Abstract

Traditional interferometric methods for measuring the vortex structure of complex wave fields suffer from many intrinsic problems and seldom yield results of any accuracy. Using the unique properties of what I call Poincaré vortices, I develop a radically different method based on Stokes parameters that offers many practical advantages. The theory of this new method is discussed, and its unique capabilities are illustrated by reconstruction with high accuracy of the vortex structure of a simulated random field containing numerous vortices, including several closely spaced vortex pairs that would be difficult, if not impossible, to resolve by traditional means.

© 2001 Optical Society of America

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).
  3. M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).
  4. Y. S. Kivshar and B. Luther-Davies, Phys. Rep.298, 81 (1998).
    [CrossRef]
  5. W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
    [CrossRef] [PubMed]
  6. G. Molina-Terriza, L. Torner, and D. V. Petrov, Opt. Lett. 24, 899 (1998).
    [CrossRef]
  7. G. P. Karman, M. W. Beijersbergen, A. van Duijl, and J. P. Woerdman, Opt. Lett. 22, 1503 (1997).
    [CrossRef]
  8. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
    [CrossRef]
  9. G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
    [CrossRef]
  10. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
    [CrossRef]
  11. A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
    [CrossRef]
  12. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).
  13. N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).
  14. J. F. Nye, Proc. R. Soc. London Ser. A 387, 105 (1983).
    [CrossRef]
  15. J. F. Nye, Proc. R. Soc. London Ser. B 389, 279 (1983).
    [CrossRef]
  16. J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
    [CrossRef]
  17. J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433, 447 (1987).
  18. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
    [CrossRef]
  19. M. V. Berry, Proc. SPIE 4403, 1 (2001), Sec.  5.
    [CrossRef]
  20. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec.  1.4.2.
  21. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), p. 21.
  22. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, Berlin, 1976), p. 43.
  23. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef] [PubMed]
  24. I. Freund, Phys. Rev. E 52, 2348 (1995).
    [CrossRef]
  25. O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).
  26. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

2001 (3)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

M. V. Berry, Proc. SPIE 4403, 1 (2001), Sec.  5.
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

2000 (1)

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

1999 (1)

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

1998 (1)

1997 (1)

1996 (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

1995 (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

1994 (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

1992 (2)

1991 (1)

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

1987 (2)

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433, 447 (1987).

1983 (2)

J. F. Nye, Proc. R. Soc. London Ser. A 387, 105 (1983).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. B 389, 279 (1983).
[CrossRef]

1981 (1)

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Allen, L.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

Baranova, N. B.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Beijersbergen, M. W.

Berry, M. V.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

M. V. Berry, Proc. SPIE 4403, 1 (2001), Sec.  5.
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Born, M.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec.  1.4.2.

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Denz, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Freund, I.

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Geisser, M.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433, 447 (1987).

Heckenberg, N. R.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Jauch, J. M.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, Berlin, 1976), p. 43.

Karman, G. P.

Kivshar, Y. S.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Y. S. Kivshar and B. Luther-Davies, Phys. Rep.298, 81 (1998).
[CrossRef]

Krolikowski, W.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Law, C. T.

G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Y. S. Kivshar and B. Luther-Davies, Phys. Rep.298, 81 (1998).
[CrossRef]

Mamaev, A. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

McCarthy, G.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Mokhun, A. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

Mokhun, I. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

Molina-Terriza, G.

Nye, J. F.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. A 387, 105 (1983).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. B 389, 279 (1983).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Ostrovskaya, E. A.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Padgett, M. J.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Petrov, D. V.

Pilipetskii, N.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Rohrlich, F.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, Berlin, 1976), p. 43.

Rubinnsztein-Dunlop, H.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Shkukov, V. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), p. 21.

Shvartsman, N.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Soskin, M.

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

Soskin, M. S.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

Staliunas, K.

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

Swartzlander, G. A.

G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef]

Tamm, Chr.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Torner, L.

van Duijl, A.

