Abstract

We study the dynamics of excitations in general two-dimensional (2D) lattices with an external spatial linear potential. Analytical expressions for the energy eigenspectrum (Wannier–Stark ladder) and the eigenstates (Wannier–Stark states) of the lattice Schrödinger equation are given, and the resulting periodic motion of localized excitations (Bloch oscillations) is investigated for arbitrary excitation, including single site and broad beam excitation.

© 2007 Optical Society of America

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  1. C. Zener, "A theory of the electrical breakdown of solid dielectrics," Proc. R. Soc. London, Ser. A 145, 523-529 (1934).
    [CrossRef]
  2. D. Dunlap and V. Kenkre, "Dynamic localization of a charged particle moving under the influence of an electric field," Phys. Rev. B 34, 3625-3633 (1986).
    [CrossRef]
  3. G. Nenciu, "Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians," Rev. Mod. Phys. 63, 92-127 (1991).
    [CrossRef]
  4. E. Mendez and G. Bastard, "Wannier-Stark ladders and Bloch oscillations in superlattices," Phys. Today 46, 34-42 (1993).
    [CrossRef]
  5. J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
    [CrossRef]
  6. G. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
    [CrossRef]
  7. E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
    [CrossRef] [PubMed]
  8. H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
    [CrossRef]
  9. J. Yellin, "Two exact lattice propagators," Phys. Rev. E 52, 2208-2215 (1995).
    [CrossRef]
  10. F. Keck and H. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A 35, 105-116 (2002).
    [CrossRef]
  11. D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
    [CrossRef]
  12. M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
    [CrossRef]
  13. C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
    [CrossRef] [PubMed]
  14. D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
    [CrossRef] [PubMed]
  15. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
    [CrossRef] [PubMed]
  16. M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
    [CrossRef]
  17. A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Int. J. Mod. Phys. B 18, 1235-1260 (2004).
    [CrossRef]
  18. A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
    [CrossRef]
  19. C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
    [CrossRef]
  20. U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
    [CrossRef]
  21. R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
    [CrossRef]
  22. T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).
  23. H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
    [CrossRef] [PubMed]
  24. A. Yariv, Optical Electronics, 4th ed. (Saunders College Publishers, 1991).
  25. A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
    [CrossRef]
  26. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
    [CrossRef]
  27. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
    [CrossRef] [PubMed]
  28. A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
    [CrossRef]
  29. N. Chiodo, G. D. Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, "Imaging of Bloch oscillations in erbium-doped curved waveguide arrays," Opt. Lett. 31, 1651-1653 (2006).
    [CrossRef] [PubMed]
  30. G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
    [CrossRef]
  31. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
    [CrossRef] [PubMed]

2007 (2)

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

2006 (3)

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

N. Chiodo, G. D. Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, "Imaging of Bloch oscillations in erbium-doped curved waveguide arrays," Opt. Lett. 31, 1651-1653 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

2005 (1)

D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
[CrossRef]

2004 (4)

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Int. J. Mod. Phys. B 18, 1235-1260 (2004).
[CrossRef]

H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
[CrossRef]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

2003 (2)

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
[CrossRef]

2002 (2)

M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
[CrossRef]

F. Keck and H. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A 35, 105-116 (2002).
[CrossRef]

1999 (2)

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

1998 (2)

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

1997 (1)

M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
[CrossRef]

1996 (1)

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

1995 (2)

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

J. Yellin, "Two exact lattice propagators," Phys. Rev. E 52, 2208-2215 (1995).
[CrossRef]

1993 (2)

E. Mendez and G. Bastard, "Wannier-Stark ladders and Bloch oscillations in superlattices," Phys. Today 46, 34-42 (1993).
[CrossRef]

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

1992 (1)

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

1991 (1)

G. Nenciu, "Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians," Rev. Mod. Phys. 63, 92-127 (1991).
[CrossRef]

1988 (1)

E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
[CrossRef] [PubMed]

1986 (1)

D. Dunlap and V. Kenkre, "Dynamic localization of a charged particle moving under the influence of an electric field," Phys. Rev. B 34, 3625-3633 (1986).
[CrossRef]

1960 (1)

G. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

1934 (1)

C. Zener, "A theory of the electrical breakdown of solid dielectrics," Proc. R. Soc. London, Ser. A 145, 523-529 (1934).
[CrossRef]

Agull-Rueda, F.

