Abstract

Each surface in an optical system reflects a small fraction of the incident flux, which thus is lost to the image; these surfaces singly and in combination form ghost images of the light source, in various locations in and about the system. The number of first-order ghosts is equal to the number of surfaces, and the number of second-order ghosts equals the number of pairs of surfaces. The total flux through the ghosts is shown to be F0ρ(1−t2n)/1−t2 for first-order ghosts, and

ρ2tn2n(i-1)t2(n-i)=ρ2tn1-t2[n-1-t2n1-t2]

for the double reflection second-order ghosts, where n is the number of surfaces at which reflection can take place, t the transmittance, and ρ the reflectance at each surface. The left member has a simple interpretation.

In the general case of s interreflections, 1<sn, combinational theory indicates an expression:

n!(n-2)!2!+2n!(n-3)!3!+3n!(n-4)!4!++(s-1)n!(n-s)!s!

for the number of ghosts, since (a) interreflection potentially can happen an unlimited number of times in any group, so that all multiplets of order less than s must be included, and (b) first reflection is prohibited at the first surface in all multiplets, because of the direction of light flow.

The relations are applied to a system of triplet condensers and projection lenses, coated and uncoated in all combinations. Even though the ghost flux is minimum in the condenser-unfilmed, objective-filmed objective, it offers no advantages over the all-filmed case, because of the lower transmission.

© 1949 Optical Society of America

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Corrections

Allen E. Murray, "Erratum: Reflected Light and Ghosts in Optical Systems," J. Opt. Soc. Am. 39, 356-356 (1949)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-39-5-356

References

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  1. L. B. Tuckerman, “On the intensity of light reflected from or transmitted through a pile of plates,” J. Opt. Soc. Am. 37, 818 (1947).
    [CrossRef] [PubMed]
  2. Frank Benford, “Interface reflections between plane parallel surfaces,” Gen. Elec. Rev. 38, 277 (1935).
  3. T. Y. Baker, “Transmission of light through a pile of plates,” Trans. Opt. Soc. 22, 88 (1920–1921).
    [CrossRef]
  4. C. Tuttle and H. E. White, “Factors which affect the contrast of a lens image in the motion picture camera,” Trans. S.M.P.E. 31, 591 (1927).
  5. E. Goldberg, Der Aufbau des Photographischen Bildes (Wilhelm Knapp, Saale, 1922), p. 24et seq.
  6. C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Company, New York, 1942), p. 788et seq.

1947 (1)

1935 (1)

Frank Benford, “Interface reflections between plane parallel surfaces,” Gen. Elec. Rev. 38, 277 (1935).

1927 (1)

C. Tuttle and H. E. White, “Factors which affect the contrast of a lens image in the motion picture camera,” Trans. S.M.P.E. 31, 591 (1927).

Baker, T. Y.

T. Y. Baker, “Transmission of light through a pile of plates,” Trans. Opt. Soc. 22, 88 (1920–1921).
[CrossRef]

Benford, Frank

Frank Benford, “Interface reflections between plane parallel surfaces,” Gen. Elec. Rev. 38, 277 (1935).

Goldberg, E.

E. Goldberg, Der Aufbau des Photographischen Bildes (Wilhelm Knapp, Saale, 1922), p. 24et seq.

Mees, C. E. K.

C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Company, New York, 1942), p. 788et seq.

Tuckerman, L. B.

Tuttle, C.

C. Tuttle and H. E. White, “Factors which affect the contrast of a lens image in the motion picture camera,” Trans. S.M.P.E. 31, 591 (1927).

White, H. E.

C. Tuttle and H. E. White, “Factors which affect the contrast of a lens image in the motion picture camera,” Trans. S.M.P.E. 31, 591 (1927).

Gen. Elec. Rev. (1)

Frank Benford, “Interface reflections between plane parallel surfaces,” Gen. Elec. Rev. 38, 277 (1935).

J. Opt. Soc. Am. (1)

Trans. Opt. Soc. (1)

T. Y. Baker, “Transmission of light through a pile of plates,” Trans. Opt. Soc. 22, 88 (1920–1921).
[CrossRef]

Trans. S.M.P.E. (1)

C. Tuttle and H. E. White, “Factors which affect the contrast of a lens image in the motion picture camera,” Trans. S.M.P.E. 31, 591 (1927).

