Abstract

Long-distance quantum communication relies on storing and retrieving photonic qubits in orthogonal field modes. The available degrees of freedom for photons are polarization, spatial-mode profile, and temporal/spectral profile. To date, methods exist for decomposing, manipulating, and analyzing photons into orthogonal polarization modes and spatial modes. Here we propose and theoretically verify the first highly efficient method to carry out analogous operations for temporally and spectrally overlapping, but field-orthogonal, temporal modes. The method relies on cascaded nonlinear-optical quantum frequency conversion.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
    [CrossRef]
  2. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
    [CrossRef]
  3. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
    [CrossRef]
  4. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
    [CrossRef]
  5. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
    [CrossRef]
  6. I. L. Chuang and Y. Yamamoto, Phys. Rev. A 52, 3489 (1995).
    [CrossRef]
  7. I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
    [CrossRef]
  8. J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, Opt. Express 21, 15959 (2013).
  9. B. J. Smith and M. G. Raymer, New J. Phys. 9, 414 (2007).
    [CrossRef]
  10. A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
    [CrossRef]
  11. T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
    [CrossRef]
  12. A. Eckstein, B. Brecht, and C. Silberhorn, Opt. Express 19, 13770 (2011).
    [CrossRef]
  13. H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
    [CrossRef]
  14. M. G. Raymer and K. Srinivasan, Phys. Today 65(11), 32 (2012).
    [CrossRef]
  15. J. Huang and P. Kumar, Phys. Rev. Lett. 68, 2153 (1992).
    [CrossRef]
  16. A. P. Vandevender and P. G. Kwiat, J. Mod. Opt. 51, 1433 (2004).
    [CrossRef]
  17. M. A. Albota and F. N. C. Wong, Opt. Lett. 29, 1449 (2004).
    [CrossRef]
  18. R. V. Roussev, C. Langrock, J. R. Kurz, and M. M. Fejer, Opt. Lett. 29, 1518 (2004).
    [CrossRef]
  19. L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, Opt. Express 20, 27454 (2012).
    [CrossRef]
  20. C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, Phys. Rev. A 85, 053829 (2012).
    [CrossRef]
  21. A. S. Clark, S. Shahnia, M. J. Collins, C. Xiong, and B. J. Eggleton, Opt. Lett. 38, 947 (2013).
    [CrossRef]
  22. A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
    [CrossRef]
  23. D. V. Reddy, M. G. Raymer, C. J. McKinstrie, L. Mejling, and K. Rottwitt, Opt. Express 21, 13840 (2013).
    [CrossRef]
  24. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
    [CrossRef]
  25. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
    [CrossRef]
  26. T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000).
    [CrossRef]
  27. A. Hayat, X. Xing, A. Feizpour, and A. M. Steinberg, Opt. Express 20, 29174 (2012).
  28. J. P. Heritage and A. M. Weiner, IEEE J. Sel. Top. Quantum Electron. 13, 1351 (2007).

2013

2012

L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, Opt. Express 20, 27454 (2012).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, Phys. Rev. A 85, 053829 (2012).
[CrossRef]

A. Hayat, X. Xing, A. Feizpour, and A. M. Steinberg, Opt. Express 20, 29174 (2012).

A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
[CrossRef]

M. G. Raymer and K. Srinivasan, Phys. Today 65(11), 32 (2012).
[CrossRef]

2011

2010

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

2008

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

2007

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

B. J. Smith and M. G. Raymer, New J. Phys. 9, 414 (2007).
[CrossRef]

J. P. Heritage and A. M. Weiner, IEEE J. Sel. Top. Quantum Electron. 13, 1351 (2007).

2004

2003

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

2002

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

2000

T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000).
[CrossRef]

1995

1992

J. Huang and P. Kumar, Phys. Rev. Lett. 68, 2153 (1992).
[CrossRef]

Abouraddy, A. F.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Albota, M. A.

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Bosenberg, W. R.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Brecht, B.

A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
[CrossRef]

A. Eckstein, B. Brecht, and C. Silberhorn, Opt. Express 19, 13770 (2011).
[CrossRef]

Brunner, N.