E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
[CrossRef] [PubMed]

Aitchinson, J.

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Bastard, G.

E. Mendez and G. Bastard, "Wannier-Stark ladders and Bloch oscillations in superlattices," Phys. Today 46, 34-42 (1993).
[CrossRef]

Bishop, A.

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

Bloemer, D.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

Braeuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

Bright, J.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Burghoff, J.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

Cai, D.

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

Castin, Y.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Cerullo, G.

Chiodo, N.

Cunningham, J.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

D. Witthaut, H. K.

H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
[CrossRef]

Dahan, M. B.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

de Sterke, C.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

de Sterke, M.

G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

Desyatnikov, A.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Dreisow, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

Dunlap, D.

D. Dunlap and V. Kenkre, "Dynamic localization of a charged particle moving under the influence of an electric field," Phys. Rev. B 34, 3625-3633 (1986).
[CrossRef]

Eisenberg, H.

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

Feldmann, J.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Glueck, M.

M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
[CrossRef]

Gronbech-Jensen, N.

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

Hammon, T.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Hong, J.

E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
[CrossRef] [PubMed]

Keck, F.

H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
[CrossRef]

F. Keck and H. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A 35, 105-116 (2002).
[CrossRef]

Kenkre, V.

D. Dunlap and V. Kenkre, "Dynamic localization of a charged particle moving under the influence of an electric field," Phys. Rev. B 34, 3625-3633 (1986).
[CrossRef]

Kivshar, Y.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Koehler, K.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Kolovsky, A.

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Int. J. Mod. Phys. B 18, 1235-1260 (2004).
[CrossRef]

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
[CrossRef]

M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
[CrossRef]

Korsch, H.

D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
[CrossRef]

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Int. J. Mod. Phys. B 18, 1235-1260 (2004).
[CrossRef]

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
[CrossRef]

M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
[CrossRef]

F. Keck and H. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A 35, 105-116 (2002).
[CrossRef]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Krug, P.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Kurz, H.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Laporta, P.

Lederer, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

Lenz, G.

G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

Leo, K.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Longhi, S.

Meier, T.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Mendez, E.

E. Mendez and G. Bastard, "Wannier-Stark ladders and Bloch oscillations in superlattices," Phys. Today 46, 34-42 (1993).
[CrossRef]

E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
[CrossRef] [PubMed]

Miller, D.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Morandotti, R.

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Morgner, U.

Mossmann, S.

H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
[CrossRef]

Nenciu, G.

G. Nenciu, "Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians," Rev. Mod. Phys. 63, 92-127 (1991).
[CrossRef]

Neshev, D.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Niu, Q.

M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
[CrossRef]

Nolte, S.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

Osellame, R.

Peik, E.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Pertsch, T.

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

Peschel, U.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, "Optical Bloch oscillations in waveguide arrays," Opt. Lett. 23, 1701-1703 (1998).
[CrossRef]

Raizen, M.

M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
[CrossRef]

Ramponi, R.

Reichel, J.

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Roskos, H.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

S. Mossmann, D. W.

D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
[CrossRef]

Salerno, M.

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

Salomon, C.

M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
[CrossRef]

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

Schmitt-Rink, S.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Schulze, A.

D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
[CrossRef]

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Schwedler, R.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Shah, J.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Silberberg, Y.

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Sukhorukov, A.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Szameit, A.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

Talanina, I.

G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

Thomas, P.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Trompeter, H.

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Tuennermann, A.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tuennermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007).
[CrossRef] [PubMed]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

Valle, G. D.

von Plessen, G.

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

Wannier, G.

G. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Waschke, C.

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

Will, M.

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tuennermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders College Publishers, 1991).

Yellin, J.

J. Yellin, "Two exact lattice propagators," Phys. Rev. E 52, 2208-2215 (1995).
[CrossRef]

Zener, C.

