Abstract

We report on results of a systematic analysis of spatial solitons in the model of one-dimensional photonic crystals, built as a periodic lattice of waveguiding channels, of width D, separated by empty channels of width LD. The system is characterized by its structural “duty cycle,” DCDL. In the case of the self-defocusing (SDF) intrinsic nonlinearity in the channels, one can predict new effects caused by competition between the linear trapping potential and the effective nonlinear repulsive one. Several species of solitons are found in the first two finite bandgaps of the SDF model, as well as a family of fundamental solitons in the semi-infinite gap of the system with the self-focusing nonlinearity. At moderate values of DC (such as 0.50), both fundamental and higher-order solitons populating the second bandgap of the SDF model suffer destabilization with an increase of the total power. Passing the destabilization point, the solitons assume a flat-top shape, while the shape of unstable solitons gets inverted with local maxima appearing in empty layers. In the model with narrow channels (around DC=0.25), fundamental and higher-order solitons exist only in the first finite bandgap, where they are stable, despite the fact that they also feature the inverted shape.

© 2008 Optical Society of America

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2008

2007

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101-202 (2007).
[CrossRef]

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, “Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices,” Phys. Lett. A 367, 149-155 (2007).
[CrossRef]

Z. Rapti, P. G. Kevrekidis, V. V. Konotop, and C. K. R. T. Jones, “Solitary waves under the competition of linear and nonlinear periodic potentials,” J. Phys. A: Math. Theor. 40, 14151-14163 (2007).
[CrossRef]

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SPECTRUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370-378 (2007).
[CrossRef]

S. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” EPL 79, 50003 (2007).
[CrossRef]

2006

Y. Sivan, G. Fibich, and M. I. Weinstein, “Waves in nonlinear lattices: ultrashort optical pulses and Bose-Einstein condensates,” Phys. Rev. Lett. 97, 193902 (2006).
[CrossRef] [PubMed]

T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
[CrossRef]

P. S. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729-4749 (2006).
[CrossRef]

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455-481 (2006).
[CrossRef]

2005

B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, 613-619 (2005).
[CrossRef]

G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of second-band vortex solitons in 2D photonic lattices,” Phys. Rev. Lett. 95, 053904 (2005).
[CrossRef] [PubMed]

B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778-1784 (2005).
[CrossRef]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, and P. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[CrossRef] [PubMed]

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

B. T. Seaman, L. D. Carr, and M. J. Holland, “Nonlinear band structure in Bose-Einstein condensates: nonlinear Schrödinger equation with a Kronig-Penney potential,” Phys. Rev. A 71, 033622 (2005).
[CrossRef]

I. M. Merhasin, B. V. Gisin, R. Driben, and B. A. Malomed, “Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity,” Phys. Rev. E 71, 016613 (2005).
[CrossRef]

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74-82 (2005).
[CrossRef]

2004

2003

P. Xie, Z. Q. Zhang, and X. D. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E 67, 026607 (2003).
[CrossRef]

A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452-459 (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. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Królikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

2002

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

G. L. Alfimov, V. V. Konotop, and M. Salerno, “Matter solitons in Bose-Einstein condensates with optical lattices,” Europhys. Lett. 58, 7-13 (2002).
[CrossRef]

B. B. Baizakov, V. V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105-5119 (2002).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. 27, 568-570 (2002).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear guided waves and spatial solitons in a periodic layered medium,” J. Opt. Soc. Am. B 19, 772-781 (2002).
[CrossRef]

2001

S. F. Mingaleev and Y. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474-5477 (2001).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

C. De Angelis, F. Gringoli, M. Midrio, D. Modotto, J. S. Aitchison, and G. F. Nalesso, “Conversion efficiency for second-harmonic generation in photonic crystals,” J. Opt. Soc. Am. B 18, 348-351 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

2000

C. Conti, S. Trillo, and G. Assanto, “Energy localization in photonic crystals of a purely nonlinear origin,” Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

1999

1998

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
[CrossRef]

N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287-2319 (1998).
[CrossRef]

1994

E. Yablonovitch, “Photonic crystals,” J. Mod. Opt. 41, 173-194 (1994).
[CrossRef]

1993

1973

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 10120 (1973) N. G. Vakhitov and A. A. Kolokolov,[Radiophys. Quantum Electron. 16, 783-789 (1973)].
[CrossRef]

Abdullaev, F.

