Abstract

The investigations of Cruikshank have brought to the attention of American scientists different methods of analyzing the optical image, methods which were also suggested by A. Kerber and the author. However, all these methods are, for the most part, restricted to the analysis of meridional rays.

In previous papers the author has shown the importance of the analysis of the diapoint configuration for the optical image. He here gives a simple formula to show how the diapoint deviations can be considered as the sum of errors originating at the single surfaces. In this way, and with the help of a series of diagrams, the lens designer can obtain an accurate picture of the effect of the single surface on all the rays traversing the optical system. Diagrams for two photographic lenses are included.

© 1948 Optical Society of America

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References

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  1. F. Cruikshank, “A system of transfer coefficients for use in the design of lens systems,” Proc. Phys. Soc. 57, 350, 362, 414, 426, 430 (1945).
    [Crossref]
  2. H. L. M’Auley, “A transfer method for deriving the effect on the image,” Proc. Phys. Soc. 57, 435 (1945).
  3. L. Seidel, “Trigonometrie der Formel für den allgemeinen Fall,” Münchener Sitzber. (2)263 (1866).
  4. A. Kerber, Beiträge zur Dioptrik (Leipzig, 1895/99).
  5. K. Schwarzschild, Über differenzen Formeln zur Durchrechnung optischer Systeme, (Göttingen, 1907).
  6. F. Nusl, “Über allgemeine differenzen Formeln,” Bull. Int. Acad. Sci. de Bohême,  12, 84 (1907).
  7. T. Smith, “The general form of the Smith-Helmholtz equation,” Trans. Opt. Soc. 31, 241 (1929).
    [Crossref]
  8. T. Smith, “Variational formulae in optics,” Proc. Phys. Soc. 57, 558 (1945).
    [Crossref]
  9. M. Herzberger, “Über die Durchrechnung von Strahlen durch optische Systeme,” Zeits. f. Physik 43, 750 (1927).
    [Crossref]
  10. M. Herzberger, “A new theory of optical image formation,” J. Opt. Soc. Am. 26, 197 (1936).
    [Crossref]
  11. M. Herzberger, “The limitations of optical image formation,” Ann. N. Y. Acad. Sci. 48, 1 (1946).
  12. F. Staeble, “Über den Zusammenhang von Koma und Sinusbedingung,” Zeits. f. Instrumenten. 27, 241 (1907).
  13. E. Lihotzky, “Verallgemeinerung der Abbe’schen Sinusbedingung,” Wiener Sitzber. 128, 85 (1919).
  14. M. Herzberger, “A direct image error theory,” Quart. App. Math. 1, 69 (1943).
  15. M. Herzberger, “Light distribution in the optical image,” J. Opt. Soc. Am. 37, 485 (1947).
    [Crossref] [PubMed]

1947 (1)

1946 (1)

M. Herzberger, “The limitations of optical image formation,” Ann. N. Y. Acad. Sci. 48, 1 (1946).

1945 (3)

T. Smith, “Variational formulae in optics,” Proc. Phys. Soc. 57, 558 (1945).
[Crossref]

F. Cruikshank, “A system of transfer coefficients for use in the design of lens systems,” Proc. Phys. Soc. 57, 350, 362, 414, 426, 430 (1945).
[Crossref]

H. L. M’Auley, “A transfer method for deriving the effect on the image,” Proc. Phys. Soc. 57, 435 (1945).

1943 (1)

M. Herzberger, “A direct image error theory,” Quart. App. Math. 1, 69 (1943).

1936 (1)

1929 (1)

T. Smith, “The general form of the Smith-Helmholtz equation,” Trans. Opt. Soc. 31, 241 (1929).
[Crossref]

1927 (1)

M. Herzberger, “Über die Durchrechnung von Strahlen durch optische Systeme,” Zeits. f. Physik 43, 750 (1927).
[Crossref]

1919 (1)

E. Lihotzky, “Verallgemeinerung der Abbe’schen Sinusbedingung,” Wiener Sitzber. 128, 85 (1919).

1907 (2)

F. Staeble, “Über den Zusammenhang von Koma und Sinusbedingung,” Zeits. f. Instrumenten. 27, 241 (1907).

F. Nusl, “Über allgemeine differenzen Formeln,” Bull. Int. Acad. Sci. de Bohême,  12, 84 (1907).

1866 (1)

L. Seidel, “Trigonometrie der Formel für den allgemeinen Fall,” Münchener Sitzber. (2)263 (1866).

Cruikshank, F.

F. Cruikshank, “A system of transfer coefficients for use in the design of lens systems,” Proc. Phys. Soc. 57, 350, 362, 414, 426, 430 (1945).
[Crossref]

Herzberger, M.

M. Herzberger, “Light distribution in the optical image,” J. Opt. Soc. Am. 37, 485 (1947).
[Crossref] [PubMed]

M. Herzberger, “The limitations of optical image formation,” Ann. N. Y. Acad. Sci. 48, 1 (1946).

M. Herzberger, “A direct image error theory,” Quart. App. Math. 1, 69 (1943).

M. Herzberger, “A new theory of optical image formation,” J. Opt. Soc. Am. 26, 197 (1936).
[Crossref]

M. Herzberger, “Über die Durchrechnung von Strahlen durch optische Systeme,” Zeits. f. Physik 43, 750 (1927).
[Crossref]

Kerber, A.

