Abstract

From a theoretical point of view, the use of anisotropic pure-phase plates (APP) is considered in order to improve the quality parameter of certain partially coherent, partially polarized beams. It is shown that, to optimize the beam-quality parameter, the phases of the two Cartesian components of the field at the output of the APP plate should be identical and should exhibit a quadratic dependence on the radial polar coordinate.

© 2003 Optical Society of America

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  1. S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
    [Crossref] [PubMed]
  2. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [Crossref]
  3. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  4. A. E. Siegman, “New developments in laser resonators,” in Laser Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [Crossref]
  5. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [Crossref]
  6. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [Crossref]
  7. P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds., Proceedings of the First Workshop on Laser Beam Characterization (Sociedad Española de Óptica, Madrid, 1993).
  8. H. Weber, N. Reng, J. Lüdtke, P. M. Mejı́as, eds., Proceedings of the Second Workshop on Laser Beam Characterization (Festkörper-Laser-Institut, Berlin, 1994).
  9. M. Morin, A. Giesen, eds., Proceedings of the Third Workshop on Laser Beam Characterization, Proc. SPIE2870 (1996).
  10. A. Giesen, M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (VDI-Technologiezentrum, Berlin, 1997).
  11. H. Laabs, H. Weber, eds., Proceedings of the Fifth International Workshop on Laser Beam and Optics Characterization (VDI-Technologiezentrum, Erice, Italy, 2000).
  12. R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distorsions,” Opt. Lett. 17, 1650–1651 (1992).
    [Crossref]
  13. R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 18–20 (1993).
    [Crossref]
  14. R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [Crossref] [PubMed]
  15. J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
    [Crossref]
  16. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [Crossref]
  17. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).
  18. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beam,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [Crossref]
  19. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
    [Crossref]
  20. Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
    [Crossref]
  21. G. Piquero, J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
    [Crossref]
  22. J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
    [Crossref]
  23. P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
    [Crossref]
  24. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
    [Crossref]
  25. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  26. D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), Vol. 4, pp. 333–485.

2002 (2)

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

2001 (2)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[Crossref]

1999 (2)

1998 (3)

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

1997 (1)

1995 (1)

Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[Crossref]

1993 (1)

R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 18–20 (1993).
[Crossref]

1992 (2)

1991 (1)

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1988 (2)

S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[Crossref] [PubMed]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Borghi, R.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Dong, S.

Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[Crossref]

Gori, F.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[Crossref]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

Keren, E.

Lavi, S.

Lü, Q.

Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[Crossref]

Marti´nez-Herrero, R.

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[Crossref]

G. Piquero, J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[Crossref]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[Crossref] [PubMed]

R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 18–20 (1993).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distorsions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Meji´as, P. M.

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[Crossref]

G. Piquero, J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[Crossref]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[Crossref] [PubMed]

R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 18–20 (1993).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distorsions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

Movilla, J. M.

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[Crossref]

G. Piquero, J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[Crossref]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[Crossref] [PubMed]

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

Pepper, D. M.

D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), Vol. 4, pp. 333–485.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

G. Piquero, J. M. Movilla, R. Martı́nez-Herrero, P. M. Mejı́as, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[Crossref]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distorsions,” Opt. Lett. 17, 1650–1651 (1992).
[Crossref]

Prochaska, R.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

Santarsiero, M.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

Serna, J.

Seshadri, S. R.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Laser Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[Crossref]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

Weber, H.

Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[Crossref]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[Crossref]

Appl. Opt. (2)

J. Eur. Opt. Soc. A Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).

J. Opt. A Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A Pure Appl. Opt. 3, 1–9 (2001).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9–16 (2002).
[Crossref]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, “Quality improvement of symmetric-intensity beams propagating through pure phase plates,” Opt. Commun. 95, 18–20 (1993).
[Crossref]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

Q. Lü, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[Crossref]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[Crossref]

Optik (Stuttgart) (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Prog. Quantum Electron. (1)

P. M. Mejı́as, R. Martı́nez-Herrero, G. Piquero, J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[Crossref]

Other (8)

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

D. M. Pepper, “Nonlinear optical phase conjugation,” in Laser Handbook, M. L. Stitch, M. Bass, eds. (North-Holland, Amsterdam, 1985), Vol. 4, pp. 333–485.

A. E. Siegman, “New developments in laser resonators,” in Laser Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[Crossref]

P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds., Proceedings of the First Workshop on Laser Beam Characterization (Sociedad Española de Óptica, Madrid, 1993).

H. Weber, N. Reng, J. Lüdtke, P. M. Mejı́as, eds., Proceedings of the Second Workshop on Laser Beam Characterization (Festkörper-Laser-Institut, Berlin, 1994).

