## Abstract

The effect of size of pupil of the eye upon the intensity distributions in retinal images of circular apertures observed in Maxwellian view is treated theoretically. Diffraction images of a small source of light are formed in the pupil and are governed by circular apertures whose angular diameters are of the order of minutes of arc. Quantitative data are presented to illustrate the manner in which sharpness of the retinal image of the aperture is increased by increasing the number of diffraction rings that are admitted to the eye. Amplitude and intensity distributions are included for a given diameter of aperture together with various sizes of pupil (including some annular pupils), and for various diameters of aperture together with one size of pupil. Equations are presented for calculating intensity distributions as modified by the Stiles-Crawford effect or by the effect of defocused eyes.

© 1959 Optical Society of America

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### Equations (11)

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(1)
$$u(r)=[K{J}_{1}({\alpha}_{0}kr)]/{\alpha}_{0}kr$$
(2)
$$\begin{array}{l}u({\alpha}^{\prime})=\frac{K}{k{\alpha}_{0}}\int \left[\frac{1}{\mathrm{\lambda}}{\int}_{0}^{2\pi}\text{exp}\{-i{\alpha}^{\prime}kr\hspace{0.17em}\text{cos}\theta \}d\theta \right]{J}_{1}({\alpha}_{0}kr)dr\\ =\frac{K}{{\alpha}_{0}}{\int}_{\text{pupil}}{J}_{0}({\alpha}^{\prime}kr){J}_{1}({\alpha}_{0}kr)dr.\end{array}$$
(3)
$$u({\alpha}^{\prime})={\int}_{0}^{\infty}{J}_{0}({\alpha}^{\prime}kr){J}_{1}({\alpha}_{0}kr)\hspace{0.17em}\tau (r)dr$$
(4)
$$\tau (r)=\{\begin{array}{rrr}\hfill 1\hfill & \hfill \text{if}& \hfill 0<r<b\hfill \\ \hfill 0& \hfill \text{if}& \hfill r>b.\end{array}$$
(5)
$$u({\alpha}^{\prime})={\int}_{0}^{b}{J}_{0}({\alpha}^{\prime}kr){J}_{1}({\alpha}_{0}kr)dr.$$
(6)
$$u({\alpha}^{\prime})={\int}_{0}^{3.26\hspace{0.17em}b}{J}_{0}({\alpha}^{\prime}t){J}_{1}(\alpha \xb7t)dt.$$
(7)
$$\tau (r)=\{\begin{array}{lll}\hfill 0\hfill & \text{if}\hfill & 0<r<{b}_{1}\hfill \\ \hfill 1\hfill & \text{if}\hfill & {b}_{1}<r<{b}_{2}\hfill \\ \hfill 0\hfill & \text{if}\hfill & \hfill r>{b}_{2}\hfill \end{array}.$$
(8)
$$\tau (r)=\{\begin{array}{lll}\eta (r)\hfill & \text{if}\hfill & 0<r<b\hfill \\ \hfill 0\hfill & \text{if}\hfill & \hfill r>b\end{array}$$
(9)
$$\tau (r)=\{\begin{array}{lll}\text{exp}\{{\scriptstyle \frac{1}{2}}ik\mathrm{\Delta}F{r}^{2}\}\hfill & \text{if}\hfill & 0<r<b\hfill \\ \hfill 0\hfill & \text{if}\hfill & \hfill r>b.\end{array}$$
(10)
$$u({\alpha}^{\prime})={\int}_{0}^{3.26\hspace{0.17em}b}{J}_{0}({\alpha}^{\prime}t){J}_{1}({\alpha}_{0}t)\hspace{0.17em}\text{exp}\{0.175i\pi {t}^{2}\mathrm{\Delta}F\}dt.$$
(11)
$$u({\alpha}^{\prime})\xb7u*({\alpha}^{\prime})={\left[{\int}_{0}^{3.26\hspace{0.17em}b}{J}_{0}({\alpha}^{\prime}t){J}_{1}({\alpha}_{0}t)\times \text{cos}(0.175\pi {t}^{2}\mathrm{\Delta}F)dt\right]}^{2}+{\left[{\int}_{0}^{3.26\hspace{0.17em}b}{J}_{0}({\alpha}^{\prime}t){J}_{1}({\alpha}_{0}t)\times \text{sin}(0.175\pi {t}^{2}\mathrm{\Delta}F)dt\right]}^{2}.$$