## Abstract

The diffraction relation between a plane and another plane that is both tilted and
translated with respect to the first one is revisited. The derivation of the result
becomes easier when the impulse function over a surface is used as a tool. Such an
approach converts the original 2D problem to an intermediate 3D problem and thus
allows utilization of easy-to-interpret Fourier transform properties due to rotation
and translation. An exact solution for the scalar monochromatic propagating waves
case when the propagation direction is restricted to be in the forward direction is
presented.

© 2011 Optical Society of America

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### Equations (19)

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(1)
$${\psi}_{z={z}_{0}}(x,y)={\mathcal{F}}_{2D}^{-1}\{{\mathcal{F}}_{2D}\{{\psi}_{z=0}(x,y)\}{e}^{j{z}_{0}\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}}\},$$
(2)
$$\psi (x,y,z)=\psi (\mathbf{x})=\underset{\mathbf{k}}{\int}{\delta}_{S}(\mathbf{k})A(\mathbf{k}){e}^{j{\mathbf{k}}^{T}\mathbf{x}}\mathrm{d}\mathbf{k}.$$
(3)
$$\u3008{\delta}_{S}(\mathbf{x}),f(\mathbf{x})\u3009=\underset{{\mathbf{R}}^{N}}{\int}{\delta}_{S}(\mathbf{x})f(\mathbf{x})\mathrm{d}\mathbf{x}=\underset{S}{\int}f(\mathbf{x})\mathrm{d}S\mathrm{.}$$
(4)
$${\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x})\stackrel{\mathrm{\Delta}}{=}\psi ({\mathbf{x}}^{\prime})=\psi (\mathbf{Rx}+\mathbf{b})\mathrm{.}$$
(5)
$${\mathrm{\Psi}}_{\mathbf{R},\mathbf{b}}(\mathbf{k})=\mathrm{\Psi}(\mathbf{Rk}){e}^{j(\mathbf{Rk}{)}^{T}\mathbf{b}}\mathrm{.}$$
(6)
$${\mathcal{F}}_{3D}\{\psi (\mathbf{x})\}=\mathrm{\Psi}(\mathbf{k})=8{\pi}^{3}{\delta}_{S}(\mathbf{k})A(\mathbf{k}).$$
(7)
$${\mathrm{\Psi}}_{\mathbf{R},\mathbf{b}}(\mathbf{k})=8{\pi}^{3}{\delta}_{S}(\mathbf{Rk})A(\mathbf{Rk}){e}^{j(\mathbf{Rk}{)}^{T}\mathbf{b}}\mathrm{.}$$
(8)
$${\mathrm{\Psi}}_{\mathbf{R},\mathbf{b}}(\mathbf{k})=8{\pi}^{3}{\delta}_{{S}_{\mathbf{R}}}(\mathbf{k})A(\mathbf{Rk}){e}^{j(\mathbf{Rk}{)}^{T}\mathbf{b}}\mathrm{.}$$
(9)
$${\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x})=\underset{\mathbf{k}}{\int}{\delta}_{{S}_{R}}(\mathbf{k})A(\mathbf{Rk}){e}^{j(\mathbf{Rk}{)}^{T}\mathbf{b}}{e}^{j{\mathbf{k}}^{T}\mathbf{x}}\mathrm{d}\mathbf{k},$$
(10)
$${\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x})=\underset{{S}_{R}}{\int}A(\mathbf{Rk}){e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b}+\mathbf{x})}\mathrm{d}S=\sum _{i}\underset{{B}_{i}}{\iint}A(\mathbf{Rk}){e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b}+\mathbf{x})}\frac{dS}{d{k}_{x}d{k}_{y}}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y},$$
(11)
$${\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x})=\underset{{B}_{1}}{\iint}A(\mathbf{Rk}){e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b}+\mathbf{x})}\frac{k}{\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}+\underset{{B}_{2}}{\iint}A(\mathbf{Rk}){e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b}+\mathbf{x})}\frac{k}{\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}.$$
(12)
$$4{\pi}^{2}A(\mathbf{k})\frac{k}{|{k}_{z}|}={\mathcal{F}}_{2D}\{{\psi}_{0}(x,y)\}={\mathrm{\Psi}}_{0}({k}_{x},{k}_{y}),$$
(13)
$$A(\mathbf{k})=A({k}_{x},{k}_{y},\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}})=\frac{1}{4{\pi}^{2}}\frac{|{k}_{z}|}{k}{\mathrm{\Psi}}_{0}({k}_{x},{k}_{y}).$$
(14)
$$A(\mathbf{Rk})=A({k}_{x}^{\prime},{k}_{y}^{\prime},{k}_{z}^{\prime})=\frac{1}{4{\pi}^{2}}\frac{|{k}_{z}^{\prime}|}{k}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime}).$$
(15)
$${\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x})=\frac{1}{4{\pi}^{2}}\underset{{B}_{1}}{\iint}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime})\frac{|{k}_{z}^{\prime}|}{|{k}_{z}|}{e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b})}{e}^{j{\mathbf{k}}^{T}\mathbf{x}}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}+\frac{1}{4{\pi}^{2}}\underset{{B}_{2}}{\iint}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime})\frac{|{k}_{z}^{\prime}|}{|{k}_{z}|}{e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b})}{e}^{j{\mathbf{k}}^{T}\mathbf{x}}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}\mathrm{.}$$
(16)
$${\psi}_{t}({x}^{\prime},{y}^{\prime})={\psi}_{\mathbf{R},\mathbf{b}}(\mathbf{x}){|}_{z=0}\phantom{\rule{0ex}{0ex}}=\frac{1}{4{\pi}^{2}}\underset{{B}_{1}}{\iint}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime})\frac{|{k}_{z}^{\prime}|}{|{k}_{z}|}{e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b})}{e}^{j({k}_{x}{x}^{\prime}+{k}_{y}{y}^{\prime})}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}\phantom{\rule{0ex}{0ex}}+\frac{1}{4{\pi}^{2}}\underset{{B}_{2}}{\iint}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime})\frac{|{k}_{z}^{\prime}|}{|{k}_{z}|}{e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b})}{e}^{j({k}_{x}{x}^{\prime}+{k}_{y}{y}^{\prime})}\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}\phantom{\rule{0ex}{0ex}}={\mathcal{F}}_{2D}^{-1}\{U({k}_{x},{k}_{y})\},$$
(17)
$$U({k}_{x},{k}_{y})\stackrel{\mathrm{\Delta}}{=}\frac{|{k}_{z}^{\prime}|}{|{k}_{z}|}{e}^{j{\mathbf{k}}^{T}({\mathbf{R}}^{T}\mathbf{b})}{\mathrm{\Psi}}_{0}({k}_{x}^{\prime},{k}_{y}^{\prime})[I({B}_{1})+I({B}_{2})],$$
(18)
$$I(B)\stackrel{\mathrm{\Delta}}{=}\{\begin{array}{cc}1& \text{if \hspace{0.17em} \hspace{0.17em}}({k}_{x},{k}_{y})\in \mathrm{B}\\ 0& \text{else}\end{array}\mathrm{.}$$
(19)
$${k}_{x}^{\prime}={k}_{x}{r}_{11}+{k}_{y}{r}_{12}+\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}{r}_{13},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{k}_{y}^{\prime}={k}_{x}{r}_{21}+{k}_{y}{r}_{22}+\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}{r}_{23},\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0ex}{0ex}}|{k}_{z}^{\prime}|=\sqrt{{k}^{2}-{k}_{x}^{\prime 2}-{k}_{y}^{\prime 2}}\mathrm{.}$$