Abstract

We propose the use of the parallel tabu search algorithm (PTS) to solve combinatorial inverse design problems in integrated photonics. To assess the potential of this algorithm, we consider the problem of beam shaping using a two-dimensional arrangement of dielectric scatterers. The performance of PTS is compared to one of the most widely used optimization algorithms in photonics design, the genetic algorithm (GA). We find that PTS can produce comparable or better solutions than the GA, while requiring less computation time and fewer adjustable parameters. For the coherent beam shaping problem as a case study, we demonstrate how PTS can tackle multiobjective optimization problems and represent a robust and efficient alternative to GA.

© 2013 Optical Society of America

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  1. L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, Opt. Lett. 29, 1623 (2004).
    [CrossRef]
  2. P. Pottier, S. Mastroiacovo, and R. M. De La Rue, Opt. Express 14, 5617 (2006).
    [CrossRef]
  3. T. Baba, Nat. Photonics 2, 465 (2008).
    [CrossRef]
  4. L. Dal Negro and S. V. Boriskina, Laser Photon. Rev. 6, 178 (2012).
    [CrossRef]
  5. Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
    [CrossRef]
  6. D. S. Wiersma, Nat. Photonics 7, 188 (2013).
    [CrossRef]
  7. E. Talbi, Metaheuristics: From Design to Implementation (Wiley, 2009).
  8. D. S. Weile and E. Michielssen, IEEE Trans. Antennas Propag. 45, 343 (1997).
    [CrossRef]
  9. S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
    [CrossRef]
  10. L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
    [CrossRef]
  11. A. Vukovic, P. Sewell, and T. M. Benson, J. Opt. Soc. Am. A 27, 2156 (2010).
    [CrossRef]
  12. P. A. Sanchez-Serrano, D. Wong-Campos, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, Opt. Lett. 37, 5040 (2012).
    [CrossRef]
  13. J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
    [CrossRef]
  14. F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
    [CrossRef]
  15. D. Gagnon, J. Dumont, and L. J. Dubé, J. Opt. Soc. Am. A 29, 2673 (2012).
    [CrossRef]
  16. F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.
  17. S. Nojima, J. Appl. Phys. 98, 043102 (2005).
    [CrossRef]
  18. G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  19. N. Olivier, D. Débarre, P. Mahou, and E. Beaurepaire, Opt. Express 20, 24886 (2012).
    [CrossRef]
  20. T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).
  21. F. Glover, ORSA J. Comput. 1, 190 (1989).
    [CrossRef]
  22. A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
    [CrossRef]
  23. S.-C. C. Chu and H. L. Fang, in Third International Conference on Knowledge-Based Intelligent Information Engineering Systems (IEEE, 1999), pp. 492–495.
  24. M. Ehrgott, Multicriteria Optimization (Springer, 2005).
  25. A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
    [CrossRef]
  26. C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
    [CrossRef]
  27. Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
    [CrossRef]
  28. A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
    [CrossRef]
  29. D. Gagnon, J. Dumont, and L. J. Dubé, “Coherent beam shaping using two-dimensional photonic crystals,” http://arxiv.org/abs/1305.0193 .

2013

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

D. S. Wiersma, Nat. Photonics 7, 188 (2013).
[CrossRef]

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

2012

2011

A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
[CrossRef]

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

2010

2008

T. Baba, Nat. Photonics 2, 465 (2008).
[CrossRef]

2007

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

2006

2005

S. Nojima, J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

2004

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, Opt. Lett. 29, 1623 (2004).
[CrossRef]

S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
[CrossRef]

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

2000

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

1997

D. S. Weile and E. Michielssen, IEEE Trans. Antennas Propag. 45, 343 (1997).
[CrossRef]

T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).

1995

F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
[CrossRef]

1989

F. Glover, ORSA J. Comput. 1, 190 (1989).
[CrossRef]

Abad, J. B.

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Agrawal, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Baba, T.

T. Baba, Nat. Photonics 2, 465 (2008).
[CrossRef]

Baets, R.

Beaurepaire, E.

Beckx, S.

Benson, T. M.

Bogaerts, W.

Borel, P. I.

Boriskina, S. V.

L. Dal Negro and S. V. Boriskina, Laser Photon. Rev. 6, 178 (2012).
[CrossRef]

Boxwell, S.