C. Zener, "A theory of the electrical breakdown of solid dielectrics," Proc. R. Soc. London, Ser. A 145, 523-529 (1934).
[CrossRef]

Appl. Phys. A (1)

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77, 109-111 (2003).
[CrossRef]

Appl. Phys. B (1)

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tuennermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B . 82, 507-512 (2006).
[CrossRef]

Int. J. Mod. Phys. B (1)

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Int. J. Mod. Phys. B 18, 1235-1260 (2004).
[CrossRef]

J. Phys. A (2)

F. Keck and H. Korsch, "Infinite-variable Bessel functions in two-dimensional Wannier-Stark systems," J. Phys. A 35, 105-116 (2002).
[CrossRef]

D. W. S. Mossmann, A. Schulze, and H. Korsch, "Two-dimensional Bloch oscillations: a Lie-algebraic approach," J. Phys. A 38, 3381-3395 (2005).
[CrossRef]

New J. Phys. (1)

H. K. D. Witthaut, F. Keck, and S. Mossmann, "Bloch oscillations in two-dimensional lattices," New J. Phys. 6, 1-18 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rep. (1)

M. Glueck, A. Kolovsky, and H. Korsch, "Wannier-Stark resonances in optical and semiconductor superlattices," Phys. Rep. 366, 103-182 (2002).
[CrossRef]

Phys. Rev. (1)

G. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960).
[CrossRef]

Phys. Rev. A (2)

A. Kolovsky and H. Korsch, "Bloch oscillations of cold atoms in two-dimensional optical lattices," Phys. Rev. A 67, 063601 (2003).
[CrossRef]

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tuennermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75, 053814 (2007).
[CrossRef]

Phys. Rev. B (2)

J. Feldmann, K. Leo, J. Shah, D. Miller, J. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, "Optical investigation of Bloch oscillations in a semiconductor superlattice," Phys. Rev. B 46, 7252-7255 (1992).
[CrossRef]

D. Dunlap and V. Kenkre, "Dynamic localization of a charged particle moving under the influence of an electric field," Phys. Rev. B 34, 3625-3633 (1986).
[CrossRef]

Phys. Rev. E (2)

J. Yellin, "Two exact lattice propagators," Phys. Rev. E 52, 2208-2215 (1995).
[CrossRef]

C. de Sterke, J. Bright, P. Krug, and T. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998).
[CrossRef]

Phys. Rev. Lett. (8)

G. Lenz, I. Talanina, and M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83, 963-966 (1999).
[CrossRef]

E. Mendez, F. Agull-Rueda, and J. Hong, "Stark localization in GaAs-GaAlAs superlattices under an electric field," Phys. Rev. Lett. 60, 2426-2429 (1988).
[CrossRef] [PubMed]

C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Koehler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993).
[CrossRef] [PubMed]

D. Cai, N. Gronbech-Jensen, A. Bishop, and M. Salerno, "Electric-field-induced nonlinear Bloch oscillations and dynamical localization," Phys. Rev. Lett. 74, 1186-1189 (1995).
[CrossRef] [PubMed]

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. Aitchinson, H. Eisenberg, and Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Braeuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4753-4755 (2004).

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Phys. Today (2)

M. Raizen, Q. Niu, and C. Salomon, "New light on quantum transport," Phys. Today 50, 30-34 (1997).
[CrossRef]

E. Mendez and G. Bastard, "Wannier-Stark ladders and Bloch oscillations in superlattices," Phys. Today 46, 34-42 (1993).
[CrossRef]

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

C. Zener, "A theory of the electrical breakdown of solid dielectrics," Proc. R. Soc. London, Ser. A 145, 523-529 (1934).
[CrossRef]

Rev. Mod. Phys. (1)

G. Nenciu, "Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians," Rev. Mod. Phys. 63, 92-127 (1991).
[CrossRef]

Other (1)

A. Yariv, Optical Electronics, 4th ed. (Saunders College Publishers, 1991).

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Figures (6)

Fig. 1
Fig. 1

Scheme of a general lattice with (a) horizontal, (b) vertical, and (c) and (d) diagonal coupling.

Fig. 2
Fig. 2

Resulting Bloch– or Wannier–Stark states for α = 0.125 and β = 0.5 . For a square lattice the amplitude of the eigenmode is shown in (a). In (b) the amplitude of the eigenstate in a hexagonal lattice is shown, and in (c) the amplitude of the eigenstate in a general lattice is shown.

