Abstract

It is theoretically demonstrated that Rabi interband oscillations are possible in waveguide arrays. Such transitions can take place in optical lattices when the unit-cell is periodically modulated along the propagation direction. Under phase-matching conditions, direct power transfer between two Floquet-Bloch modes can occur. In the nonlinear domain, periodic oscillations between two different lattice solitons are also possible.

© 2008 Optical Society of America

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  1. X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
    [CrossRef]
  2. M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
    [CrossRef]
  3. J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
    [CrossRef]
  4. M. Skorobogatiy and J. D. Joannopoulos, "Rigid vibrations of a photonic crystal and induced interband transitions," Phys. Rev. B 61, 5293-5302 (2000).
    [CrossRef]
  5. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  6. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  7. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically induced Zener tunneling in one-dimensional lattices," Opt. Lett. 31, 790-792 (2006).
    [CrossRef] [PubMed]
  8. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
    [CrossRef] [PubMed]
  9. V. S. Shchesnovich and S. Chavez-Cerda, "Bragg-resonance-induced Rabi oscillations in photonic lattices," Opt. Lett. 32, 1920-1922 (2007).
    [CrossRef] [PubMed]
  10. G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
    [CrossRef] [PubMed]
  11. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  12. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
    [CrossRef] [PubMed]
  13. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
    [CrossRef] [PubMed]
  14. K. Shandarova, C. E. Rüter, R. Dong, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev "Experimental observation of Rabi oscillations in a one dimensional photonic lattice," paper QFG4, CLEO/QELS 2008, May 4-9, San Jose, California.

2007 (1)

2006 (3)

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically induced Zener tunneling in one-dimensional lattices," Opt. Lett. 31, 790-792 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

2003 (3)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

2000 (1)

M. Skorobogatiy and J. D. Joannopoulos, "Rigid vibrations of a photonic crystal and induced interband transitions," Phys. Rev. B 61, 5293-5302 (2000).
[CrossRef]

1999 (1)

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

1998 (2)

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1996 (1)

X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
[CrossRef]

1988 (1)

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Assanto, G.

Bartal, G.

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
[CrossRef] [PubMed]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Brzdakiewicz, K. A.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

Chavez-Cerda, S.

Christodoulides, D. N.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Fan, S.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

Fischer, M. C.

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

Fratalocchi, A.

Georgakis, G. A.

X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
[CrossRef]

Ippen, E. P.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

Joannopoulos, J. D.

M. Skorobogatiy and J. D. Joannopoulos, "Rigid vibrations of a photonic crystal and induced interband transitions," Phys. Rev. B 61, 5293-5302 (2000).
[CrossRef]

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

Joseph, R. I.

Karpierz, M. A.

Kivshar, Y. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Lederer, F.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Madison, K. W.

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

Manela, O.

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
[CrossRef] [PubMed]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Neshev, D. N.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Niu, Q.

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
[CrossRef]

Pertsch, T.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Peschel, U.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Raizen, M. G.

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

Segev, M.

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Shchesnovich, V. S.

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Skorobogatiy, M.

M. Skorobogatiy and J. D. Joannopoulos, "Rigid vibrations of a photonic crystal and induced interband transitions," Phys. Rev. B 61, 5293-5302 (2000).
[CrossRef]

Sukhorukov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Trompeter, H.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

Winn, J. N.

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

Zhao, X.

X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
[CrossRef]

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. A (1)

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, "Observation of Rabi oscillations between Bloch bands in an optical potential," Phys. Rev. A 58, R2648-R2651 (1998).
[CrossRef]

Phys. Rev. B (3)

J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, "Interband transitions in photonic crystals," Phys. Rev. B 59, 1551-1554 (1999).
[CrossRef]

M. Skorobogatiy and J. D. Joannopoulos, "Rigid vibrations of a photonic crystal and induced interband transitions," Phys. Rev. B 61, 5293-5302 (2000).
[CrossRef]

X. Zhao, G. A. Georgakis, and Q. Niu, "Rabi oscillations between Bloch bands," Phys. Rev. B 54, R5235-R5238 (1996).
[CrossRef]

Phys. Rev. Lett. (4)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-023905 (2003).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. S. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903-053906 (2006).
[CrossRef] [PubMed]

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97, 073906-073909 (2006).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays,"Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Other (1)

K. Shandarova, C. E. Rüter, R. Dong, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev "Experimental observation of Rabi oscillations in a one dimensional photonic lattice," paper QFG4, CLEO/QELS 2008, May 4-9, San Jose, California.

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Figures (5)

Fig. 1.
Fig. 1.

Allowed all-optical direct transition between the first and third band. The inset depicts a top view of a periodically modulated optical lattice.

Fig. 2.
Fig. 2.

Selection rule diagram for direct transitions between the first three bands.

Fig. 3.
Fig. 3.

(a) Intensity evolution pattern of a strong direct transition from the second to the third band at Bloch wave number k=-π/D, and (b) the corresponding total energy in second (red line) and the third (blue line) band, respectively, as a function of propagation distance. The arrow in (a) indicates the location where maximum energy exchange occurs.

Fig. 4.
Fig. 4.

(a) Intensity evolution pattern of a weak direct transition from the first to the third band at Bloch wave number k=0, and (b) the corresponding total energy in the first (red line) and the third (blue line) band, respectively, as a function of propagation distance. The arrow in (a) indicates the location where maximum energy exchange occurs.

Fig. 5.
Fig. 5.

(a) Diffraction dynamics in a periodically modulated lattice under wide beam excitation, and (b) nonlinear power exchange between a soliton residing at the semi-infinite gap and a soliton in the second gap of the corresponding lattice, under the same excitation conditions.

Equations (4)

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i U z + 2 U x 2 + V ( x , z ) U = 0
i d c n dz + M mn c m exp [ i ( Δ β nm 2 π Λ ) z ] = 0
i dc m dz + M nm c n exp [ i ( Δ β nm 2 π Λ ) z ] = 0
i U z + 2 U x 2 + V ( x , z ) U + U 2 U = 0

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