T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Byer, R. L.

Cargill, D. S.

Christ, A.

A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
[CrossRef]

A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
[CrossRef]

Chuang, I. L.

I. L. Chuang and Y. Yamamoto, Phys. Rev. A 52, 3489 (1995).
[CrossRef]

Clark, A. S.

Collins, M. J.

Courtial, J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

de Riedmatten, H.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Dridi, K. H.

T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000).
[CrossRef]

Eckardt, R. C.

Eckstein, A.

Eggleton, B. J.

Feizpour, A.

Fejer, M. M.

Gisin, N.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Hayat, A.

Heritage, J. P.

J. P. Heritage and A. M. Weiner, IEEE J. Sel. Top. Quantum Electron. 13, 1351 (2007).

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Huang, J.

J. Huang and P. Kumar, Phys. Rev. Lett. 68, 2153 (1992).
[CrossRef]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Kumar, P.

J. Huang and P. Kumar, Phys. Rev. Lett. 68, 2153 (1992).
[CrossRef]

Kurz, J. R.

Kwiat, P. G.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

A. P. Vandevender and P. G. Kwiat, J. Mod. Opt. 51, 1433 (2004).
[CrossRef]

Langrock, C.

Lavery, M. P. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Lupo, C.

A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
[CrossRef]

Marcikic, I.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Mauerer, W.

A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
[CrossRef]

McGuinness, H. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

McKinstrie, C. J.

D. V. Reddy, M. G. Raymer, C. J. McKinstrie, L. Mejling, and K. Rottwitt, Opt. Express 21, 13840 (2013).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, Phys. Rev. A 85, 053829 (2012).
[CrossRef]

L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, Opt. Express 20, 27454 (2012).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

Mejling, L.

Moore, R. O.

Myers, L. E.

Nunn, J.

Okamoto, M.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Padgett, M. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Pierce, J. W.

Pironio, S.

T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Radic, S.

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Raymer, M. G.

D. V. Reddy, M. G. Raymer, C. J. McKinstrie, L. Mejling, and K. Rottwitt, Opt. Express 21, 13840 (2013).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, Phys. Rev. A 85, 053829 (2012).
[CrossRef]

M. G. Raymer and K. Srinivasan, Phys. Today 65(11), 32 (2012).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

B. J. Smith and M. G. Raymer, New J. Phys. 9, 414 (2007).
[CrossRef]

Reddy, D. V.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Rottwitt, K.

Roussev, R. V.

Saleh, B. E. A.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Sasada, H.

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Scarani, V.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Shahnia, S.

Silberhorn, C.

A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
[CrossRef]

A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
[CrossRef]

A. Eckstein, B. Brecht, and C. Silberhorn, Opt. Express 19, 13770 (2011).
[CrossRef]

Smith, B. J.

Söller, C.

Søndergaard, T.

T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000).
[CrossRef]

Srinivasan, K.

M. G. Raymer and K. Srinivasan, Phys. Today 65(11), 32 (2012).
[CrossRef]

Steinberg, A. M.

Teich, M. C.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Tittel, W.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

van Enk, S. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

Vandevender, A. P.

A. P. Vandevender and P. G. Kwiat, J. Mod. Opt. 51, 1433 (2004).
[CrossRef]

Vértesi, T.

T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Walmsley, I. A.

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Weiner, A. M.

J. P. Heritage and A. M. Weiner, IEEE J. Sel. Top. Quantum Electron. 13, 1351 (2007).

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Wong, F. N. C.

Wright, L. J.

Xing, X.

Xiong, C.

Yamamoto, Y.

I. L. Chuang and Y. Yamamoto, Phys. Rev. A 52, 3489 (1995).
[CrossRef]

Yarnall, T.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Zbinden, H.

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

Zhang, L.

IEEE J. Sel. Top. Quantum Electron.