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, “Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices,” Phys. Lett. A 367, 149-155 (2007).
[CrossRef]

Abdumalikov, A.

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, “Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices,” Phys. Lett. A 367, 149-155 (2007).
[CrossRef]

Adhikari, S.

S. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” EPL 79, 50003 (2007).
[CrossRef]

Aitchison, J. S.

Akozbek, N.

N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287-2319 (1998).
[CrossRef]

Alfimov, G. L.

G. L. Alfimov, V. V. Konotop, and M. Salerno, “Matter solitons in Bose-Einstein condensates with optical lattices,” Europhys. Lett. 58, 7-13 (2002).
[CrossRef]

Assanto, G.

C. Conti, S. Trillo, and G. Assanto, “Energy localization in photonic crystals of a purely nonlinear origin,” Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

Atai, J.

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

Baets, R.

Baizakov, B. B.

B. B. Baizakov, V. V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105-5119 (2002).
[CrossRef]

Bang, O.

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

Bartal, G.

G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of second-band vortex solitons in 2D photonic lattices,” Phys. Rev. Lett. 95, 053904 (2005).
[CrossRef] [PubMed]

Bergé, L.

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
[CrossRef]

Bhat, N. A. R.

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Bienstman, P.

Binosi, D.

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Cai, T.

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74-82 (2005).
[CrossRef]

Campbell, S.

Carr, L. D.

B. T. Seaman, L. D. Carr, and M. J. Holland, “Nonlinear band structure in Bose-Einstein condensates: nonlinear Schrödinger equation with a Kronig-Penney potential,” Phys. Rev. A 71, 033622 (2005).
[CrossRef]

Carter, J. D.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SPECTRUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370-378 (2007).
[CrossRef]

Christodoulides, D. N.

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]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. 27, 568-570 (2002).
[CrossRef]

Chu, P. L.

Cohen, O.

G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of second-band vortex solitons in 2D photonic lattices,” Phys. Rev. Lett. 95, 053904 (2005).
[CrossRef] [PubMed]

Conti, C.

C. Conti, S. Trillo, and G. Assanto, “Energy localization in photonic crystals of a purely nonlinear origin,” Phys. Rev. Lett. 85, 2502-2505 (2000).
[CrossRef] [PubMed]

Corney, J. F.

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

De Angelis, C.

De la Rue, R. M.

T. F. Krauss and R. M. De la Rue, “Photonic crystals in the optical regime--past, present and future,” Prog. Quantum Electron. 23, 51-96 (1999).
[CrossRef]

Deconinck, B.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SPECTRUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370-378 (2007).
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Figures (13)

Fig. 1
Fig. 1

Bandgap structure, as a function of modulation depth U [see Eq. (2)], found from the linearization of Eq. (3) for (a) D = 3 π 2 , (b) D = π , and (c) D = π 2 , the respective DC values being DC = 0.75 , 0.50 , 0.25 . Here and in other figures that display ranges of k shaded areas are occupied by Bloch bands. The spectrum was not computed in the black area.

Fig. 2
Fig. 2

Total power Q versus propagation constant k for fundamental and higher-order GS families in the defocusing model is shown at three values of the structural DC, DC D L = ( a ) 0.75, (b) 0.5, and (c) 0.25. Families of stable and unstable solutions are shown, severally, by black and red (gray in the black-and-white version) curves.

Fig. 3
Fig. 3

Examples of stable fundamental and double-peak solitons (a) and their three- and four-peak counterparts (b) found in the first bandgap of the defocusing model with DC = 0.75 for a fixed value of the propagation constant, k = 2 . In each panel (here and in similar figures below), the right plot shows the spectral plane of stability eigenvalues for the respective soliton, λ λ r + i λ i (the stability is implied by λ i = 0 ). Here and in other figures, the background pattern represents the underlying structure of the layered medium.

Fig. 4
Fig. 4

Same as in Fig. 3, but in the second bandgap, for k = 1.3 .

