Abstract

By a single lens (two refracting surfaces, one dispersive medium) an achromatic real point image can be obtained. We show how such a lens brings a large aperture bundle of axial-parallel rays to the same sharp focus for two wavelengths, that is, for two refractive-index values. The normal of one of the two refracting surfaces is, in general, discontinuous at the axis. The numerical aperture which is practically attainable can be enhanced by off-axis steps in the refracting surfaces. The design principles for such lenses, computational formulas, and examples of results are given. Limitations and further possibilities are discussed.

© 1983 Optical Society of America

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References

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  1. See, for example, H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 80;M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 231;W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 173;C. Hofmann, Die optische Abbildung (Geest & Portig, Leipzig, 1980), p. 241.
  2. R. Riekher, Fernrohre und ihre Meister (Verlag Technik, Berlin, 1957), p. 87;M. Herzberger, Modern Geometrical Optics (Inter-science, New York, 1958), p. 112.
  3. K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 11, 14 (1905).
  4. R. D. Sigler, Appl. Opt. 21, 2804 (1982).
    [CrossRef] [PubMed]
  5. V. I. Kreopalov, N. L. Sokolov, Opt. Spektrosk. 53, 743 (1982)[Opt. Spectrosc. 53, 441 (1982)].
  6. S. Czapski, O. Eppenstein, Grundzüge der Theorie der optischen Instrumente (Johann Ambrosius Barth, Leipzig, 1924), p. 290.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 197.
  8. G. Schulz, Opt. Commun. 41, 315 (1982).
    [CrossRef]
  9. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of splines and Their Applications (Academic, New York, 1967), p. 11.

1982

R. D. Sigler, Appl. Opt. 21, 2804 (1982).
[CrossRef] [PubMed]

V. I. Kreopalov, N. L. Sokolov, Opt. Spektrosk. 53, 743 (1982)[Opt. Spectrosc. 53, 441 (1982)].

G. Schulz, Opt. Commun. 41, 315 (1982).
[CrossRef]

1905

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 11, 14 (1905).

Ahlberg, J. H.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of splines and Their Applications (Academic, New York, 1967), p. 11.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 197.

Czapski, S.

S. Czapski, O. Eppenstein, Grundzüge der Theorie der optischen Instrumente (Johann Ambrosius Barth, Leipzig, 1924), p. 290.

Eppenstein, O.

S. Czapski, O. Eppenstein, Grundzüge der Theorie der optischen Instrumente (Johann Ambrosius Barth, Leipzig, 1924), p. 290.

Hopkins, H. H.

See, for example, H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 80;M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 231;W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 173;C. Hofmann, Die optische Abbildung (Geest & Portig, Leipzig, 1980), p. 241.

Kreopalov, V. I.

V. I. Kreopalov, N. L. Sokolov, Opt. Spektrosk. 53, 743 (1982)[Opt. Spectrosc. 53, 441 (1982)].

Nilson, E. N.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of splines and Their Applications (Academic, New York, 1967), p. 11.

Riekher, R.

R. Riekher, Fernrohre und ihre Meister (Verlag Technik, Berlin, 1957), p. 87;M. Herzberger, Modern Geometrical Optics (Inter-science, New York, 1958), p. 112.

Schulz, G.

G. Schulz, Opt. Commun. 41, 315 (1982).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 11, 14 (1905).

Sigler, R. D.

Sokolov, N. L.

V. I. Kreopalov, N. L. Sokolov, Opt. Spektrosk. 53, 743 (1982)[Opt. Spectrosc. 53, 441 (1982)].

Walsh, J. L.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of splines and Their Applications (Academic, New York, 1967), p. 11.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 197.

Appl. Opt.

Astr. Mitt. Kgl. Sternw. Göttingen

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 11, 14 (1905).

Opt. Commun.

G. Schulz, Opt. Commun. 41, 315 (1982).
[CrossRef]

Opt. Spektrosk.

V. I. Kreopalov, N. L. Sokolov, Opt. Spektrosk. 53, 743 (1982)[Opt. Spectrosc. 53, 441 (1982)].

Other

S. Czapski, O. Eppenstein, Grundzüge der Theorie der optischen Instrumente (Johann Ambrosius Barth, Leipzig, 1924), p. 290.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 197.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of splines and Their Applications (Academic, New York, 1967), p. 11.

See, for example, H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 80;M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964), p. 231;W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 173;C. Hofmann, Die optische Abbildung (Geest & Portig, Leipzig, 1980), p. 241.

R. Riekher, Fernrohre und ihre Meister (Verlag Technik, Berlin, 1957), p. 87;M. Herzberger, Modern Geometrical Optics (Inter-science, New York, 1958), p. 112.

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Figures (11)

Fig. 1
Fig. 1

Schematic representation for constructing an axisymmetric single lens which images the axial points O and O′ into each other sharply and really for two wavelengths λ*, λ**. Construction of the represented surface elements in the succession (1) or (2) (Sec. II).

Fig. 2
Fig. 2

Axisymmetric single lens which brings an axial-parallel incident bundle to a sharp focus O′ for two wavelengths λ*, λ** (schematic representation). Construction of the surface elements at the discrete points Pμ and P μ (μ being any integer) in the succession (1) or (2) (Sec. II). (a) The normal of the right surface is discontinuous at the axis. (b) The normal of the left surface is discontinuous at the axis.

