Abstract

In general, as diffractive optical elements formed by use of self-repeating patterns possess beneficial characteristics such as scratch resistance, low design effort, ease of fabrication, and natural formation of large panels, an efficient design methodology that was developed with a modified preserving-the-best strategy of genetic algorithms is presented. Both genetic algorithms and simulated annealing are examined by the Markov-chain stochastic process to create the insight needed to use these two heuristic algorithms efficiently. It was found that adding the preserving-the-best strategy to traditional genetic algorithms guarantees the possibility of locating the global optimum. Combining this sufficient and necessary condition for locating a global optimum for genetic algorithms with the built-in chromosome crossover searching mechanism and its neighborhood identification makes this newly developed genetic algorithm an effective method for designing diffractive optical elements. In our study, a prototype was fabricated based on our case study with the modified genetic algorithm. The performance of this prototype was measured and analyzed. Experimental results are shown to agree well with theoretical predictions.

© 1997 Optical Society of America

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1996

1995

1993

1991

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

1990

1989

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

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

M. R. Feldman, C. C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 279–481 (1989).
[CrossRef]

1988

1987

1986

1985

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983

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

1981

1978

1973

N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1973).

1972

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

Aarts, E.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann machines, (Wiley, New York, 1989), Chap. 2, pp. 13–14.

Abushagar, M. A. G.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

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]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

Beletic, J. W.

D. C. O’Shea, J. W. Beletic, M. Poutous, “Binary mask generation for diffractive optical elements using microcomputers,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 114–116.

Brown, D.

D. Brown, A. Kathman, “Multi-element diffractive optical designs using evolutionary programming,” in Diffractive and Holographic Optics Technology II, Ivan Cindrich, S. H. Lee, eds., Proc. SPIE2404, 17–27 (1995).
[CrossRef]

Brown, D. R.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, 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]

Cook, A. L.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

Davis, L.

L. Davis, Genetic Algorithms and Simulated Annealing (Pitman, London, 1987), Chap. 1, pp. 1–11.

Delaney, B.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

Domash, L. H.

P. S. Levin, L. H. Domash, “MacBeep: a desktop system for binary optics,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 120–122.

Feldman, M. R.

M. R. Feldman, C. C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 279–481 (1989).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1973).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Recipes (Cambridge U. Press, New York, 1986), Chap. 10, pp. 326–334.

Gallagher, N. C.

Galytsis, E. N.

Gaylord, T. K.

Gelatt, C. D.

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

Geman, D.

D. Geman, “Random fields and inverse problems in imaging,” in Proceedings of the École d’Été de Probabilitiés de Saint-Flour XVIII-1988, Lecture Notes in Mathematics, Vol. 1427, (Springer-Verlag, New York, 1991), pp. 113–193.

Gerchberg, R. W.

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

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989), Chap. 4, pp. 125–129.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1992), Chap. 4, pp. 57–70.

Granet, G.

Guest, C. C.

M. S. Kim, C. C. Guest, “Simulated annealing algorithms for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 279–481 (1989).
[CrossRef]

Guizal, B.

Hochmuth, D. H.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT, Cambridge, Mass., 1992), Chap. 1, pp. 1–19.

J. H. Holland, Adaptation in Natural and Artificial Systems, (MIT, Cambridge, Mass., 1992), Chap. 3, pp. 49–52.

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, D. E. G.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

Johnson, E.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

Kathman, A.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

D. C. O’Shea, T. K. Gaylord, J. N. Mait, A. Kathman, “Course notes,” presented at the Diffractive Optics workshop, Georgia Institute of Technology, Atlanta, Georgia, 21–24 March 1995.

D. Brown, A. Kathman, “Multi-element diffractive optical designs using evolutionary programming,” in Diffractive and Holographic Optics Technology II, Ivan Cindrich, S. H. Lee, eds., Proc. SPIE2404, 17–27 (1995).
[CrossRef]

Kathman, A. D.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

Kim, M. S.

