Abstract

For an imaging optical system it is, in general, desirable to transform a collection of point sources of light into point images distributed over the focal plane with the appropriate magnification. In practice, this is achieved by varying the lens system parameters such that the spread of a bundle of rays from each object point over the image plane has been minimized. In this study, caustic surfaces are used to construct a merit function that describes the spread of the caustic surfaces from an ideal image point. This caustic merit function has been used to optimize a large collection of three- and four-element lens systems. The performance of the optimized lenses has been evaluated by comparing the rms blur circle radii vs field angle to that of similar lenses designed by conventional techniques. The average rms improvement has been calculated for optimized systems. Results indicate that minimizing the caustic merit function reduces the rms blur radii over the field of view and the total aberrations of the lens systems, particularly for systems with large apertures and wide fields of view.

© 1990 Optical Society of America

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References

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  1. R. S. Chang, O. N. Stavroudis, “Generalized Ray Tracing, Caustic Surfaces, Generalized Bending, and the Construction of a Novel Merit Function for Optical Design,” J. Opt. Soc. Am. 70, 976–985 (1980).
    [CrossRef]
  2. A. M. Kassim, “Caustic Wave Aberration Theory for General Optical Systems,” Ph.D. Dissertation, U. Alabama at Birmingham (1985).
  3. A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic Merit Function for Optical Design,” Appl. Opt. 28, 601–606 (1989).
    [CrossRef] [PubMed]
  4. D. L. Shealy, D. G. Burkhard, “Caustic Surface Merit Function in Optical Design,” J. Opt. Soc. Am. 66, 1122 (1976).
  5. I. H. Al-Ahdali, “Optimization of Three and Four-Element Lens Systems by Minimizing the Caustic Merit Function,” Ph.D. Dissertation, U. of Alabama at Birmingham (1989).
  6. D. G. Burkhard, D. L. Shealy, “Simplified Formula for Illuminance in an Optical System,” Appl. Opt. 20, 897–909 (1980).
    [CrossRef]
  7. Military Standardization Handbook: Optical Design, MILHDBK-141.5 (U.S. Government Printing Office, Washington, DC, 1962).
  8. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1976), pp. 286–295.
  9. R. E. Hopkins, “Third-Order and Fifth-Order Analysis of the Triplet,” J. Opt. Soc. Am. 52, 389–394 (1962).
    [CrossRef]
  10. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 341–346.
  11. A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientist and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1976), p. 30.
  12. T. H. Jamieson, Optimization Technique in Lens Design (American Elsevier, New York, 1971), pp. 16–20.
  13. M. J. Kidger, C. G. Wynne, “Author, Add Title to the Galleys,” Opt. Acta 14, 279 (1967).
    [CrossRef]
  14. J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983), p. 15.
  15. The IMSL Library, Ed. 8 (IMSL, Inc., Houston, 1980).
  16. O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Nat. Bur. Stand. (U.S.) Monogr.93 (1965), p. 59.

1989 (1)

1980 (2)

1976 (1)

D. L. Shealy, D. G. Burkhard, “Caustic Surface Merit Function in Optical Design,” J. Opt. Soc. Am. 66, 1122 (1976).

1967 (1)

M. J. Kidger, C. G. Wynne, “Author, Add Title to the Galleys,” Opt. Acta 14, 279 (1967).
[CrossRef]

1965 (1)

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Nat. Bur. Stand. (U.S.) Monogr.93 (1965), p. 59.

1962 (1)

Al-Ahdali, I. H.

I. H. Al-Ahdali, “Optimization of Three and Four-Element Lens Systems by Minimizing the Caustic Merit Function,” Ph.D. Dissertation, U. of Alabama at Birmingham (1989).

Burkhard, D. G.

Chang, R. S.

Dennis, J. E.

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983), p. 15.

Hopkins, R. E.

Jamieson, T. H.