Vasnetsov, M.

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Weilnau, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Weiss, C. O.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

White, A. G.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

Woerdman, J. P.

Wolf, E. W.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec.  1.4.2.

Zel’dovich, B. Y.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

J. Mod. Opt. (1)

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinnsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, J. Mod. Opt. 38, 2531 (1991).
[CrossRef]

JETP Lett. (2)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Opt. Lett. (3)

Photon. Sci. News (1)

M. Soskin and M. Vasnetsov, Photon. Sci. News 4(4), 21 (1999).

Phys. Rev. A (2)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

Phys. Rev. Lett. (2)

G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (5)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. A 387, 105 (1983).
[CrossRef]

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433, 447 (1987).

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

J. F. Nye, Proc. R. Soc. London Ser. B 389, 279 (1983).
[CrossRef]

Proc. SPIE (2)

M. V. Berry, Proc. SPIE 4403, 1 (2001), Sec.  5.
[CrossRef]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, Proc. SPIE 4403, 116 (2001).

Other (6)

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarizations singularities,” Phys. Rev. E (to be published).

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec.  1.4.2.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), p. 21.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, Berlin, 1976), p. 43.

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

Y. S. Kivshar and B. Luther-Davies, Phys. Rep.298, 81 (1998).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Proposed experimental apparatus. The reference (Ref.)  and sample beams have orthogonal linear polarizations that are oriented perpendicular x and parallel y to the plane of the figure. These beams are combined by a polarizing cube (PC) that serves as a monolithic beam-combining interferometer. The intensities transmitted by the Stokes filters (S) are recorded by the CCD camera. Since these optical components can be translated along the beam path without the need for realignment, while interferometer PC remains stationary, easy measurement of the evolution of the sample beam under propagation is possible. Also, as shown in Fig.  2, neither noise nor the quality of the reference wave is especially important, and even a quasi-random reference beam does not degrade accuracy.

Fig. 2
Fig. 2

Poincaré vortices: (a), (b) quasi-random reference wave. (a) Random amplitude fluctuations 0-1 and (b) random phase variations 0-π are coded black-white. (c) Phase map of Gaussian random sample beam, showing the vortex structure. (d) Phase map of the Poincaré field, showing Poincaré vortices. In contour maps (c) and (d), positive (negative) vortices are shown by filled (open) circles. Note the precise one-to-one correspondence in both the positions and the signs between the sample beam vortices and Poincaré vortices and the presence of several closely spaced vortex pairs that would be difficult if not impossible to measure by traditional interferometric methods. Adding 5% white noise to the Stokes parameters does not noticeably change the vortex structure in (d). For ExEy, alignment errors in θ [Eq.  (2)] should not exceed 5° if one wishes to avoid mixing sample beam Ey vortices into the reference beam Ex.

Fig. 3
Fig. 3

Phase maps of (a) a negative C vortex and (c) a positive C vortex. The phase is coded 02π for black–white. (b), (d) Corresponding arrangement of ellipses around the central C point (filled circles). The field components in (a), (b), the “star”, are Ex=1-iy, Ey=i1-x; those in (c), (d), the “lemon”, are Ex=1+iy, Ey=i1-x. For the star (lemon) the ellipses rotate by π in the opposite (same) direction as a counterclockwise path that encircles the C point, and IC=-1/2 +1/2, in accord with Eq.  (6). The seemingly subtle differences between the star and the lemon are extremely difficult to measure in a random field, in which the orientations of the surrounding, nearly circular ellipses are almost always changing rapidly.

Equations (6)

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

s0=Ex2+Ey2,S1=s0-1Ex2-Ey2,S2=2s0-1ReEx*Ey,S3=2s0-1ImEx*Ey,
Ex=Excosθ-Eysinθ,Ey=Exsinθ+Eycosθ.
P=S2+iS3.
P=APexpiΦP=2s0-1ExEyexpiφy-φx.
C=S1+iS2,
σ=2IC.

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