Abstract

The distortion of the image formed by a prism, principally an ophthalmic prism, is shown to be adequately described quantitatively by five parameters. Expressions are derived which relate the physical constants of the prism and the position of the prism before the eye to these five parameters. Comparisons of measurements and theory are given. This theory is useful for the study of this distortion and for designing prisms with a minimum of certain aspects of this distortion.

© 1951 Optical Society of America

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References

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  1. H. Hartinger, Z. Ophth. Optik. 15–17, 129–146 (1928–1929).
  2. Adelbert Ames, Am. J. Psychol. 59, 333–357 (1946).
    [CrossRef] [PubMed]
  3. K. N. Ogle, Researches in Binocular Vision (W. B. Saunders Company, Philadelphia, 1950).
  4. W. B. Lancaster, Trans. Am. Ophth. Soc. 46, 262–283 (1948).
  5. P. W. Miles, Am. J. Ophth. 34, 87–93 (1951).
  6. A. C. Hardy and F. H. Perrin, The Principles of Optics (Mc-Graw-Hill Book Company, Inc., New York, 1932), first edition, p. 551.
  7. Ames, Ogle, and Gliddon, J. Opt. Soc. Am. 22, 538, 575 (1932).
    [CrossRef]
  8. Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).
  9. L. G. Hardy, Arch. Ophthalmol. 34, 16–23 (1945).
    [CrossRef]
  10. K. N. Ogle and A. Ames, J. Opt. Soc. Am. 33, 137–142 (1943).
    [CrossRef]

1951 (1)

P. W. Miles, Am. J. Ophth. 34, 87–93 (1951).

1948 (1)

W. B. Lancaster, Trans. Am. Ophth. Soc. 46, 262–283 (1948).

1946 (1)

Adelbert Ames, Am. J. Psychol. 59, 333–357 (1946).
[CrossRef] [PubMed]

1945 (1)

L. G. Hardy, Arch. Ophthalmol. 34, 16–23 (1945).
[CrossRef]

1943 (1)

1932 (2)

Ames, Ogle, and Gliddon, J. Opt. Soc. Am. 22, 538, 575 (1932).
[CrossRef]

Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).

Ames,

Ames, Ogle, and Gliddon, J. Opt. Soc. Am. 22, 538, 575 (1932).
[CrossRef]

Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).

Ames, A.

Ames, Adelbert

Adelbert Ames, Am. J. Psychol. 59, 333–357 (1946).
[CrossRef] [PubMed]

Gliddon,

Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).

Ames, Ogle, and Gliddon, J. Opt. Soc. Am. 22, 538, 575 (1932).
[CrossRef]

Hardy, A. C.

A. C. Hardy and F. H. Perrin, The Principles of Optics (Mc-Graw-Hill Book Company, Inc., New York, 1932), first edition, p. 551.

Hardy, L. G.

L. G. Hardy, Arch. Ophthalmol. 34, 16–23 (1945).
[CrossRef]

Hartinger, H.

H. Hartinger, Z. Ophth. Optik. 15–17, 129–146 (1928–1929).

Lancaster, W. B.

W. B. Lancaster, Trans. Am. Ophth. Soc. 46, 262–283 (1948).

Miles, P. W.

P. W. Miles, Am. J. Ophth. 34, 87–93 (1951).

Ogle,

Ames, Ogle, and Gliddon, J. Opt. Soc. Am. 22, 538, 575 (1932).
[CrossRef]

Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).

Ogle, K. N.

K. N. Ogle and A. Ames, J. Opt. Soc. Am. 33, 137–142 (1943).
[CrossRef]

K. N. Ogle, Researches in Binocular Vision (W. B. Saunders Company, Philadelphia, 1950).

Perrin, F. H.

A. C. Hardy and F. H. Perrin, The Principles of Optics (Mc-Graw-Hill Book Company, Inc., New York, 1932), first edition, p. 551.

