Abstract

On the basis of the irradiance-moments formalism, four matrices are proposed whose elements, defined for any partially coherent field, are closely related with the second-order measurable parameters handled in the ISO standard 11146. These matrices are shown to exhibit a number of properties concerning the orientation of the transverse profile of a partially coherent beam. This behavior is described by the rotation of the principal axes of the field around its propagation axis. In addition, these matrices provide information about the spatial structure of the field (beam spread product and orbital part of the angular momentum). A new overall parameter is introduced in terms of the above matrices, which remains invariant under rotation of the Cartesian coordinate axes and upon propagation through rotationally-symmetric first-order optical systems.

© 2006 Optical Society of America

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  1. R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988).
    [CrossRef]
  2. S. Lavi, R. Prochaska, and E. Keren, "Generalized beam parameters and transformation law for partially coherent light," Appl. Opt. 27, 3696-3703 (1988).
    [CrossRef] [PubMed]
  3. M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).
  4. A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  5. J. Serna, R. Martínez-Herrero, and 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. ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).
  8. J. A. Arnaud and H. Kogelnik, "Light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
    [CrossRef] [PubMed]
  9. J. Serna and G. Nemes, "Decoupling of coherent Gaussian beams with general astimatism," Opt. Lett. 18, 1174-1175 (1993).
    [CrossRef]
  10. G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
    [CrossRef]
  11. F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).
  12. G. Nemes and J. Serna, "Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document," in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29-49.
  13. J. Serna, F. Encinas and G. Nemes, "Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens," J. Opt. Soc. Am. A 18, 1726-1733 (2001).
    [CrossRef]
  14. P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
    [CrossRef]
  15. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).
  16. G. Nemes, private communication.
  17. J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
    [CrossRef]
  18. F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
    [CrossRef]
  19. P. M. Mejías, R. Martínez-Herrero, G. Piquero and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002), and references therein.
    [CrossRef]

2002 (1)

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

2001 (1)

1999 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

1998 (2)

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

1994 (2)

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

1993 (1)

1992 (2)

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

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

1991 (1)

1990 (1)

A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1989 (1)

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

1988 (2)

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

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

1969 (1)

Arnaud, J. A.

Bagini, V.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Bastiaans, M. J.

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

Encinas, F.

Encinas-Sanz, F.

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

Frezza, F.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Friberg, A. T.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Gori, F.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Honkanen, M.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Keren, E.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Kogelnik, H.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Kuittinen, M.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Lautanen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Lavi, S.

Martínez, C.

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

Martínez-Herrero, R.

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

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

J. Serna, R. Martínez-Herrero, and 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.

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

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

J. Serna, R. Martínez-Herrero, and 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]

Movilla, J. M.

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

Mukunda, N.

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

Nemes, G.

Pääkkönen, P.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Piquero, G.

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

Prochaska, R.

Santarsiero, M.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Schettini, G.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Schirripa Spagnolo, G.

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Serna, J.

Siegman, A. E.

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Simon, R.

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

Soife, V. A.

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

Sudarshan, E. C. G.

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

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

Weber, H.

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

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998).
[CrossRef]

J. Mod. Opt. (2)

P. Pääkkönen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soife and A. T. Friberg, "Rotating optical fields: exeperimental demonstartion with diffractive optics," J. Mod. Opt. 45, 2355-2369 (1998).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen and J. Turunen, "Generation of rotating Gauss-Laguerre modes with bibary-phase diffractive optics," J. Mod. Opt. 46, 227-238 (1999).

J. Opt. Soc. Am A (1)

G. Nemes and A. E. Siegman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am A 11, 2257-2264 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

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

Opt. Lett. (1)

Opt. Quantum Electron. (2)

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

J. Serna, P. M. Mejías and R. Martínez-Herrero, "Rotation of partially coherent beams through free space," Opt. Quantum Electron. 24, 873-880 (1992).
[CrossRef]

Opt. Review (1)

F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini and G. Schirripa Spagnolo, "Coherent and partially coherent twisting beams," Opt. Review 1, 143-145 (1994).

Optik (1)

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

Proc. SPIE (1)

A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Prog. Quantum Electron. (1)

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

Other (3)

G. Nemes, private communication.

ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).

G. Nemes and J. Serna, "Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document," in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29-49.

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Equations (55)

