Abstract

In practical photography a lens must form a high-quality image at a variety of magnifications. It has, however, been known for more than a century that no lens can be designed that is perfect at more than one conjugate. It follows that lenses designed for a range of magnifications must have residual aberrations that cannot be corrected no matter how complicated the lens is made. In this paper it is shown how these residual aberrations can be calculated.

© 1989 Optical Society of America

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References

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  1. T. Smith, “The ideal photographic lens,” Nat. Phys. Lab. (UK) Collect Res. 10, 231–238 (1913).
  2. M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931).
    [CrossRef]
  3. C. Caratheodory, Geometrische Optik (Springer-Verlag, Berlin, 1937).
    [CrossRef]
  4. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).
  5. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).
  6. A. Walther, “Mock ray tracing,”J. Opt. Soc. Am. 60, 918–920 (1970).
    [CrossRef]
  7. J. C. Maxwell, “On the general laws of optical instruments,”Q. J. Pure Appl. Math. II, 271–285 (1858).
  8. H. Boegehold, M. Herzberger, “Kan man zwei verschiedene Flächen durch dieselbe Folge von Umdrehungsflächen scharf abbilden?” Compos. Math. 1, 488 (1935). See also M. Herzberger, Modern Geometrical Optics (Wiley, New York, 1952).
  9. H. Boegehold, M. Herzberger, “Kugelsymmetrische Systeme,” Z. Angew. Math. Mech. 15, 157–187 (1935).
    [CrossRef]
  10. F. Hekker, “On concentric optical systems,” doctoral dissertation (Delft Institute of Technology, Delft, The Netherlands, 1947).
  11. H. Bruns, “Das Eikonal,” Leipz. Sitzgsber. 21, 321–436 (1895).
  12. C. Carathéodory, “Über den Zusammenhang der Theorie der absoluten optischen Instrumenten met einem Satze der Variationsrechnung,” Sitzgsber. Bayer. Akad. Wiss. Math-Naturwiss.56, 1–18 (1926).
  13. E. Abbe, “Über die Bedingungen des Aplanatismus der linsensysteme,” Gesammelte Abh.Fischer, Jena, 1904–1906.
  14. J. F. W. Herschel, “On the aberrations of compound lenses and object glasses,” Philos. Trans. 111, 222–266 (1821).
    [CrossRef]
  15. See Ref. 2, pp. 163 – 166 .
  16. T. Smith, “The changes in aberration when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921–1922).
  17. One might also specify L and M and solve for L′ and M′. One advantage of solving for L and M is that the problem can be phrased as an optimization problem (see Ref. 18). It also allows us to choose L′ and M′ to be equally spaced, which helps to simplify diffraction calculations (see, e.g., Ref. 19).
  18. A. Walther, “Numerical techniques in eikonal function theory,” J. Opt. Soc. Am. A 5, 511–515 (1988).
    [CrossRef]
  19. A. Walther, “Lenses, wave optics, and eikonal functions,”J. Opt. Soc. Am. 59, 1325–1333 (1969).
    [CrossRef]
  20. W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

1988 (1)

1970 (1)

1969 (1)

1935 (2)

H. Boegehold, M. Herzberger, “Kan man zwei verschiedene Flächen durch dieselbe Folge von Umdrehungsflächen scharf abbilden?” Compos. Math. 1, 488 (1935). See also M. Herzberger, Modern Geometrical Optics (Wiley, New York, 1952).

H. Boegehold, M. Herzberger, “Kugelsymmetrische Systeme,” Z. Angew. Math. Mech. 15, 157–187 (1935).
[CrossRef]

1913 (1)

T. Smith, “The ideal photographic lens,” Nat. Phys. Lab. (UK) Collect Res. 10, 231–238 (1913).

1895 (1)

H. Bruns, “Das Eikonal,” Leipz. Sitzgsber. 21, 321–436 (1895).

1858 (1)

J. C. Maxwell, “On the general laws of optical instruments,”Q. J. Pure Appl. Math. II, 271–285 (1858).

1821 (1)

J. F. W. Herschel, “On the aberrations of compound lenses and object glasses,” Philos. Trans. 111, 222–266 (1821).
[CrossRef]

Abbe, E.

E. Abbe, “Über die Bedingungen des Aplanatismus der linsensysteme,” Gesammelte Abh.Fischer, Jena, 1904–1906.

Boegehold, H.

