## Abstract

Gated, laser night-viewing systems are briefly described, and a method of calculation is presented for determining apparent illuminance as a function of target distance. A general equation for gated viewing system is developed,

$$L(x)={\mathit{\int}}_{{\text{Q}}_{0}}^{\infty}I(x,ct)\xb7\mathrm{\Upsilon}(x,ct)dt,$$ where

*L*(

*x*) is apparent illuminance,

*I*(

*x,ct*) is illuminance as a function of time, and ϒ(

*x,ct*) is optical transmittance of a shutter. Its practical use is discussed and illustrated, and it is shown that optimum performance is achieved by using a light pulse of duration equal to the shutter open period. Methods are outlined for calculation of integrated beam backscatter. Variations in the apparent illuminance profile for various laser pulse shapes and shutter transmittance functions are discussed. Several specific cases, including one highly practical case apt to occur in actual systems, are treated quantitatively.

© 1966 Optical Society of America

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### Equations (34)

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(1)
$$L(x)={\mathit{\int}}_{{\text{Q}}_{0}}^{\infty}I(x,ct)\xb7\mathrm{\Upsilon}(x,ct)dt,$$
(2)
$$L({x}_{a})={I}_{0}({t}_{b}-{t}_{a}),$$
(3)
$$\begin{array}{cc}\begin{array}{ll}{t}_{a}\hfill & ={t}_{2}-x/c\hfill \\ {t}_{b}\hfill & ={t}_{1}+x/c\hfill \end{array}\}& \text{for}\hspace{0.17em}c({t}_{2}-{t}_{1})/2<x<c{t}_{2}/2\end{array}$$
(4)
$$\begin{array}{cc}\begin{array}{ll}{t}_{a}\hfill & =x/c\hfill \\ {t}_{b}\hfill & ={t}_{3}-x/c\hfill \end{array}\}& \text{for}\hspace{0.17em}c{t}_{2}/2<x<c{t}_{3}/2\end{array}.$$
(5)
$$\begin{array}{ccc}L(x)={I}_{0}[{t}_{1}-{t}_{2}+2(x/c)]& \text{for}& c({t}_{2}-{t}_{1})/2<x<c{t}_{2}/2\end{array}$$
(6)
$$\begin{array}{ccc}L(x)={I}_{0}[{t}_{3}-2(x/c)]& \text{for}& c{t}_{2}/2<x<c{t}_{3}/2\end{array}.$$
(7)
$$x=c({t}_{2}-{t}_{1})/2$$
(9)
$$x=c({t}_{2}-{t}_{1})/2+{x}^{\prime},$$
(10)
$$2{I}_{0}\alpha {x}^{\prime}/c.$$
(11)
$${\mathit{\int}}_{c({t}_{2}-{t}_{1})/2}^{c({t}_{2}-{t}_{1})/2+{x}^{\prime}}(2{I}_{0}/c)[x-c({t}_{2}-{t}_{1})/2]\beta dx={I}_{0}\beta {x}^{\prime 2}/c.$$
(12)
$$\frac{\text{target illuminance}}{\text{beam backscatter}}=\frac{2\alpha {I}_{0}({x}^{\prime}/c)}{{I}_{0}\beta ({x}^{\prime}/c)}=\frac{2\alpha}{\beta {x}^{\prime}}.$$
(13)
$$\begin{array}{llll}L(x)\hfill & ={I}_{0}({t}_{1}-{t}_{2}+x/c)\hfill & \text{for}\hfill & c({t}_{2}-{t}_{1})/2<x<c{t}_{2}/2\hfill \\ L(x)\hfill & ={I}_{0}{t}_{1}\hfill & \text{for}\hfill & c{t}_{2}/2<x<c({t}_{3}-{t}_{2})/2\hfill \\ L(x)\hfill & ={I}_{0}[{t}_{3}-2(x/c)]\hfill & \text{for}\hfill & c({t}_{3}-{t}_{1})/2<x<c{t}_{3}/2\hfill \end{array}$$