Fig. 3
Fig. 3

Width of the output pattern as a function of the potential α at z = π α with β = 0.25 . Compared are the predicted values using Eq. (19) (solid curve) and the numerically determined widths.

Fig. 4
Fig. 4

Resulting output patterns for α = 0.125 and β = 0.5 after a propagation length of z = π β 6.283 are shown for (a) square, (b) hexagonal, and (c) general lattices.

Fig. 5
Fig. 5

Experimental setup for the investigation of 2D Bloch oscillations in a tilted square lattice using curved waveguides with a constant curvature. The tilting angle φ determines the direction of the linear potential.

Fig. 6
Fig. 6

Left: Bloch oscillation of a Gaussian excitation on a 2D cubic lattice with a linear potential. The solid curve corresponds to an analytic approximation. α = 0.2, β = 0.35 , a = b = 1 , c = d = 1 2 . Right: phase structure of the moving light bundle at different points on the propagation path.

Equations (35)

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

[ i d d z + α m + β n ] φ m , n ( z ) + a [ φ m + 1 , n ( z ) + φ m 1 , n ( z ) ] + b [ φ m , n + 1 ( z ) + φ m , n 1 ( z ) ] + c [ φ m + 1 , n + 1 ( z ) + φ m 1 , n 1 ( z ) ] + d [ φ m + 1 , n 1 ( z ) + φ m 1 , n + 1 ( z ) ] = 0 ,
[ i z + i α μ + i β ν + 2 a cos ( μ ) + 2 b cos ( ν ) + 2 c cos ( μ + ν ) + 2 d cos ( μ ν ) ] φ ̃ ( μ , ν , z ) ,
φ m , n ( z ) = π π φ ̃ ( μ , ν , z ) exp { i ( μ m + ν n ) } d μ d ν ,
φ ̃ ( μ , ν , z ) = 1 ( 2 π ) 2 m , n φ m , n ( z ) exp { i ( μ m + ν n ) } ,
φ ̃ m , n j , k = u ̃ m , n j , k ( μ , ν ) exp { i λ j , k z } ,
u ̃ j , k ( μ , ν ) = 1 ( 2 π ) 2 exp { i ( 2 a α sin ( μ ) + 2 b β sin ( ν ) + 2 c α + β sin ( μ + ν ) + 2 d α β sin ( μ ν ) λ j α μ λ k β ν ) } ,
u ̃ m , n j , k ( π , ν ) = u ̃ m , n j , k ( π , ν ) ,
u ̃ m , n j , k ( μ , π ) = u ̃ m , n j , k ( μ , π )
λ j = j α ,
λ k = k β ,
φ m , n j , k ( z ) = φ m j + j , n k + k j , k ( z ) exp { i z [ ( j j ) α + ( k k ) β ] } .
u m , n = 1 ( 2 π ) 2 π π exp { 2 i ( a α sin ( μ ) + b β sin ( ν ) + c α + β sin ( μ + ν ) + d α β sin ( μ ν ) ) } exp { μ m + ν n } d μ d ν = 1 ( 2 π ) 2 p , r , s , t J p ( 2 a α ) J r ( 2 b β ) J s ( 2 c α + β ) J t ( 2 d α β ) × π π exp { i μ [ p + s + t + m ] + i ν [ r + s t + n ] } d μ d ν ,
k J k ( x ) e i β k = e i x sin ( β )
π π e i μ [ p + s + t + m ] + i ν [ r + s t + n ] d μ d ν = ( 2 π ) 2 δ 0 , p + s + t + m δ 0 , r + s t + n .
p = ( s + t + m )
r = ( s t + n )
u m , n = s , t J ( s + t + m ) ( 2 a α ) J ( s t + n ) ( 2 b β ) J s ( 2 c α + β ) J t ( 2 d α β )