Kirkpatrick, S.

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

Korst, J.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann machines, (Wiley, New York, 1989), Chap. 2, pp. 13–14.

Levin, P. S.

P. S. Levin, L. H. Domash, “MacBeep: a desktop system for binary optics,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 120–122.

Liu, B.

Magnuson, R.

Mait, J. N.

J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
[CrossRef]

D. C. O’Shea, T. K. Gaylord, J. N. Mait, A. Kathman, “Course notes,” presented at the Diffractive Optics workshop, Georgia Institute of Technology, Atlanta, Georgia, 21–24 March 1995.

Millane, P.

Moharam, M. G.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 10, pp. 341–376.

O’Shea, D. C.

T. J. Suleski, D. C. O’Shea, “Fidelity of PostScript-generated masks for diffractive optics fabrication,” Appl. Opt. 34, 627–634 (1995).
[CrossRef] [PubMed]

D. C. O’Shea, T. K. Gaylord, J. N. Mait, A. Kathman, “Course notes,” presented at the Diffractive Optics workshop, Georgia Institute of Technology, Atlanta, Georgia, 21–24 March 1995.

D. C. O’Shea, J. W. Beletic, M. Poutous, “Binary mask generation for diffractive optical elements using microcomputers,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 114–116.

Poutous, M.

D. C. O’Shea, J. W. Beletic, M. Poutous, “Binary mask generation for diffractive optical elements using microcomputers,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 114–116.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Recipes (Cambridge U. Press, New York, 1986), Chap. 10, pp. 326–334.

Ross, S. M.

S. M. Ross, Stochastic Process (Wiley, New York, 1983), pp. 100–111.

Saxton, W. D.

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

Seldowitz, M. A.

Suleski, T. J.

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]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Recipes (Cambridge U. Press, New York, 1986), Chap. 10, pp. 326–334.

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Vecchi, M. P.

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

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Recipes (Cambridge U. Press, New York, 1986), Chap. 10, pp. 326–334.

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Wyrowski, F.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1973).

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

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Opt. Lett.

M. R. Feldman, C. C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 279–481 (1989).
[CrossRef]

Optik

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

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Rep. Prog. Phys.

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

Science

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

Other

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. 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 SPIE Critical Reviews, Bellingham, Wash., 1993), pp. 54–74.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989), Chap. 4, pp. 125–129.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT, Cambridge, Mass., 1992), Chap. 1, pp. 1–19.

L. Davis, Genetic Algorithms and Simulated Annealing (Pitman, London, 1987), Chap. 1, pp. 1–11.

D. Brown, A. Kathman, “Multi-element diffractive optical designs using evolutionary programming,” in Diffractive and Holographic Optics Technology II, Ivan Cindrich, S. H. Lee, eds., Proc. SPIE2404, 17–27 (1995).
[CrossRef]

D. C. O’Shea, T. K. Gaylord, J. N. Mait, A. Kathman, “Course notes,” presented at the Diffractive Optics workshop, Georgia Institute of Technology, Atlanta, Georgia, 21–24 March 1995.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Recipes (Cambridge U. Press, New York, 1986), Chap. 10, pp. 326–334.

D. C. O’Shea, J. W. Beletic, M. Poutous, “Binary mask generation for diffractive optical elements using microcomputers,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 114–116.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann machines, (Wiley, New York, 1989), Chap. 2, pp. 13–14.

J. H. Holland, Adaptation in Natural and Artificial Systems, (MIT, Cambridge, Mass., 1992), Chap. 3, pp. 49–52.

Ref. 32, Chap. 3, pp. 49–52.

S. M. Ross, Stochastic Process (Wiley, New York, 1983), pp. 100–111.

D. Geman, “Random fields and inverse problems in imaging,” in Proceedings of the École d’Été de Probabilitiés de Saint-Flour XVIII-1988, Lecture Notes in Mathematics, Vol. 1427, (Springer-Verlag, New York, 1991), pp. 113–193.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1992), Chap. 4, pp. 57–70.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 10, pp. 341–376.

