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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18165–18172
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Dissipative soliton acceleration in nonlinear optical lattices

Yannis Kominis, Panagiotis Papagiannis, and Sotiris Droulias  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18165-18172 (2012)
http://dx.doi.org/10.1364/OE.20.018165


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Abstract

An effective mechanism for dissipative soliton acceleration in nonlinear optical lattices under the presence of linear gain and nonlinear loss is presented. The key idea for soliton acceleration consists of the dynamical reduction of the amplitude of the effective potential experienced by the soliton so that its kinetic energy eventually increases. This is possible through the dependence of the effective potential amplitude on the soliton mass, which can be varied due to the presence of gain and loss mechanisms. In contrast to the case where either the linear or the nonlinear refractive index is spatially modulated, we show that when both indices are modulated with the same period we can have soliton acceleration and mass increasing as well as stable soliton propagation with constant non-oscillating velocity. The acceleration mechanism is shown to be very robust for a wide range of configurations.

© 2012 OSA

1. Introduction

Optical lattices have been a subject of continuously increasing research interest in the past few years. Such inhomogeneous photonic structures provide efficient mechanisms for the control of the light propagation dynamics that have no counterpart in bulk media [1

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

4

4. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: [CrossRef] .

]. The linear diffraction properties of these configurations can be managed so that the propagation characteristics of optical wave fronts can be engineered. Several linear phenomena such as optical Bloch oscillations [5

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

, 6

6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

], Rabi oscillations [7

7. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008). [CrossRef] [PubMed]

, 8

8. K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009). [CrossRef] [PubMed]

] and Anderson localization [9

9. Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72, 026606 (2005). [CrossRef]

, 10

10. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef] [PubMed]

] have been studied with some of them, although classical, being in direct analogy to quantum phenomena in solid state structures [11

11. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009). [CrossRef]

]. The presence of nonlinearity in these photonic structures introduces the existence of spatially localized waves due to the interplay between the linear diffraction and nonlinearity. Therefore soliton formation and dynamics can be controlled by tailoring the geometric and material properties of the configuration, resulting in interesting functionality which is promising for all-optical control concepts and applications. In parallel, Bose-Einstein condensates with either attractive or repulsive atom interactions have been successfully loaded in optical lattices and the properties of the respective lattice solitons have been investigated [12

12. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science 293, 843–846 (2001). [CrossRef] [PubMed]

15

15. F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A 64, 436061–4360610 (2001). [CrossRef]

].

In this work we present a novel mechanism for soliton acceleration in transversely periodic structures with modulated linear and nonlinear refractive indices under the presence of dissipative mechanisms. The key idea is that, although the medium is longitudinally homogeneous, a soliton with varying mass, due to gain and/or loss mechanisms, ”experiences” a longitudinally varying effective potential through the dependence of the latter on the soliton mass. In these systems the soliton formation results not only from the balance between nonlinearity and diffraction but also from the competition between gain and loss mechanisms of different physical origin. In contrast to the conservative cases, the localized modes of the system have no longer neutral stability properties and become attractors of enhanced stability which can be excited by a sufficiently larger class of initial conditions [32

32. N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons, Lect. Notes Phys. (Springer, 2005), Vol. 661. [CrossRef]

34

34. F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010). [CrossRef]

].

2. Model and results

The underlying model of soliton propagation is the NonLinear Schrödinger (NLS) equation with linear and nonlinear dissipative terms and spatially modulated linear and nonlinear refractive index
iuz+2ux2+2|u|2u=n(x)(α1+α2|u|2)ui(σ1+σ2|u|2)u
(1)
where z and x are the normalized longitudinal and transverse coordinates, respectively; n(x) describes the normalized transverse modulation of the linear and nonlinear refractive index which, for simplicity, is considered to vary as n(x) = sin(Kx); αi(i = 1, 2) are the modulation amplitudes of the linear and the nonlinear refractive index, σi(i = 1, 2) are the normalized coefficients of linear gain (σ1 < 0), and two- photon absorption (σ2 > 0), respectively. In order to study soliton dynamics, we utilize a perturbative approach [23

23. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978). [CrossRef]

, 24

24. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989). [CrossRef]

