Abstract

We develop a genetic algorithm (GA) optimization method and use it in the design of a refractive-beam profile-shaping system. In this application, we employ the GA to determine the shape of one surface of the primary beam profile-shaping element in our system. The GA is instructed to vary the shape of this surface such that the output intensity profile is flat on a spherical surface some distance away. The GA does this while insuring that only a specified area of the output surface is illuminated. The calculation of the intensity profile is based on geometrical optics and is accomplished exclusively through ray tracing, giving this method broad applicability.

© 1998 Optical Society of America

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  1. A fitness landscape, or, using more conventional optics terminology, a multidimensional merit function that quantifies the worth of an optical system in terms of the various parameters that define the system.
  2. M. Gell-Mann, The Quark and the Jaguar (W. H. Freeman, New York, 1994) Chap. 20, p. 312.
  3. J. Bengtsson, “Kinoform-only Gaussian-to-rectangle beam shaper for a semiconductor laser,” Appl. Opt. 35, 3807–3814 (1996).
    [CrossRef] [PubMed]
  4. C. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef]
  5. X. Tan, B. Gu, G. Yang, B. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
    [CrossRef] [PubMed]
  6. F. M. Dickey, S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
    [CrossRef]
  7. P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  8. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  9. P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance redistribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).
    [CrossRef] [PubMed]
  10. M. Kuittinen, P. Vahimaa, M. Honkanen, J. Turunen, “Beam shaping in the nonparaxial domain of diffractive optics,” Appl. Opt. 36, 2034–2041 (1997).
    [CrossRef] [PubMed]
  11. E. Betensky, “Postmodern lens design,” Opt. Eng. 32, 1750–1756 (1993).
    [CrossRef]
  12. K. Nemoto, T. Nayuki, T. Fujii, N. Goto, Y. Kanai, “Optimum control of the laser beam intensity profile with a deformable mirror,” Appl. Opt. 36, 7689–7695 (1997).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970) Chap. 3, pp. 115–116.
  14. J. H. McDermit, T. E. Horton, “Reflective optics for obtaining prescribed irradiative distributions from collimated sources,” Appl. Opt. 13, 1444–1450 (1974).
    [CrossRef] [PubMed]
  15. V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
    [CrossRef]
  16. T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaptation,” Appl. Opt. 35, 6865–6874 (1996).
    [CrossRef] [PubMed]
  17. C. Wang, D. L. Shealy, “Design of gradient-index lens systems for laser beam reshaping,” Appl. Opt. 32, 4763–4769 (1993).
    [CrossRef] [PubMed]
  18. W. Jiang, D. L. Shealy, J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2000, 64–75 (1993).
    [CrossRef]
  19. K. M. Baker, D. L. Shealy, W. Jiang, “Directional light filters: three-dimensional azo-dye-formed microhoneycomb images with optical resins,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 144–158 (1995).
    [CrossRef]
  20. Ken Baker of Optimetrix Co. provided us with the initial specifications and requirements for this system. Optimetrix is located at 13659 Victory Blvd., Van Nuys, Calif. 91401. The final system will be fabricated by Optimetrix as part of a holographic projection system.
  21. Our GA code is based on ga164.f, D. L. Carroll’s FORTRAN Genetic Algorithm Driver. Contact authors for information regarding the availability of ga165.f, which may be distributed freely.
  22. CODE V is a registered trademark of Optical Research Associates, 3280 E. Foothill Blvd., Pasadena, Calif. 91107.
  23. K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, Proc. SPIE1196, 289–296 (1989).
    [CrossRef]
  24. P. Mouroulis, J. Macdonald, Geometrical Optics and Optical Design (Oxford University, New York, 1997) Chap. 9, pp. 248–250.
  25. W. Jiang, Application of a Laser Beam Profile Reshaper to Enhance Performance of Holographic Projection Systems (University of Alabama at Birmingham Department of Physics, Birmingham, Ala., 1993) Chap. 2, pp. 30–32.

1997 (2)

1996 (3)

1995 (1)

1993 (2)

1992 (1)

1983 (1)

1982 (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

1980 (1)

1974 (1)

Baker, K. M.

K. M. Baker, D. L. Shealy, W. Jiang, “Directional light filters: three-dimensional azo-dye-formed microhoneycomb images with optical resins,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 144–158 (1995).
[CrossRef]

Bengtsson, J.

Betensky, E.

E. Betensky, “Postmodern lens design,” Opt. Eng. 32, 1750–1756 (1993).
[CrossRef]

Beyerlein, M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970) Chap. 3, pp. 115–116.