Ref. 14, Chap. 2, pp. 27–57.

P. S. Levin, L. H. Domash, “MacBeep: a desktop system for binary optics,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 120–122.

“Lasers and instruments guide,” B.5, 12–16, Melles Griot Corp., 4665 Nautilus Court, South Boulder, Colorado (1994).

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Figures (7)

Fig. 1
Fig. 1

Sketch of the evolution of each generation in the GA.

Fig. 2
Fig. 2

Graphic representation of the probability distribution of mutation for different GA’s: (a) pure GA, (b) modified GA, (c) modified GA with a random-search procedure.

Fig. 3
Fig. 3

Modified preserving-the-best strategy GA with a random-search procedure.

Fig. 4
Fig. 4

Individual N × N unit cell with an M-level phase for a self-repeating DOE.

Fig. 5
Fig. 5

Numerical results obtained with a GA to design a fan-out grating: (1) modified GA, two-level element; (2) pure GA, two-level element; (3) modified GA with a random-search procedure: two-level element; (4) modified GA, eight-level element.

Fig. 6
Fig. 6

Fan-out grating designed with a GA.

Fig. 7
Fig. 7

Layout for diffractive power measurement.

Tables (5)

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Table 1 Pseudocode of a SA Method

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Table 2 Pseudocode of a GA

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Table 3 7 × 7 Diffraction Image Arraya

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Table 4 Numerical Results of the Diffraction Image Intensity Obtained with a GA

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Table 5 Intensity Distribution of the Designed DOE (in Nanowatts)

Equations (27)

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vn+1x=yΩvnypy, x.
μx=yΩμypy, x
μypy, x=μxpx, y.
μx=μxypy, x=yμxpy, x=yμypy, x.
px, y=pTx, y=min1, μTy/μTx/#Ω  for every yx,
px, x=pTx, x=1-yxpTx, y,
μTx=exp-Hx/T/Z.
μ=1for the best configuration x00for other configurations.
p0x, y=1,  if μy>μx,
p0x, y=0,  otherwise;
px, y=μy/μx+μy/#Ω  for every yx,
px, x=pTx, x=1-yxpTx, y.
X=xi1, xi2, xi3, , xikΩk.
Mxi=XΩk|the best element in X is xi.
μx=yΩμypMy, Mx,
μx=μx0pMx0, Mx.
pMx0, Mx=0.
tx, y=m=-n=-δx-mN, y-nNp=0N-1q=0N-1 rectx-p, y-qexpiϕpq,
rectx=1x0.50otherwise.
Tm, n=sincm/N, n/Np=0N-11=0N-1 expiϕpq×exp-i2πmp/N+nq/N,
MSE=1#Dm, nDIm, n-γTm, n2,
γ=m, nDI-IavgT2-Tavg2m, nDI-Iavg21/2m, nDT2-Tavg221/2,
Fkx, ky=--Ux, yexpiϕx, y×exp-ikxx+kyydxdy,
i=1Nq1-qi-1=q1-1-qN/1-1-q=1-1-qN
Fkx, ky=A expi0exp-ikx+kydxdy+B expiπexp-ikx+kydxdy=A exp-ikx+kydxdy-B exp-ikx+kydxdy.
F˜kx, ky=A expiΔ/2exp-ikxx+kyydxdy+B expiπ-Δ/2×exp-ikxx+kyydxdy=cosΔ/2{A1exp-ikxx+kyydxdy+B-1exp-ikxx+kyydxdy}+i sinΔ/2A exp-ikxx+kyydxdy+B exp-ikxx+kyydxdy=cosΔ/2Fkx, ky+i sinΔ/2δkx, ky,
F˜kx, ky=cosΔ/2Fkx, ky  except when kx=ky=0,

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