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)=Q0I(x,ct)·ϒ(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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Figures (12)

F. 1
F. 1

Space-time locus of points illuminated by a laser pulse initiated at time t0 and terminated at time t1.

F. 2
F. 2

Space-time locus of points capable of reflecting flux acceptable to a shutter which opens at time t2 and closes at time t3.

F. 3
F. 3

Space-time locus of points illuminated by laser pulse and capable of reflecting flux acceptable to shutter. From an observation point located behind the shutter, a target located at point xa appears to be illuminated from time ta until time tb.

F. 4
F. 4

Apparent illuminance as a function of range for square-pulse illumination and shutter profiles of equal duration. x1 =c(t2t1)/2, x2=ct2/2, x3=ct3/2.

F. 5
F. 5

Space-time locus of points apparently illuminated for square-pulse illumination and shutter profiles of unequal duration. x1=c(t2t2)/2, x2=ct2/2, x3=c(t3t1)/2, x4=ct3/2.

F. 6
F. 6

Apparent illuminance as a function of range for square laser and shutter pulses of unequal duration.

F. 7
F. 7

Three-dimensional surface representing illuminance from a triangular pulse of light as a function of distance and time.

F. 8
F. 8

Zones of integration over which the Iϒ product can be expressed as a function of time and distance. Diagonal lines are limits of integration.

F. 9
F. 9

Apparent illuminance as a function of distance for a triangular laser pulse and a triangular shutter profile. Width of apparent illumination profile, from zero point to zero point, is equal to the duration of the laser pulse, from zero point to zero point, multiplied by the velocity of the light. Half-width of apparent illumination profile is less than the half-width of the laser profile multiplied by the velocity of light.

F. 10
F. 10

Direct copy from oscilloscope trace of Q-switched laser pulse, using TRG, Inc. laser equipped with Lummer–Gehrke plate and rotating prism.

F. 11
F. 11

Lorentz distribution function, with time as the independent variable. Owing to the limitation in scope of the present work, the similarity between the Lorentz function and the Q-switched laser pulse profile must be assumed coincidental.

F. 12
F. 12

Apparent illumination profile for a square shutter pulse and a Lorentz-type laser pulse.

Equations (34)

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L ( x ) = Q 0 I ( x , c t ) · ϒ ( x , c t ) d t ,
L ( x a ) = I 0 ( t b t a ) ,
t a = t 2 x / c t b = t 1 + x / c } for c ( t 2 t 1 ) / 2 < x < c t 2 / 2
t a = x / c t b = t 3 x / c } for c t 2 / 2 < x < c t 3 / 2 .
L ( x ) = I 0 [ t 1 t 2 + 2 ( x / c ) ] for c ( t 2 t 1 ) / 2 < x < c t 2 / 2
L ( x ) = I 0 [ t 3 2 ( x / c ) ] for c t 2 / 2 < x < c t 3 / 2 .
x = c ( t 2 t 1 ) / 2
x = c t 2 / 2 .
x = c ( t 2 t 1 ) / 2 + x ,
2 I 0 α x / c .
c ( t 2 t 1 ) / 2 c ( t 2 t 1 ) / 2 + x ( 2 I 0 / c ) [ x c ( t 2 t 1 ) / 2 ] β d x = I 0 β x 2 / c .
target illuminance beam backscatter = 2 α I 0 ( x / c ) I 0 β ( x / c ) = 2 α β x .
L ( x ) = I 0 ( t 1 t 2 + x / c ) for c ( t 2 t 1 ) / 2 < x < c t 2 / 2 L ( x ) = I 0 t 1 for c t 2 / 2 < x < c ( t 3 t 2 ) / 2 L ( x ) = I 0 [ t 3 2 ( x / c ) ] for c ( t 3 t 1 ) / 2 < x < c t 3 / 2
I ( x , c t ) = 0 for c t x < 0
I ( x , c t ) = ( I 0 / c t 1 ) ( c t x ) for 0 < c t x < c t 1
I ( x , c t ) = 2 I 0 ( I 0 / c t 1 ) ( c t x ) for c t 1 < c t x < 2 c t 1
I ( x , c t ) = 0 for c t x > 2 c t 1 .
ϒ ( x , c t ) = 0 for c ( t T ) + x < 0
ϒ ( x , c t ) = ( ϒ 0 / c t 1 ) c ( t T ) + x for 0 < c ( t T ) + x < c t 1
ϒ ( x , c t ) = 2 ϒ 0 ( ϒ 0 / c t 1 ) c ( t T ) + x for c t 1 < c ( t T ) + x < 2 c t 1
ϒ ( x , c t ) = 0 for c ( t T ) + x > 2 c t 1 .
L ( x ) = 0 I ( x , c t ) · ϒ ( x , c t ) d t .
( ϒ I ) b 3 = ϒ b I 3 = ( ϒ 0 / c t 3 ) [ c ( t T ) + x ] · [ 2 I 0 ( I 0 / c t 1 ) ( c t x ) ]
( ϒ I ) b 2 = ϒ b I 2 = ( ϒ 0 / c t 1 ) [ c ( t T ) + x ] · [ ( I 0 / c t 1 ) ( c t x ) ] .
( c / 2 ) ( T 2 t 1 ) < x < ( c / 2 ) ( ϒ t 1 ) , L ( x ) = T ( x / c ) 2 t 1 + ( x / c ) ( ϒ I ) b 3 · d t ,
L ( x ) = ( x / c ) + t 1 t 1 + T ( x / c ) ( ϒ I ) b 3 · d t + 2 T ( x / c ) ( x / c ) + t 1 ( ϒ I ) b 2 · d t .
L ( x ) = ( I 0 ϒ 0 / 6 c 3 t 1 2 ) ( 2 x + 2 c t 1 c T ) 3
( c / 2 ) ( T 2 t 1 ) < x < ( c / 2 ) ( T t 1 ) .
L ( x ) = ( I 0 ϒ 0 / c 3 t 1 2 ) [ 4 x 3 + ( 6 T 4 t 1 ) c x 2 + ( 4 t 1 T 3 T 2 ) c 2 x + c 3 ( 1 2 T 3 t 1 T 2 + 2 3 t 1 3 ) ]
( c / 2 ) ( T t 1 ) < x < ( c T / 2 ) .
I ( x , c t ) = I 0 / [ ( c t x ) 2 + h 2 ] ,
ϒ = ϒ 0 for c t 2 + c t 0 x > c t > c t 2 x ϒ = 0 for all other values of ct ,
L ( x ) = t 2 ( x / c ) t 2 + t 0 ( x / c ) d t / [ ( c t x ) 2 + h 2 ]
L ( x ) = ( 1 / h ) ( tan 1 c t 2 2 x h tan 1 c t 2 2 x c t 0 h ) .