Dickey, F. M.

F. M. Dickey, S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

Dong, B.

Dresel, T.

Fujii, T.

Gell-Mann, M.

M. Gell-Mann, The Quark and the Jaguar (W. H. Freeman, New York, 1994) Chap. 20, p. 312.

Goto, N.

Gu, B.

Han, C.

Holswade, S. C.

F. M. Dickey, S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

Honkanen, M.

Horton, T. E.

Ishii, Y.

Jiang, W.

W. Jiang, D. L. Shealy, J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2000, 64–75 (1993).
[CrossRef]

K. M. Baker, D. L. Shealy, W. Jiang, “Directional light filters: three-dimensional azo-dye-formed microhoneycomb images with optical resins,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 144–158 (1995).
[CrossRef]

W. Jiang, Application of a Laser Beam Profile Reshaper to Enhance Performance of Holographic Projection Systems (University of Alabama at Birmingham Department of Physics, Birmingham, Ala., 1993) Chap. 2, pp. 30–32.

Kanai, Y.

Krishnakumar, K.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, Proc. SPIE1196, 289–296 (1989).
[CrossRef]

Kuittinen, M.

Macdonald, J.

P. Mouroulis, J. Macdonald, Geometrical Optics and Optical Design (Oxford University, New York, 1997) Chap. 9, pp. 248–250.

Malyak, P. H.

Martin, J. C.

W. Jiang, D. L. Shealy, J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2000, 64–75 (1993).
[CrossRef]

McDermit, J. H.

Mirov, S.

V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
[CrossRef]

Mouroulis, P.

P. Mouroulis, J. Macdonald, Geometrical Optics and Optical Design (Oxford University, New York, 1997) Chap. 9, pp. 248–250.

Murata, K.

Nayuki, T.

Nemoto, K.

Oliker, V.

V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
[CrossRef]

Prussner, L.

V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
[CrossRef]

Rhodes, P. W.

Schwider, J.

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Shealy, D. L.

C. Wang, D. L. Shealy, “Design of gradient-index lens systems for laser beam reshaping,” Appl. Opt. 32, 4763–4769 (1993).
[CrossRef] [PubMed]

P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
[CrossRef] [PubMed]

V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
[CrossRef]

W. Jiang, D. L. Shealy, J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2000, 64–75 (1993).
[CrossRef]

K. M. Baker, D. L. Shealy, W. Jiang, “Directional light filters: three-dimensional azo-dye-formed microhoneycomb images with optical resins,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 144–158 (1995).
[CrossRef]

Tan, X.

Turunen, J.

Vahimaa, P.

Wang, C.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970) Chap. 3, pp. 115–116.

Yang, G.

Appl. Opt. (10)

J. H. McDermit, T. E. Horton, “Reflective optics for obtaining prescribed irradiative distributions from collimated sources,” Appl. Opt. 13, 1444–1450 (1974).
[CrossRef] [PubMed]

P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
[CrossRef] [PubMed]

C. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef]

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance redistribution of a laser beam,” Appl. Opt. 31, 4377–4383 (1992).
[CrossRef] [PubMed]

C. Wang, D. L. Shealy, “Design of gradient-index lens systems for laser beam reshaping,” Appl. Opt. 32, 4763–4769 (1993).
[CrossRef] [PubMed]

K. Nemoto, T. Nayuki, T. Fujii, N. Goto, Y. Kanai, “Optimum control of the laser beam intensity profile with a deformable mirror,” Appl. Opt. 36, 7689–7695 (1997).
[CrossRef]

M. Kuittinen, P. Vahimaa, M. Honkanen, J. Turunen, “Beam shaping in the nonparaxial domain of diffractive optics,” Appl. Opt. 36, 2034–2041 (1997).
[CrossRef] [PubMed]

X. Tan, B. Gu, G. Yang, B. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
[CrossRef] [PubMed]

J. Bengtsson, “Kinoform-only Gaussian-to-rectangle beam shaper for a semiconductor laser,” Appl. Opt. 35, 3807–3814 (1996).
[CrossRef] [PubMed]

T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaptation,” Appl. Opt. 35, 6865–6874 (1996).
[CrossRef] [PubMed]

Opt. Eng. (2)

F. M. Dickey, S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35, 3285–3295 (1996).
[CrossRef]

E. Betensky, “Postmodern lens design,” Opt. Eng. 32, 1750–1756 (1993).
[CrossRef]

Opt. Laser Technol. (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Other (12)

W. Jiang, D. L. Shealy, J. C. Martin, “Design and testing of a refractive reshaping system,” in Current Developments in Optical Design and Optical Engineering III, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2000, 64–75 (1993).
[CrossRef]

K. M. Baker, D. L. Shealy, W. Jiang, “Directional light filters: three-dimensional azo-dye-formed microhoneycomb images with optical resins,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 144–158 (1995).
[CrossRef]

Ken Baker of Optimetrix Co. provided us with the initial specifications and requirements for this system. Optimetrix is located at 13659 Victory Blvd., Van Nuys, Calif. 91401. The final system will be fabricated by Optimetrix as part of a holographic projection system.

