Abstract

Hybrid approaches that combine genetic algorithms (GA’s) with traditional gradient-based local search techniques are proposed for the optimization design of diffractive phase elements (DPE’s) for laser beam shaping. These hybrid methods exploit the global nature of the GA’s as well as the local improvement capabilities of the gradient-based local search techniques and will perform a more improved search in comparison with each of the individual approaches. The incorporated local search technique that we used here is the Davidon–Fletcher–Powell method. A cost function that can directly control the performance of the final solutions is also used. By performing the DPE design with different desired diffraction efficiencies, we obtain a set of results that approximately reflect the trade-off between the design objectives, namely, signal-to-noise ratio (SNR) and diffraction efficiency. Reasonable solutions can be chosen on the basis of the knowledge of the problem. Simulation computations are detailed for two rotationally symmetric beam-shaping systems, in which an incident Gaussian profile laser beam is converted into a uniform beam and a zero-order Bessel beam. Numerical results demonstrate that the proposed algorithm is highly efficient and robust. DPE’s that have high diffraction efficiency and excellent SNR can be achieved by using the algorithm that we propose.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  9. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
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    [CrossRef] [PubMed]
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  33. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

1999

1998

1997

1996

J. M. Renders, S. P. Flasse, “Hybrid methods using genetic algorithms for global optimization,” IEEE Trans. Syst. Man Cybern. 26, 243–258 (1996).
[CrossRef]

1995

1994

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

1993

1991

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1989

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

1988

1987

J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1984

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

1983

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1982

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

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

1977

1974

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1957

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
[CrossRef]

1953

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Arif, M.

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Awwal, A. A. S.

Brown, D. R.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

Chen, N. X.

Chen, Y.

Cong, W. X.

Cook, A.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Cordingley, J.

Delaney, B.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Dixit, S. N.

Dong, B. Z.

Durnin, J. E.

J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Flasse, S. P.

J. M. Renders, S. P. Flasse, “Hybrid methods using genetic algorithms for global optimization,” IEEE Trans. Syst. Man Cybern. 26, 243–258 (1996).
[CrossRef]

Friberg, A. T.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989), Chap 5, pp. 148–214.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 63–90.

Gu, B. Y.

Henesian, M. A.

Hochmuth, D. H.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Honkanen, M.

Hopkins, H. H.

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. London Sect. B 70, 1002–1005 (1957).
[CrossRef]

Hossain, M. M.

Islam, M. N.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Johnson, E. G.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Kastner, C. J.

Kathman, A. D.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Review Series (SPIE Press, Bellingham, Wash.,1993), pp. 54–74.

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kuittinen, M.

Mait, J. N.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Morgan, A. J.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Palmieri, F.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

Powell, H. T.

Qi, X.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

Renders, J. M.

J. M. Renders, S. P. Flasse, “Hybrid methods using genetic algorithms for global optimization,” IEEE Trans. Syst. Man Cybern. 26, 243–258 (1996).
[CrossRef]

Rhodes, P. W.

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Shealy, D. L.

Siegman, A. E.

Song, H.

Swanson, G. J.

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Tan, X.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Thomas, I. M.

Turunen, J.

Vahimaa, P.

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Veldkamp, W. B.

Wang, Z.

Wegner, P. J.

Weldkamp, W. B.

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

Wolfe, M. A.

M. A. Wolfe, Numerical Methods for Unconstrained Optimization: An Introduction (Van Nostrand Reinhold, New York, 1978), Chap. 6, pp. 161–167.

Woods, B. W.

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Yang, G. Z.

Zhou, G.

Appl. Opt.

M. Arif, M. M. Hossain, A. A. S. Awwal, M. N. Islam, “Refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37, 649–652 (1998).
[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]

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2542 (1993).
[CrossRef] [PubMed]

G. Zhou, Y. Chen, Z. Wang, H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. 38, 4281–4290 (1999).
[CrossRef]

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

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

S. N. Dixit, I. M. Thomas, B. W. Woods, A. J. Morgan, M. A. Henesian, P. J. Wegner, H. T. Powell, “Random phase plates for beam smoothing on the Nova laser,” Appl. Opt. 32, 2543–2554 (1993).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

IEEE Trans. Neural Netw.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

J. M. Renders, S. P. Flasse, “Hybrid methods using genetic algorithms for global optimization,” IEEE Trans. Syst. Man Cybern. 26, 243–258 (1996).
[CrossRef]

J. Chem. Phys.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

Opt. Eng.

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Configuration of laser beam-shaping system with DPE’s.

Fig. 2
Fig. 2

Chromosome used in hybrid algorithm for DPE design.

Fig. 3
Fig. 3

Brief flow chart of the hybrid algorithm.

Fig. 4
Fig. 4

Flow chart of the DFP method.

Fig. 5
Fig. 5

Optimization design results of DPE’s for the conversion of a Gaussian beam into a uniform beam.

Fig. 6
Fig. 6

Convergence properties of the proposed hybrid algorithm and a canonical GA for optimization design of DPE’s for Gaussian-to-uniform beam shaping when ηd=0.95 and C=500.

