Abstract

The propagation of a wave packet in a honeycomb photonic lattice has been studied using the time-dependent wave packet dynamics. It is found that the wave packet, superposed from the positive and negative energy modes at the vicinity of the two inequivalent Dirac points, can transform into a double-ring structure, which is caused by the interference between the two positive and negative energy modes around the Dirac points and is closely related to the Zitterbewegung (ZB). Also, a possible way to detect the ZB effect is proposed in the honeycomb photonic lattice.

© 2011 Optical Society of America

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  1. E. Shrödinger, Sitzungsb. Preuss. Akac. Wiss. Phys. Math. KI. 24, 418 (1930).
  2. T. Rusin and W. Zawadzki, Phys. Rev. B 80, 045416 (2009).
    [Crossref]
  3. M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
    [Crossref]
  4. O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
    [Crossref]
  5. X. Zhang, Phys. Rev. Lett. 100, 113903 (2008).
    [Crossref] [PubMed]
  6. X. Zhang and Z. Liu, Phys. Rev. Lett. 101, 264303 (2008).
    [Crossref]
  7. F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
    [Crossref]
  8. S. Longhi, Opt. Lett. 35, 235 (2010).
    [Crossref] [PubMed]
  9. F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
    [Crossref]
  10. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
    [Crossref]
  11. Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
    [Crossref] [PubMed]
  12. If the wave packet is constructed by the components from two inequivalent DPs, the choice of c1 and c2 will not cause a difference because of the time-reversal symmetry linking the two DPs. However, if the wave packet is constructed by waves from one DP, the choice of c1 and c2 is relevant and setting c1=1 and c2=0 will make sure the wave packet expands into a ring in the x‒y plane.
  13. M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
    [Crossref]

2010 (2)

S. Longhi, Opt. Lett. 35, 235 (2010).
[Crossref] [PubMed]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

2009 (1)

T. Rusin and W. Zawadzki, Phys. Rev. B 80, 045416 (2009).
[Crossref]

2008 (3)

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

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

X. Zhang and Z. Liu, Phys. Rev. Lett. 101, 264303 (2008).
[Crossref]

2007 (2)

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[Crossref]

2000 (1)

Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
[Crossref] [PubMed]

1998 (1)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

1990 (1)

F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
[Crossref]

1930 (1)

E. Shrödinger, Sitzungsb. Preuss. Akac. Wiss. Phys. Math. KI. 24, 418 (1930).

Battal, G.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[Crossref]

Cannata, F.

F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
[Crossref]

Christodoulides, D. N.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Dreisow, F.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

Elliott, S. R.

Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
[Crossref] [PubMed]

Ferrari, L.

F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
[Crossref]

Freedman, B.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Heinrich, M.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[Crossref]

Juzeliunas, G.

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

Keil, R.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

Liu, Z.

X. Zhang and Z. Liu, Phys. Rev. Lett. 101, 264303 (2008).
[Crossref]

Loh, Y. L.

Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
[Crossref] [PubMed]

Longhi, S.

S. Longhi, Opt. Lett. 35, 235 (2010).
[Crossref] [PubMed]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Manela, O.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Merkl, M.

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

Nolte, S.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Öhberg, P.

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

Peleg, O.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Rusin, T.

T. Rusin and W. Zawadzki, Phys. Rev. B 80, 045416 (2009).
[Crossref]

Russo, G.

F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
[Crossref]

Segev, M.

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

Shrödinger, E.

E. Shrödinger, Sitzungsb. Preuss. Akac. Wiss. Phys. Math. KI. 24, 418 (1930).

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

Szameit, A.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Taraskin, S. N.

Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
[Crossref] [PubMed]

Tünnermann, A.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

Zawadzki, W.

T. Rusin and W. Zawadzki, Phys. Rev. B 80, 045416 (2009).
[Crossref]

Zhang, X.

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

X. Zhang and Z. Liu, Phys. Rev. Lett. 101, 264303 (2008).
[Crossref]

Zimmer, F. E.

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

Europhys. Lett. (1)

M. Merkl, F. E. Zimmer, G. Juzeliūnas, and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (1)

T. Rusin and W. Zawadzki, Phys. Rev. B 80, 045416 (2009).
[Crossref]

Phys. Rev. Lett. (6)

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, Phys. Rev. Lett. 105, 143902(2010).
[Crossref]

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 81, 1405 (1998).
[Crossref]

Y. L. Loh, S. N. Taraskin, and S. R. Elliott, Phys. Rev. Lett. 84, 2290 (2000).
[Crossref] [PubMed]

O. Peleg, G. Battal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 10391 (2007).
[Crossref]

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

X. Zhang and Z. Liu, Phys. Rev. Lett. 101, 264303 (2008).
[Crossref]

Prog. Opt. (1)

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[Crossref]

Sitzungsb. Preuss. Akac. Wiss. Phys. Math. KI. (1)

E. Shrödinger, Sitzungsb. Preuss. Akac. Wiss. Phys. Math. KI. 24, 418 (1930).

Solid State Commun. (1)

F. Cannata, L. Ferrari, and G. Russo, Solid State Commun. 74, 309 (1990).
[Crossref]

Other (1)

If the wave packet is constructed by the components from two inequivalent DPs, the choice of c1 and c2 will not cause a difference because of the time-reversal symmetry linking the two DPs. However, if the wave packet is constructed by waves from one DP, the choice of c1 and c2 is relevant and setting c1=1 and c2=0 will make sure the wave packet expands into a ring in the x‒y plane.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic show of a photonic honeycomb lattice. The two arrows denote its two basic unit vectors. (b) HPC’s energy band, calculated from the TB Hamiltonian. The K and K represent two inequivalent DPs of the band. (c) The z dependence of the outer-ring thickness (denoted by the FWHM).(d)–(f) Shows three snapshots of the wave packet at z = 0 , 60, and 120, respectively.

Fig. 2
Fig. 2

(a1)–(a4) One-dimensional intensity pattern I ( x , y c ) of a Gaussian wave packet constructed from both the positive and negative energy modes near the DPs at z = 15 , 45, 75, and 150, respectively. (b1)–(b4) Plots of the 2D I ( x , y ) corresponding to those in (a1)–(a4), respectively.

Fig. 3
Fig. 3

(a) One-dimensional intensity pattern I ( x , y c ) at z = 200 , computed from Eq. (6), where the open square (blue), solid circle (red) and solid triangle (black) represent those obtained from merely the first, the second term, and both terms, respectively, at the right of Eq. (6). All curves are scaled to their maximum for comparation. The parameter g is set to be g = 35 . (b) Double-cone structure in the corresponding three-dimensional stationary intensity pattern I ( x , y , z ) .

Equations (6)

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H ^ = ν F [ 0 k x + i k y k x i k y 0 ] .
i ψ z + 2 ψ V 0 ψ 1 + I 0 ( x , y ) + | ψ | 2 = 0.
i ψ z = H ^ 0 ψ ,
| ψ ( 0 ) = i Φ ( R i ) [ c 1 c 2 ] e i K · R i + i Φ ( R i ) [ c 1 c 2 ] e i K · R i ,
| ψ ( x , y , z ) = d k x d k y e g k e i α k z [ 1 e i ϕ ] e i k · R + d k x d k y e g k e i α k z [ 1 e i ϕ ] e i k · R ,
| ψ ( x , y , z ) = [ 1 i r e i β g i α z ] g i α z [ ( g i α z ) 2 + r 2 ] 3 2 + [ 1 i r e i β g + i α z ] g + i α z [ ( g + i α z ) 2 + r 2 ] 3 2 ,

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