] starting from Eq. (1) with zero rhs corresponding to the integrable NLS equation with the soliton solution u = ηsech[η(xx0)] exp[i(vx/2 + 2ϕ)], with /dz = η2/2 − v2/8. The soliton is treated as an effective particle moving under the effect of the perturbation R(x, u; α1, α2, σ1, σ2) = n(x)(α1 + α2 |u|2)ui(σ1 + σ2 |u|2)u corresponding to the rhs of Eq. (1). The effective particle mass m = ∫ |u|2dx = 2η, momentum p = mv and position x0, vary according to the equations
dmdz=2+Im{uR*(u,x)}dx=2m(σ1+σ26m2)
(2)
dpdz=4+Re{uxR*(u,x)}dx=Veffx0
(3)
dx0dz=pm
(4)
with
Veff=[2α1+α26(K2+m2)]πKsinh(πK/m)sin(Kx0).
(5)
In the absence of dissipation (σi = 0, i = 1, 2) the effective particle motion, given by Eqs. (2)(4), describes two distinct dynamical behaviors corresponding to trapped solitons oscillating around a potential minimum and traveling solitons moving transversely across the potential, depending on the initial energy of the effective particle. From Eq. (5) it is obvious that, in the same periodic medium, solitons of different mass move in potentials having different amplitude. The dependence of the potential strength on the soliton mass reflects the relation between the spatial width of soliton (∼ η−1) and the spatial period of the lattice (∼ K−1) [35

35. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. A 72, 046610 (2005).

, 36

36. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010). [CrossRef]

]. This dependence is shown in Fig. 1, for cases where the linear, the nonlinear or both refractive indices are spatially modulated. The presence of dissipation mechanisms results in soliton mass variation according to Eq. (2). The key idea for soliton acceleration is the dynamical reduction of its potential energy, through the dependence of the potential amplitude on the soliton mass, so that its kinetic energy increases.

Fig. 1 Amplitude of the effective potential versus soliton mass, as given by Eq. (5), for the cases where either the linear, the nonlinear or both refractive indices are spatially modulated with a period 2π (K = 1). α1 = 0.05, α2 = 0 (blue line), α1 = 0, α2 = 0.05 (green line), α1 = 0.05, α2 = −0.12 (red, thick line).

Let us consider a soliton of mass min launched at a local maximum of the effective potential with zero initial velocity. Firstly we investigate the case where only the linear or the nonlinear refractive index is modulated (α2 = 0 or α1 = 0). In the absence of linear gain (σ1 = 0, σ2 > 0), the soliton will evolve with continuously decreasing mass in a potential of continuously decreasing amplitude (Fig. 1), and as a result will have a continuously increasing velocity. In that case we can only achieve acceleration of solitons having an increasing width and decreasing amplitude during propagation; such a mechanism would be useful only for a restricted propagation distance so that a specific degree of localization is retained. Under the presence of linear gain (σ1 ≠ 0), there is fixed mass value m0'=6σ1/σ2 for which the rhs of Eq. (2) becomes zero. The gain and loss mechanisms are exactly balanced for m0, resulting in stable soliton propagation. A soliton with initial mass min > m0 will evolve with decreasing mass until its mass reaches the value m0 and then propagate with constant shape. During the transient phase, the amplitude of effective potential has been decreased and soliton has been accelerated. Subsequently, the soliton propagates with a velocity which is highly oscillating around a nonzero mean value, as shown in Fig. 2(a).

Fig. 2 Soliton velocity evolution versus propagation distance for the case of lattice period 2π (K = 1), under the presence of nonlinear dissipation (σ2 = 0.005) and linear gain σ1 = − 0.0033, for which zero dissipation mass is m0 = 2. (a) Velocity evolution of a soliton having min = 2.4 for the case where only the linear (blue line) or the nonlinear (green line) refractive index is modulated; (α1 = 0.05, α2 = 0) and (α1 = 0, α2 = 0.05), respectively. (b) Velocity evolution of a soliton having min = 2.4 or min = 1.6 for the case where both the linear and the nonlinear refractive indices are modulated (α1 = 0.05, α2 = −0.12). Results obtained under the effective particle approach [Eqs. (2)(4)].