A. Kerber, Beiträge zur Dioptrik (Leipzig, 1895/99).

Lihotzky, E.

E. Lihotzky, “Verallgemeinerung der Abbe’schen Sinusbedingung,” Wiener Sitzber. 128, 85 (1919).

M’Auley, H. L.

H. L. M’Auley, “A transfer method for deriving the effect on the image,” Proc. Phys. Soc. 57, 435 (1945).

Nusl, F.

F. Nusl, “Über allgemeine differenzen Formeln,” Bull. Int. Acad. Sci. de Bohême,  12, 84 (1907).

Schwarzschild, K.

K. Schwarzschild, Über differenzen Formeln zur Durchrechnung optischer Systeme, (Göttingen, 1907).

Seidel, L.

L. Seidel, “Trigonometrie der Formel für den allgemeinen Fall,” Münchener Sitzber. (2)263 (1866).

Smith, T.

T. Smith, “Variational formulae in optics,” Proc. Phys. Soc. 57, 558 (1945).
[Crossref]

T. Smith, “The general form of the Smith-Helmholtz equation,” Trans. Opt. Soc. 31, 241 (1929).
[Crossref]

Staeble, F.

F. Staeble, “Über den Zusammenhang von Koma und Sinusbedingung,” Zeits. f. Instrumenten. 27, 241 (1907).

Ann. N. Y. Acad. Sci. (1)

M. Herzberger, “The limitations of optical image formation,” Ann. N. Y. Acad. Sci. 48, 1 (1946).

Bull. Int. Acad. Sci. de Bohême (1)

F. Nusl, “Über allgemeine differenzen Formeln,” Bull. Int. Acad. Sci. de Bohême,  12, 84 (1907).

J. Opt. Soc. Am. (2)

Münchener Sitzber. (2) (1)

L. Seidel, “Trigonometrie der Formel für den allgemeinen Fall,” Münchener Sitzber. (2)263 (1866).

Proc. Phys. Soc. (3)

F. Cruikshank, “A system of transfer coefficients for use in the design of lens systems,” Proc. Phys. Soc. 57, 350, 362, 414, 426, 430 (1945).
[Crossref]

H. L. M’Auley, “A transfer method for deriving the effect on the image,” Proc. Phys. Soc. 57, 435 (1945).

T. Smith, “Variational formulae in optics,” Proc. Phys. Soc. 57, 558 (1945).
[Crossref]

Quart. App. Math. (1)

M. Herzberger, “A direct image error theory,” Quart. App. Math. 1, 69 (1943).

Trans. Opt. Soc. (1)

T. Smith, “The general form of the Smith-Helmholtz equation,” Trans. Opt. Soc. 31, 241 (1929).
[Crossref]

Wiener Sitzber. (1)

E. Lihotzky, “Verallgemeinerung der Abbe’schen Sinusbedingung,” Wiener Sitzber. 128, 85 (1919).

Zeits. f. Instrumenten. (1)

F. Staeble, “Über den Zusammenhang von Koma und Sinusbedingung,” Zeits. f. Instrumenten. 27, 241 (1907).

Zeits. f. Physik (1)

M. Herzberger, “Über die Durchrechnung von Strahlen durch optische Systeme,” Zeits. f. Physik 43, 750 (1927).
[Crossref]

Other (2)

A. Kerber, Beiträge zur Dioptrik (Leipzig, 1895/99).

K. Schwarzschild, Über differenzen Formeln zur Durchrechnung optischer Systeme, (Göttingen, 1907).

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Figures (4)

Fig. 1
Fig. 1

Vignetting diagram for an optical system.

Fig. 2
Fig. 2

Plot of diamagnification contribution of four surfaces.

Fig. 3
Fig. 3

Plot of asymmetry contribution for two radii and two thicknesses.

Fig. 4
Fig. 4

Diamagnification and asymmetry error for one system: System A (upper curves) is the final form, and system B (lower curve) is a form before final corection.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

y = m y , z = m z + S ,
y = f η ζ z = f + S ,
S = α i c i .
y 1 = m 1 y 1 , z = m 1 z 1 ,
τ = y η + z ζ , Q 2 = μ 2 ( r 2 - x 2 - y 2 ) + τ 2 , Q 2 = μ 2 ( r 2 - x 2 - y 2 ) + τ 2 ,
λ = 1 μ 2 ( Q - τ ) , Φ = 1 r 2 ( Q - Q ) ,
m = ( 1 ) / ( 1 + λ Φ ) .
y 2 = m 2 y 2 , and y 2 = y 1 , z 2 = m 2 z 2 , z 2 = z 1 - c 1 ,
m k m k - 1 m λ = M λ .
y = M 1 y 1 , z = M 1 z 1 + S ,
S = M 2 c 1 + M 3 c 2 + c k - 1 = 1 k - 1 M v + 1 c v .
c i = t i + r i + 1 - r i ,
S = M v + 1 t v + ( m v - 1 ) r v M v + 1 .
( m v - 1 ) r v = - ( Q - Q ) 1 + γ Φ γ ,
lim r ( m - 1 ) r - λ ( μ - μ ) ,