Abstract

For light fields propagating through rotationally symmetric first-order optical systems, the possibility of improvement of the beam-propagation factor is shown to arise when the vectorial behavior is taken into account. For partially polarized beams, we find the optimized value of the beam-quality parameter that can be attained by using this kind of system. This value is given in terms of the beam qualities associated with the transverse polarization components of the vector field. On the basis of the so-called intensity-moment formalism, the general conditions that should be fulfilled at some plane to reach such an optimized value are determined. A procedure to experimentally get the optimization conditions is also proposed.

© 2005 Optical Society of America

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References

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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 functions 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).
  5. J. Serna, R. Martínez-Herrero, P. M. Mejías, “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. Mejías, H. Weber, R. Martínez-Herrero, and A. González-Ureña, eds., Laser Beam Characterization (SEDO, Madrid, Spain, 1993).
  8. H. Weber, N. Reng, J. Lüdtke, and P. M. Mejías, eds., Proceedings of the 2nd Workshop on Laser Beam Characterization (Festkörper-Laser-Institut, Berlin, 1994).
  9. M. Morin and A. Giesen, eds., Third International Workshop on Laser Beam and Optics Characterization, Proc. SPIE2870, (1996).
  10. A. Giesen and M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (VDI-Technologie Zentrum, Düsseldorf, Germany, 1998).
  11. H. Laabs and H. Weber, eds., Proceedings of the Fifth International Workshop on Laser Beam and Optics Characterization (VDI-Technologie Zentrum, Düsseldorf, Germany, 2000).
  12. A. Giesen and H. Weber, eds., Seventh International Workshop on Laser Beam and Optics Characterization, Proc. SPIE4932, (2002).
  13. ISO/DIS 11146, “Optics and optical instruments—laser and laser related equipment—test methods for laser beam parameters: beam widths, divergence angle and beam propagation factor” (International Organization for Standardization, Geneva, Switzerland, 1999).
  14. R. Martínez-Herrero, P. M. Mejías, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [CrossRef]
  15. J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
    [CrossRef]
  16. J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
    [CrossRef]
  17. 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]
  18. 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]
  19. P. M. Mejías, R. Martínez-Herrero, G. Piquero, J. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
    [CrossRef]
  20. R. Martínez-Herrero, P. M. Mejías, G. Piquero, “Anisotropic pure-phase plates for quality improvement of partially coherent, partially polarized beams,” J. Opt. Soc. Am. A 20, 577–581 (2003).
    [CrossRef]
  21. R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
    [CrossRef]
  22. 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]
  23. J. M. Movilla, R. Martínez-Herrero, P. M. Mejías, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
    [CrossRef]
  24. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1999).
    [CrossRef]

2004

R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
[CrossRef]

2003

2002

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

2001

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. Martínez-Herrero, P. M. Mejías, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[CrossRef]

2000

J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

1998

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

1997

1995

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]

1992

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

1991

1989

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

1988

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 functions 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]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1999).
[CrossRef]

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]

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]

Martínez-Herrero, R.

R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, “Anisotropic pure-phase plates for quality improvement of partially coherent, partially polarized beams,” J. Opt. Soc. Am. A 20, 577–581 (2003).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, J. 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. Martínez-Herrero, P. M. Mejías, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

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

R. Martínez-Herrero, P. M. Mejías, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

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

Mejías, P. M.

R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, “Anisotropic pure-phase plates for quality improvement of partially coherent, partially polarized beams,” J. Opt. Soc. Am. A 20, 577–581 (2003).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, J. 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. Martínez-Herrero, P. M. Mejías, “Quality improvement of partially polarized beams,” Appl. Opt. 40, 6098–6101 (2001).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

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

R. Martínez-Herrero, P. M. Mejías, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

J. Serna, R. Martínez-Herrero, P. M. Mejías, “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.

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

Movilla, J. M.

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

J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

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

R. Martínez-Herrero, P. M. Mejías, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

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]

Piquero, G.

R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, G. Piquero, “Anisotropic pure-phase plates for quality improvement of partially coherent, partially polarized beams,” J. Opt. Soc. Am. A 20, 577–581 (2003).
[CrossRef]

P. M. Mejías, R. Martínez-Herrero, G. Piquero, J. 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]

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, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

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

Serna, J.

Siegman, A. E.

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

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]

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]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1999).
[CrossRef]

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

R. Martínez-Herrero, G. Piquero, P. M. Mejías, “Parametric characterization of the spatial structure of partially coherent and partially polarized beams,” J. Opt. A, Pure Appl. Opt. 6, 67–71 (2004).
[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]

J. Opt. Soc. Am. A

Opt. Commun.

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]

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

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

Opt. Lett.

Opt. Quantum Electron.

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]

J. M. Movilla, G. Piquero, R. Martínez-Herrero, P. M. Mejías, “On the measurement of the generalized degree of polarization,” Opt. Quantum Electron. 32, 1333–1342 (2000).
[CrossRef]

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

Optik (Stuttgart)

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

Prog. Quantum Electron.

