Abstract

We address Anderson localization in disordered honeycomb photonic lattices and show that the localization process is strongly affected by the spectral position of the input wavepacket within the first Brillouin zone of the lattice. In spite of the fact that in regular lattices expansion of the beam is much stronger for excitation near the Dirac points—where light exhibits conical diffraction—than for excitation at the center of the Brillouin zone—where light exhibits normal diffraction—we found that disorder leads to pronounced Anderson localization even around the Dirac points. We found that for the same disorder level the width of the averaged output intensity distribution for excitations around the Dirac points may be substantially larger than that for excitations at the center of the Brillouin zone.

© 2013 Optical Society of America

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  1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  2. H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
    [CrossRef]
  3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  4. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  5. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef]
  6. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
    [CrossRef]
  7. U. Naether, Y. V. Kartashov, V. A. Vysloukh, S. Nolte, A. Tünnermann, L. Torner, and A. Szameit, Opt. Lett. 37, 593 (2012).
    [CrossRef]
  8. S. Stützer, Y. V. Kartashov, V. A. Vysloukh, A. Tünnermann, S. Nolte, M. Lewenstein, L. Torner, and A. Szameit, Opt. Lett. 37, 1715 (2012).
    [CrossRef]
  9. D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
    [CrossRef]
  10. D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
    [CrossRef]
  11. D. M. Jovic, C. Denz, and M. R. Belic, Opt. Lett. 37, 4455 (2012).
    [CrossRef]
  12. M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
    [CrossRef]
  13. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
    [CrossRef]
  14. A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
    [CrossRef]
  15. M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).
  16. A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
    [CrossRef]
  17. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
    [CrossRef]
  18. J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

2013 (4)

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

2012 (3)

2011 (3)

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

2010 (1)

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

2007 (2)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

1989 (1)

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

1958 (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Bahat-Treidel, O.

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

Bahrampour, A. R.

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Belic, M. R.

D. M. Jovic, C. Denz, and M. R. Belic, Opt. Lett. 37, 4455 (2012).
[CrossRef]

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Buljan, H.

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

Capeta, D.

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Corrielli, G.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

Crespi, A.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

de Raedt, H.

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

de Vries, P.

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Denz, C.

D. M. Jovic, C. Denz, and M. R. Belic, Opt. Lett. 37, 4455 (2012).
[CrossRef]

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Golshani, M.

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

Heinrich, M.

Jovic, D. M.

D. M. Jovic, C. Denz, and M. R. Belic, Opt. Lett. 37, 4455 (2012).
[CrossRef]

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Langari, A.

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

Lewenstein, M.

Longhi, S.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Naether, U.

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

U. Naether, Y. V. Kartashov, V. A. Vysloukh, S. Nolte, A. Tünnermann, L. Torner, and A. Szameit, Opt. Lett. 37, 593 (2012).
[CrossRef]

S. Stützer, Y. V. Kartashov, V. A. Vysloukh, A. Tünnermann, S. Nolte, M. Lewenstein, L. Torner, and A. Szameit, Opt. Lett. 37, 1715 (2012).
[CrossRef]

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
[CrossRef]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

Osellame, R.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

Peleg, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Radic, J.

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Stützer, S.

Szameit, A.

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

S. Stützer, Y. V. Kartashov, V. A. Vysloukh, A. Tünnermann, S. Nolte, M. Lewenstein, L. Torner, and A. Szameit, Opt. Lett. 37, 1715 (2012).
[CrossRef]

U. Naether, Y. V. Kartashov, V. A. Vysloukh, S. Nolte, A. Tünnermann, L. Torner, and A. Szameit, Opt. Lett. 37, 593 (2012).
[CrossRef]

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1172 (2010).
[CrossRef]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

Torner, L.

Tünnermann, A.

Valle, G. D.

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

Vysloukh, V. A.

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Zeil, P.

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

Nat. Photonics (1)

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153 (2013).

Nature (3)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

New J. Phys. (1)

A. Crespi, G. Corrielli, G. D. Valle, R. Osellame, and S. Longhi, New J. Phys. 15, 013012 (2013).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. A (4)

D. Capeta, J. Radic, A. Szameit, M. Segev, and H. Buljan, Phys. Rev. A 84, 011801(R) (2011).
[CrossRef]

D. M. Jovic, Y. S. Kivshar, C. Denz, and M. R. Belic, Phys. Rev. A 83, 033813 (2011).
[CrossRef]

M. Golshani, A. R. Bahrampour, A. Langari, and A. Szameit, Phys. Rev. A 87, 033817 (2013).
[CrossRef]

A. Szameit, M. C. Rechtsman, O. Bahat-Treidel, and M. Segev, Phys. Rev. A 84, 021806(R) (2011).
[CrossRef]

Phys. Rev. Lett. (3)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef]

H. de Raedt, A. Lagendijk, and P. de Vries, Phys. Rev. Lett. 62, 47 (1989).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Other (1)

J. M. Zeuner, M. C. Rechtsman, S. Nolte, and A. Szameit, “Edge states protected by chiral symmetry in disordered photonic graphene,” arXiv: 1304.6911 (2013).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Honeycomb lattice and (b) first two allowed bands from its Floquet–Bloch spectrum at p=8, a=0.5, d=2.

Fig. 2.
Fig. 2.

Intensity distribution at different distances illustrating diffraction in regular lattice for the excitation at (a) the top of the first allowed band and (b) for the excitation in one of the Dirac points. The width of the input envelope is w=10.

Fig. 3.
Fig. 3.

Averaged intensity distributions at ξ=500 for different disorder levels for the excitation at (a) the top of the first allowed band and (b) for the excitation at one of the Dirac points. The width of the input envelope is w=10.

Fig. 4.
Fig. 4.

Cross-sections of averaged intensity distributions at ζ=0 for pd=0.3 (black outer curves) and pd=0.8 (red inner curves) in the logarithmic scale. Panel (a) corresponds to the excitation at the top of the first allowed band, panel (b) corresponds to the excitation at the Dirac point. The width of the input envelope is w=10.

Fig. 5.
Fig. 5.

(a) Averaged integral form-factor versus disorder level pd at w=10. (b) Inverse averaged form-factor versus width of the input envelope w at pd=0.4. The lines marked with letter “n” correspond to the excitation at the top of the first allowed band, while lines marked with “d” correspond to the excitation in the Dirac point. Circles correspond to the averaged intensity distributions shown in Fig. 3.

Equations (2)

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

iqξ=12(2qη2+2qζ2)R(η,ζ)q,
Iav(η,ζ)=1Qi=1,Q|qi(η,ζ)|2,χav2=1QU2i=1,Q|qi(η,ζ)|4dηdζ,U=|qi(η,ζ)|2dηdζ,

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