Abstract

The paper presents a theoretical development of the optical characteristics of photometers employing lenses. Both in-focus and out-of-focus operation are analyzed in detail for three types of photometers: a photometer employing fixed apertures, a photometer employing a simple lens, and a photometer employing a combination of a simple lens and a fixed aperture. The range of satisfactory operation of such instruments is carefully defined, as well as the gross errors which may occur in special applications.

© 1965 Optical Society of America

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References

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  1. P. Moon and D. E. Spencer, “Photometry,” Encyclopedia Britannica, (Encyclopedia Britannica Press, New York, 1948), Vol. 17, pp. 480–485.
  2. P. Moon, Scientific Basis of Illuminating Engineering (Dover Press, New York, 1961), p. 334.

Moon, P.

P. Moon and D. E. Spencer, “Photometry,” Encyclopedia Britannica, (Encyclopedia Britannica Press, New York, 1948), Vol. 17, pp. 480–485.

P. Moon, Scientific Basis of Illuminating Engineering (Dover Press, New York, 1961), p. 334.

Spencer, D. E.

P. Moon and D. E. Spencer, “Photometry,” Encyclopedia Britannica, (Encyclopedia Britannica Press, New York, 1948), Vol. 17, pp. 480–485.

Other (2)

P. Moon and D. E. Spencer, “Photometry,” Encyclopedia Britannica, (Encyclopedia Britannica Press, New York, 1948), Vol. 17, pp. 480–485.

P. Moon, Scientific Basis of Illuminating Engineering (Dover Press, New York, 1961), p. 334.

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Figures (16)

Fig. 1
Fig. 1

Aperture photometer, (P) photoelectric receiver.

Fig. 2
Fig. 2

Minimum radius R0 of object for measurement at distance p0 with aperture photometer. (Ra) radius of aperture in photometer.

Fig. 3
Fig. 3

Simple-lens photometer.

Fig. 4
Fig. 4

Focus condition for simple-lens photometer.

Fig. 5
Fig. 5

Calibration curve for lens photometer.

Fig. 6
Fig. 6

Distance of intersection of extreme rays from in-focus source, as function of distance.

Fig. 7
Fig. 7

Angular radius of field of lens photometer, as function of distance.

Fig. 8
Fig. 8

Radius of in-focus source (dashed line). Radii of out-of-focus sources, at distances (p) shown (solid lines).

Fig. 9
Fig. 9

Derivation of radii of out-of-focus sources.

Fig. 10
Fig. 10

Fixed-focus operation of lens photometer.

Fig. 11
Fig. 11

Fixed-aperture, simple-lens photometer.

Fig. 12
Fig. 12

Calibration curve for fixed-aperture, simple-lens photometer.

Fig. 13
Fig. 13

Location of image of fixed aperture, as function of distance from lens of in-focus source; f=16, δ=8.

Fig. 14
Fig. 14

Location of intersection of extreme rays for typical fixed-aperture photometer.

Fig. 15
Fig. 15

Angular radius of field of typical fixed-aperture photometer.

Fig. 16
Fig. 16

Out-of-focus field of view of typical fixed-aperture, simple-lens photometer.

Equations (67)