Other (2)

E. Goldberg, Der Aufbau des Photographischen Bildes (Wilhelm Knapp, Saale, 1922), p. 24et seq.

C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Company, New York, 1942), p. 788et seq.

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Figures (2)

Fig. 1
Fig. 1

First-order ghosts. The arrows indicate the direction of light flow.

Fig. 2
Fig. 2

Second-order ghosts. The n−1 ghosts whose initial reflection occurs at the last surface. The arrows indicate the direction of flow of the flux.

Tables (2)

Tables Icon

Table I Ghost flux in combination filmed and unfilmed triplet systems.

Tables Icon

Table II Reciprocal ghost flux, transmittance, and contrast relative to unfilmed optical systems.

Equations (18)

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ρ 2 t n 2 n ( i - 1 ) t 2 ( n - i ) = ρ 2 t n 1 - t 2 [ n - 1 - t 2 n 1 - t 2 ]
n ! ( n - 2 ) ! 2 ! + 2 n ! ( n - 3 ) ! 3 ! + 3 n ! ( n - 4 ) ! 4 ! + + ( s - 1 ) n ! ( n - s ) ! s !
1 - ρ / 1 + ( n - 1 ) ρ ,
( 1 - ρ ) / ( 1 + ( n - 1 ) ρ ) - ( 1 - ρ ) n = ( t ) / ( 1 + ( n - 1 ) ρ ) - t n ,
F 0 ρ [ t 2 ( n - 1 ) + + t 4 + t 2 + 1 ] = F 0 ρ i n t 2 ( k - 1 ) .
( 1 - x s + 1 ) / ( 1 - x ) = 1 + x + x 2 + + x s             ( x 2 < 1 ) ,
F 0 ρ 1 n t 2 ( k - 1 ) = F 0 ρ ( 1 - t 2 n ) / ( 1 - t 2 ) .
n ! / ( n - 2 ) ! 2 ! = 1 2 n ( n - 1 ) ,
ρ 2 t n [ 1 + t 2 + t 4 + + t 2 ( n - 2 ) ] = ρ 2 t n 0 n - 2 t 2 i = ρ 2 t n [ 1 - t 2 ( n - 1 ) 1 - t 2 ]
ρ 2 t 4 [ 1 + t 2 + t 4 + + t 2 ( n - 3 ) ] = ρ 2 t n 0 n - 3 t 2 i = ρ 2 t n [ 1 - t 2 ( n - 2 ) 1 - t 2 ] .
ρ 2 t n [ 1 + t 2 + t 4 + + t 2 ( n - p - 1 ) ] = ρ 2 t n 0 n - ( p + 1 ) t 2 i = ρ 2 t n [ 1 - t 2 ( n - p ) 1 - t 2 ] .
ρ 2 t n p = 1 n - 1 i = 0 n - ( p + 1 ) t 2 i = ρ 2 t n 1 n - 1 1 - t 2 ( n - p ) 1 - t 2 = ρ 2 t n 1 - t 2 [ ( n - 1 ) - 1 n - 1 t 2 ( n - p ) ] ,
ρ 2 t n / ( 1 - t 2 ) [ n - ( 1 - t 2 n ) / ( 1 - t 2 ) ]
2 n ( i - 1 ) t 2 ( n - i ) .
ρ 2 t n 2 n ( i - 1 ) t 2 ( n - i ) = ρ 2 t n 1 - t 2 [ n - 1 - t 2 n 1 - t 2 ] .
ρ 2 t 6 2 6 ( i - 1 ) t 2 ( 6 - i ) = ρ 2 t 6 [ t 8 + 2 t 6 + 3 t 4 + 4 t 2 + 5 ] .
n ! ( n - 2 ) ! 2 ! + 2 n ! ( n - 3 ) ! 3 ! + 3 n ! ( n - 4 ) ! 4 ! + + ( s - 1 ) n ! ( n - s ) ! s ! . .
ρ = 0.04 , t = 0.96 , ρ = 0.01 , t = 0.99 ,