Abstract

The first-order derivative matrix of a function with respect to a variable vector is referred to as the Jacobian matrix in mathematics. Current commercial software packages for the analysis and design of optical systems use a finite difference (FD) approximation methodology to estimate the Jacobian matrix of the wavefront aberration with respect to all of the independent system variables in a single raytracing pass such that the change of the wavefront aberration can be determined simply by computing the product of the developed Jacobian matrix and the corresponding changes in the system variables. The proposed method provides an ideal basis for automatic optical system design applications in which the merit function is defined in terms of wavefront aberration. The validity of the proposed approach is demonstrated by means of two illustrative examples. It is shown that the proposed method requires fewer iterations than the traditional FD approach and yields a more reliable and precise optimization performance. However, the proposed method incurs an additional CPU overhead in computing the Jacobian matrix of the merit function. As a result, the CPU time required to complete the optimization process is longer than that required by the FD method.

© 2012 Optical Society of America

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References

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  1. H. H. Hopkins, Wave Theory of Aberrations (OxfordU. Press, 1950), Chap. 1.
  2. T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).
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    [CrossRef]
  4. J. Meiron, “The use of merit functions on wave-front aberrations in automatic lens design,” Appl. Opt. 7, 667–672 (1968).
    [CrossRef]
  5. W. T. Welford, “A new total aberration Formula,” Opt. Acta 19, 719–727 (1972).
    [CrossRef]
  6. D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18–22October1976).
    [CrossRef]
  7. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
    [CrossRef]
  8. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459–8466 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
    [CrossRef]
  19. C. C. Hsueh and P. D. Lin, “Computationally efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009).
    [CrossRef]
  20. S. K. Gupta and R. Hradaynath, “Angular tolerance on Dove prisms,” Appl. Opt. 22, 3146–3147 (1983).
    [CrossRef]
  21. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).
  22. J. S. Arora, Introduction to Optimum Design (McGraw-Hill, 1989), p. 313.

2010 (1)

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

2009 (1)

2002 (1)

A. Mikŝ, “Dependence of the wave-front aberration on the radius of the reference sphere,” J. Opt. Soc. Am. 19, 1187–1190 (2002).
[CrossRef]

1997 (1)

1989 (1)

1983 (1)

1978 (1)

D. S. Grey, “The inclusion of tolerance sensitivities in the merit function for lens optimization,” SPIE Rev. 147, 63–65 (1978).

1976 (1)

D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18–22October1976).
[CrossRef]

1972 (1)

W. T. Welford, “A new total aberration Formula,” Opt. Acta 19, 719–727 (1972).
[CrossRef]

1968 (2)

1966 (1)

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

1965 (1)

1964 (1)

1963 (2)

1958 (1)

T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).

1957 (1)

1947 (1)

A. Maréchal, “Etude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux,” Revue d'Optique, Theorique et Instrumentale 26, 257–277 (1947).

Arora, J. S.

J. S. Arora, Introduction to Optimum Design (McGraw-Hill, 1989), p. 313.

Barakat, R.

Braat, J.

Burkhard, D. G.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18–22October1976).
[CrossRef]

Feder, D. P.

Grey, D. S.

D. S. Grey, “The inclusion of tolerance sensitivities in the merit function for lens optimization,” SPIE Rev. 147, 63–65 (1978).

Gupta, S. K.

Hopkins, H. H.

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (OxfordU. Press, 1950), Chap. 1.

Houston, A.

Hradaynath, R.

Hsueh, C. C.

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Computationally efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009).
[CrossRef]

Kassim, A. M.

Kneisly, J. A.

Lin, P. D.

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

C. C. Hsueh and P. D. Lin, “Computationally efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009).
[CrossRef]

Maréchal, A.

A. Maréchal, “Etude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux,” Revue d'Optique, Theorique et Instrumentale 26, 257–277 (1947).

Meiron, J.

Miks, A.

A. Mikŝ, “Dependence of the wave-front aberration on the radius of the reference sphere,” J. Opt. Soc. Am. 19, 1187–1190 (2002).
[CrossRef]

Paul, R. P.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

Shealy, D. L.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18–22October1976).
[CrossRef]

Suzuki, T.

T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).

Uwoki, I.

T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).

Welford, W. T.

W. T. Welford, “A new total aberration Formula,” Opt. Acta 19, 719–727 (1972).
[CrossRef]

Wormell, P.

Wynne, C. G.

Appl. Opt. (7)

Appl. Phys. B (1)

C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010).
[CrossRef]

J. Opt. Soc. Am. (6)

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

T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).