Our GA code is based on ga164.f, D. L. Carroll’s FORTRAN Genetic Algorithm Driver. Contact authors for information regarding the availability of ga165.f, which may be distributed freely.

CODE V is a registered trademark of Optical Research Associates, 3280 E. Foothill Blvd., Pasadena, Calif. 91107.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, Proc. SPIE1196, 289–296 (1989).
[CrossRef]

P. Mouroulis, J. Macdonald, Geometrical Optics and Optical Design (Oxford University, New York, 1997) Chap. 9, pp. 248–250.

W. Jiang, Application of a Laser Beam Profile Reshaper to Enhance Performance of Holographic Projection Systems (University of Alabama at Birmingham Department of Physics, Birmingham, Ala., 1993) Chap. 2, pp. 30–32.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970) Chap. 3, pp. 115–116.

V. Oliker, L. Prussner, D. L. Shealy, S. Mirov, “Optical design of a two-mirror symmetrical reshaping system and its application in superbroadband color center laser,” in Current Developments in Optical Design and Optical Engineering IV, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 10–18 (1994).
[CrossRef]

A fitness landscape, or, using more conventional optics terminology, a multidimensional merit function that quantifies the worth of an optical system in terms of the various parameters that define the system.

M. Gell-Mann, The Quark and the Jaguar (W. H. Freeman, New York, 1994) Chap. 20, p. 312.

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Figures (8)

Fig. 1
Fig. 1

Beam expander with input plane and output surface. The beam profile is shaped to be flat on the output surface.

Fig. 2
Fig. 2

Determination of dA i ; dP i = P i - P i-1; and dS i =dP i /cos(χ i out).

Fig. 3
Fig. 3

Merit function versus μ and P N . N = 200 in our system.

Fig. 4
Fig. 4

Example of genetic material for a single individual. Values (Real*8) for each parameter are converted into binary strings, which are in turn concatenated into one long string: the genetic material for that individual.

Fig. 5
Fig. 5

Beam-shaping system with ray trace showing that the density of rays increases at the periphery of the output surface, as one would expect, to compensate for the Gaussian nature of the input beam. Both the thin-lens and the shaping element are shown. The shaping element is determined by the GA.

Fig. 6
Fig. 6

Shaping element showing aspherical surface, which is determined by the GA. The axial thickness of this element is 6 mm.

Fig. 7
Fig. 7

Input beam intensity profile. The 1/e 2 diameter of the input beam is 7.882 mm. Integrating σ(ρ) over the input plane yields 21.1 units, a quantity that must be conserved according to Eq. (1).

Fig. 8
Fig. 8

Beam profile on the output surface. The radius (P N ) of the output surface is 52.5 mm. The mean value of the profile, , is 2.13 × 10-3 rays/mm2, with a standard deviation of 3.78 × 10-5. Integrating this mean value [u(P) = constant = ̅u] over the output surface yields a value of 20.7 units.

Tables (3)

Tables Icon

Table 1 Constraints on Surface Parametersa

Tables Icon

Table 2 Beam-Shaping System Parameters

Tables Icon

Table 3 Lens Element Parameters

Equations (11)

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

E = I   σ ρ d a = O   u P d A ,
u P = σ ρ d a d A .
σ ρ = exp - α ρ 2 .
ρ i = r N i ,     i = 0 N .
d ρ i = ρ i - ρ i - 1 .
d S i = d P i cos   χ i out .
u P i = exp - α ρ i 2 ρ i ρ i - ρ i - 1 cos χ i out P i P i - P i - 1 ,
z h = ch 2 1 + 1 - 1 + k c 2 h 2 1 / 2 + j = 2 5   A 2 j h 2 j ,
M = 1 μ exp - 0.01 50 - P N 2 ,
μ = 1 N i = 1 N u P i - u ¯ 2 1 / 2 ,
u ¯ = 1 N i = 1 N   u P i .

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