Abstract

We show that three refracting surfaces (two aspherical and one spherical surface) are sufficient to form, in air, a real image having any lateral magnification, and free from all five monochromatic primary (Seidel) aberrations. The way of finding such systems is described, and examples of them are given.

© 1980 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 2nd ed (Pergamon, New York, 1964), p. 150.
  2. M. Herzberger, Modern Geometrical Optics (Interscience, New York, London, 1958), Sec. 22;Ref. 1, pp. 149 and 169;G. Schulz, Paradoxa aus der Optik (Barth, Leipzig, 1974), pp. 30–33 and 74–76.
  3. Reference 1, Sec. 5.3.
  4. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, New York, San Francisco, 1974), p. 201.
  5. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), pp. 137–138.
  6. G. Kirchhof, “Über die gleichzeitige Korrektion aller Seidelschen Bildfehler bei Triplets aus dunnen Linsen,” Optik 13, 79–80 (1956),“Ein Beitrag zur Seidelschen Theorie der Triplets aus dunnen Linsen,” Optik 14, 388–398 (1957).
  7. A. Cox, A System of Optical Design (Focal, London, New York, 1964), p. 164.
  8. O. N. Stavroudis and R. I. Mercado, “Canonical properties of optical-design modules,” J. Opt. Soc. Am. 65, 509–517 (1975).
    [CrossRef]
  9. D. Korsch, “Closed-form solutions for imaging systems, corrected for third-order aberrations,” J. Opt. Soc. Am. 63, 667–672 (1973).
    [CrossRef]
  10. P. N. Robb, “Three-mirror telescopes: design and optimation,” Appl. Opt. 17, 2677–2685 (1978).
    [CrossRef] [PubMed]
  11. J. Dyson, “Unit magnification optical system without seidel aberrations,” J. Opt. Soc. Am. 49, 713–716 (1959).
    [CrossRef]
  12. C. G. Wynne, “A Unit-Power Telescope for Projection Copying,” in Optical Instruments and Techniques 1969, edited by J. Home Dickson (Oriel, Newcastle, 1970), pp. 429–434;S. Rosin, “Unit magnification optical system,” Appl. Opt. 16, 2568–2571 (1977).
    [CrossRef] [PubMed]
  13. Reference 5, pp.133 and 135.
  14. Reference 1, Sec. 5.5.
  15. L. van der Waerden, Moderne Algebra II, (Springer, Berlin, 1931), Secs. 73–74.
  16. Reference 15, Sec. 71;O. Th. Bürklen and F. Ringleb, Mathematische Formelsammlung, Sammlung Göschen Band 51 (Gruyter, Berlin, 1939), pp. 35–36.
  17. Some of these parameters, especially s1 could be varied such that one or another of the properties of the solution is optimized. However, the purpose of this paper is not to find a relative or an absolute optimum, but to show the existence of an optical system having the property mentioned in the abstract. Therefore, in the following, only examples of such systems are shown.
  18. a4bi/8ri3 is the distance between surface i and the ri sphere in axis direction at the height a from the axis; therefore, bi/8ri3 is a measure of the asphericity of surface i.

1978 (1)

1975 (1)

1973 (1)

1959 (1)

1956 (1)

G. Kirchhof, “Über die gleichzeitige Korrektion aller Seidelschen Bildfehler bei Triplets aus dunnen Linsen,” Optik 13, 79–80 (1956),“Ein Beitrag zur Seidelschen Theorie der Triplets aus dunnen Linsen,” Optik 14, 388–398 (1957).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed (Pergamon, New York, 1964), p. 150.

Bürklen, O. Th.

Reference 15, Sec. 71;O. Th. Bürklen and F. Ringleb, Mathematische Formelsammlung, Sammlung Göschen Band 51 (Gruyter, Berlin, 1939), pp. 35–36.

Cox, A.

A. Cox, A System of Optical Design (Focal, London, New York, 1964), p. 164.

Dyson, J.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, London, 1958), Sec. 22;Ref. 1, pp. 149 and 169;G. Schulz, Paradoxa aus der Optik (Barth, Leipzig, 1974), pp. 30–33 and 74–76.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), pp. 137–138.

Kirchhof, G.

G. Kirchhof, “Über die gleichzeitige Korrektion aller Seidelschen Bildfehler bei Triplets aus dunnen Linsen,” Optik 13, 79–80 (1956),“Ein Beitrag zur Seidelschen Theorie der Triplets aus dunnen Linsen,” Optik 14, 388–398 (1957).

Korsch, D.

Mercado, R. I.

Ringleb, F.

Reference 15, Sec. 71;O. Th. Bürklen and F. Ringleb, Mathematische Formelsammlung, Sammlung Göschen Band 51 (Gruyter, Berlin, 1939), pp. 35–36.

Robb, P. N.

Stavroudis, O. N.

van der Waerden, L.

L. van der Waerden, Moderne Algebra II, (Springer, Berlin, 1931), Secs. 73–74.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, New York, San Francisco, 1974), p. 201.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed (Pergamon, New York, 1964), p. 150.