S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
[CrossRef]

Chu, S.-C. C.

S.-C. C. Chu and H. L. Fang, in Third International Conference on Knowledge-Based Intelligent Information Engineering Systems (IEEE, 1999), pp. 492–495.

Chutinan, A.

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Cluzel, B.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

Crainic, T. G.

T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).

Dal Negro, L.

L. Dal Negro and S. V. Boriskina, Laser Photon. Rev. 6, 178 (2012).
[CrossRef]

De Fornel, F.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

De La Rue, R. M.

Débarre, D.

Dehesa, J. S.

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Dickey, F.

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

Dubé, L. J.

D. Gagnon, J. Dumont, and L. J. Dubé, J. Opt. Soc. Am. A 29, 2673 (2012).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Coherent beam shaping using two-dimensional photonic crystals,” http://arxiv.org/abs/1305.0193 .

Dumon, P.

Dumont, J.

D. Gagnon, J. Dumont, and L. J. Dubé, J. Opt. Soc. Am. A 29, 2673 (2012).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Coherent beam shaping using two-dimensional photonic crystals,” http://arxiv.org/abs/1305.0193 .

Ehrgott, M.

M. Ehrgott, Multicriteria Optimization (Springer, 2005).

Fang, H. L.

S.-C. C. Chu and H. L. Fang, in Third International Conference on Knowledge-Based Intelligent Information Engineering Systems (IEEE, 1999), pp. 492–495.

Fang, J. A.

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Fox, S. G.

S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
[CrossRef]

Frandsen, L. H.

Gagnon, D.

D. Gagnon, J. Dumont, and L. J. Dubé, J. Opt. Soc. Am. A 29, 2673 (2012).
[CrossRef]

D. Gagnon, J. Dumont, and L. J. Dubé, “Coherent beam shaping using two-dimensional photonic crystals,” http://arxiv.org/abs/1305.0193 .

Gendreau, M.

T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).

Glover, F.

F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
[CrossRef]

F. Glover, ORSA J. Comput. 1, 190 (1989).
[CrossRef]

Gutiérrez-Vega, J. C.

Harpøth, A.

Hofler, A.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Holdsworth, C.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Holswade, S.

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

Jarvis, C.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Kansson, A. H.

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Kelly, J. P.

F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
[CrossRef]

Kim, M.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Kramer, M.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Kristensen, M.

Kurths, J.

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Laguna, M.

F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
[CrossRef]

Liao, J.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Lin, F.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Lopez-Aguayo, S.

Mahou, P.

Marques-Hueso, J.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

Martínez-Pastor, J. P.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

Mastroiacovo, S.

Michielssen, E.

D. S. Weile and E. Michielssen, IEEE Trans. Antennas Propag. 45, 343 (1997).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Morozov, V.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Nahata, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Noda, S.

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Nojima, S.

S. Nojima, J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

Olivier, N.

Phillips, M. H.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Pottier, P.

Roblin, Y.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Roman, J. F.

S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
[CrossRef]

Sanchez-Serrano, P. A.

Sanchis, L.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Sedaghat, R.

A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
[CrossRef]

Sengupta, A.

A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
[CrossRef]

Sewell, P.

Stewart, R. D.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Talbi, E.

E. Talbi, Metaheuristics: From Design to Implementation (Wiley, 2009).

Tang, Y.

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Terzic, B.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Thorhauge, M.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Toulouse, M.

T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).

Vardeny, Z. V.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Vukovic, A.

Wang, Z.

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Weile, D. S.

D. S. Weile and E. Michielssen, IEEE Trans. Antennas Propag. 45, 343 (1997).
[CrossRef]

Wiaux, V.

Wiersma, D. S.

D. S. Wiersma, Nat. Photonics 7, 188 (2013).
[CrossRef]

Wong, W. K.

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Wong-Campos, D.

Wouters, J.

Zanón, D. L.

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Zeng, Z.

A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
[CrossRef]

Zhuang, Y. X.

Zvezdin, A.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Appl. Phys. Lett.

L. Sanchis, A. H. Kansson, D. L. Zanón, J. B. Abad, and J. S. Dehesa, Appl. Phys. Lett. 84, 4460 (2004).
[CrossRef]

Chaos

Y. Tang, Z. Wang, W. K. Wong, J. Kurths, and J. A. Fang, Chaos 21, 025114 (2011).
[CrossRef]

Comput. Oper. Res.