J. P. Heritage and A. M. Weiner, IEEE J. Sel. Top. Quantum Electron. 13, 1351 (2007).

J. Mod. Opt.

A. P. Vandevender and P. G. Kwiat, J. Mod. Opt. 51, 1433 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Phys.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

New J. Phys.

B. J. Smith and M. G. Raymer, New J. Phys. 9, 414 (2007).
[CrossRef]

A. Christ, C. Lupo, and C. Silberhorn, New J. Phys. 14, 083007 (2012).
[CrossRef]

A. Christ, B. Brecht, W. Mauerer, and C. Silberhorn, New J. Phys. 15, 053038 (2013).
[CrossRef]

Opt. Commun.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, Opt. Commun. 283, 747 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, Phys. Rev. A 85, 053829 (2012).
[CrossRef]

I. L. Chuang and Y. Yamamoto, Phys. Rev. A 52, 3489 (1995).
[CrossRef]

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, Phys. Rev. A 66, 062308 (2002).
[CrossRef]

H. Sasada and M. Okamoto, Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Phys. Rev. B

T. Søndergaard and K. H. Dridi, Phys. Rev. B 61, 15688 (2000).
[CrossRef]

Phys. Rev. Lett.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 99, 250502 (2007).
[CrossRef]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

T. Vértesi, S. Pironio, and N. Brunner, Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

J. Huang and P. Kumar, Phys. Rev. Lett. 68, 2153 (1992).
[CrossRef]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, Phys. Rev. Lett. 105, 093604 (2010).
[CrossRef]

Phys. Today

M. G. Raymer and K. Srinivasan, Phys. Today 65(11), 32 (2012).
[CrossRef]

Science

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Temporal-mode interferometer using two nonlinear media (QFC 1 and QFC 2) with identical (DC) or opposite-sign (RC) dispersion. Appropriate choices for pump-pulse shapes, pump powers, and the phase shift θ will selectively frequency convert a specific s (green) TM into an r (blue) TM at a different central frequency, while not affecting temporally orthogonal s-input TMs. The pump q is used only for χ(3) implementations. WDM, wavelength-division multiplexer; PC, prechirp modules, which are necessary for χ(3) implementations. The coupler C contains frequency dependent delays for the DC case.

Fig. 2.
Fig. 2.

(a) Dominant Schmidt-mode CEs |ρn|2 for single-stage TWM. (b) Corresponding s-channel input Schmidt modes for a Gaussian shaped pump p. Selectivity is maximum (0.81) at γ=1.0.

Fig. 3.
Fig. 3.

CEs |ρn|2 and Schmidt modes for TWM-TMI with Gaussian pump. The first two Schmidt modes are shown for (b) r input, (c) r output, (d) s input, and (e) s output. Plot (a) shows the CEs of the first four Schmidt modes, with selectivity S in the inset.

Fig. 4.
Fig. 4.

(a) CEs |ρn|2; (b)–(e) Schmidt modes; and (g)–(j) first Schmidt-mode phase profiles for FWM-TMI, with Gaussian pumps. The first two Schmidt modes are shown for (b) r input, (c) r output, (d) s input, and (e) s output. (f) shows the first-stage pump prechirps used for FWM-TMI.

Equations (4)

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

(z+βpt)Ap=i(γ/2)δF[|Ap|2+2|Aq|2]Ap,(z+βqt)Aq=i(γ/2)δF[2|Ap|2+|Aq|2]Aq,(z+βrt)Ar=iγApAq*As+iγδF[|Ap|2+|Aq|2]Ar,(z+βst)As=iγAp*AqAr+iγδF[|Ap|2+|Aq|2]As,
Aj(L,t)=dtk=r,sGjk(t,t)Ak(0,t).
Grr(t,t)=n=1τnΨn(t)ψn*(t),Grs(t,t)=n=1ρnΨn(t)ϕn*(t),Gsr(t,t)=n=1ρnΦn(t)ψn*(t),Gss(t,t)=n=1τnΦn(t)ϕn*(t).
Grs(t,t)=dt[Grs(2)(t,t)Gss(1)(t,t)+eiθGrr(2)(t,t)Grs(1)(t,t)],

Metrics