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

Other

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

P. M. Mejías, H. Weber, R. Martínez-Herrero, and A. González-Ureña, eds., Laser Beam Characterization (SEDO, Madrid, Spain, 1993).

H. Weber, N. Reng, J. Lüdtke, and P. M. Mejías, eds., Proceedings of the 2nd Workshop on Laser Beam Characterization (Festkörper-Laser-Institut, Berlin, 1994).

M. Morin and A. Giesen, eds., Third International Workshop on Laser Beam and Optics Characterization, Proc. SPIE2870, (1996).

A. Giesen and M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (VDI-Technologie Zentrum, Düsseldorf, Germany, 1998).

H. Laabs and H. Weber, eds., Proceedings of the Fifth International Workshop on Laser Beam and Optics Characterization (VDI-Technologie Zentrum, Düsseldorf, Germany, 2000).

A. Giesen and H. Weber, eds., Seventh International Workshop on Laser Beam and Optics Characterization, Proc. SPIE4932, (2002).

ISO/DIS 11146, “Optics and optical instruments—laser and laser related equipment—test methods for laser beam parameters: beam widths, divergence angle and beam propagation factor” (International Organization for Standardization, Geneva, Switzerland, 1999).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, UK, 1999).
[CrossRef]

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Equations (54)

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E ( r , z ) = ( E s ( r ; z ) , E p ( r ; z ) ) ,
Q = r 2 η 2 r η 2 ,
r 2 = I s I r 2 s + I p I r 2 p ,
r η = I s I r η s + I p I r η p ,
η 2 = I s I η 2 s + I p I η 2 p .
ξ ζ j = I I j k 2 4 π 2 ξ ζ E j * ( r + s 2 , z ) E j ( r s 2 , z ) ¯ exp ( i k s η ) d s d r d η ,
j = s , p ; ξ , ζ = x , y , u , v ,
I j = E j ( r ) 2 ¯ d r , j = s , p ,
Q = ( I s I ) 2 Q s + ( I p I ) 2 Q p + ( I s I p I 2 ) Q s p ,
Q j = r 2 j η 2 j r η j 2 , j = s , p ,
Q s p = r 2 s η 2 p + r 2 p η 2 s 2 r η s r η p ,
θ = 1 2 { arctan [ I s I p 2 Re ( I s p ) ] } ,
I s p = E s ( r ) E p * ( r ) ¯ d r .
Q = 1 4 ( Q s + Q p + Q s p ) .
r η s = 0 ,
Q = 1 4 ( Q s + Q p + r 2 s η 2 p + r 2 p η 2 s ) .
F 1 = Q s + r 2 s η 2 p ,
F 2 = Q p + r 2 p η 2 s .
F 1 + F 2 2 F 1 F 2 ,
Q 1 2 F 1 F 2 .
F 1 = G 1 + J 1 ,
F 2 = G 2 + J 2 ,
F 1 = G 1 + J 1 2 G 1 J 1 ,
F 2 = G 2 + J 2 2 G 2 J 2 ,
Q 2 1 4 F 1 F 2 G 1 J 1 G 2 J 2 .
G 1 J 1 G 2 J 2 = Q s Q p r 2 s η 2 s r 2 p η 2 p = Q s 2 Q p r 2 p η 2 p Q s 2 Q p ( r 2 p η 2 p r η p 2 ) = Q s 2 Q p 2 ,
Q 2 Q s Q p ,
S ( χ , β ) = 4 ( Q Q s Q p ) = ( Q s Q p ) 2 + β a + χ b 2 Q s Q p ,
a b = Q s 1 k 2 ,
χ β γ 2 = Q p 1 k 2 ,
T ( χ , γ ) = ( Q s Q p ) 2 + a χ ( Q p + γ 2 ) + b χ 2 Q s Q p .
T χ = 0 , T γ = 0 .
T χ = b a χ 2 ( Q P + γ 2 ) = 0 ,
T γ = 2 γ a χ = 0 ,
χ c = a Q p Q s ,
γ c = 0 ,
r 2 p = r 2 s Q p Q s ,
η 2 p = η 2 s Q p Q s ,
r η p = 0 .
( Q ) optim = 1 4 ( Q s + Q p ) 2 .
min ( Q s , Q p ) Q optim max ( Q s , Q p ) .
( z R ) j = [ ( r 2 j ) w η 2 j ] 1 2 , j = s , p ,
Q s = r 2 s η 2 s = r 2 p η 2 p = Q p ,
μ = r 2 s r 2 p = η 2 p η 2 s .
Q = Q s 4 ( 2 + μ + 1 μ ) .
Δ rel Q = Q Q optim Q ,
Δ rel Q = ( 1 μ ) 2 ( 1 + μ ) 2 .
r 2 s = r 2 p ,
η 2 s η 2 p ,
r η s = r η p = 0 .
Q = ( 1 + σ ) Q s 2 ,
σ = η 2 p η 2 s ,
Q optim = ( 1 + σ + 2 σ ) Q s 4 .
Δ rel Q = Q Q optim Q = ( 1 σ ) 2 2 ( 1 + σ ) .

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