Fig. 3
Fig. 3

Derivation of the basic relations. At P there is an element of the left and at P′ an element of the right surface of the lens (surface elements shown as short thick lines). If P is a point Pμ of Fig. 2, P′ is the point P μ or P μ 1 . The focus O′ is the coordinate origin.

Fig. 4
Fig. 4

Result of the computation of a lens which brings an axial-parallel bundle to the same sharp focus for two wavelengths λ*, λ** with n(λ*) = n* = 1.50, n(λ**) = n** = 1.51. (For further data see Sec. III.B.) The figure shows the lens in a geometrically similar representation.

Fig. 5
Fig. 5

Result of the computation of a lens as in Fig. 4, now with n* = 1.78,n** = 1.84.

Fig. 6
Fig. 6

Construction with arbitrary choice of a* and a**, a* determines the optical path length [ P μ P μ ], a** determines [ P μ + 1 P μ ]. Here a** is too large in comparison with a*, no reasonable surfaces are obtained.

Fig. 7
Fig. 7

Construction when a** is balanced against a* in such a way that for one pair of adjacent surface elements P μ M ¯ = MP ¯ μ + 1.

Fig. 8
Fig. 8

Enlarged section from Fig. 2(b). The surface part between P2 and P3 (e.g., Q2) has been determined by interpolation between P2 and P3 according to the first paragraph of Sec. III.B. Then the other surface parts (e.g., Q 2 , Q3,…) can or could be calculated from that part according to Sec. III.A [repeated application of relations (5)–(14)].

Fig. 9
Fig. 9

Making the curvatures continuous everywhere. Shown is a part of the left surface with one a* and three different a** values: (a) a** is too large (similar to Fig. 6). The curvature is positive directly above Pμ+1 and negative directly below it, that is, at Pμ+1 the curvature jumps from a larger to a smaller value. (b) a** is too small. At Pμ+1 the curvature jumps from a smaller to a larger value. (c) a** has been appropriately chosen so that the curvature at Pμ+1 is continuous (here slightly positive).

Fig. 10
Fig. 10

Result of the computation of a seven-step lens (in a geometrically similar representation). The lens brings an axial-parallel bundle of diameter d to the same sharp focus for two wavelengths λ*, λ** with n(λ*) = n* = 1.78, n(λ**) = n** = 1.84. tanθ = 0.4. In the upper part of the figure, the beam path through two adjacent steps is hatched. Within each beam, any λ* ray and its adjacent λ** ray are so close to each other that they are not distinguishable in this representation.

Fig. 11
Fig. 11

Paraxial ray bundle which is incident parallel to the axis and is to be brought to the same focus F for two wavelengths λ*, λ**. The lens with the normalized thickness d = 1 has the two paraxial radii of curvature, r1 and r2, and is surrounded by air having the refractive index 1.

Equations (34)

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P 3 P 3 P 4 P 4
P 3 P 3 P 2 . P 2
[ P P O ] = z + a ,
a = { a * ( n = n * ) , a * * ( n = n * * ) .
sin α = n sin ( α ω ) , ω = α arcsin ( sin α / n ) .
n ( z z ) / cos ω + [ h ( z z ) tan ω ] 2 + z 2 = z + a .
A = ( n 2 1 ) / cos 2 ω ,
B = 2 [ h tan ω ( a z ) n / cos ω z ] ,
C = a 2 2 a z h 2 ,
z = [ B B 2 4 A C ] / ( 2 A ) + z .
h = h ( z z ) tan ω ,
σ = arctan ( h / z ) .
α = arctan sin σ n sin ω cos σ n cos ω .
ω = α arcsin [ sin ( α σ ) / n ] [ σ from Eq . ( 9 ) ] ;
z = ( n z / cos ω + h 2 + z 2 a ) / ( n / cos ω 1 ) ;
h = h + ( z z ) tan ω ;
α = arctan n sin ω n cos ω 1 .
P μ M ¯ = M P μ + 1 ¯ .
a * = n * ( z 0 z 0 ) + h 0 2 + z 0 2 + z 0 ,
σ 0 = arctan ( h 0 / z 0 ) ,
α 0 = arctan sin σ 0 cos σ 0 n * ,
ω 01 = α 0 arcsin [ sin ( α 0 σ 0 ) / n * * ] ,
α 1 = arctan n * * sin ω 01 n * * cos ω 01 1 ,
a * * = a * + ( n * * / cos ω 01 n * ) ( z 0 z 0 ) ( n * * cos ω 01 ) ( z 0 z 0 ) tan ω 01 sin α 1 cos ( α 1 ω 01 ) + cos ω 01 .
1 / r 1 = n * ( 1 / r 1 1 / s 1 * ) = n * * ( 1 / r 1 1 / s 1 * * )
1 / r 2 1 / s 2 = n * ( 1 / r 2 1 / s 2 * ) = n * * ( 1 / r 2 1 / s 2 * * ) ,
1 = s 1 * s 2 * = s 1 * * s 2 * * .
1 = 1 N * u 1 N * υ + c / n * = 1 N * * u 1 N * * υ + c / n * * ,
N * = 1 1 / n * , N * * = 1 1 / n * * ,
c = 1 / s 2 , u = 1 / r 1 , υ = 1 / r 2 .
N * ( 1 + c ) u 2 ( 1 + N * + / N * * ) c u + c / N * * = 0.
( 1 + N * / N * * ) 2 c 2 4 N * ( 1 + c ) c / N * * 0.
s 2 ( 1 / q 2 + q ) / 4 ,
q = 1 1 / n * 1 1 / n * * > 0 ( n * , n * * > 1 ) .

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