## 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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### References

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- X. Zhao, G. A. Georgakis, and Q. Niu, “Rabi oscillations between Bloch bands,” Phys. Rev. B 54, R5235–R5238 (1996).

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

[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] -
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] -
V. S. Shchesnovich and S. Chavez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. 32, 1920–1922 (2007).

[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] - 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.

#### 2007 (1)

V. S. Shchesnovich and S. Chavez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. 32, 1920–1922 (2007).

[Crossref]
[PubMed]

#### 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.

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]

#### 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.

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]

#### 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.

V. S. Shchesnovich and S. Chavez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. 32, 1920–1922 (2007).

[Crossref]
[PubMed]

#### 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]

[Crossref]
[PubMed]

#### Eisenberg, H. S.

[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.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Fratalocchi, A.

[Crossref]
[PubMed]

#### 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]

[Crossref]

#### Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).

[Crossref]
[PubMed]

#### Karpierz, M. A.

[Crossref]
[PubMed]

#### 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.

[Crossref]
[PubMed]

#### Lederer, F.

[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.

#### 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.

[Crossref]

#### Neshev, D. N.

[Crossref]
[PubMed]

#### Niu, Q.

[Crossref]

X. Zhao, G. A. Georgakis, and Q. Niu, “Rabi oscillations between Bloch bands,” Phys. Rev. B 54, R5235–R5238 (1996).

[Crossref]

#### Peleg, O.

#### Pertsch, T.

[Crossref]
[PubMed]

#### Peschel, U.

[Crossref]
[PubMed]

#### Raizen, M. G.

[Crossref]

#### Rüter, C. E.

#### Segev, M.

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Shandarova, K.

#### Shchesnovich, V. S.

V. S. Shchesnovich and S. Chavez-Cerda, “Bragg-resonance-induced Rabi oscillations in photonic lattices,” Opt. Lett. 32, 1920–1922 (2007).

[Crossref]
[PubMed]

#### Silberberg, Y.

[Crossref]
[PubMed]

[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.

[Crossref]
[PubMed]

#### Trompeter, H.

[Crossref]
[PubMed]

#### Winn, J. N.

[Crossref]

#### Zhao, X.

[Crossref]

#### Nature (2)

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Opt. Lett. (3)

[Crossref]
[PubMed]

[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]

#### Phys. Rev. A (1)

[Crossref]

#### Phys. Rev. B (3)

[Crossref]

[Crossref]

[Crossref]

#### Phys. Rev. Lett. (4)

[Crossref]
[PubMed]

[Crossref]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

#### Other (1)

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### Figures (5)

**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.**

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

**Fig. 3.**

(a) Intensity evolution pattern of a strong direct transition from the second to the third band at Bloch wave number

**Fig. 4.**

(a) Intensity evolution pattern of a weak direct transition from the first to the third band at Bloch wave number

**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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