Am. J. Ophth. (1)

P. W. Miles, Am. J. Ophth. 34, 87–93 (1951).

Am. J. Psychol. (1)

Adelbert Ames, Am. J. Psychol. 59, 333–357 (1946).
[CrossRef] [PubMed]

Ann. Distinguished Serv. Found. Optom. (1)

Ames, Ogle, and Gliddon, Ann. Distinguished Serv. Found. Optom. 1, 61–70 (1932).

Arch. Ophthalmol. (1)

L. G. Hardy, Arch. Ophthalmol. 34, 16–23 (1945).
[CrossRef]

J. Opt. Soc. Am. (2)

Trans. Am. Ophth. Soc. (1)

W. B. Lancaster, Trans. Am. Ophth. Soc. 46, 262–283 (1948).

Z. Ophth. Optik. (1)

H. Hartinger, Z. Ophth. Optik. 15–17, 129–146 (1928–1929).

Other (2)

K. N. Ogle, Researches in Binocular Vision (W. B. Saunders Company, Philadelphia, 1950).

A. C. Hardy and F. H. Perrin, The Principles of Optics (Mc-Graw-Hill Book Company, Inc., New York, 1932), first edition, p. 551.

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Figures (6)

Fig. 1
Fig. 1

The image pattern depicted in this drawing must not be taken literally but to indicate only a directional pattern and not a specific spatially located image. This “directional” image pattern will change with distance and orientation of the prism before the eye. The object pattern will be considered to be at a great distance from the eye.

Fig. 2
Fig. 2

The principal section of a prism.

Fig. 3
Fig. 3

Geometric relations involved in determining the course of a ray through the back surface of the prism.

Fig. 4
Fig. 4

Geometric relations involved in determining the course of a ray through the front surface of the prism.

Fig. 5
Fig. 5

Magnitude of the unsymmetric angular magnification B, for afocal prisms of various prism power for a distance of eye to lens of 15 mm, which have different flexures as defined by the surface power of the front surface.

Fig. 6
Fig. 6

A comparison of measured angular magnification for different azimuth angles, and that calculated from theory, for various vertex distances for a 5.1 prism-diopter afocal prism, ground on a 9 diopter front surface curve. The open circles represent actual measurements.

Tables (1)

Tables Icon

Table I Comparative calculated and measured values for an afocal 5 prism-diopter prism ground with a back surface power of 6.25 diopters.

Equations (40)

