Abstract

A new type of waveguide lattice that relies on the effect of bandgap guidance, rather than total internal reflection, in the regions between the waveguide defects is proposed. Two different setting, for low index and high index defects are suggested. We analyze the linear bandgap and diffraction properties of such lattices. In the nonlinear regime the Kerr effect can counteract diffraction leading to the formation of gap lattice solitons. Interestingly enough, in the case of low index defects, stable soliton solutions are localized in the low index areas. This finding challenges the widely accepted idea that stable solitons can be sustained in high refractive index regions. In addition, in the case of high index defects, the coupling coefficient can become negative. Physical settings where the linear and nonlinear properties for bandgap lattices can be experimentally realized are presented.

© 2005 Optical Society of America

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Appl. Phys. Lett. (2)

A. Y. Cho, A. Yariv, and P. Yeh, "Observation of confined propagation in Bragg waveguides," Appl. Phys. Lett. 30, 471 (1977).
[CrossRef]

S. Linden, J. P. Mondia, H. M. V. Driel, T. C. Kleckner, C. R. Stanley, D. Modotto, A. Locatelli, C. D. Angelis, R. Morandotti, and J. S. Aitchison, "Nonlinear transmission properties of a deep-etched microstructured waveguide," Appl. Phys. Lett. 84, 5437-5439 (2004).
[CrossRef]

Eur. Phys. J. D (1)

P. Kevrekidis, B. Malomed, and Z. Musslimani, "Discrete gap solitons in a diffraction-managed waveguide array," Eur. Phys. J. D 23, 421-436 (2003).
[CrossRef]

Europhys. Lett. (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Multidimensional solitons in periodic potentials," Europhys. Lett. 63, 642-648 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Phys. A (1)

S. Theodorakis and E. Leontidis, "Bound states in a nonlinear Kronig-Penney model," J. Phys. A 30, 4835-4849 (1997).
[CrossRef]

Nature (2)

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]

Opt. Commun. (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

P. Roy. Soc. Lond. A Mat. (1)

R. d. L. Kronig and W. G. Penney, "Quantum Mechanics of Electrons in Crystal Lattices," P. Roy. Soc. Lond. A Mat. 814, 499-513 (1931).
[CrossRef]

Phys. Lett. A (2)

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).
[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]

Phys. Rev. A (2)

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, 013,602-1-9 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, "Lattice solitons in Bose-Einstein condensates," Phys. Rev. A 67, 063,608-1-9 (2003).
[CrossRef]

Phys. Rev. B (1)

N. Stefanou and A. Modinos, "Impurity bands in photonic insulators," Phys. Rev. B 57, 12,127-12,133 (1998).
[CrossRef]

Phys. Rev. E (5)

B. A. Malomed and P. G. Kevrekidis, "Discrete vortex solitons," Phys. Rev. E 64, 026,601-1-6 (2001).

N. K. Efremidis, D. N. Christodoulides, S. Sears, J. W. Fleischer, and M. Segev, "Discrete Solitons in Photorefractive Optically-Induced Photonic Lattices," Phys. Rev. E 66, 046,602-1-5 (2002).
[CrossRef]

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

W. Li and A. Smerzi, "Nonlinear Krönig-Penney model," Phys. Rev. E 70, 016,605-1-4 (2004).
[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, 016,613-1-12 (2005).
[CrossRef]

Phys. Rev. Lett. (24)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, "Discrete Solitons and Soliton-Induced Dislocations in Partially Coherent Photonic Lattices," Phys. Rev. Lett. 92, 123,902-1-4 (2004).
[CrossRef]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, "Two-dimensional optical lattice solitons," Phys. Rev. Lett. 91, 213,906-1-4 (2003).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, "Observation of Vortex-Ring "Discrete" Solitons in 2D Photonic Lattices," Phys. Rev. Lett. 92, 123,904-1-4 (2004).
[CrossRef]

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, "Observation of Discrete Vortex Solitons in Optically Induced Photonic Lattices," Phys. Rev. Lett. 92, 123,903-1-4 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary Solitons in Bessel Optical Lattices," Phys. Rev. Lett. 93, 093,904-1-4 (2004).
[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]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, "Slow Bragg Solitons in Nonlinear Periodic Structures," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonic Crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Dynamics of Discrete Solitons in Optical Waveguide Arrays," Phys. Rev. Lett. (1999).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. ederer, Y. Min, and W. Sohler, "Observation of Discrete Quadratic Solitons," Phys. Rev. Lett. 93, 113,902-1-4 (2004).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Discrete Vector Solitons in Kerr Nonlinear Waveguide Arrays," Phys. Rev. Lett. 91, 143,907-1-4 (2003).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053,902-1-4 (2003).
[CrossRef]

