,y){e}^{j{\phi}_{3}(x,y)}=\mathrm{IFrT}[{U}_{2}(x,y)],$$
(3)
$${h}^{\prime}(x,y;z)=-\frac{{e}^{-jkz}}{j\lambda z}\{\mathrm{exp}[-\frac{jk}{2z}({x}^{2}+{y}^{2})\left]\right\}.$$
(4)
$${\mathrm{DOE}}_{2}(x,y)={e}^{-j{\phi}_{3}(x,y)},$$
(5)
$${U}_{4}(x,y,{z}_{0})={A}_{4}(x,y){e}^{j{\phi}_{4}(x,y)}=\mathrm{IFrT}[{U}_{3}^{\prime}(x,y)].$$
(6)
$${\mathrm{DOE}}_{1}(x,y)={A}_{4}(x,y).$$
(7)
$${U}_{0}(x,y)={\mathrm{DOE}}_{1}(x,y){u}_{\text{incident}}(x,y).$$
(8)
$${U}_{1}(x,y,{z}_{0})={A}_{1}(x,y){e}^{j{\phi}_{1}(x,y)}=\mathrm{FrT}[{U}_{0}(x,y)],$$
(9)
$$h(x,y;z)=\frac{{e}^{jkz}}{j\lambda z}\{\mathrm{exp}[\frac{jk}{2z}({x}^{2}+{y}^{2})\left]\right\}.$$
(10)
$${U}_{1}^{\prime}(x,y;z)={U}_{1}(x,y){\mathrm{DOE}}_{2}(x,y).$$
(11)
$${U}_{2}(x,y;z)={A}_{2}(x,y){e}^{j{\phi}_{2}(x,y)}=\mathrm{FrT}[{U}_{1}^{\prime}(x,y;z)].$$
(12)
$${\overline{\mathrm{DOE}}}_{2}(x,y)={\mathrm{DOE}}_{2}(x,y){e}^{-j{\phi}_{1}(x,y)},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{U}_{1}^{\prime}(x,y;z)={U}_{1}(x,y){\overline{\mathrm{DOE}}}_{2}(x,y).$$
(13)
$$\mathrm{MSE}=\frac{\iint |{I}_{\text{target}}(x,y)-k{I}_{\text{output}}(x,y){|}^{2}\mathrm{d}x\mathrm{d}y}{\iint |{I}_{\text{target}}(x,y){|}^{2}\mathrm{d}x\mathrm{d}y},$$
(14)
$$\mathrm{NDE}=1-\frac{{\iint}_{s}{I}_{\text{output}}(x,y)\mathrm{d}x\mathrm{d}y}{\iint {I}_{\text{output}}(x,y)\mathrm{d}x\mathrm{d}y},$$
(15)
$$\mathrm{NU}=\mathrm{std}[{I}_{\text{output}}(x,y){]}_{S},$$