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α = A α + B α 2 + C β 2 β = D β + E α β .
sin η = c sin θ = sin Δ / [ n 2 + 1 - 2 n cos Δ ] 1 2 ,
cos γ = cos ω cos , cos γ = cos ω cos ,
sin sin γ = sin sin γ .
sin i = ( R 2 - h ) sin γ / R 2 ,             n sin i = sin r , γ = γ - ( i - r )             and             s = E C 2 = ( R 2 - h ) sin γ / n sin γ .
γ = a γ
a = [ R 2 + ( n - 1 ) h ] / n R 2 .
ω = a ω ,             = a             and             s = ( R 2 - h ) / n a .
tan ρ = q sin θ / ( s - q cos θ ) ,
cot ρ = d / a k - cot θ ,
cos χ = cos ( ρ - ω ) cos
sin i = p sin χ / R 1 ,             in which             p = C 1 E
sin r = n sin i
χ = χ + ( r - i ) ,
sin = sin sin χ / sin χ
cos ( ρ - ω x ) cos = cos χ .
ω = ω x + Δ .
cos χ = sin ( 90° - χ ) = cos ρ + sin ρ [ a ω - 1 2 N a 2 ω 2 - 1 2 N a 2 2 + ] .
sin y = x + δ ,
y = arcsin ( x + δ ) = arcsin x + δ [ 1 - x 2 ] - 1 2 + 1 2 x δ 2 [ 1 - x 2 ] - 3 2 + .
χ = ρ - [ a ω - 1 2 N a 2 2 ] .
sin χ = sin ρ [ 1 - N a ω - 1 2 a 2 ω 2 + 1 2 N 2 a 2 2 ] .
sin i = sin η [ 1 - N a ω - 1 2 a 2 ω 2 + 1 2 N 2 a 2 2 ] .
χ = ( ρ + Δ ) - a ( 1 + N Q ) ω + 1 2 a 2 ( N 2 P - Q ) ω 2 + 1 2 a 2 N [ 1 + N Q ] 2 .
= S a + a 2 [ S N - T ( 1 + N Q ) ] ω .
S = cos Δ + N sin Δ ;             and             T = N cos Δ - sin Δ .
cos χ = cos ( ρ + Δ ) + u [ sin ( ρ + Δ ) ] ω - [ v sin ( ρ + Δ ) + 1 2 u 2 cos ( ρ + Δ ) ] ω 2 - w sin ( ρ + Δ ) 2 .
cos ( ρ - ω x ) = cos ( ρ + Δ ) - { u sin ( ρ + Δ ) ω - [ v sin ( ρ + Δ ) + 1 2 u 2 cos ( ρ + Δ ) ] ω 2 + [ w sin ( ρ + Δ ) - 1 2 S 2 a 2 cos ( ρ + Δ ) ] 2 } .
ρ - ω x = ( ρ + Δ ) + u ω + v ω 2 + [ w - 1 2 S 2 a 2 cot 2 ( ρ + Δ ) ] 2 .
ω = a ( 1 + N Q ) ω - 1 2 a 2 ( N 2 P - Q ) ω 2 - 1 2 a 2 [ N ( 1 + N Q ) - S T ] 2 .
A = 1 / a ( 1 + N Q ) B = 1 2 a 2 A [ N 2 P - Q ] C = 1 2 D a [ N D - T A ] D = 1 / a S E = a D [ T D - N A ] .
A = ( n cos Δ - 1 ) / ( n cos Δ ) ;             B = 1 2 A ( n 2 - A 2 ) ( n sin Δ ) / n 2 ( n cos Δ - 1 ) ; C = 1 2 ( n 2 - 1 ) sin Δ / n ( n - cos Δ ) ;             D = 1 ;             E = 0.
Δ e = A Δ + B Δ 2 .
α = α Δ + Δ ,             and             α = α Δ + Δ
α e = ( A + 2 B Δ ) α e + B α e 2 + C β 2 β e = ( D + E Δ ) β + E α e β .
a = [ R 2 - ( n - 1 ) h ] / n R 2 N = cot θ + [ R 2 ( R 2 + h ) ] / R 1 [ R 2 - ( n - 1 ) h ] k k = sin Δ / [ 1 + n 2 - 2 n cos Δ ] 1 2 S = cos Δ + N sin Δ T = N cos Δ - sin Δ Q = tan η - tan η P = tan 3 η - tan 3 η η - η = Δ sin η = n sin η
cot θ = [ ( R 2 2 - r 2 ) 1 2 + t e - R 1 ( 1 - L 2 ) 1 2 ] / R 1 k ,
R 1 [ ( 1 - L 2 ) 1 2 - k ( Q / P ) 1 2 ] = ( R 2 2 - r 2 ) 1 2 + t e + R 2 ( R 2 + h ) / [ R 2 - ( n - 1 ) h ] R 1 [ k ( Q / P ) 1 2 + n / ( n - 1 ) - ( 1 - k 2 ) 1 2 ] = R 2 [ 1 - n / ( n - 1 - R 2 V 0 ) - ( R 2 + h ) / { R 2 - ( n - 1 ) h } ]
V 0 = D 1 / ( 1 - D 1 t / n ) + D 2 t = R 1 ( 1 - k 2 ) 1 2 + R 1 k cot θ + R 2 .
α / α = ( A + 2 B Δ ) + B α .