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< x m y n u p v q > 1 I o + x m y n u p v q h r η z d r d η ,
M 0 = ( < x 2 + y 2 > < xu + yv > < xu + yv > < u 2 + v 2 > ) ,
M 1 = ( < x 2 y 2 > < xu yv > < xu yv > < u 2 v 2 > ) ,
M 2 = ( 2 < xy > < xv + yu > < xv + yu > 2 < uv > ) ,
M 3 = ( 0 < xv yu > < yu xv > 0 ) ,
tan 2 θ ( z ) = 2 < xy > + 2 z ( < xv > + < yu > ) + 2 z 2 < uv > < x 2 > < y 2 > + 2 z ( < xu > < yv > ) + z 2 ( < u 2 > < v 2 > ) ,
M 1 = 0 tan 2 θ ( z ) = ,
M 2 = 0 tan 2 θ ( z ) = 0 ,
M 3 = c M 1 tan 2 θ ( z ) = c ,
M 1 = M 1 cos 2 α + M 2 sin 2 α ,
M 2 = M 1 sin 2 α + M 2 cos 2 α ,
M 1 = ( cos 2 α + c sin 2 α ) M 1 ,
M 2 = ( sin 2 α + c cos 2 α ) M 1 .
tan 2 α 0 = c .
( M i ) output = S ( M i ) input S t , i = 0,1,2,3 ,
B output = P B input P t ,
B = ( < x 2 > < xy > < xu > < xv > < xy > < y 2 > < yu > < yv > < xu > < yu > < u 2 > < uv > < xv > < yv > < uv > < v 2 > ) .
< xv > output = ac < xy > input + a d < xv > input + b c < yu > input + b d < uv > input ,
( M 3 ) output = S ( M 3 ) input S t ,
< xv > rot = sin α cos α < xu > cos 2 α < xv > sin 2 α < yu > + sin α cos α < yv > ,
< yu > rot = sin α cos α < xu > sin 2 α < xv > + cos 2 α < yu > + sin α cos α < yv > ,
< xv yu > rot = < xv yu > .
g = tr [ ( M 0 1 M 1 ) 2 ] tr [ ( M 0 1 M 2 ) 2 ] [ tr ( M 0 1 M 1 M 0 1 M 2 ) ] 2 ,
( M 0 1 M i ) output = ( S t ) 1 ( M 0 1 ) input S 1 S ( M i ) input S t = ( S t ) 1 ( M 0 1 M i ) input S t , i = 1,2
tr [ ( M 0 1 M i ) output 2 ] = tr [ ( M 0 1 M i ) input 2 ] , i = 1,2 ,
tr [ ( M 0 1 M 1 M 0 1 M 2 ) output ] = tr [ ( M 0 1 M 1 M 0 1 M 2 ) input ] .
2 J = i = 0 3 det M i .
( M 0 1 M 1 ) = M 0 1 M 1 cos 2 α + M 0 1 M 2 sin 2 α ,
( M 0 1 M 2 ) = M 0 1 M 1 sin 2 α + M 0 1 M 2 cos 2 α ,
tr { [ ( M 0 1 M 1 ) ] 2 } =
= tr [ ( M 0 1 M 1 ) 2 ] cos 2 2 α + tr [ ( M 0 1 M 2 ) 2 ] sin 2 2 α +
+ 2 tr ( M 0 1 M 1 M 0 1 M 2 ) sin 2 α cos 2 α
tr { [ ( M 0 1 M 2 ) ] 2 } =
= tr [ ( M 0 1 M 1 ) 2 ] sin 2 2 α + tr [ ( M 0 1 M 2 ) 2 ] cos 2 2 α
2 tr ( M 0 1 M 1 M 0 1 M 2 ) sin 2 α cos 2 α
tr [ ( M 0 1 M 1 M 0 1 M 2 ) ] =
= { tr [ ( M 0 1 M 1 ) 2 ] + tr [ ( M 0 1 M 2 ) 2 ] } sin 2 α cos 2 α +
+ 2 tr ( M 0 1 M 1 M 0 1 M 2 ) ( cos 2 2 α sin 2 2 α )
T 1 tr [ ( M 0 1 M 1 ) 2 ] ,
T 2 tr [ ( M 0 1 M 2 ) 2 ] ,
T 3 tr ( M 0 1 M 1 M 0 1 M 2 ) ,
g after rotation = T 1 T 2 T 3 2 = g before rotation ,
M 0 = ( < r 2 > 0 0 < η 2 > ) ,
M 2 = ( 2 < xy > < xv + yu > < xv + yu > 0 ) ,
M 0 1 M 1 = ( < x 2 y 2 > < r 2 > < xu yv > < r 2 > < xu yv > < η 2 > < u 2 v 2 > < η 2 > ) ,
M 0 1 M 2 = ( 2 < x y > < r 2 > < x v + y u > < r 2 > < x v + y u > < η 2 > 0 ) .
tr ( M 0 1 M 1 M 0 1 M 1 ) = < x 2 y 2 > 2 < r 2 > 2 + 2 < xu yv > 2 < r 2 > < η 2 > + < u 2 v 2 > 2 < η 2 > 2 ,
tr ( M 0 1 M 2 M 0 1 M 2 ) = 4 < xy > 2 < r 2 > 2 + 2 < xv + yu > 2 < r 2 > < η 2 > ,
tr ( M 0 1 M 1 M 0 1 M 2 ) = 2 < xy > < x 2 y 2 > < r 2 > 2 + 2 < xv yu > < xu yv > < r 2 > < η 2 > .
g = 2 < r 2 > 3 < η 2 > ( < xv + yu > < x 2 y 2 > 2 < xy > < xu yv > ) 2 +
+ 4 < xy > 2 < u 2 v 2 > 2 < r 2 > 2 < η 2 > 2 + 2 < xv + yu > 2 < u 2 v 2 > 2 < r 2 > < η 2 > 2 0 , Q . E . D .
< xv + yu > < x 2 y 2 > = 2 < xy > < xu yv > ,
< xy > < u 2 v 2 > = 0 ,
< xv + yu > < u 2 v 2 > = 0 .
< xy > = < xv + yu > = 0 ,

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