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F = ( π H / 2 ) { s 2 + R a 2 + R r 2 - [ ( s 2 + R a 2 + R r 2 ) 2 - 4 R a 2 R r 2 ] 1 2 } .
4 R a 2 R r 2 s 2 + R a 2 + R r 2 ,
F = π H [ R a 2 R r 2 / ( s 2 + R a 2 + R r 2 ) ] .
H = F [ ( s 2 + R a 2 + R r 2 ) / π R a 2 R r 2 ] ,
R a / ( s - a ) = R r / a ,
a = s R r / ( R a + R r ) .
tan α = R r / a .
α = tan - 1 [ ( R a + R r ) / s ] .
R 0 = R a + p 0 tan α ,
R 0 = R a + p 0 [ ( R a + R r ) / s ] .
q r = p f / ( p - f ) .
p f ,             q r f .
f q r 2 f ,
2 f p < ,
F = ( π τ H / 2 ) × { q r 2 + R 2 + R r 2 - [ ( q r 2 + R 2 + R r 2 ) 2 - 4 R 2 R r 2 ] 1 2 } ,
F ( p ) = π τ H 2 ( ( p f p - f ) 2 + R 2 + R r 2 - { [ ( p f p - f ) 2 + R 2 + R r 2 ] 2 - ( 4 R 2 r R 2 ) } 1 2 ) .
F ( ) = ( π τ H / 2 ) × { f 2 + R 2 + R r 2 - [ ( f 2 + R 2 + R r 2 ) 2 - 4 R 2 R r 2 ] 1 2 } .
F ( p ) F ( ) = ( f / R 1 - f / p ) 2 + ( R r R ) 2 + 1 - { [ ( f / R 1 - f / p ) 2 + ( R r R ) 2 + 1 ] 2 - 4 ( R r R ) 2 } 1 2 ( f / R ) 2 + ( R r / R ) 2 + 1 - { [ ( f / R ) 2 + ( R r / R ) 2 + 1 ] 2 - 4 ( R r / R ) 2 } 1 2 .
R r / R 1 + ( R r / R ) 2 + [ f / R / ( 1 - f / p ) ] 2 ,
F ( p ) F ( ) = 1 + ( R r / R ) 2 + ( f / R ) 2 1 + ( R r / R ) 2 + [ f / R / ( 1 - f / p ) ] 2 .
tan α = R / a = R s / ( p - a ) ,
R r / q r = R s / p .
R / a = p R r / ( p - a ) q r .
R / a = R r ( p - f ) / f ( p - a ) .
a = R f p / [ ( R - R r ) f + R r p ] .
f a ( R / R r ) f .
α = tan - 1 [ ( R - R r ) / p + R r / f ] .
α = tan - 1 R r / f ,
R s = p R r / f - R r .
( R 0 + R ) / p 0 = R / a
R 0 = ( R / a ) p 0 - R .
R 0 = [ ( R - R r ) p + R r f ] p 0 - R ,             p 0 > p .
( R - R 0 ) / p 0 = ( R - R s ) / p
R 0 = ( p 0 / p ) R s - R ) + R .
R 0 = [ R r f - ( R + R r ) p ] p 0 + R ,             p 0 < p .
( R - R a ) / q ac = ( R + R r ) / q rc .
R = ( R a q rc + R r q ac ) / ( q rc - q ac ) .
q rc = p c f / ( p c - f )
q ac = q rc - s = [ p c f / ( p c - f ) ] - s ,
R = p c [ f ( R a + R r ) - s R r ] + f s R r s ( p c - f ) ,
p c = f s ( R + R r ) s ( R + R r ) - f ( R a + R r ) .
F = ( π r H / 2 ) { s 2 + R a 2 + R r 2 - [ ( s 2 + R a 2 + R r 2 ) 2 - 4 R a 2 R r 2 ] 1 2 } .
p a = q a f / ( q a - f )
q a = p f / ( p - f ) - s ,
p a = [ p f ( f - s ) + f 2 s ] / [ f ( f + s ) - p s ] .
p = f ( f + s ) / s .
p a = - [ f ( f - s ) / s ] .
tan α = R ia / ( p a - a ) = R s / ( a - p ) ,
R ia = ( p a / q a ) R a .
a = ( p R ia + p a R s ) / ( R ia + R s )
R s = ( p / f - 1 ) R r
R ia = R a ( p a / f - 1 ) .
a = f 2 s R r + f [ f R a + ( f - s ) R r ] p f [ f R a + ( f + s ) R r ] - s R r p .
a - ( f / s R r ) [ f R a + ( f - s ) R r ] .
α = tan - 1 { f [ f R a + ( f + s ) R r ] - s R r p f s ( p - f ) } .
p = f [ f R a + ( f + s ) R r ] / s R r ,
tan α = ( R 0 - R ia ) / ( p 0 - p a ) = R s / ( a - p ) ,
R 0 = R ia + R s [ ( p 0 - p a ) / ( a - p ) ] .
R 0 = p 0 [ f [ f R a + ( f + s ) R r ] - s R r p f s ( p - f ) ] - ( p f R a + [ f ( s + p ) - s p ] R r ) s ( p - f ) ,
tan α = R s / ( a - p ) = R 0 / ( a - p 0 ) ,
R 0 = R s ( a - p 0 ) / ( a - p ) .
R 0 = f 2 s R r + f [ f R a + ( f - s ) R r ] p - p 0 { f [ f R a + ( f + s ) R r ] - s R r p } f s ( p - f ) ,
( R ia - R 0 ) / ( p a - p 0 ) = ( R ia - R s ) / ( p a - p ) .
R 0 = R ia + ( R s - R ia ) ( p a - p 0 p a - p ) .
R 0 = 1 s ( p - f ) { R r [ p ( f - s ) + f s ] - R a p f - p 0 [ R r f ( f + s ) - p s f - R a f ] } ,
p a - [ f ( f - s ) / s ] ,
R 0 - p 0 ( R r / f ) - [ f R a + ( f - s ) R r ] / s .