H. Boegehold, M. Herzberger, “Kan man zwei verschiedene Flächen durch dieselbe Folge von Umdrehungsflächen scharf abbilden?” Compos. Math. 1, 488 (1935). See also M. Herzberger, Modern Geometrical Optics (Wiley, New York, 1952).

H. Boegehold, M. Herzberger, “Kugelsymmetrische Systeme,” Z. Angew. Math. Mech. 15, 157–187 (1935).
[CrossRef]

Bruns, H.

H. Bruns, “Das Eikonal,” Leipz. Sitzgsber. 21, 321–436 (1895).

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

Caratheodory, C.

C. Caratheodory, Geometrische Optik (Springer-Verlag, Berlin, 1937).
[CrossRef]

Carathéodory, C.

C. Carathéodory, “Über den Zusammenhang der Theorie der absoluten optischen Instrumenten met einem Satze der Variationsrechnung,” Sitzgsber. Bayer. Akad. Wiss. Math-Naturwiss.56, 1–18 (1926).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Hekker, F.

F. Hekker, “On concentric optical systems,” doctoral dissertation (Delft Institute of Technology, Delft, The Netherlands, 1947).

Herschel, J. F. W.

J. F. W. Herschel, “On the aberrations of compound lenses and object glasses,” Philos. Trans. 111, 222–266 (1821).
[CrossRef]

Herzberger, M.

H. Boegehold, M. Herzberger, “Kugelsymmetrische Systeme,” Z. Angew. Math. Mech. 15, 157–187 (1935).
[CrossRef]

H. Boegehold, M. Herzberger, “Kan man zwei verschiedene Flächen durch dieselbe Folge von Umdrehungsflächen scharf abbilden?” Compos. Math. 1, 488 (1935). See also M. Herzberger, Modern Geometrical Optics (Wiley, New York, 1952).

M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

Maxwell, J. C.

J. C. Maxwell, “On the general laws of optical instruments,”Q. J. Pure Appl. Math. II, 271–285 (1858).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Smith, T.

T. Smith, “The changes in aberration when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921–1922).

T. Smith, “The ideal photographic lens,” Nat. Phys. Lab. (UK) Collect Res. 10, 231–238 (1913).

Teukolski, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Walther, A.

Compos. Math. (1)

H. Boegehold, M. Herzberger, “Kan man zwei verschiedene Flächen durch dieselbe Folge von Umdrehungsflächen scharf abbilden?” Compos. Math. 1, 488 (1935). See also M. Herzberger, Modern Geometrical Optics (Wiley, New York, 1952).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Leipz. Sitzgsber. (1)

H. Bruns, “Das Eikonal,” Leipz. Sitzgsber. 21, 321–436 (1895).

Nat. Phys. Lab. (UK) Collect Res. (1)

T. Smith, “The ideal photographic lens,” Nat. Phys. Lab. (UK) Collect Res. 10, 231–238 (1913).

Philos. Trans. (1)

J. F. W. Herschel, “On the aberrations of compound lenses and object glasses,” Philos. Trans. 111, 222–266 (1821).
[CrossRef]

Q. J. Pure Appl. Math. (1)

J. C. Maxwell, “On the general laws of optical instruments,”Q. J. Pure Appl. Math. II, 271–285 (1858).

Trans. Opt. Soc. (1)

T. Smith, “The changes in aberration when the object and stop are moved,” Trans. Opt. Soc. 23, 139–153 (1921–1922).

Z. Angew. Math. Mech. (1)

H. Boegehold, M. Herzberger, “Kugelsymmetrische Systeme,” Z. Angew. Math. Mech. 15, 157–187 (1935).
[CrossRef]

Other (10)

F. Hekker, “On concentric optical systems,” doctoral dissertation (Delft Institute of Technology, Delft, The Netherlands, 1947).

M. Herzberger, Strahlenoptik (Springer-Verlag, Berlin, 1931).
[CrossRef]

C. Caratheodory, Geometrische Optik (Springer-Verlag, Berlin, 1937).
[CrossRef]

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

One might also specify L and M and solve for L′ and M′. One advantage of solving for L and M is that the problem can be phrased as an optimization problem (see Ref. 18). It also allows us to choose L′ and M′ to be equally spaced, which helps to simplify diffraction calculations (see, e.g., Ref. 19).

