Abstract

An optical analog of Zitterbewegung (ZB), i.e., of the trembling motion of Dirac electrons caused by the interference between positive and negative energy states, is proposed for spatial beam propagation in binary waveguide arrays. In this optical system the ZB is simply observable as a quiver spatial oscillatory motion of the beam center of mass around its mean trajectory.

© 2010 Optical Society of America

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L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, Opt. Lett. 34, 1510 (2009).
[CrossRef] [PubMed]

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL 86, 47008 (2009).
[CrossRef]

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

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

2008

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251 (2008).
[CrossRef] [PubMed]

X. Zhang, Phys. Rev. Lett. 100, 113903 (2008).
[CrossRef] [PubMed]

J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002 (2008).
[CrossRef] [PubMed]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

2007

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

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

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

T. M. Rusin and W. Zawadzki, Phys. Rev. B 76, 195439 (2007).
[CrossRef]

2006

J. Cserti and G. David, Phys. Rev. B 74, 172305 (2006).
[CrossRef]

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

2004

2003

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

2002

1993

1952

K. Huang, Am. J. Phys. 20, 479 (1952).
[CrossRef]

Aitchison, J. S.

Bahat-Treidel, O.

Bartal, G.

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

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Bräuer, A.

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

Christodoulides, D. N.

Clark, C. W.

J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002 (2008).
[CrossRef] [PubMed]

Cserti, J.

J. Cserti and G. David, Phys. Rev. B 74, 172305 (2006).
[CrossRef]

David, G.

J. Cserti and G. David, Phys. Rev. B 74, 172305 (2006).
[CrossRef]

Della Valle, G.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Dreisow, F.

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

Duo, L.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Feng, J.

Finazzi, M.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Foglietti, V.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Freedman, B.

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

Greiner, W.

W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990).

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Heinrich, M.

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

Huang, K.

K. Huang, Am. J. Phys. 20, 479 (1952).
[CrossRef]

Kivshar, Y. S.

Lamata, L.

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

Laporta, P.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Lederer, F.

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

León, J.

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

Longhi, S.

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

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

Mandelik, D.

Manela, O.

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

Michaelis, D.

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

Morandotti, R.

Nolte, S.

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

Ornigotti, M.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Peleg, O.

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251 (2008).
[CrossRef] [PubMed]

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

Pertsch, T.

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

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

Peschel, U.

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

Raghu, S.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

Rusin, T. M.

T. M. Rusin and W. Zawadzki, Phys. Rev. B 76, 195439 (2007).
[CrossRef]

Savoini, M.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

Schätz, T.

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

Segev, M.

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251 (2008).
[CrossRef] [PubMed]

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

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Silberberg, Y.

Solano, E.

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

Sorel, M.

Streppel, U.

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

Sukhorukov, A. A.

Szameit, A.

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

Trompeter, H.

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

Tünnermann, A.

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

Vaishnav, J. Y.

J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002 (2008).
[CrossRef] [PubMed]

Wang, L.-G.

Wang, Z.-G.

Zawadzki, W.

T. M. Rusin and W. Zawadzki, Phys. Rev. B 76, 195439 (2007).
[CrossRef]

Zhang, J.-X.

Zhang, X.

X. Zhang, Phys. Rev. Lett. 100, 113903 (2008).
[CrossRef] [PubMed]

Zhu, S.-Y.

Am. J. Phys.

K. Huang, Am. J. Phys. 20, 479 (1952).
[CrossRef]

EPL

L.-G. Wang, Z.-G. Wang, and S.-Y. Zhu, EPL 86, 47008 (2009).
[CrossRef]

Laser Photonics Rev.

S. Longhi, Laser Photonics Rev. 3, 243 (2009).
[CrossRef]

Opt. Lett.

Phys. Rev. A

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Phys. Rev. B

J. Cserti and G. David, Phys. Rev. B 74, 172305 (2006).
[CrossRef]

T. M. Rusin and W. Zawadzki, Phys. Rev. B 76, 195439 (2007).
[CrossRef]

Phys. Rev. Lett.

J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002 (2008).
[CrossRef] [PubMed]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef] [PubMed]

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

L. Lamata, J. León, T. Schätz, and E. Solano, Phys. Rev. Lett. 98, 253005 (2007).
[CrossRef] [PubMed]

X. Zhang, Phys. Rev. Lett. 100, 113903 (2008).
[CrossRef] [PubMed]

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[CrossRef] [PubMed]

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

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

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

Other

W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of a one-dimensional binary array (left) and refractive index profile (right). (b) Dispersion curves of the first two minibands of a tight-binding binary array (solid curves) and corresponding dispersion curves of the Dirac equation (4) (dotted curves). (c) Broad-beam excitation geometry of the array at a tilting angle θ.

Fig. 2
Fig. 2

(a) Evolution of | c n ( z ) | 2 in a binary array excited by a broad Gaussian beam for σ = 1 and δ = 0.6 and (b) corresponding behavior of beam trajectory n ( z ) (solid curve) as obtained by numerical analysis of Eq. (2). The dotted curve in (b), almost overlapped with the solid curve, reproduces the behavior of 2 ξ ( z ) + 1 / 2 , where ξ ( z ) is calculated according to Eq. (5). (c), (d) Same as (a) and (b), but for δ = 1 .

Fig. 3
Fig. 3

(a) Beam propagation (snapshot of | E ( x , z ) | 2 ) in a binary array, as obtained by numerical analysis of Eq. (1), for broad Gaussian beam excitation tilted at the Bragg angle; parameter values are given in the text. (b) Corresponding behavior of x ( z ) .

Equations (7)

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i z E = [ λ / ( 4 π n s ) ] x 2 E + ( 2 π / λ ) [ n s n ( x ) ] E ,
i ( d c n / d z ) = σ ( c n + 1 + c n 1 ) + ( 1 ) n δ c n ,
ω ( q ) = ± δ 2 + 4 σ 2 cos 2 ( q a )
i z ψ + i σ α ξ ψ δ β ψ = 0 ,
α = ( 0 1 1 0 ) ,     β = ( 1 0 0 1 ) .
ξ ( z ) = ξ ( 0 ) + 4 π σ 3 z d k ( k / ϵ ) 2 | G ̂ ( k ) | 2 + 2 π σ δ 2 d k ( 1 / ϵ 3 ) sin ( 2 ϵ z ) | G ̂ ( k ) | 2 .
n = 2 ξ + 1 2 4 π δ σ d k ( k / ϵ 2 ) | G ̂ ( k ) | 2 sin 2 ( ϵ z ) .

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