## Abstract

For a confocal fluorescent microscope with a finite-sized circular detector, the three-dimensional optical transfer function (OTF) for a thick object has been developed without the use of Stockseth’s approximation. The results show that the OTF has negative values when the radius of the detector exceeds certain magnitudes. The two-dimensional OTF derived from the three-dimensional OTF is also given.

© 1992 Optical Society of America

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### Equations (9)

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(1)
$${h}_{e}(v,u)=\mid h(v,u){\mid}^{2}[\mid h(v,u){\mid}^{2}{\otimes}_{2}D(v)],$$
(2)
$$C(l,s)={\mathcal{F}}_{3}\{\mid h(v,u){\mid}^{2}[\mid h(v,u){\mid}^{2}{\otimes}_{2}D(v)]\},$$
(3)
$$C(l,s)={\mathcal{F}}_{3}[\mid h(v,u){\mid}^{2}]{\otimes}_{3}\{{\mathcal{F}}_{3}[\mid h(v,u){\mid}^{2}]{\mathcal{F}}_{2}[D(v)]\},$$
(4)
$${\mathcal{F}}_{3}[\mid h(v,u){\mid}^{2}]=\frac{1}{\mid l\mid}\text{Re}\left\{{\left[1-{\left(\frac{\mid s\mid}{l}+\frac{l}{2}\right)}^{2}\right]}^{1/2}\right\},$$
(5)
$${\mathcal{F}}_{2}[D(v)]={v}_{d}[{J}_{1}(l{v}_{d})/l],$$
(6)
$$C(l,s)={v}_{d}{\iiint}_{v}\frac{1}{{{l}_{1}}^{2}}\text{Re}\left\{{\left[{{l}_{1}}^{2}-{(\left|{s}^{\prime}-\frac{s}{2}\right|+\frac{{{l}_{1}}^{2}}{2})}^{2}\right]}^{1/2}\right\}\frac{1}{{{l}_{2}}^{2}}\times \text{Re}\left\{{\left[{{l}_{2}}^{2}-{\left(\left|{s}^{\prime}+\frac{s}{2}\right|+\frac{{{l}_{2}}^{2}}{2}\right)}^{2}\right]}^{1/2}\right\}\frac{{J}_{1}({l}_{2}{v}_{d})}{{l}_{2}}\text{d}m\text{d}n\text{d}{s}^{\prime},$$
(7)
$${l}_{1}={[{(m-l/2)}^{2}+{n}^{2}]}^{1/2},$$
(8)
$${l}_{2}={[{(m+l/2)}^{2}+{n}^{2}]}^{1/2},$$
(9)
$${C}_{2}(l)={\int}_{-1}^{1}C(l,s)\text{d}s.$$