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

See Ref. 2, pp. 163 – 166 .

C. Carathéodory, “Über den Zusammenhang der Theorie der absoluten optischen Instrumenten met einem Satze der Variationsrechnung,” Sitzgsber. Bayer. Akad. Wiss. Math-Naturwiss.56, 1–18 (1926).

E. Abbe, “Über die Bedingungen des Aplanatismus der linsensysteme,” Gesammelte Abh.Fischer, Jena, 1904–1906.

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Figures (8)

Fig. 1
Fig. 1

Full-field meridional trace, in micrometers, for a lens corrected perfectly for G = −0.2, at five different magnifications: 0, −0.125, −0.25, −.375, and −0.5 for curves A, B, C, D, and E, respectively. (This denotation of the curves holds for all subsequent figures.) The horizontal axis represents the pupil coordinate, as usual.

Fig. 2
Fig. 2

On-axis trace for the optimized system.

Fig. 3
Fig. 3

Optimized system: meridional trace for an image point 5 mm off axis.

Fig. 4
Fig. 4

Optimized system: meridional trace for an image point 10 mm off axis.

Fig. 5
Fig. 5

Optimized system: meridional trace for an image point 15 mm off axis.

Fig. 6
Fig. 6

Optimized system: meridional trace for an image point at full field, i.e., 20 mm off axis.

Fig. 7
Fig. 7

Optimized system: x component of the sagittal trace at full field.

Fig. 8
Fig. 8

Optimized system: y component of the sagittal trace at full field.

Tables (2)

Tables Icon

Table 1 Third-Order Spherical Aberration and Coma, as Functions of the Magnification for a Lens Perfectly Corrected at G = −0.2

Tables Icon

Table 2 Coefficients of akblcm in the Series Development of the Focal-Point Angle Eikonal for a Unit-Focal-Length Lens Perfect at a Magnification Ga

Equations (33)

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

( x 2 + y 2 ) - 2 r z + α z 2 = 0
r 1 = - r 2 = - f ( 1 - G 2 ) G ( 1 + G ω ) ,
r 1 = - r 2 = - f 1 - G 2 G + ω ,
α 1 = α 2 = 1 + G 2 ( 1 - ω 2 ) ( 1 + G ω ) 2 ,
α 1 = α 2 = 1 + 1 - ω 2 ( G + ω ) 2 .
W par = - f ( L L + M M ) .
x F = W L ,             y F = W M .
x F = - W L ,             y F = - W M .
x - L z / N = W L ,
y - M z / N = W M .
x = - W L + z L N ,
y = - W M + z M N .
z = f / G ,             z = - G f ,
x - G x = - ( W L + f L N ) - G ( W L + f L N )
a = ( L 2 + M 2 ) / 2 ,
b = L L + M M ,
c = ( L 2 + M 2 ) / 2.
W ( a , b , c ) parax = - f b .
K = a - G b + G 2 c .
W ( L , M , L , M ) = F ( K ) + f G ( 1 - 2 a ) 1 / 2 + f G ( 1 - 2 c ) 1 / 2 ,
F par = f K / G = f G a - f b + f G c .
F ( K ) = - ( 1 - G ) 2 G f [ 1 - 2 K ( 1 - G ) 2 ] 1 / 2 .
COMA = - 3 x r p 2 2 f 2 ( 1 - G ) ( 1 - G ) ( G - G ) ,
SPHR = r p 3 2 f 2 G 3 ( 2 - G ) - 3 G 2 + 3 G G - 2 G + 1 ( 1 - G ) 2 ( 1 - G ) 3 ( G - G ) .
x = W L + L z / N ,
y = W M + M z / N ,
x = - W L + z L / N ,
y = - W M + z M / N .
x = W L + f L / G N .
x par - G W L - f L / N = 0.
y par - G M M - f M / N = 0.
( L 2 - L 1 ) L A ( L 1 , M 1 ) + ( M 2 - M 1 ) M A ( L 1 , M 1 ) = - A ( L 1 , M 1 ) ,
( L 2 - L 1 ) L B ( L 1 , M 1 ) + ( M 2 - M 1 ) M B ( L 1 , M 1 ) = - B ( L 1 , M 1 ) .

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