Fig. 7
Fig. 7

Convergence property of the DFP method for optimization design of DPE’s for Gaussian-to-uniform beam shaping when ηd=0.95 and C=500.

Fig. 8
Fig. 8

Optimal design result obtained by the hybrid algorithm with ηd=0.95 and C=500 for Gaussian-to-uniform beam shaping. The inset shows the phase profile of the optimal DPE1.

Fig. 9
Fig. 9

Beam-shaping result obtained by the geometrical transformation technique for the conversion of a Gaussian beam into a uniform beam. The inset shows the phase profile of the corresponding DPE1.

Fig. 10
Fig. 10

Optimization design results of DPE’s for the conversion of a Gaussian beam into a zero-order Bessel beam.

Fig. 11
Fig. 11

Convergence properties of the proposed hybrid algorithm and a canonical GA for optimization design of DPE’s for Gaussian-to-Bessel beam shaping when ηd=0.94 and C=500.

Fig. 12
Fig. 12

Convergence properties of the DFP method for optimization design of DPE’s for Gaussian-to-Bessel beam shaping when ηd=0.94 and C=500.

Fig. 13
Fig. 13

Optimal design result obtained by the hybrid algorithm with ηd=0.94 and C=500 for Gaussian-to-Bessel beam shaping. The inset shows the phase profile of the optimal DPE1.

Fig. 14
Fig. 14

Beam-shaping result obtained by the geometrical transformation technique for the conversion of a Gaussian beam into a J0 Bessel beam. The inset shows the phase profile of the corresponding DPE1.

Equations (38)

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U(x, y)=Uin(x, y)exp[iϕDPE1(x, y)],
U(x, y)=G(x, y, x, y; z)U(x, y)dxdy,
G(x, y, x, y; z)=1iλz expi2πzλ×expiπλz[(x-x)2+(y-y)2],
Uout(x, y)=U(x, y)exp[iϕDPE2(x, y)].
Upq=U(xp, yq)=m=1Mn=1NGpqmn exp(iϕmn),
Gpqmn=SmnG(xp, yq, x, y; z)Uin(x, y)dxdy,
η=EoutEin=p=1Pq=1Q|Upq|2Spqm=1Mn=1N|Umn|2Smn,
SNR=p=1Pq=1Q|Updd|2Spqp=1Pq=1Q(|Upqd|-γ|Upq|)2Spq,
γ=p=1Pq=1Q|Upq||Upqd|Spqp=1Pq=1Q|Upq|2Spq.
F=1SNR+CR(η),
R(η)=0ηηd(ηd-η)2η<ηd,
xi=ui+vi-ui2l-1 j=1laij2j-1,
pi=Fitness(i)iFitness(i),
Φk+1=Φk+skdk,
F(Φk+skdk)=mins[F(Φk+sdk)].
dk=-HkF(Φk),H0=I,
Hk+1=Hk+Ak,
yk=Φk+1-Φk,
zk=F(Φk+1)-F(Φk);
Ak=ykykTykTzk-HkzkzkTHkTzkTHkzk.
Ujk=m=1Mn=1NGjkmn exp(iϕmn).
Fϕmn=j=1Pk=1Q F|Ujk| |Ujk|ϕmn.
|Ujk|ϕmn=ϕmn(|Ujk|2)=12|Ujk| ϕmn(UjkUjk*)=12|Ujk| Ujk* Ujkϕmn+Ujk Ujk*ϕmn,
Ujkϕmn=iGjkmn exp(iϕmn),
Ujk*ϕmn=-iGjkmn* exp(-iϕmn),
|Ujk|ϕmn=ImGjkmn*Ujk exp(-iϕmn)|Ujk|.
Fϕmn=Imexp(-iϕmn)j=1Pk=1QGjkmn* Ujk|Ujk| F|Ujk|.
F|Ujk|=|Ujk| 1SNR+C R|Ujk|.
R|Ujk|=0,ηηd2(η-ηd) η|Ujk|,η<ηd,
η|Ujk|=2|Ujk|Sjkm=1Mn=1N|Umn|2Smn.
|Ujk| 1SNR=-2p=1Pq=1Q|Upqd|2Spq×p=1Pq=1Q(|Upqd|-γ|Upq|)Spq×γ |Upq||Ujk|+|Upq| γ|Ujk|,
γ|Ujk|=(|Ujkd|-2γ|Ujk|)Skp=1Pq=1Q|Upq|2Spq,
|Upq||Ujk|=δpjδqk,
δpj=1,p=j0,pj,δqk=1,q=k0,qk.
|Ujk| 1SNR=-2p=1Pq=1Q(|Upqd|-γ|Upq|)|Upq|Spqp=1Pq=1Q|Upqd|2Spqp=1Pq=1Q|Upq|2Spq×(|Ujkd|-2γ|Ujk|)Sjk-2(|Ujkd|-γ|Ujk|)γSjkp=1Pq=1Q|Upqd|2Spq.
0ϕm2π,m=1, 2 ,, M.
ϕm=ϕm+2kπ,
U(ρ, z)=exp(iβz)J0(αρ),

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