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, 023,902-1-4 (2003).
[CrossRef]

D. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap Solitons inWaveguide Arrays," Phys. Rev. Lett. 92, 093,904-1-4 (2004).
[CrossRef]

D. Neshev, A. A. Sukhorukov, B. Hanna,W. Krolikowski, and Y. S. Kivshar, "Controlled Generation and Steering of Spatial Gap Solitons," Phys. Rev. Lett. 93, 083,905-1-4 (2004).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High Transmission through Sharp Bends in Photonic Crystal Waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave Propagation in Two-Dimensional Dielectric Lattices," Phys. Rev. Lett. 67, 2017-2020 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and Acceptor Modes in Photonic Band Strucure," Phys. Rev. Lett. 67, 3380-3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong Localization of Photons in Certain Disordered Dielectric Superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Progress in Optics (1)

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed., vol. XXXIII, p. 203 (North-Holland, Amsterdam, 1994).

Science (2)

J. Knight, J. Broeng, T. A. Birks, and P. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other (2)

J. Joannopoulos, R. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986).

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Figures (10)

Fig. 1.
Fig. 1.

Different physical settings where BGL can be implemented. These settings can be (a) photonic crystal fiber arrays, (b) optically induced lattices, and (c) waveguide arrays. Notice that higher index areas are shown in darker colors. In these configurations the intensity of light is expected to be localized in the low refractive index defects of the lattice.

Fig. 2.
Fig. 2.

Different physical settings where bandgap lattices can be implemented. These settings can be (a) photonic crystal fiber arrays, (b) an optically induced lattices, and (c) waveguide arrays. Notice that higher index areas are shown in darker colors. In these configurations the intensity of light is expected to be localized in the high refractive index defects of the lattice.

Fig. 3.
Fig. 3.

On the left panel the band structure corresponding to the potential of Eq. (8) is shown. The period of this potential is L = 1 and its depth is V 0 = 9. The discrete spectrum is depicted with dots and the arrow indicates the band from which this mode bifurcates. The inset shows an enlarged view of the bifurcation. The first two bands of the spectrum have eigenvalues in the domains [2.62 5.31] and [-14.99 - 3.05], respectively. The eigenvalue of the first defect mode is q = -1.59. On the right panel the amplitude of the defect mode and a schematic illustration of the refractive index contrast across the lattice are shown.

Fig. 4.
Fig. 4.

The exact band structure (left panel) of a periodic superlattice as described by Eq. (10) for V 0 = 9 and L = 5. The inset shows an expanded view of the defect band. This band extends in the region [-1.68 - 1.51] while the bands above and bellow have spatial frequencies [3.04 3.09] and [-4.38 -3.32], respectively. On the right panel the Bloch modes at the base and the edge of the Brillouin zone are depicted. On the bottom of the figure a schematic illustration of the refractive index modulation along the lattice is presented.

Fig. 5.
Fig. 5.

Numerical simulations of low power (linear) beams propagating according to Eq. (1) where the lattice is given by Eq. (10). In (a) only two defects are involved, thus, forming a directional bandgap coupler, (b) diffraction of a single waveguide excitation, and (c) propagation with minimum diffraction when the beam is launched with a Bloch momentum k = π/2L that corresponds to the middle of the Brillouin zone. (a) and (b) the initial condition is the localized mode of a single low index defect and in (c) it is a Gaussian superposition of single defect modes having the proper phase difference. The propagation length is normalized to the coupling length zc = π/κ. Because the maximum beam intensity reduces rapidly as it diffracts, a highly nonlinear scheme in the color range (b) able to capture the diffraction pattern for several diffraction lengths is utilized.

Fig. 6.
Fig. 6.

On the left panel we can see the dependence of the width of the defect band from the spacing between the defects. On the right panel we plot the same data in a logarithmic scale for the width of the defect band. The data are presented with dots and the solid curve show the fitting function.

Fig. 7.
Fig. 7.

Self-focusing soliton solutions of the low index defect lattice given by Eq. (10) with period L = 5 and index contrast V 0 = 9. In the top left panel the existence curve of this family of soliton solutions (total power vs. the eigenvalue) is presented. The bandgaps are shown in gray color and the bands in white. The two defect modes corresponding to points (a) and (b) of the existence curve, with eigenvalues q = -1.33 and q = 1.61 respectively, are depicted in the bottom panel. The stable evolution of mode (a) over 5 diffraction lengths is presented in the top right panel.

