Abstract

We have studied the optical oscillation and tunneling of light waves in optical waveguide ladders (OWLs) formed by two coupled planar optical waveguide arrays. For the band structure, a midzone gap is formed owing to band hybridization, and its wavenumber position can be tuned throughout the whole Brillouin zone, which is different from the Bragg gap. By imposing a gradient in the propagation constant in each array, Bloch–Zener oscillation (BZO) is realized with Zener tunneling between the bands occurring at the midzone, which is contrary to the common BZO with tunneling at the center or edge of the Brillouin zone. The occurrence of BZO is demonstrated by using the field-evolution analysis. The tunable hybridization at the midzone enhances the tunability of BZO in the OWLs. This Letter may offer new insights into the coherent phenomena in optical lattices.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2010 (2)

M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, Y. S. Chan, and K. W. Yu, J. Opt. Soc. Am. B 27, 1299 (2010).
[CrossRef]

2009 (1)

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

2008 (2)

S. Longhi, Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

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

2006 (6)

X. Fan and G. P. Wang, Opt. Lett. 31, 1322 (2006).
[CrossRef] [PubMed]

N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, Opt. Lett. 31, 1651 (2006).
[CrossRef] [PubMed]

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

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

B. M. Breid, D. Witthaut, and H. J. Korsch, New J. Phys. 8, 110 (2006).
[CrossRef]

S. Longhi, Europhys. Lett. 76, 416 (2006).
[CrossRef]

2003 (1)

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

1999 (1)

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

1998 (1)

1934 (1)

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

1929 (1)

F. Bloch, Z. Phys. 52, 555 (1929).
[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]

Bloch, F.

F. Bloch, Z. Phys. 52, 555 (1929).
[CrossRef]

Brauer, A.

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

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

Breid, B. M.

B. M. Breid, D. Witthaut, and H. J. Korsch, New J. Phys. 8, 110 (2006).
[CrossRef]

Cerullo, G.

Chan, Y. S.

Chiodo, N.

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]

Dannberg, P.

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

Dreisow, F.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Elflein, W.

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

Fan, X.

Fratalocchi, A.

Heinrich, M.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Korsch, H. J.

B. M. Breid, D. Witthaut, and H. J. Korsch, New J. Phys. 8, 110 (2006).
[CrossRef]

Laporta, P.

Lederer, F.

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

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

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

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

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

Longhi, S.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

S. Longhi, Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, Opt. Lett. 31, 1651 (2006).
[CrossRef] [PubMed]

S. Longhi, Europhys. Lett. 76, 416 (2006).
[CrossRef]

Michaelis, D.

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

Morgner, U.

Nolte, S.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Osellame, R.

Pertsch, T.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

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

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, 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.

Ramponi, R.

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]

Stegeman, G. I.

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

Streppel, U.

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

Szameit, A.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Trompeter, H.

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

Tunnermann, A.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Valle, G. Della

Wang, G. P.

Witthaut, D.

B. M. Breid, D. Witthaut, and H. J. Korsch, New J. Phys. 8, 110 (2006).
[CrossRef]

Xiao, J. J.

M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A 81, 033829 (2010).
[CrossRef]

Yu, K. W.

M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, Y. S. Chan, and K. W. Yu, J. Opt. Soc. Am. B 27, 1299 (2010).
[CrossRef]

Zener, C.

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

Zheng, M. J.

M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, Y. S. Chan, and K. W. Yu, J. Opt. Soc. Am. B 27, 1299 (2010).
[CrossRef]

Europhys. Lett. (1)

S. Longhi, Europhys. Lett. 76, 416 (2006).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

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

New J. Phys. (1)

B. M. Breid, D. Witthaut, and H. J. Korsch, New J. Phys. 8, 110 (2006).
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

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

Phys. Rev. A (1)

M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A 81, 033829 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

S. Longhi, Phys. Rev. Lett. 101, 193902 (2008).
[CrossRef] [PubMed]

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

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

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, Phys. Rev. Lett. 102, 076802(2009).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

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

Z. Phys. (1)

F. Bloch, Z. Phys. 52, 555 (1929).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic diagram of a cross section of OWL. The coupling constants for the PNC case are κ 1 = 1 , κ 2 = 0.5 , and κ = 0.2 . Band structures for (b) δ β = 0 , (c) δ β = 1 , and (d) δ β = 1 . Band hybridization occurs at k = 0.5 π , 0.4 π , and 0.6 π , respectively. (e) Wavenumber position of band hybridization as a function of δ β .

Fig. 2
Fig. 2

Evolution of a broad Gaussian beam centered at n low = 50 with spatial width of σ = 3 . PNC case ( κ 2 = 0.5 , κ = 0.2 ): (a) overlap of contour plots of | ψ ( n , z ) | 2 in the upper and lower graded waveguide array and (b) visual band. PPC case ( κ 2 = 0.2 , κ = 0.1 ): (c) and (d) are similar to (a) and (b), respectively. The common parameters are N up = N low = 100 , κ 1 = 1.0 , β 0 = 2 , δ β = 0 , α = 0.08 , k 0 = 0 , and σ = 3 .

Fig. 3
Fig. 3

Evolution of single waveguide excitation starting at n low = 50 . PNC case: (a) contour plots of | ψ ( n , z ) | 2 in the upper graded waveguide array and (b) in the lower graded waveguide array, respectively. PPC case: (c) and (d) are similar to (a) and (b), respectively. Parameters are the same as those in Fig. 2 except for σ = 0.02 .

Equations (3)

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( i d d z + β u ) a n ( z ) + κ 1 [ a n + 1 ( z ) + a n 1 ( z ) ] + κ b n ( z ) = 0 , ( i d d z + β l ) b n ( z ) + κ 2 [ b n + 1 ( z ) + b n 1 ( z ) ] + κ a n ( z ) = 0 ,
β ± = β 0 + δ β 2 + ( κ 1 + κ 2 ) cos k ± Δ ,
ψ ( n , 0 ) = ( 2 π σ 2 ) 1 / 4 exp [ ( n n 0 ) 2 4 σ 2 i k 0 ( n n 0 ) ] ,

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