## Abstract

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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 31,
- Issue 5,
- pp. 362-368
- (1941)
- •doi: 10.1364/JOSA.31.000362

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This conventional ray diagram shows three rays,

The front of this lens mount is shown occulting the stop opening, and the illumination at the image is therefore reduced below that indicated by the

The lens in the center of the holder has been photographed through a wire mesh in order to determine the relative area presented by the lens to a point on the axis.

The effective area, as measured by counting openings in the wire mesh, is here reduced by an end ring cutting across the opening of the lens.

The end rings here determine the speed of the lens and the diaphragm has no influence on the brightness of the image.

Illumination curves of the six fastest of the 36 lenses tested. Rated speed from

Illumination curves of six lenses with rated speeds from

Illumination curves of six lenses with rated speeds from

Illumination curves of six lenses all with rated speeds of

Illumination curves of six lenses all with rated speeds of

Illumination curves of the six slowest lenses with rated speed from

The angles at which occultation begins and ends are here plotted to see if there are any general rules among practical lenses. The uniformity of illumination and width of field are roughly proportional to the

Ray diagram showing a lens with a spherical principal surface and a sphere cutting through the periphery of the surface and tangent to the center of the focal plane.

The curves on the chart show the departure from rated speed when the image plane is back of the principal focal plane. The relative brightness of image is given for various speeds and focal distances, and across the top of the chart is a scale of magnification corresponding to various object distances.

This ray diagram shows how the principal surface is determined for the Schmidt lens, and how the effective diameter of the reflector lens combination is found.

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$${E}_{i}=BW/S.$$

$$\begin{array}{ll}\hfill (S-W)/R=& R/W\\ \hfill R=& S\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\phi /W\end{array}$$

$$S=({W}^{2}+{R}^{2})/W.$$

$$F\hspace{0.17em}\text{cos}\hspace{0.17em}\theta =S\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\phi -{X}_{i},$$

$$F\hspace{0.17em}\text{sin}\hspace{0.17em}\theta =R,$$

$${F}^{2}={R}^{2}+{S}^{2}\hspace{0.17em}{\text{cos}}^{4}\hspace{0.17em}\phi -2{X}_{i}S\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\phi +{{X}_{i}}^{2}.$$

$$W=\frac{{R}^{2}(1-({F}^{2}-{R}^{2}{)}^{{\scriptstyle \frac{1}{2}}})}{{{X}_{i}}^{2}-({F}^{2}-{R}^{2})}.$$

$${E}_{i}=\frac{{R}^{2}B}{{{X}_{i}}^{2}+2{X}_{i}({F}^{2}-{R}^{2}{)}^{{\scriptstyle \frac{1}{2}}}+{F}^{2}}$$

$$F=1$$

$${E}_{i}=\frac{{R}^{2}B}{{{X}_{i}}^{2}+2{X}_{i}(1-{R}^{2}{)}^{{\scriptstyle \frac{1}{2}}}+1}.$$

$${X}_{0}={F}^{2}/{X}_{i}$$

$$u={X}_{0}+F,$$

$$M={X}_{i}/F$$

$$L={E}_{i}A\hspace{0.17em}\text{lumens}$$

$${E}_{i}A=(A/{M}^{2})BC,$$

$$C=\frac{{R}^{2}}{{F}^{2}+2{X}_{0}({F}^{2}-{R}^{2}{)}^{{\scriptstyle \frac{1}{2}}}+{{X}_{0}}^{2}},$$

$$L=\frac{{R}^{2}{A}_{0}{B}_{0}}{1+2{X}_{0}(1-{R}^{2}{)}^{{\scriptstyle \frac{1}{2}}}+{{X}_{0}}^{2}}$$

$$C=10/1000.$$

$$L=(10/1000)3\xb74000=120\hspace{0.17em}\text{lumens}.$$

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