Wynne, C. G.

C. G. Wynne, “A Unit-Power Telescope for Projection Copying,” in Optical Instruments and Techniques 1969, edited by J. Home Dickson (Oriel, Newcastle, 1970), pp. 429–434;S. Rosin, “Unit magnification optical system,” Appl. Opt. 16, 2568–2571 (1977).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Optik (1)

G. Kirchhof, “Über die gleichzeitige Korrektion aller Seidelschen Bildfehler bei Triplets aus dunnen Linsen,” Optik 13, 79–80 (1956),“Ein Beitrag zur Seidelschen Theorie der Triplets aus dunnen Linsen,” Optik 14, 388–398 (1957).

Other (13)

A. Cox, A System of Optical Design (Focal, London, New York, 1964), p. 164.

M. Born and E. Wolf, Principles of Optics, 2nd ed (Pergamon, New York, 1964), p. 150.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, London, 1958), Sec. 22;Ref. 1, pp. 149 and 169;G. Schulz, Paradoxa aus der Optik (Barth, Leipzig, 1974), pp. 30–33 and 74–76.

Reference 1, Sec. 5.3.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, New York, San Francisco, 1974), p. 201.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), pp. 137–138.

C. G. Wynne, “A Unit-Power Telescope for Projection Copying,” in Optical Instruments and Techniques 1969, edited by J. Home Dickson (Oriel, Newcastle, 1970), pp. 429–434;S. Rosin, “Unit magnification optical system,” Appl. Opt. 16, 2568–2571 (1977).
[CrossRef] [PubMed]

Reference 5, pp.133 and 135.

Reference 1, Sec. 5.5.

L. van der Waerden, Moderne Algebra II, (Springer, Berlin, 1931), Secs. 73–74.

Reference 15, Sec. 71;O. Th. Bürklen and F. Ringleb, Mathematische Formelsammlung, Sammlung Göschen Band 51 (Gruyter, Berlin, 1939), pp. 35–36.

Some of these parameters, especially s1 could be varied such that one or another of the properties of the solution is optimized. However, the purpose of this paper is not to find a relative or an absolute optimum, but to show the existence of an optical system having the property mentioned in the abstract. Therefore, in the following, only examples of such systems are shown.

a4bi/8ri3 is the distance between surface i and the ri sphere in axis direction at the height a from the axis; therefore, bi/8ri3 is a measure of the asphericity of surface i.

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Figures (5)

FIG. 1
FIG. 1

Optical system (the ray shown is a paraxial ray, c is a proportionality factor).

FIG. 2
FIG. 2

Quantities of optical systems with three refracting surfaces and all five Seidel aberrations being zero, corresponding to a solution of Eqs. (1)(9). The quantities of such systems, namely, r1r2, r3, d2, s 3 , b 1 / 8 r 1 3 and b 3 / 8 r 3 3 (Ref. 18), are shown as functions of the magnification M for n1 = 1.7, n2 = 1.5, and s1 = −2. Here each M value corresponds to an optical system without Seidel aberrations. These systems are normalized such that the first lens thickness d1 = 1.

FIG. 3
FIG. 3

Quantities as in Fig. 2, but shown as functions of s1 for M = −⅓. Again n1 = 1.7 and n2 = 1.5. Here each s1 value corresponds to an optical system without Seidel aberrations.

FIG. 4
FIG. 4

Example of an optical system with all five Seidel aberrations being zero; M = −⅓, n1 = 1.7, n2 = 1.5, s1 = −2.

FIG. 5
FIG. 5

Single thick lens with two aspheric surfaces, 1 and 2, (for simplicity, the subscripts of n1 and h2 are omitted).

Equations (29)