Fig. 5
Fig. 5

Examples of stable solitons in the second bandgap of the defocusing model with DC = 0.5 for a fixed propagation constant k = 0.4 . (a) Fundamental soliton with Q = 3.20 and a double-peak one with Q = 6.69 . (b) Three- and four-peak solitons with Q = 10.17 and Q = 13.65 , respectively.

Fig. 6
Fig. 6

Weakly unstable nearly flat-top solitons found at k = 0 in the second bandgap of the defocusing model with DC = 0.50 . (a) Fundamental soliton with Q = 5.08 and flat-top counterpart of the double-peak soliton with Q = 11.33 . (b) Flat-top counterparts of three- and four-peak solitons with Q = 17.58 and 23.83, respectively.

Fig. 7
Fig. 7

Unstable solitons with the inverted shape (i.e., with local power maxima in empty layers). All the examples are shown for k = 0.22 . (a) Top and bottom plots display, respectively, a fundamental soliton with Q = 9.17 and the former double-peak soliton (which actually features a single-peak structure as a result of the inversion) with Q = 21.06 . (b) Top and bottom plots display, respectively, inverted counterparts of former three- and four-peak solitons with powers Q = 30.88 and Q = 42.74 .

Fig. 8
Fig. 8

(a) Evolution of an unstable fundamental soliton with k = 0.15 and Q = 17.58 in the SDF model with DC = 0.5 . (b) Same for an unstable flat-top soliton belonging to the three-peak family with k = 0 and Q = 17.58 .

Fig. 9
Fig. 9

Examples of stable GSs in the first bandgap of the SDF model with DC = 0.25 . (a) Top and bottom plots represent, respectively, a fundamental soliton (i.e., the one localized in a single channel) with k = 0.1 and Q = 11.77 , and a soliton occupying two channels with k = 0.05 and Q = 15.26 . (b) Higher-order solitons occupying three and four channels with Q = 23.18 and 31.60, respectively, both pertaining to k = 0.05 . Note that these stable solitons feature local maxima of the power in empty channels.

Fig. 10
Fig. 10

(a) Total power of antisymmetric (twisted) bound states of fundamental GSs versus the propagation constant in the defocusing model with DC = 0.50 . The same characteristic is also shown for the family of subfundamental solitons in the second bandgap. (b) Typical examples of stable ( k = 1.2 and Q = 3.19 ) and unstable ( k = 0.1 and Q = 8.17 ) antisymmetric bound states in the first and second bandgaps.

Fig. 11
Fig. 11

(a) Total power of the antisymmetric bound state versus the propagation constant in the first bandgap of the defocusing model with DC = 0.25 . (b) Examples of these stable antisymmetric bound states, (top panel) k = 0.5 and Q = 2.96 ; (bottom panel) k = 0 and Q = 11.14 .

Fig. 12
Fig. 12

(a) Evolution of an unstable twisted soliton (antisymmetric bound state of two fundamental GSs) in the second bandgap for k = 0.1 and Q = 8.17 in the SDF model with DC = 0.5 . (b) Example of a weakly unstable subfundamental soliton (for k = 0.1 and Q = 2.43 ) found in the second bandgap of the model with DC = 0.50 .

Fig. 13
Fig. 13

Top plot shows the total power versus the propagation constant for families of solitons found in the semi-infinite gap of the SF model. The bottom plot displays a typical example of a stable soliton with k = 6 and Q = 2.21 , trapped in a narrow channel corresponding to DC = 0.25 .

Equations (4)

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i Ψ z + Ψ x x + W ( x ) ( 1 + σ Ψ 2 ) Ψ = 0 .
W ( x ) = { 0 , D + L n < x < L ( 1 + n ) U , L n < x < D + L n } , n = 0 , ± 1 , ± 2 .
k Φ + Φ x x + W ( x ) ( Φ + σ Φ 3 ) = 0 .
( ( d 2 d x 2 k ) W ( x ) ( 1 + 2 σ Φ 0 2 ) σ W ( x ) Φ 0 2 σ W ( x ) Φ 0 2 ( d 2 d x 2 k ) + W ( x ) ( 1 + 2 σ Φ 0 2 ) ) ( u v ) = λ ( u v ) ,

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