M. Morin, A. Giesen, eds., Proceedings of the Third Workshop on Laser Beam Characterization, Proc. SPIE2870 (1996).

A. Giesen, M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (VDI-Technologiezentrum, Berlin, 1997).

H. Laabs, H. Weber, eds., Proceedings of the Fifth International Workshop on Laser Beam and Optics Characterization (VDI-Technologiezentrum, Erice, Italy, 2000).

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Equations (40)

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

Γjj(r1, r2)=Ej*(r1)Ej(r2)¯=Aj(r1, r2)exp{i[αj(r2)-αj(r1)]},
j=s,p
As(r1, r2)=As(r2, r1),
Ap(r1, r2)=Ap(r2, r1),
As(r, r)0,Ap(r, r)0.
Ij=Γjj(r, r)dr,j=s,p.
T(r)=exp[iΔs(r)]00exp[iΔp(r)].
Γsso(r1, r2)=As(r1, r2)exp{i[φs(r2)-φs(r1)]},
Γppo(r1, r2)=Ap(r1, r2)exp{i[φp(r2)-φp(r1)]},
φs(r)=αs(r)+Δs(r),
φp(r)=αp(r)+Δp(r).
Q=r2η2-r·η2,
r2=IsIr2s+IpIr2p,
η2=IsIη2s+IpIη2p,
r·η=IsIr·ηs+IpIr·ηp,
αβj=1Ijk24π2×αβEj*(r+s/2, z)Ej(r-s/2, z)¯×exp(iks·η)drdηds,
j=s, p,α,β=x, y, u, v,
Qo=r2G+F,
G=1k2I2Asx1x2+2Asy1y2x1=x2=x;y1=y2=ydxdy+2Apx1x2+2Apy1y2x1=x2=x;y1=y2=ydxdy
F=r2k2I|φs|2As(r, r)dxdy+|φp|2Ap(r, r)dxdy-1kI(r·φs)As(r, r)dxdy+1kI(r·φp)Ap(r, r)dxdy2
Δs(r)+αs(r)=Δp(r)+αp(r)=μr2,
r2o=r2i,
η2o=IsIη2so+IpIη2po.
η2so=-12k2I2Γsso(r1, r2)x12+2Γsso(r1, r2)x22-22Γsso(r1, r2)x1x2r1=r2=r=(x,y)dxdy+2Γsso(r1, r2)y12+2Γsso(r1, r2)y22-22Γsso(r1, r2)y1y2r=r2=r=(x,y)dxdy,
η2so=-12k2I2As(r1, r2)x12+2As(r1, r2)x22-22As(r1, r2)x1x2r1=r2=r=(x,y)dxdy+2As(r1, r2)y12+2As(r1, r2)y22-22As(r1, r2)y1y2r1=r2=r=(x,y)dxdy+12k2Iφs(r1)x12+φs(r2)x22+2φs(r1)x1φs(r2)x2r1=r2=r=(x,y)As(r, r)dxdy+φs(r1)y12+φs(r2)y22+2φs(r1)y1×φs(r2)y2r1=r2=r=(x,y)As(r, r)dxdy,
η2so=2k2I2As(r1, r2)x1x2+2As(r1, r2)y1y2r1=r2=r=(x,y)dxdy+2k2I|φs(r)|2As(r, r)dxdy,
η2o=1k2I2(As+Ap)x1x2+2(As+Ap)y1y2r1=r2=r=(x,y)dxdy+1k2I|φs(r)|2As(r, r)+|φp(r)|2Ap(r, r)dxdy.
r·ηso=-1ikIxΓsso(r1, r2)x1-Γsso(r1, r2)x2r1=r2=r=(x,y)dxdy-yΓsso(r1, r2)y1-Γsso(r1, r2)y2r1=r2=r=(x,y)dxdy.
r·ηso=2kI[r·|φs(r)|]As(r, r)dxdy,
r·ηo=1kI{[r·|φs(r)|]As(r, r)+[r·|φp(r)|]Ap(r, r)}dxdy.
mj=1k2I|φj|2Aj(r, r)dxdy,j=s,p,
M=1kIr·φs Asdxdy+1kIr·φpApdxdy2.
M1kIr·φsAsdxdy+1kIr·φpApdxdy2,
signr·φsAsdxdy=signr·φpApdxdy.
M(r2sms+r2pmp)2,
φs=ρr,
φp=μr,
F=r2(ms+mp)-M,
F(r2s+r2p)(ms+mp)-r2sms-r2pmp-2r2sr2pmsmp=(r2pms-r2smp)20,Q.E.D.
r2pms=r2smp.

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