F. Glover, J. P. Kelly, and M. Laguna, Comput. Oper. Res. 22, 111 (1995).
[CrossRef]

IEEE Trans. Antennas Propag.

D. S. Weile and E. Michielssen, IEEE Trans. Antennas Propag. 45, 343 (1997).
[CrossRef]

INFORMS J. Comput.

T. G. Crainic, M. Toulouse, and M. Gendreau, INFORMS J. Comput. 9, 61 (1997).

J. Appl. Phys.

S. Nojima, J. Appl. Phys. 98, 043102 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Laser Photon. Rev.

L. Dal Negro and S. V. Boriskina, Laser Photon. Rev. 6, 178 (2012).
[CrossRef]

Med. Phys.

C. Holdsworth, R. D. Stewart, M. Kim, J. Liao, and M. H. Phillips, Med. Phys. 38, 2964 (2011).
[CrossRef]

Microprocess. Microsyst.

A. Sengupta, R. Sedaghat, and Z. Zeng, Microprocess. Microsyst. 35, 392 (2011).
[CrossRef]

Nat. Photonics

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

D. S. Wiersma, Nat. Photonics 7, 188 (2013).
[CrossRef]

T. Baba, Nat. Photonics 2, 465 (2008).
[CrossRef]

Nat. Phys.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Opt. Eng.

J. Marques-Hueso, L. Sanchis, B. Cluzel, F. De Fornel, and J. P. Martínez-Pastor, Opt. Eng. 52, 091710 (2013).
[CrossRef]

S. Boxwell, S. G. Fox, and J. F. Roman, Opt. Eng. 43, 1643 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

ORSA J. Comput.

F. Glover, ORSA J. Comput. 1, 190 (1989).
[CrossRef]

Phys. Rev. B

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Phys. Rev. Spec. Top.

A. Hofler, B. Terzić, M. Kramer, A. Zvezdin, V. Morozov, Y. Roblin, F. Lin, and C. Jarvis, Phys. Rev. Spec. Top. 16, 010101 (2013).
[CrossRef]

Other

D. Gagnon, J. Dumont, and L. J. Dubé, “Coherent beam shaping using two-dimensional photonic crystals,” http://arxiv.org/abs/1305.0193 .

S.-C. C. Chu and H. L. Fang, in Third International Conference on Knowledge-Based Intelligent Information Engineering Systems (IEEE, 1999), pp. 492–495.

M. Ehrgott, Multicriteria Optimization (Springer, 2005).

E. Talbi, Metaheuristics: From Design to Implementation (Wiley, 2009).

F. Dickey and S. Holswade, Laser Beam Shaping Applications (Taylor & Francis, 2005), Chap. 8.

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Figures (3)

Fig. 1.
Fig. 1.

Basic photonic lattice configuration for the beam shaping problem. The dotted line indicates the plane used for the computation of the desired beam profile.

Fig. 2.
Fig. 2.

Comparison of the GA and PTS algorithms applied to the incoherent beam shaping problem. Each simulation represents 5000 generations/iterations, with a similar computational cost. 100 simulations are shown for each algorithm.

Fig. 3.
Fig. 3.

Multiobjective optimization results. (a) Sampling of the Pareto front for the coherent beam shaping problem. The dotted lines indicate the best possible value for each of the two objectives separately. (b) Hermite–Gauss beam profile with the lowest possible value of f1 (f1=0.0167, f2=0.1019). (c) Best possible trade-off between the two objectives (f1=0.0255, f2=0.0833). Since the phase is controlled, the Hermite–Gauss profile shape is preserved over a greater propagation distance. This can be seen in the number of ridges in the transverse profile (arbitrary intensity units). The lattice configurations producing these beams are made of Ns=56 (b) and Ns=52 (c) scattering sites whose explicit positions can be found in [29].

Equations (3)

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

g1=||u(x0,y)|2|u¯(x0,y)|2|dy|u¯(x0,y)|2dy,
g2=|Im[u(x0,y)eiϕ(x0,0)]|2dy|u¯(x0,y)|2dy,
minξΞi=1pαifi(ξ),

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