Abstract

We show that optical Bloch oscillations can emerge in waveguide arrays with linearly varying propagation constants. The existence of localized modes (Wannier–Stark states) with equidistant wave-number spacing (Wannier–Stark ladder) that do not undergo diffraction is analytically proved. The evolution of arbitrary initial excitations is described, and potential applications are suggested.

© 1998 Optical Society of America

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References

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  1. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
    [CrossRef]
  2. J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.
  3. C. Zener, Proc. R. Soc. London Ser. A 145, 523 (1932).
    [CrossRef]
  4. C. Waschke, H. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Phys. Rev. Lett. 70, 3319 (1993).
    [CrossRef] [PubMed]
  5. D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
    [CrossRef] [PubMed]
  6. C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
    [CrossRef]

1998 (1)

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

1996 (1)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

1995 (1)

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

1993 (1)

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

1932 (1)

C. Zener, Proc. R. Soc. London Ser. A 145, 523 (1932).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Aitchison, J. S.

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

Barad, Y.

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

Bishop, A.

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

Bright, J.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Cai, D.

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

de Sterke, C.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Eisenberg, H.

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

Gronbech-Jensen, N.

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

Hammon, T.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Köhler, K.

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

Krug, P.

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Kurz, H.

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

Lederer, F.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Leo, K.

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

Morandotti, R.

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Roskos, H.

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

Salerno, M.

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

Schwedler, R.

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

Silberberg, Y.

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

Trillo, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Waschke, C.

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

Zener, C.

C. Zener, Proc. R. Soc. London Ser. A 145, 523 (1932).
[CrossRef]

Phys. Rev. E (2)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

C. de Sterke, J. Bright, P. Krug, and T. Hammon, Phys. Rev. E 57, 2365 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

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

D. Cai, A. Bishop, N. Gronbech-Jensen, and M. Salerno, Phys. Rev. Lett. 74, 1186 (1995).
[CrossRef] [PubMed]

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

C. Zener, Proc. R. Soc. London Ser. A 145, 523 (1932).
[CrossRef]

Other (1)

J. S. Aitchison, R. Morandotti, H. Eisenberg, Y. Barad, and Y. Silberberg, in International Quantum Electronics Conference, Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 19.

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Figures (4)

Fig. 1
Fig. 1

Electro-optic waveguide array sandwiched between two current-leading electrodes. The changing voltage ΔU produces a linear variation of the propagation constants of the waveguides.

Fig. 2
Fig. 2

Periodic motion (Bloch oscillation) of a discrete Gaussian beam in a waveguide array. The insets show the intensity and the respective phase distribution at three different distances α=0.5.

Fig. 3
Fig. 3

Field shape of a localized linear eigenmode (Wannier–Stark state) of a waveguide array with a linearly varying propagation constant.

Fig. 4
Fig. 4

Evolution of a single waveguide excitation that shows consecutive diffraction and focusing. Inset, field and phase distribution at the turning point α=0.2.

Equations (11)

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

iddz+αnanz+an+1z+an-1z=0,
iz+iακ+2 cosκa˜κ,z=0,
a˜κ,z=12πnanzexp-iκn,anz=12π-ππa˜κ,zexpiκndκ.
u˜mκ=exp-iαβmκ-2 sin κ.
anmz=an-m+mmzexpim-mαz,
un0=12π-ππexp2iαsin κ+iκndκ=2πJ-n2α,
a˜κ,z=a˜0κ-αzexp-2iαsinκ-αz-sinκ.
anz=12π-ππexpiκn-2iαsinκ-αz-sinκdκ=J-n4αsinα2zexpin2αz-π.
a˜κ,za˜0κ-αzexp-2iαsinκ0-sinκ0+αz×exp-2iαcosκ0-cosκ0+αz×κ-κ0+αz+.
anzexpiαzn+iϕzan-δn z=0,
δn=2/αcosκ0-cosκ0+αz.

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