Opt. Acta (2)

W. T. Welford, “A new total aberration Formula,” Opt. Acta 19, 719–727 (1972).
[CrossRef]

H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966).
[CrossRef]

Revue d'Optique, Theorique et Instrumentale (1)

A. Maréchal, “Etude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux,” Revue d'Optique, Theorique et Instrumentale 26, 257–277 (1947).

SPIE Rev. (1)

D. S. Grey, “The inclusion of tolerance sensitivities in the merit function for lens optimization,” SPIE Rev. 147, 63–65 (1978).

Other (3)

H. H. Hopkins, Wave Theory of Aberrations (OxfordU. Press, 1950), Chap. 1.

R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).

J. S. Arora, Introduction to Optimum Design (McGraw-Hill, 1989), p. 313.

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Figures (9)

Fig. 1.
Fig. 1.

Wavefront aberration in axis-symmetrical optical system with n boundary surfaces.

Fig. 2.
Fig. 2.

Schematic representation of unit directional vector ¯0 originating from source pointP¯0.

Fig. 3.
Fig. 3.

Illustrative optical system with three elements and five boundary surfaces.

Fig. 4.
Fig. 4.

Illustrative optical system with six elements and eleven boundary surfaces.

Fig. 5.
Fig. 5.

Transformation of axis-symmetrical system shown in Fig. 1 to nonaxially symmetric system by considering reference sphere rref as virtual boundary surface with radius Rref centered at Gaussian imaging point.

Fig. 6.
Fig. 6.

Sagittal and meridional aberration curves at field of view of 2° in system shown in Fig. 3.

Fig. 7.
Fig. 7.

Variation of merit function with number of iterations for system shown in Fig. 3.

Fig. 8.
Fig. 8.

Sagittal and meridional aberration curves at field of view of 10° in system shown in Fig. 4.

Fig. 9.
Fig. 9.

Variation of merit function with number of iterations for system shown in Fig. 4.

Tables (3)

Tables Icon

Table 1. Results Obtained for ΔΦ/Δξe1 by Two FD Methods Given Different Values of Δα0 (Percentage Values Indicate the Errors of the Two FD Methods Relative to the Invariant Result of Φ/ξe1=0.97169214 Obtained Using the Proposed Method)

Tables Icon

Table 2. Optimization Results Obtained from the DBCONF and DBCONG Subroutines for the System Shown in Fig. 3 (units: mm)

Tables Icon

Table 3. Optimization Results obtained from the DBCONF and DBCONG Subroutines for the System Shown in Fig. 4 (units: mm)

Equations (12)

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

W(α0,β0)=OPL(P¯0,P¯ref)OPL(P¯0,P¯),
W(α0,β0)=OPL(P¯0,P¯ref)OPL(P¯0,P¯ref/chief)
X¯sys=[R1ξe1qe1R2ve3]T,
X¯sys=[R1ξe1qe1R2ve2R3ξe2qe2R4ve4R7ξe4qe4R8ve5R9ξe5qe5R10ve6]T.
W(α0,β0)X¯sys=OPL(P¯0,P¯ref)X¯sysOPL(P¯0,P¯ref/chief)X¯sys.
ΔW(α0,β0)=ΔOPL(P¯0,P¯ref)ΔOPL(P¯0,P¯ref/chief)=W(α0,β0)X¯sysΔX¯sys=OPL(P¯0,P¯ref)X¯sysΔX¯sysOPL(P¯0,P¯ref/chief)X¯sysΔX¯sys.
W(α0,β0)=OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief).
W(α0,β0)X¯sys=OPL(P¯0,P¯n)X¯sysOPL(P¯0,P¯n/chief)X¯sys.
W(α0,β0)X¯sys=[0.459132.07050.10040.30520.45700.07683.27490.01950.00030.04720.00090.71610.03440.00010.03220.00210.72080.02060.00240.0027]T.
Φ=1q1mW2(α0,β0)=1q1m[OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief)]2.
ΔΦΔX¯sys=21q1mW(α0,β0)ΔW(α0,β0)ΔX¯sys=21q1m[OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief)][ΔOPL(P¯0,P¯n)ΔX¯sysΔOPL(P¯0,P¯n/chief)ΔX¯sys],
ΦX¯sys=21q1mW(α0,β0)W(α0,β0)X¯sys=21q1m[OPL(P¯0,P¯n)OPL(P¯0,P¯n/chief)][OPL(P¯0,P¯n)X¯sysOPL(P¯0,P¯n/chief)X¯sys].

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