T. H. Jamieson, Optimization Technique in Lens Design (American Elsevier, New York, 1971), pp. 16–20.

Kassim, A. M.

A. M. Kassim, D. L. Shealy, D. G. Burkhard, “Caustic Merit Function for Optical Design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef] [PubMed]

A. M. Kassim, “Caustic Wave Aberration Theory for General Optical Systems,” Ph.D. Dissertation, U. Alabama at Birmingham (1985).

Kidger, M. J.

M. J. Kidger, C. G. Wynne, “Author, Add Title to the Galleys,” Opt. Acta 14, 279 (1967).
[CrossRef]

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1976), pp. 286–295.

Nussbaum, A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientist and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1976), p. 30.

Phillips, R. A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientist and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1976), p. 30.

Shealy, D. L.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 341–346.

Stavroudis, O. N.

R. S. Chang, O. N. Stavroudis, “Generalized Ray Tracing, Caustic Surfaces, Generalized Bending, and the Construction of a Novel Merit Function for Optical Design,” J. Opt. Soc. Am. 70, 976–985 (1980).
[CrossRef]

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Nat. Bur. Stand. (U.S.) Monogr.93 (1965), p. 59.

Sutton, L. E.

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Nat. Bur. Stand. (U.S.) Monogr.93 (1965), p. 59.

Wynne, C. G.

M. J. Kidger, C. G. Wynne, “Author, Add Title to the Galleys,” Opt. Acta 14, 279 (1967).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Nat. Bur. Stand. (U.S.) Monogr. (1)

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Nat. Bur. Stand. (U.S.) Monogr.93 (1965), p. 59.

Opt. Acta (1)

M. J. Kidger, C. G. Wynne, “Author, Add Title to the Galleys,” Opt. Acta 14, 279 (1967).
[CrossRef]

Other (9)

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983), p. 15.

The IMSL Library, Ed. 8 (IMSL, Inc., Houston, 1980).

Military Standardization Handbook: Optical Design, MILHDBK-141.5 (U.S. Government Printing Office, Washington, DC, 1962).

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1976), pp. 286–295.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 341–346.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientist and Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1976), p. 30.

T. H. Jamieson, Optimization Technique in Lens Design (American Elsevier, New York, 1971), pp. 16–20.

A. M. Kassim, “Caustic Wave Aberration Theory for General Optical Systems,” Ph.D. Dissertation, U. Alabama at Birmingham (1985).

I. H. Al-Ahdali, “Optimization of Three and Four-Element Lens Systems by Minimizing the Caustic Merit Function,” Ph.D. Dissertation, U. of Alabama at Birmingham (1989).

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Figures (14)

Fig. 1
Fig. 1

Meridional intersection of the tangential and sagittal sheets of the caustic surfaces of a singlet lens.

Fig. 2
Fig. 2

Comparison of the rms blur radii vs the field angle for the initial Hopkins lens no. 1 and the optimized lenses for design angles 0°, 10°, and 20°.

Fig. 3
Fig. 3

Meridional caustic surfaces without optimizing triplet lens for the Hopkins initial lens no. 1 for different field angles and for the design angle 0°.

Fig. 4
Fig. 4

Meridional caustic surface for the optimized Hopkins lens no. 1 for different field angles and for the design angle 0°.

Fig. 5
Fig. 5

Meridional caustic surface for the optimized Hopkins lens no. 1 for different field angles and for the design angle 10°.

Fig. 6
Fig. 6

Meridional caustic surface for the optimized Hopkins lens no. 1 for different field angles and for the design angle 20°.

Fig. 7
Fig. 7

Average rms improvement for the optimized Hopkins lenses for different design angles.

Fig. 8
Fig. 8

Comparison of rms blur radii vs the field angle for the Kingslake lens and the optimized lenses for design angles 0°, 10°, and 20°.

Fig. 9
Fig. 9

Meridional intersection of the caustic surfaces for the Kingslake lens for different field angles.

