## Abstract

This paper describes a new Fourier propagator for computing the impulse response of an optical system with a curved focal plane array, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh-Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics on a spherical surface. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh-Sommerfeld diffraction formula calculation and extends Fourier optics to cases where the receiver plane is a curved surface. Bounds on the phase error introduced by the approximations are derived, which show it should be more widely applicable than traditional Fresnel propagators. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of the diffraction patterns computed by the new technique compared to those computed using the Rayleigh-Sommerfeld technique in order to demonstrate the utility of the propagator.

Full Article |

PDF Article
### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

### Equations (26)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$U({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}},z)=\frac{z}{j\lambda}\underset{x}{\int}\underset{y}{\int}U(x,y,0)\frac{{e}^{j2\pi R(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})/\lambda}}{{R}^{2}(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})}dxdy$$
(2)
$$R(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})=\sqrt{{(x-{x}^{\mathrm{\prime}})}^{2}+{(y-{y}^{\mathrm{\prime}})}^{2}+z{({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})}^{2}}$$
(3)
$$z({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})=\sqrt{{(S)}^{2}-x{}^{\mathrm{\prime}2}-y{}^{\mathrm{\prime}2}}-(S-{z}_{o}),$$
(4)
$$R(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})=\sqrt{{(S)}^{2}+{x}^{2}+{y}^{2}-2x{x}^{\mathrm{\prime}}-2y{y}^{\mathrm{\prime}}}$$
(5)
$${R}_{o}^{2}(x,y)=(S{)}^{2}+{x}^{2}+{y}^{2}$$
(6)
$$R(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})={R}_{o}^{}(x,y)\sqrt{1+\frac{-2x{x}^{\mathrm{\prime}}-2y{y}^{\mathrm{\prime}}}{{R}_{o}^{2}(x,y)}}$$
(7)
$$R(x,y,{x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}})\approx {R}_{o}^{}(x,y)\left(1+\frac{-2x{x}^{\mathrm{\prime}}-2y{y}^{\mathrm{\prime}}}{2{R}_{o}^{2}(x,y)}\right)$$
(8)
$$U({x}^{\mathrm{\prime}},{y}^{\mathrm{\prime}},{z}_{o})=\frac{1}{j\lambda (S)}\underset{x}{\int}\underset{y}{\int}U(x,y,0){e}^{j2\pi {R}_{o}(x,y)/\lambda}{e}^{\frac{-j2\pi ({x}^{\mathrm{\prime}}x+{y}^{\mathrm{\prime}}y)}{\lambda {R}_{o}(x,y)}}dxdy$$
(9)
$$\frac{2\pi}{\lambda}\left(\frac{(x{x}^{\mathrm{\prime}}+y{y}^{\mathrm{\prime}})}{{R}_{o}(x,y)}-\frac{(x{x}^{\mathrm{\prime}}+y{y}^{\mathrm{\prime}})}{S}\right)$$
(10)
$$\frac{2\pi (x{x}^{\mathrm{\prime}}+y{y}^{\mathrm{\prime}})}{\lambda}\left(\frac{1}{{R}_{o}(x,y)}-\frac{1}{S}\right)$$
(11)
$${\theta}_{err}\le \frac{2\pi (x{x}^{\mathrm{\prime}}+y{y}^{\mathrm{\prime}})}{\lambda {(S)}^{2}}|{R}_{o}({x}_{2},{y}_{2})-S|$$
(12)
$${R}_{o}(x,y)\approx (S)\left(1+\frac{{x}^{2}+{y}^{2}}{2{(S)}^{2}}\right)$$
(13)
$$\left|\frac{2\pi (x{x}^{\mathrm{\prime}}+y{y}^{\mathrm{\prime}})}{\lambda}\right|\left|\frac{{x}^{2}+{y}^{2}}{2{(S)}^{3}}\right|<<1$$
(14)
$$\frac{\pi {n}_{max}{D}^{2}}{4{(S)}^{2}}<<1$$
(15)
$${n}_{max}<<(F\mathrm{\#}{)}^{2}$$
(16)
$${U}_{r}(n,m)=\frac{{\mathrm{\Delta}}_{s}^{2}}{j\lambda (S)}\sum _{k,l}^{}{U}_{s}(k,l){e}^{\frac{j2\pi {R}_{o}(k,l)}{\lambda}}{e}^{\frac{-j2\pi (nk+ml)}{N}}$$
(17)
$$\sqrt{1+b}\approx 1+b/2$$
(18)
$$2\pi {b}^{2}/8\lambda =\frac{2\pi {(2x{x}^{\mathrm{\prime}}+2y{y}^{\mathrm{\prime}})}^{2}}{8\lambda {R}_{o}^{4}(x,y)}<<1$$
(19)
$$\frac{\pi {n}_{max}^{2}\lambda {(x+y)}^{2}}{4{D}^{2}(S)}<<1$$
(20)
$$(F\mathrm{\#}{)}^{4}\lambda <<(S)$$
(21)
$$\frac{\pi {D}^{4}}{16\lambda}<<{z}^{3}$$
(22)
$$(F\mathrm{\#})<<(S/\lambda {)}^{1/4}$$
(23)
$${\left(\frac{\pi D}{16\lambda}\right)}^{1/3}<<(F\mathrm{\#})$$
(24)
$${\theta}_{tot}(k,l)={\theta}_{lens}(k,l)+2\pi {R}_{o}(k,l)/\lambda ,$$
(25)
$${\mathrm{\Delta}}_{r}=\frac{\lambda {S}_{2}}{L},$$
(26)
$$\mathrm{\%}Error=\frac{\sum _{x=1}^{N}\sum _{y=1}^{N}{({I}_{RS}(x,y)-{I}_{FT}(x,y))}^{2}}{\sum _{x=1}^{N}\sum _{y=1}^{N}{I}_{RS}^{2}(x,y)}\times 100$$