## Abstract

The optical properties of an aberration-free defocused optical system used to image incoherently illuminated objects are analyzed. The exact diffraction optical transfer function (OTF) and point spread function (PSF) are compared to the geometrical OTF and PSF, respectively, for both small and large amounts of defocusing. It follows that geometrical optics does not describe diffraction optics very well for any reasonable amount of defocusing. Calculation of the diffraction OTF is complicated and time consuming, even with a large computer. An empirically derived analytic approximation to the diffraction OTF which is much easier to calculate, is formulated. It is also shown, and examples are presented to demonstrate, that the exact OTF and PSF are different in planes at equal distances on the two sides of the in-focus plane.

© 1969 Optical Society of America

Full Article |

PDF Article
### Equations (29)

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

(1)
$$T(s)=\frac{\underset{-\infty}{\overset{\infty}{\mathit{\iint}}}f\left(x+\frac{s}{2},y\right)f*\left(x-\frac{s}{2},y\right)\mathit{\text{dxdy}}}{\underset{-\infty}{\overset{\infty}{\mathit{\iint}}}{|f(x,y)|}^{2}\mathit{\text{dxdy}}},$$
(2)
$$s=(\mathrm{\lambda}/nsin\alpha )f,$$
(3)
$$f(x,y)=\{\begin{array}{ll}exp[i(2\pi /\mathrm{\lambda})w({x}^{2}+{y}^{2})],\hfill & \text{if}\hspace{0.17em}{x}^{2}+{y}^{2}\leqq 1\hfill \\ 0,\hfill & \text{if}\hspace{0.17em}{x}^{2}+{y}^{2}>1,\hfill \end{array}$$
(4)
$$\begin{array}{ll}{T}_{E}(s)\hfill & =\frac{4}{\pi a}cos[\frac{1}{2}as]\times \left\{\beta {J}_{1}(a)+\text{\u2211}_{n=1}^{\infty}{(-1)}^{n+1}\frac{sin2n\beta}{2n}[{J}_{2n-1}(a)-{J}_{2n+1}(a)]\right\}\hfill \\ \hfill & -\frac{4}{\pi a}sin[\frac{1}{2}as]\text{\u2211}_{n=0}^{\infty}{(-1)}^{n}\frac{sin(2n+1)\beta}{2n+1}\times [{J}_{2n}(a)-{J}_{2(n+1)}(a)],\hfill \end{array}$$
(5)
$$\begin{array}{ll}a\hfill & =(4\pi /\mathrm{\lambda})ws,\hfill \\ \beta \hfill & ={cos}^{-1}(s/2),\hfill \end{array}$$
(6)
$$\begin{array}{ll}{T}_{G}(s)=2{J}_{1}(a)/a,\hfill & a=(4\pi /\mathrm{\lambda})ws\hfill \end{array}.$$
(7)
$${T}_{E}(s)\to {\pi}^{-1}\{2arccos(s/2)-sin[2arccos(s/2)]\}={\pi}^{-1}(2\beta -sin2\beta ),$$
(8)
$$\gamma =3.3(w/\mathrm{\lambda}).$$
(9)
$${T}_{A}(s)=2{J}_{1}[a-y(s,w)]/[a-y(s,w)].$$
(10)
$$y(s,w)=0.5as=(2\pi /\mathrm{\lambda})w{s}^{2}.$$
(11)
$${T}_{A}(1+s)={T}_{A}(1-s).$$
(12)
$${T}_{A}(s)=\{\begin{array}{ll}2(1-0.69s+0.0076{s}^{2}+0.043{s}^{3})\times [{J}_{1}(a-0.5as)/(a-0.5as)],\hfill & \text{when}|s|<2\hfill \\ 0,\hfill & \text{when}|s|\ge 2,\hfill \end{array}$$
(13)
$${(AP)}^{2}={(AI)}^{2}+{(IP)}^{2}-2(AI)(IP)cos(\pi -\alpha ).$$
(14)
$${(r+z)}^{2}={(r+w)}^{2}+{z}^{2}-2(r+w)zcos(\pi -\alpha ),$$
(15)
$$w=-r-zcos\alpha +{({r}^{2}+2rz+{z}^{2}{cos}^{2}\alpha )}^{\frac{1}{2}}.$$
(16)
$$\mathrm{\Delta}{w}_{R}=\frac{2[|w(-{z}^{\prime})|-|w(-{z}^{\prime})|]}{|w(-{z}^{\prime})|+|w({z}^{\prime})|},$$
(17)
$$\mathrm{\Delta}{w}_{R}=2({z}^{\prime}/r).$$
(18)
$$w=\frac{1}{2}z{\alpha}^{2}[1-z/(r+z)]$$
(19)
$$w=z(1-cos\alpha ).$$
(20)
$$w=\frac{1}{2}z{sin}^{2}\alpha ,$$
(21)
$${T}_{G}(f)=\text{const}[{J}_{1}(2\pi {r}_{G}f)/2\pi {r}_{G}f].$$
(22)
$$\begin{array}{ll}A\hfill & =2\pi {r}_{G}f\hfill \\ \hfill & =4\pi ws/\mathrm{\lambda}cos\alpha \hfill \\ \hfill & =(4\pi /\mathrm{\lambda})ws\hfill \\ \hfill & =a.\hfill \end{array}$$
(23)
$${T}_{G}(s)=2{J}_{1}(a)/a$$
(24)
$${d}_{G}=2ztan\alpha .$$
(25)
$${d}_{G}=4w/tan\alpha ,$$
(27)
$$(f/)={(2tan\alpha )}^{-1}.$$
(28)
$${d}_{A}=2\xb71.22\xb7\mathrm{\lambda}\xb7(f/).$$
(29)
$$\gamma ={d}_{G}/{d}_{A}=3.3w/\mathrm{\lambda},$$