Abstract

Under the influence of the next-nearest-neighbor interaction, we theoretically investigate the occurrence of Bloch oscillations in zigzag waveguide arrays. Because of the special topological configuration of the lattice itself, the second-order coupling (SOC) can be enhanced significantly and leads to the band alteration beyond the nearest-neighbor model, i.e., the offset of minimum value from the band edge. Contrary to the behavior in the vanishing SOC, the oscillation patterns exhibit new features, namely, a double turning-back occurs when the beam approaches the band edge. Our results can be applied to some ordered-lattice systems.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
    [CrossRef]
  2. S. Longhi, Laser Photon. Rev. 3, 243 (2009), and references therein.
    [CrossRef]
  3. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
    [CrossRef] [PubMed]
  4. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
    [CrossRef] [PubMed]
  5. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 83, 4752 (1999).
    [CrossRef]
  6. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
    [CrossRef]
  7. D. Hennig, Eur. Phys. J. B 20, 419 (2001).
    [CrossRef]
  8. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  9. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  10. N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (2002).
    [CrossRef]
  11. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 33, 2689 (2008).
    [CrossRef] [PubMed]
  12. A. Szameit, R. Keil, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 34, 2838 (2009).
    [CrossRef] [PubMed]
  13. A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
    [CrossRef]
  14. U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
    [CrossRef]
  15. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
    [CrossRef]
  16. S. Longhi, Phys. Rev. B 76, 195117 (2007).
    [CrossRef]
  17. From the Hamiltonian H=σx+2κ1cos⁡(kx)+2κ2cos⁡(2kx), the motion equations are given: dx/dz=dH/dkx, dkx/dz=−dH/dx. With appropriate initial conditions, solutions take the form kx=σz, x=x0−(2/σ)(κ1+κ2)+(2κ1/σ)cos⁡(σz)+(2κ2/σ)cos⁡(2σz). The BO path directly reflects the corresponding dispersion relation magnified by 1/σ. The Hamiltonian optics offer a good approximation of the field evolution.
  18. A. Fratalocchi and G. Assanto, Opt. Lett. 31, 3351 (2006).
    [CrossRef] [PubMed]
  19. S. Longhi, Opt. Lett. 33, 473 (2008).
    [CrossRef] [PubMed]

2009

2008

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

S. Longhi, Opt. Lett. 33, 473 (2008).
[CrossRef] [PubMed]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 33, 2689 (2008).
[CrossRef] [PubMed]

2007

S. Longhi, Phys. Rev. B 76, 195117 (2007).
[CrossRef]

2006

A. Fratalocchi and G. Assanto, Opt. Lett. 31, 3351 (2006).
[CrossRef] [PubMed]

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

2002

N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (2002).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

2001

D. Hennig, Eur. Phys. J. B 20, 419 (2001).
[CrossRef]

1999

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]

1998

1996

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

1993

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[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]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

A. Fratalocchi and G. Assanto, Opt. Lett. 31, 3351 (2006).
[CrossRef] [PubMed]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

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

Castin, Y.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

Christodoulides, D.

N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (2002).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Dahan, M. B.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

Dannberg, P.

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

Dreisow, F.

Efremidis, N.

N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (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]

Fratalocchi, A.

Heinrich, M.

Hennig, D.

D. Hennig, Eur. Phys. J. B 20, 419 (2001).
[CrossRef]

Keil, R.

Köhler, K.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Kurz, H.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

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

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Leo, K.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, Laser Photon. Rev. 3, 243 (2009), and references therein.
[CrossRef]

S. Longhi, Opt. Lett. 33, 473 (2008).
[CrossRef] [PubMed]

S. Longhi, Phys. Rev. B 76, 195117 (2007).
[CrossRef]

Morandotti, R.

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

Morsch, O.

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

Nolte, S.

Oberthaler, M.

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

Peik, E.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

Pertsch, T.

A. Szameit, R. Keil, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 34, 2838 (2009).
[CrossRef] [PubMed]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Lett. 33, 2689 (2008).
[CrossRef] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

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

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

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

U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[CrossRef]

Reichel, J.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

Roskos, H. G.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Salomon, C.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[CrossRef] [PubMed]

Schwedler, R.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

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

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Szameit, A.

Tünnermann, A.

Waschke, C.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

Appl. Phys. Lett.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Appl. Phys. Lett. 80, 3247 (2002).
[CrossRef]

Eur. Phys. J. B

D. Hennig, Eur. Phys. J. B 20, 419 (2001).
[CrossRef]

Laser Photon. Rev.

S. Longhi, Laser Photon. Rev. 3, 243 (2009), and references therein.
[CrossRef]

Nature

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. A

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

Phys. Rev. B

S. Longhi, Phys. Rev. B 76, 195117 (2007).
[CrossRef]

Phys. Rev. E

N. Efremidis and D. Christodoulides, Phys. Rev. E 65, 056607 (2002).
[CrossRef]

Phys. Rev. Lett.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
[CrossRef] [PubMed]

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[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]

Rev. Mod. Phys.

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

Other

From the Hamiltonian H=σx+2κ1cos⁡(kx)+2κ2cos⁡(2kx), the motion equations are given: dx/dz=dH/dkx, dkx/dz=−dH/dx. With appropriate initial conditions, solutions take the form kx=σz, x=x0−(2/σ)(κ1+κ2)+(2κ1/σ)cos⁡(σz)+(2κ2/σ)cos⁡(2σz). The BO path directly reflects the corresponding dispersion relation magnified by 1/σ. The Hamiltonian optics offer a good approximation of the field evolution.

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

Band structures of waveguide arrays with (without) next-nearest-neighbors coupling. The inset gives the waveguide array. The zigzag arrangement is designed to enhance the SOC. Here x is defined as the transverse dimension and z as the propagation dimension. The distances between the guides and between the layers are denoted with d and d L , respectively, and κ 1 ( 2 ) denotes the first (second)-order coupling resulting from the field overlap. The parameter Λ = κ 2 / κ 1 is introduced to describe the ratio between second- and first-order coupling strength.

Fig. 2
Fig. 2

Band diagram of waveguide arrays superimposed; the linear potential σ = 0.2 . Crossed areas correspond to the gaps, while the white areas indicate the band. Clearly, the band is separated into two sections by the band edge. Parameter: Λ = 0.7 .

Fig. 3
Fig. 3

For the same linear potential n σ , the orbits n k x in phase space for lattices with various SOCs.

Fig. 4
Fig. 4

Evolution of the launched Gaussian beam. Compared with the typical BO [Fig. 4a], the dynamics with the SOC [Fig. 4b] clearly behaves in a different pattern. The solid curve is the dispersion relation of the homogeneous waveguide arrays, which is scaled by 1 / σ . Parameters: (a) Λ = 0.0 , (b) Λ = 0.7 .

Equations (2)

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

ι d a n d z = σ n a n + κ 1 ( a n + 1 + a n 1 ) + κ 2 ( a n + 2 + a n 2 ) .
W exp ( ( n n 0 ) 2 W 0 2 ) exp ( ι k x d )

Metrics