(14)
$$\begin{array}{ccc}I(x,ct)=0& \text{for}& ct-x<0\end{array}$$
(15)
$$\begin{array}{ccc}I(x,ct)=({I}_{0}/c{t}_{1})(ct-x)& \text{for}& 0<ct-x<c{t}_{1}\end{array}$$
(16)
$$\begin{array}{ccc}I(x,ct)=2{I}_{0}-({I}_{0}/c{t}_{1})(ct-x)& \text{for}& c{t}_{1}<ct-x<2c{t}_{1}\end{array}$$
(17)
$$\begin{array}{ccc}I(x,ct)=0& \text{for}& ct-x>2c{t}_{1}\end{array}.$$
(18)
$$\begin{array}{ccc}\mathrm{\Upsilon}(x,ct)=0& \text{for}& c(t-T)+x<0\end{array}$$
(19)
$$\begin{array}{ccc}\mathrm{\Upsilon}(x,ct)=({\mathrm{\Upsilon}}_{0}/c{t}_{1})c(t-T)+x& \text{for}& 0<c(t-T)+x<c{t}_{1}\end{array}$$
(20)
$$\begin{array}{ccc}\mathrm{\Upsilon}(x,ct)=2{\mathrm{\Upsilon}}_{0}-({\mathrm{\Upsilon}}_{0}/c{t}_{1})c(t-T)+x& \text{for}& c{t}_{1}<c(t-T)+x<2c{t}_{1}\end{array}$$
(21)
$$\begin{array}{ccc}\mathrm{\Upsilon}(x,ct)=0& \text{for}& c(t-T)+x>2c{t}_{1}\end{array}.$$
(22)
$$L(x)={\mathit{\int}}_{0}^{\infty}I(x,ct)\xb7\mathrm{\Upsilon}(x,ct)dt.$$
(23)
$${(\mathrm{\Upsilon}I)}_{b3}={\mathrm{\Upsilon}}_{b}{I}_{3}=({\mathrm{\Upsilon}}_{0}/c{t}_{3})[c(t-T)+x]\xb7[2{I}_{0}-({I}_{0}/c{t}_{1})(ct-x)]$$
(24)
$${(\mathrm{\Upsilon}I)}_{b2}={\mathrm{\Upsilon}}_{b}{I}_{2}=({\mathrm{\Upsilon}}_{0}/c{t}_{1})[c(t-T)+x]\xb7[({I}_{0}/c{t}_{1})(ct-x)].$$
(25)
$$\begin{array}{c}(c/2)(T-2{t}_{1})<x<(c/2)(\mathrm{\Upsilon}-{t}_{1}),\\ L(x)={\mathit{\int}}_{T-(x/c)}^{2{t}_{1}+(x/c)}{(\mathrm{\Upsilon}I)}_{b3}\xb7dt,\end{array}$$
(26)
$$L(x)={\mathit{\int}}_{(x/c)+{t}_{1}}^{{t}_{1}+T-(x/c)}{\left(\mathrm{\Upsilon}I\right)}_{b3}\xb7dt+2{\mathit{\int}}_{T-(x/c)}^{(x/c)+{t}_{1}}{(\mathrm{\Upsilon}I)}_{b2}\xb7dt.$$
(27)
$$L(x)=({I}_{0}{\mathrm{\Upsilon}}_{0}/6{c}^{3}{{t}_{1}}^{2}){(2x+2c{t}_{1}-cT)}^{3}$$
(28)
$$(c/2)(T-2{t}_{1})<x<(c/2)(T-{t}_{1}).$$
(29)
$$L(x)=({I}_{0}{\mathrm{\Upsilon}}_{0}/{c}^{3}{{t}_{1}}^{2})[-4{x}^{3}+(6T-4{t}_{1})c{x}^{2}+(4{t}_{1}T-3{T}^{2}){c}^{2}x+{c}^{3}(\frac{1}{2}{T}^{3}-{t}_{1}{T}^{2}+\frac{2}{3}{{t}_{1}}^{3})]$$
(30)
$$(c/2)(T-{t}_{1})<x<(cT/2).$$
(31)
$$I(x,ct)={I}_{0}/[{(ct-x)}^{2}+{h}^{2}],$$
(32)
$$\begin{array}{ll}\mathrm{\Upsilon}={\mathrm{\Upsilon}}_{0}\hfill & \text{for}\hspace{0.17em}c{t}_{2}+c{t}_{0}-x>ct>c{t}_{2}-x\hfill \\ \mathrm{\Upsilon}=0\hfill & \text{for all other values of}\hspace{0.17em}\mathit{\text{ct}},\hfill \end{array}$$
(33)
$$L(x)={\mathit{\int}}_{{t}_{2}-(x/c)}^{{t}_{2}+{t}_{0}-(x/c)}dt/[{(ct-x)}^{2}+{h}^{2}]$$
(34)
$$L(x)=(1/h)\left({tan}^{-1}\frac{c{t}_{2}-2x}{h}-{tan}^{-1}\frac{c{t}_{2}-2x-c{t}_{0}}{h}\right).$$