Abstract

We study Zener tunneling in two-dimensional photonic lattices and derive, for the case of hexagonal symmetry, the generalized Landau–Zener–Majorana model describing resonant interaction between high-symmetry points of the photonic spectral bands. We demonstrate that this effect can be employed for the generation of Floquet–Bloch modes and verify the model by direct numerical simulations of the tunneling effect.

© 2007 Optical Society of America

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  1. F. Bloch, Z. Phys. 52, 555 (1928).
  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. Bräuer, 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. H. Nguyen, P. Domachuk, B. Eggleton, M. Steel, M. Straub, M. Gu, and M. Sumetsky, Opt. Express 12, 1528 (2004).
    [Crossref] [PubMed]
  11. C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
    [Crossref]
  12. L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).
  13. E. Majorana, Nuovo Cimento 9, 43 (1932).
    [Crossref]
  14. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).
  15. S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993).
    [Crossref]
  16. A. V. Shytov, Phys. Rev. A 70, 052708 (2004).
    [Crossref]

2006 (2)

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, 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 (4)

H. Nguyen, P. Domachuk, B. Eggleton, M. Steel, M. Straub, M. Gu, and M. Sumetsky, Opt. Express 12, 1528 (2004).
[Crossref] [PubMed]

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]

A. V. Shytov, Phys. Rev. A 70, 052708 (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]

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]

1993 (1)

S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993).
[Crossref]

1934 (1)

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

1932 (3)

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

L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).

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

1928 (1)

F. Bloch, Z. Phys. 52, 555 (1928).

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. 52, 555 (1928).

Bräuer, A.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, 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]

Brundobler, S.

S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993).
[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.

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]

Domachuk, P.

Eggleton, B.

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]

Elser, V.

S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993).
[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]

Gu, M.

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.

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. Sowjetunion 2, 46 (1932).

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. Bräuer, 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. Bräuer, 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]

Nguyen, H.

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. Bräuer, 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. Bräuer, 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]

Shytov, A. V.

A. V. Shytov, Phys. Rev. A 70, 052708 (2004).
[Crossref]

Silberberg, Y.

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

Steel, M.

Straub, M.

Streppel, U.

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, 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]

Sumetsky, M.

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. Bräuer, 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]

J. Phys. A (1)

S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993).
[Crossref]

Nuovo Cimento (1)

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

Opt. Express (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. A (1)

A. V. Shytov, Phys. Rev. A 70, 052708 (2004).
[Crossref]

Phys. Rev. Lett. (6)

H. Trompeter, Th. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, 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]

Phys. Z. Sowjetunion (1)

L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).

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

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

Proc. R. Soc. London, Ser. A (1)

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

Z. Phys. (1)

F. Bloch, Z. Phys. 52, 555 (1928).

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

(a) Intensity of the lattice, Eq. (2), with b = 2 ; (b) the first Brillouin zone; (c) band spectra of the Floquet–Bloch waves of the lattice, Eq. (2), for V 0 = 0.1 .

Fig. 2
Fig. 2

Examples of the solutions to the LZM, Eq. (4), system with c 1 ( 100 ) 2 0.986 and c 2 , 3 ( 100 ) 2 0.007 with seed tunneled beams (a) in and (b) out of phase with the main beam c 1 ( 100 ) . The dashed line at 0.442 in both (a) and (b) shows the asymptotic value for the initial condition c 1 ( ) = 1 .

Fig. 3
Fig. 3

Symmetric threefold resonance with the index gradient shown by an arrow, α = 0.05 . Intensity (left) and Fourier images (right) are shown for the propagation distances (a) z = 0 , (b) z = 9 , and (c) z = 16 . The initial beam is taken as exp ( r 2 2 w 2 + i q r ) with w = 20 , and we choose its initial position in the Fourier domain q such that the accelerating beam moves through the M point.

Fig. 4
Fig. 4

(a) Evolution of the powers of the input beam P 1 (red) and two resonantly tunneled beams P 2 , 3 (green and blue) for the tunneling in Fig. 3. (b) Same as in (a) but for an initial position of the beam in the Fourier domain closer to the M point than for (a).

Equations (5)

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i E z = 1 2 ( 2 E x 2 + 2 E y 2 ) + [ V ( r ) + α r ] E ,
V ( r ) = 8 V 0 cos 2 ( b 1 r 2 ) cos 2 ( b 2 r 2 ) cos 2 ( b 3 r 2 ) ,
i d C ( k , z ) d z = β 0 ( k α z ) C ( k , z ) + Q V ̂ Q C ( k Q , z ) ,
i d c j d z = ( ν j z ) c j + V 0 ( c j + 1 + c j + 2 ) ,
P = exp [ 2 π V 0 2 ( ν 1 ν 2 1 + ν 1 ν 3 1 ) ] .

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