Fig. 8.
Fig. 8.

Self-defocusing soliton solutions of the low index defect lattice given by Eq. (10) with period L = 5 and index contrast V 0 = 9. In the top left panel the existence curve of this family of soliton solution (total power vs. the eigenvalue) is presented. The bandgaps are show in gray color and the bands in white. The two defect mode corresponding to points (a) and (b) of the existence curve, and with eigenvalues q = 1.77 and q = -2.51 respectively, are depicted in the bottom panel. The stable evolution of mode (a) over 5 diffraction lengths is presented in the top right panel.

Fig. 9.
Fig. 9.

On the top left panel the band structure corresponding to the potential which is expressed by Eq. (22) is shown. The continuous part of the spectrum (the bands) of this potential is exactly the same as that of Eq. (10) (shown in Fig. 4). The period of this potential outside the defect is L = 1 and its depth is V 0 = 9. The first two modes of the discrete spectrum (which are associated with the defect) are depicted with dots, and the arrows indicate the bifurcations of these modes from the continuous bands. The top right panel shows enlarged views of these bifurcations. On the bottom panel the first two defect modes [(a) and (b)] are shown. The first two bands of the spectrum have eigenvalues in the regions [2.62 5.31] and [-14.99 - 3.05], respectively. The eigenvalue of the first defect mode (located in the TIR region) is q = 6.84 and of the second mode (located in the first bandgap between the first two bands) is q = 0.53.

Fig. 10.
Fig. 10.

Expanded view of the first two defect bands [(a) and (b)] inside the band structure (left panel) of a periodic superlattice as described by Eq. (23) for V 0 = 9 and L = 5.5. Bands (a) and (b) originate from the corresponding defect modes of Fig. 9. The narrow defect band (a) spans the region [6.8425 6.8429] while the band bellow has spatial frequencies in the domain [5.11 5.16]. Defect band (b) spans the region [0.49 0.57] while the bands above and bellow have spatial frequencies in the domains [3 3.16] and [-3.35 -4], respectively. On the right panel the Bloch modes of bands (a) and (b) at the base of the Brillouin zone are shown.

Equations (24)

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i ψ z + 1 2 ψ xx + V ( x ) ψ + γ ψ 2 = 0 .
ψ ( x , z ) = u ( x ) e iqz
qu + 1 2 u xx + V ( x ) u + γ u 3 = 0 .
u ( x ) = g ( x ) exp ( ikx )
V g ( x ; V 0 ) = { V 0 x < 1 4 0 1 4 < x 1 2 V 0 2 x = 1 4
V g ( x ) = V g ( x = x n )
i ψ z + 1 2 ψ x x + V ( x ; V 0 ) ψ + γ ψ 2 ψ = 0 .
V t ( x ) = V t ( x ; V 0 ) = V g ( x ; V 0 ) V d ( x , V 0 )
V d ( x ; V 0 ) = { V 0 x < 1 4 V 0 2 x = 1 4 0 otherwise
V ( x ; V 0 ) = { V g ( x ; V 0 ) V d ( x ; V 0 ) x L 2 V g ( x = x n L ) V d ( x = x n L ) x > L 2 and x L 2
q ϕ + 1 2 ϕ xx + V t ( x ) ϕ = 0 .
ψ ( x , z ) = e iqz n c n ( z ) ϕ n ( x ) .
i d c n d z + Δ q c n + κ ( c n + 1 + c n 1 ) + λ c n 2 c n = 0
κ = 1 2 ϕ n * d 2 ϕ n + 1 d x 2 dx + ϕ n * ϕ n + 1 V ( x ) dx
Δ q = ϕ n 2 ( V ( x ) V t ( x ) ) dx
λ = γ ϕ n 4 dx
q = Δ q + 2 κ cos k .
c n ( z ) = J n ( 2 κ z ) exp ( i π n 2 + i Δ q z )
ψ ( x , z = 0 ) = n ϕ n ( x ) e ( n A ) 2 e i k n ,
P = ψ 2 dx ,
H = [ ψ x 2 2 V ( x ) ψ 2 ψ 4 ] .
ψ ( x , z ) = u ( x ) exp ( iqz )
V t ( x ) = V t ( x ; V 0 ) = V g ( x 1 4 )
V ( x ) = V ( x ; V 0 ) = V t ( x = x Ln ; V 0 ) ,

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