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0 = 2 B ( n 1 1 ) b 1 / r 1 3 + K 1 2 S 1 + h 2 4 K 2 2 S 2 + h 3 4 ( 1 n 2 ) b 3 / r 3 3 + h 3 4 K 3 2 S 3 ,
0 = 2 C S 1 + ( 1 + h 2 2 k 2 K 2 ) 2 S 2 + h 3 4 k 3 2 ( 1 n 2 ) b 3 / r 3 3 + ( 1 + h 3 2 k 3 K 3 ) 2 S 3 ,
0 = 2 ( C D ) ( 1 / n 1 1 ) / r 1 + ( 1 / n 2 1 / n 1 ) / r 2 + ( 1 1 / n 2 ) / r 3 ,
0 = 2 E ( 1 / n 1 2 1 ) + k 2 ( 1 + h 2 2 k 2 K 2 ) ( 2 + h 2 2 k 2 K 2 ) S 2 ( 1 + h 2 2 k 2 K 2 ) ( 1 / n 2 2 1 / n 1 2 ) / h 2 2 + h 3 4 k 3 3 ( 1 n 2 ) b 3 / r 3 3 + k 3 ( 1 + h 3 2 k 3 K 3 ) ( 2 + h 3 2 k 3 K 3 ) S 3 ( 1 + h 3 2 k 3 K 3 ) ( 1 1 / n 2 2 ) / h 3 2 ,
0 = 2 F K 1 S 1 + h 2 2 K 2 ( 1 + h 2 2 k 2 K 2 ) S 2 + h 3 4 k 3 ( 1 n 2 ) b 3 / r 3 3 + h 3 3 K 3 ( 1 + h 3 2 k 3 K 3 ) S 3 ,
K i n i ( 1 / r i 1 / s i ) ; S i ( 1 / n i s i 1 / n i 1 s i ) ; k 2 1 / n 1 h 2 ; k 3 1 / n 1 h 2 + d 2 / n 2 h 2 h 3 .
n i 1 ( 1 r i 1 s i ) = n i ( 1 r i 1 s i ) ( i = 1 , 2 , 3 ) ;
s 2 = s 1 1 , s 3 = s 2 d 2 ;
h 2 = s 2 / s 1 , h 3 / h 2 = s 3 / s 2 .
M = s 3 / s 1 h 3 .
M , n 1 , n 2 ,
b 1 , b 3 , h 2 , h 3 , d 2 , r i , s i , s i ( i = 1 , 2 , 3 ) .
0 = 2 C k 3 2 E ,
0 = 2 C 2 F k 3 ,
0 = S 1 k 3 + ( 1 + h 2 2 k 2 K 2 ) 2 S 2 k 3 + ( 1 + h 3 2 k 3 K 3 ) 2 S 3 k 3 + ( 1 / n 1 2 1 ) k 2 ( 1 + h 2 2 k 2 K 2 ) ( 2 + h 2 2 k 2 K 2 ) S 2 + ( 1 + h 2 2 k 2 K 2 ) ( 1 / n 2 2 1 / n 1 2 ) / h 2 2 k 3 ( 1 + h 3 2 k 3 K 3 ) ( 2 + h 3 2 k 3 K 3 ) S 3 + ( 1 + h 3 2 k 3 K 3 ) ( 1 1 / n 2 2 ) / h 3 2 ,
0 = S 1 + ( 1 + h 2 2 k 2 K 2 ) 2 S 2 + ( 1 + h 3 2 k 3 K 3 ) 2 S 3 K 1 S 1 k 3 h 2 2 K 2 ( 1 + h 2 2 k 2 K 2 ) S 2 k 3 h 3 2 K 3 ( 1 + h 3 2 k 3 K 3 ) S 3 k 3 ,
K 1 = ( 1 + h 2 1 / s 1 ) n 1 / ( n 1 1 ) ,
K 2 = ( 1 + 1 / r 2 + 1 / h 2 ) n 1 ,
K 3 = [ ( 1 / r 2 1 / s 1 ) n 2 / n 1 + n 2 ( 1 + h 2 ) 1 / r 2 ] × ( 1 n 2 1 + 1 M s 1 [ ( n 2 n 1 ) h 2 / r 2 n 1 ( 1 + h 2 ) ] + 1 ) ,
S 1 = ( 1 + h 2 ) / n 1 1 / s 1 ,
S 2 = ( 1 / n 2 n 1 / n 2 2 ) / r 2 + ( 1 + 1 / h 2 ) ( 1 / n 1 n 1 / n 2 2 ) ,
S 3 = ( 1 / r 2 1 / s 1 ) / n 1 + 1 + h 2 1 / n 2 r 2 ( n 1 n 2 ) h 2 / r 2 + n 1 ( 1 + h 2 ) 1 / M s 1 × [ n 2 / M s 1 + ( 1 n 1 / n 2 ) h 2 / r 2 ( 1 + h 2 ) n 1 / n 2 ] ,
k 2 = 1 / n 1 h 2 ,
k 3 = 1 ( n 2 n 1 ) h 2 / r 2 n 1 ( 1 + h 2 ) × ( ( 1 / r 2 1 / s 1 ) n 2 / n 1 + n 2 ( 1 + h 2 ) 1 / r 2 ( n 2 n 1 ) h 2 / r 2 n 1 ( 1 + h 2 ) + 1 / M s 1 1 h 2 ) 1 n 1 h 2 ,
h 3 = ( n 2 n 1 ) h 2 / r 2 n 1 ( 1 + h 2 ) + 1 / M s 1 ( 1 / r 2 1 / s 1 ) n 2 / n 1 + n 2 ( 1 + h 2 ) 1 / r 2 .
0 = 2 C k 2 2 E = ( 1 1 / n 2 ) ( 1 + 1 / r + 1 / h ) + ( 1 1 / n ) ( 1 + 1 / r 1 / h ) / r ,
0 = 2 C 2 F k 2 = [ ( n 1 / n ) ( 1 + h ) ( n 1 ) / r ] ( 1 1 / r ) / h + [ ( n 1 / n ) ( 1 + 1 / h ) + ( n 1 ) / r ] ( 1 + 1 / r ) h ;
h = 1 + 1 / n 1 / r ( 1 + 1 / n + 1 / r ) ( 1 + 1 / r ) ,
0 = ( 1 + 1 / n + 1 / r ) / r 2 ,