φ ̃ ( μ , ν , z ) = exp { 2 i a α [ sin { μ α z } sin μ ] } exp { 2 i b β [ sin { ν β z } sin ν ] } exp { 2 i c α + β [ sin { μ + ν ( α + β ) z } sin { μ + ν } ] } exp { 2 i d α β [ sin { μ ν ( α β ) z } sin { μ ν } ] } .
k i k J k ( x ) e i β k = e i x cos ( β ) .
φ m , m , n , n = e i z ( α m + β n ) s , t J ( s + t + m m ) ( 4 a α sin { α z 2 } ) × exp { i m m + s + t 2 ( α z π ) } J ( s t + n n ) ( 4 b β sin { β z 2 } ) exp { i n n + s t 2 ( β z π ) } × J s ( 4 c α + β sin { ( α + β ) z 2 } ) exp { i s 2 ( ( α + β ) z π ) } J t ( 4 d α β sin { ( α β ) z 2 } ) exp { i t 2 ( ( α β ) z π ) } .
P 1 = 2 π α , P 2 = 2 π β , P 3 = 2 π α + β , P 4 = 2 π α β .
s = 4 c α + β sin { ( α + β ) z 2 } ,
t = 4 d α β sin { ( α β ) z 2 } .
Δ m 4 a α sin { α z 2 } + 4 c α + β sin { ( α + β ) z 2 } + 4 d α β sin { ( α β ) z 2 }
Δ n 4 b β sin { β z 2 } + 4 c α + β sin { ( α + β ) z 2 } + 4 d α β sin { ( α β ) z 2 } .
φ ̃ ( μ , ν , z ) = exp { 2 i a α [ sin μ 0 sin { μ 0 + α z } ] } × exp { 2 i b β [ sin ν 0 sin { ν 0 + β z } ] } × exp { 2 i c α + β [ sin { μ 0 + ν 0 } sin { μ 0 + ν 0 + ( α + β ) z } ] } × exp { 2 i d α β [ sin { μ 0 ν 0 } sin { μ 0 ν 0 + ( α β ) z } ] } × exp { 2 i a α [ cos μ 0 cos { μ 0 + α z } ] [ μ μ 0 α z ] } × exp { 2 i b β [ cos ν 0 cos { ν 0 + β z } ] [ ν ν 0 β z ] } × exp { 2 i c α + β [ cos { μ 0 + ν 0 } ( μ μ 0 α z + ν ν 0 β z ) ] } × exp { 2 i c α + β [ cos { μ 0 + ν 0 + ( α + β ) z } ( μ μ 0 α z + ν ν 0 β z ) ] } × exp { 2 i d α β [ cos { μ 0 ν 0 } ( μ μ 0 α z ν + ν 0 + β z ) ] } × exp { 2 i d α β [ cos { μ 0 ν 0 + ( α β ) z } ( μ μ 0 α z ν + ν 0 + β z ) ] } .
φ m , n ( z ) = π π φ ̃ ( μ , ν , z ) exp { i ( μ m + ν n ) } d μ d ν ,
φ m , n ( z ) = π π e i μ ( m m ) + i ν ( ν ν ) e i α z m + i β z n e i μ 0 m + i ν 0 n d μ d ν ,
m = 2 a α [ cos μ 0 cos { μ 0 + α z } ] + 2 c α + β [ cos { μ 0 + ν 0 } cos { μ 0 + ν 0 + ( α + β ) z } ] + 2 d α β [ cos { μ 0 ν 0 } cos { μ 0 ν 0 + ( α β ) z } ] ,
n = 2 b β [ cos ν 0 cos { ν 0 + β z } ] + 2 c α + β [ cos { μ 0 + ν 0 }
cos { μ 0 + ν 0 + ( α + β ) z } ] 2 d α β [ cos { μ 0 ν 0 } cos { μ 0 ν 0 + ( α β ) z } ] .
J k ( x ) = i k 2 π π π e i x cos θ e i θ k d θ ,
φ m , n ( z ) e i α z m + i β z n + i Φ ( z ) φ m m , n n ( 0 )
Φ ( z ) = μ 0 m + ν 0 n δ ,
δ = 2 a α [ sin μ 0 sin { μ 0 + α z } ] + 2 b β [ sin ν 0 sin { ν 0 + β z } ] + 2 c α + β [ sin { μ 0 + ν 0 } sin { μ 0 + ν 0 + ( α + β ) z } ] + 2 d α β [ sin { μ 0 ν 0 } sin { μ 0 ν 0 + ( α β ) z } ] .

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