Abstract

We show that propagation of optical beams in periodic lattices induces power oscillations between the Fourier spectrum peaks, with the indices related by the Bragg resonance condition. In the spatial coordinates, this is reflected in the beam position oscillations. A simple resonant theory explains the phenomenon. The effect can be used for controlled generation of the Floquet–Bloch modes in photonic lattices.

© 2007 Optical Society of America

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  1. F. Bloch, Z. Phys. A: Hadrons Nucl. 52, 555 (1928) (in German).
  2. C. Zener, Proc. R. Soc. London, Ser. A 145, 523 (1934).
    [CrossRef]
  3. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
    [CrossRef]
  4. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
    [CrossRef]
  5. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
    [CrossRef]
  6. V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
    [CrossRef] [PubMed]
  7. B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
    [CrossRef]
  8. H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
    [CrossRef] [PubMed]
  9. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
    [CrossRef] [PubMed]
  10. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).
  11. V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
    [CrossRef]
  12. A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
    [CrossRef] [PubMed]
  13. L. D. Landau, Phys. Z. USSR 2, 46 (1932) (in German).
  14. C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).
    [CrossRef]
  15. E. Majorana, Nuovo Cimento 9, 43 (1932).
    [CrossRef]
  16. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]

2007 (1)

2006 (3)

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

2004 (2)

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

2003 (1)

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

1999 (2)

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

1976 (1)

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

1934 (1)

C. Zener, Proc. R. Soc. London, Ser. A 145, 523 (1934).
[CrossRef]

1932 (3)

L. D. Landau, Phys. Z. USSR 2, 46 (1932) (in German).

C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).
[CrossRef]

E. Majorana, Nuovo Cimento 9, 43 (1932).
[CrossRef]

1928 (1)

F. Bloch, Z. Phys. A: Hadrons Nucl. 52, 555 (1928) (in German).

Agarwal, V.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Bloch, F.

F. Bloch, Z. Phys. A: Hadrons Nucl. 52, 555 (1928) (in German).

Brauer, A.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Bräuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Cavalcanti, S. B.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

Christodoulides, D. N.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Coquillat, D.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Costantino, P.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

del Río, J. A.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Desyatnikov, A. S.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Fleischer, J. W.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Ghulinyan, M.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Gil, B.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Golant, K. M.

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

Hickmann, J. M.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

Kavokin, A.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau, Phys. Z. USSR 2, 46 (1932) (in German).

Lederer, F.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Majorana, E.

E. Majorana, Nuovo Cimento 9, 43 (1932).
[CrossRef]

Malpuech, G.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Michaelis, D.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Neshev, D. N.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Nirligareev, J. Kh.

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

Oton, C. J.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pavesi, L.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pertsch, T.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

Pertsch, Th.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Peschel, U.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Sapienza, R.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Scalbert, D.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Shchesnovich, V. S.

A. S. Desyatnikov, Yu. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, Opt. Lett. 32, 325 (2007).
[CrossRef] [PubMed]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Streppel, U.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

Sukhorukov, A. A.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Sychugov, V. A.

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

Trompeter, H.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Usievich, B. A.

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

Vladimirova, M.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Wiersma, D.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Zamfirescu, M.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Zener, C.

C. Zener, Proc. R. Soc. London, Ser. A 145, 523 (1934).
[CrossRef]

C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).
[CrossRef]

Nuovo Cimento (1)

E. Majorana, Nuovo Cimento 9, 43 (1932).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

B. A. Usievich, V. A. Sychugov, J. Kh. Nirligareev, and K. M. Golant, Opt. Spectrosc. 97, 790 (2004).
[CrossRef]

Phys. Rev. E (2)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, and Yu. S. Kivshar, Phys. Rev. E 74, 056602 (2006).
[CrossRef]

Phys. Rev. Lett. (6)

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, Th. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Phys. Rev. Lett. 92, 097401 (2004).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (2)