The final velocity value vout as a function of the initial soliton mass min is shown in Fig. 3(a) for spatial modulations having different periods, as obtained from the effective particle Eqs. (2)(4). Note that the initial soliton position located at the maxima of the respective effective potential differs by 2π/K for the cases min > m0 and min < m0, due to sign reversal of the potential. The maxima of the potential are saddle points for soliton dynamics in the absence of gain and loss. The mass dependence on z, due to the presence of gain and loss, results in an additional degree of freedom of the effective particle system, which becomes nonintegrable. Complex dependence of the soliton evolution on the initial conditions occurs in the region around a saddle point corresponding to a potential maximum, so that a stationary soliton can be accelerated to either positive or negative velocity. However, this region, corresponding to a chaotic separatrix layer is very narrow. Therefore, we can always choose an initial soliton position which is close enough to a potential maximum corresponding to high initial particle energy but slightly displaced from it in order to determine the sign of the final soliton velocity. Such a displacement from the potential maximum of 1% with respect to the lattice period has been used in all results. As it can be seen in Fig. 3(a), the modulation period K determines the value mass m0 for which the soliton ”experiences” no effective potential as well as the maximum output velocity for given modulation amplitudes (αi, i = 1, 2). On the other hand, the output velocity scales with the square root of αi so that an increased modulation amplitude results in increased output velocities. In Fig. 3(a) and (b), the two cases of soliton acceleration with decreasing and increasing mass, respectively, are shown, as given by the effective particle Eqs. (2)(4) and numerical simulations of the original NLS Eq. (1). The results are shown to be in remarkable agreement with respect to the initial stage of the acceleration process as well as to the final velocities given by the slopes of the respective lines.

Fig. 3 (a) Soliton final velocity versus initial mass for a lattice with α1 = 0.05, α2 = −0.12, σ2 = 0.005, σ1 = −0.0033 and K = 0.5, 1, 1.5 as obtained from the effective particle approach. (b) and (c) Transverse soliton position under propagation for the cases where min > m0 (b) and min < m0 (c). The lattice parameters correspond to the case m0 = m0 and the initial soliton masses correspond to the points depicted in (a). Results from the effective particle approach (green line) are compared to results from numerical simulation of the original NLS Eq. (1) (blue line). A noise level of 1% with respect to soliton amplitude has been superimposed in the initial soliton profiles. The insets show the initial and final soliton shapes (amplitude and width).

Fig. 4 Numerical simulation of the original NLS Eq. (1) for the case of a stronger lattice having α1 = 0.4, α2 = −0.96, K = 0.5 and higher gain/loss coefficients σ2 = 0.01 and σ1 = −0.0067 (m0 = m0 = 2.18) for solitons having min = 2.9 (a) and min = 1.5 (b). A noise level of 1% with respect to soliton amplitude has been superimposed in the initial soliton profiles.

3. Discussion and conclusions

The importance of modulating both the linear and the nonlinear refractive index has also been presented. In contrast to the case where either the linear or the nonlinear refractive index is spatially modulated, we have shown that when both indices are modulated with the same period we can have soliton acceleration and mass increasing as well as stable soliton propagation with constant non-oscillating velocity. The acceleration mechanism has been shown to persist under a wide range of lattice and gain/loss parameters while being robust under the presence of noise.

Acknowledgments

Discussions with Prof. K. Hizanidis and Prof. T. Bountis are kindly acknowledged. Y.K and P.P. were supported by the Research Project NWDCCPS implemented within the framework of the Action Supporting Postdoctoral Researchers of the Operational Program Education and Lifelong Learning (Actions Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. S.D. was supported by the Research Project ANEMOS co-financed by the European Union (European Social Fund-ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) Research Funding Program: Thales. Investing in knowledge society through the European Social Fund.

References and links

1.

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

2.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

3.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011). [CrossRef]

4.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: [CrossRef] .

5.

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

6.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

7.

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16, 10309–10314 (2008). [CrossRef] [PubMed]

8.

K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102, 123905 (2009). [CrossRef] [PubMed]

9.

Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72, 026606 (2005). [CrossRef]

10.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef] [PubMed]

11.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009). [CrossRef]

12.

F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science 293, 843–846 (2001). [CrossRef] [PubMed]

13.

