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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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/QELS2008, May 4–9, San Jose, California.

2007 (1)

2006 (3)

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]

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

2003 (3)

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]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (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.

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, 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 and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[Crossref] [PubMed]

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/QELS2008, May 4–9, San Jose, California.

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]

Dong, R.

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/QELS2008, May 4–9, San Jose, California.

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, 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]

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.

Kip, D.

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/QELS2008, May 4–9, San Jose, California.

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]

Makris, K. G.

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/QELS2008, May 4–9, San Jose, California.

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]

Peleg, O.

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/QELS2008, May 4–9, San Jose, California.

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]

Rüter, C. E.

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/QELS2008, May 4–9, San Jose, California.

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, 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]

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/QELS2008, May 4–9, San Jose, California.

Shandarova, K.

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/QELS2008, May 4–9, San Jose, California.

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)

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]

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]

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]

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/QELS2008, May 4–9, San Jose, California.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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

Metrics