C. Zener, Proc. R. Soc. London, Ser. A 137, 696 (1932).
[CrossRef]

C. Zener, Proc. R. Soc. London, Ser. A 145, 523 (1934).
[CrossRef]

Z. Phys. A: Hadrons Nucl. (1)

F. Bloch, Z. Phys. A: Hadrons Nucl. 52, 555 (1928) (in German).

Other (2)

L. D. Landau, Phys. Z. USSR 2, 46 (1932) (in German).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

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Figures (7)

Fig. 1
Fig. 1

Square lattice and its first Brillouin zone. We have set V 0 = 0.05 and ε = 1 . The base reciprocal lattice vectors are denoted by Q 1 and Q 2 .

Fig. 2
Fig. 2

Fourier peak powers P 1 at k = ( k B , 0 ) (solid line) and P 2 at k = ( k B , 0 ) (dashed line), their average values (straight lines), and the sum P 1 + P 2 (dashed–dotted line). Here V 0 = 0.05 , ε = 1 and E ( r , 0 ) = 1 π σ exp { i k B x r 2 ( 2 σ 2 ) } with σ = 50 .

Fig. 3
Fig. 3

Numerically obtained E (upper row) and its Fourier spectrum F ( E ) (lower row). We display the absolute values and just a part of the spatial domain centered at r = 0 , for better visibility. The parameters are as in Fig. 2. The only visible nonresonant Fourier peaks are at k = { ( k B , 2 k B ) , ( k B , 2 k B ) , ( k B , 2 k B ) , ( k B , 2 k B ) } .

Fig. 4
Fig. 4

Average position of the beam corresponding to Fig. 2 (solid line) and the theoretical result (5) (dashed line) with C j 2 substituted with the numerical Fourier power P j of Fig. 2.

Fig. 5
Fig. 5

Fourier peak powers for (a) the full system (1), and (b) the model (6). We give P M 1 (solid line), P M 2 (dashed line), P M 3 , and P M 4 (dashed–dotted line). The rest of Fourier peaks have negligible powers. Here V 0 = 0.1 , ε = 0.5 and E ( r , 0 ) = 1 π σ exp { i k B ( x + y ) r 2 ( 2 σ 2 ) } with σ = 50 .

Fig. 6
Fig. 6

Average positions x and y of the beam corresponding to Fig. 5 (solid line, y = x in this case) and the theoretical result [Eqs. (7, 8)] with C j 2 substituted with P M j from Fig. 5 (dashed line).

Fig. 7
Fig. 7

Rabi oscillations of the Fourier powers P M 1 and P M 2 (solid line) and P M 3 and P M 4 (dashed line) in the notations and for the parameters of Fig 5. We have used E ( r , 0 ) = 2 π σ cos [ k B ( x + y ) ] exp { r 2 ( 2 σ 2 ) } with σ = 50 .

Equations (8)

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

i E z = 1 2 ( 2 E x 2 + 2 E y 2 ) + V ( r ) E ,
V ( r ) = V 0 ( cos x + cos y + ε cos x cos y ) .
i d C ( k , z ) d z = β ( k ) C ( k , z ) + Q V ̂ Q C ( k Q , z ) ,
i d C 1 d z = β X C 1 + V 0 2 C 2 , i d C 2 d z = β X C 2 + V 0 2 C 1 ,
d x d z = k B ( C 1 2 C 2 2 ) ,
i d d z ( C 1 C 2 C 3 C 4 ) = ( β ε V 0 4 V 0 2 V 0 2 ε V 0 4 β V 0 2 V 0 2 V 0 2 V 0 2 β ε V 0 4 V 0 2 V 0 2 ε V 0 4 β ) ( C 1 C 2 C 3 C 4 ) ,
d x d z = k B ( C 1 2 + C 4 2 C 2 2 C 3 2 ) ,
d y d z = k B ( C 1 2 + C 3 2 C 2 2 C 4 2 ) .

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