S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chiofalo, and M. P. Tosi, “Superfluid and dissipative dynamics of a Bose–Einstein condensate in a periodic optical potential,” Phys. Rev. Lett. 86, 4447–4450 (2001). [CrossRef] [PubMed]

14.

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001). [CrossRef] [PubMed]

15.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A 64, 436061–4360610 (2001). [CrossRef]

16.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]

17.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]

18.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef] [PubMed]

19.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

20.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

21.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011). [CrossRef] [PubMed]

22.

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84, 023842 (2011). [CrossRef]

23.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978). [CrossRef]

24.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989). [CrossRef]

25.

V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Dynamics of matter solitons in weakly modulated optical lattices,” Phys. Rev. A 70, 043604 (2004). [CrossRef]

26.

R. Scharf and A. R. Bishop, “Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials,” Phys. Rev. E 47, 1375–1383 (1993). [CrossRef]

27.

D. Polleti, T. J. Alexander, E. A. Ostrovskaya, B. Li, and Y. S. Kivshar, “Dynamics of matter-wave solitons in a ratchet potential,” Phys. Rev. Lett. 101, 150403 (2008). [CrossRef]

28.

G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104, 053903 (2010). [CrossRef] [PubMed]

29.

M. Rietmann, R. Carretero-Gonzalez, and R. Chacon, “Controlling directed transport of matter wave solitons using the ratchet effect,” Phys. Rev. A 83, 053617 (2011). [CrossRef]

30.

P. Papagiannis, Y. Kominis, and K. Hizanidis, “Power- and momentum-dependent soliton dynamics in lattices with longitudinal modulation,” Phys. Rev. A 84, 013820 (2011). [CrossRef]

31.

S. Flach, O. Yevtushenko, and Y. Zolotaryuk, “Directed current due to broken time-space symmetry,” Phys. Rev. Lett. 84, 2358–2361 (2000). [CrossRef] [PubMed]

32.

N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons, Lect. Notes Phys. (Springer, 2005), Vol. 661. [CrossRef]

33.

F. Kh. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A 76, 043611 (2007). [CrossRef]

34.

F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010). [CrossRef]

35.

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. A 72, 046610 (2005).

36.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010). [CrossRef]

37.

B. A. Malomed, “Evolution of nonsoliton and ’quasi-classical’ wavetrains in nonlinear Schrödinger and Korteweg–De Vries equations with dissipative perturbations,” Physica D 29, 155–172 (1987). [CrossRef]

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 13, 2012
Revised Manuscript: July 16, 2012
Manuscript Accepted: July 16, 2012
Published: July 23, 2012

Citation
Yannis Kominis, Panagiotis Papagiannis, and Sotiris Droulias, "Dissipative soliton acceleration in nonlinear optical lattices," Opt. Express 20, 18165-18172 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18165


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References

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003). [CrossRef] [PubMed]
  2. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008). [CrossRef]
  3. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys.83, 247–305 (2011). [CrossRef]
  4. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: . [CrossRef]
  5. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998). [CrossRef]
  6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999). [CrossRef]
  7. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express16, 10309–10314 (2008). [CrossRef] [PubMed]
  8. K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett.102, 123905 (2009). [CrossRef] [PubMed]
  9. Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E72, 026606 (2005). [CrossRef]
  10. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007). [CrossRef] [PubMed]
  11. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev.3, 243–261 (2009). [CrossRef]
  12. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science293, 843–846 (2001). [CrossRef] [PubMed]
  13. S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chiofalo, and M. P. Tosi, “Superfluid and dissipative dynamics of a Bose–Einstein condensate in a periodic optical potential,” Phys. Rev. Lett.86, 4447–4450 (2001). [CrossRef] [PubMed]
  14. A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett.86, 2353–2356 (2001). [CrossRef] [PubMed]
  15. F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A64, 436061–4360610 (2001). [CrossRef]
  16. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651–654 (1987). [CrossRef]
  17. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
  18. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
  19. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007). [CrossRef]
  20. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4, 103–106 (2010). [CrossRef]
  21. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011). [CrossRef] [PubMed]
  22. R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A84, 023842 (2011). [CrossRef]
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