Fig. 10
Fig. 10

Meridional intersection of the caustic surfaces for the optimized Kingslake lens at the design angle of 10° at different field angles.

Fig. 11
Fig. 11

Plot of rms blur radii vs the field angle for the optimized lens with design angle of 10°, the Stavroudis off-axis design lens, the Kassim off-axis optimized lens, and the initial Stavroudis lens for a 10-mm aperture radius.

Fig. 12
Fig. 12

Plot of rms blur radii vs the field angle for the optimized lens with design angle of 10°, the Stavroudis off-axis design lens, the Kassim off-axis optimized lens, and the initial Stavroudis lens for an 18-mm aperture radius.

Fig. 13
Fig. 13

Plot of rms blur radii vs the field angle for the optimized lens with design angle of 10°, the Stavroudis off-axis design lens, the Kassim off-axis optimized lens, and the initial Stavroudis lens for a 20-mm aperture radius.

Fig. 14
Fig. 14

Comparison of rms blur radii vs the field angle for the initial four-element wide angle, aerial camera lens and the optimized lens for design angles 0°, 10°, and 20°.

Tables (4)

Tables Icon

Table I List of Triplet Lens Parameters Designed by Using the Caustic Merit Function Optimization Technique for Design Angles αD and Based on Hopkins Lenses

Tables Icon

Table II List of Triplet Lens Parameters Designed by Kingslake (Lens No. 1) and by Using the Caustic Merit Function Optimization Technique for Design Angles αD = 0°,10°,20° (Lens Nos. 2, 3, and 4, Respectively)

Tables Icon

Table III List of Triplet Lens Parameters for the Chang and Stavroudis Initial Lens (No. 1), Chang and Stavroudis Off-Axis Design (No. 2), Kassim Off-Axis Design (No. 3), and Caustic Merit Function Optimization Technique for Design Angle αD = 10° (No. 4)

Tables Icon

Table IV List of Four-Element Lens Parameters of the Initial System (Lens No. 1) and Optimized System for Design Angle αD = 0°, 10°, and 20° (Lens Nos. 2, 3, and 4, Respectively)

Equations (12)

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Φ = X c - X F + X c - X F ,
Φ = 1 N ρ N ϕ i = 1 N ϕ j = 1 N ρ Φ ( i , j ) ,
P T = P 1 + P 2 + P 3 - X P 3 ( P 1 + P 2 ) - P 3 Y ( P 1 + P 2 ) - X P 2 ( P 1 - P 3 ) + X Y P 1 P 2 P 3 + X t 2 P 1 P 2 P 3 - ( P 1 + P 2 ) t 2 P 3 .
Y = P T - ( P 1 + P 2 + P 3 ) - X ( P 2 P 3 t 2 - P 3 - P 2 ) P 1 + ( P 1 + P 2 ) P 3 t 2 P 1 P 2 P 3 X - ( P 1 + P 2 ) P 3 ,
P 1 = ( n 1 - 1 ) [ 1 R 1 - 1 R 2 + t 1 ( n 1 - 1 ) n 1 R 1 R 2 ] ,
P 2 = ( n 2 - 1 ) [ 1 R 3 - 1 R 4 + t 2 ( n 2 - 1 ) n 2 R 3 R 4 ] ,
P 3 = ( n 3 - 1 ) [ 1 R 5 - 1 R 6 + t 3 ( n 3 - 1 ) n 3 R 5 R 6 ] .
Φ ( R 1 , , R 6 ) R m = 0 for m = 1 , 2 , , 6.
Φ = Φ T + Φ s ,
Φ T = X c - X F = [ ( X c - X F ) 2 + ( Y c - Y F ) 2 + ( Z c - Z F ) 2 ] 1 / 2 ,
Φ s = X c - X F = [ ( X c - X F ) 2 + ( Y c - Y F ) 2 + ( Z c - Z F ) 2 ] ,
rms improvement = rms ( initial ) - rms ( optimized ) rms ( initial ) × 100.

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