Diffraction management of focused light beams in optical lattices with a quadratic frequency modulation

We reveal that the effective diffraction experienced by light beams launched along the central guiding channel of optical lattice with a quadratic frequency modulation can be tuned in strength and sign. Complete suppression of the linear diffraction in the broad band of spatial frequencies is shown to be possible, thus profoundly affecting properties of nonlinear self-sustained beams as well. In particular, we report on the properties of a new class of stable solitons supported by such lattices in defocusing media. © 2005 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.5530) Pulse propagation and solitons References and links 1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides” Opt. Lett. 13, 794 (1988). 2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000). 3. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices” Nature 424, 817 (2003). 4. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. 29, 268 (2004). 5. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. 29, 468 (2004). 6. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901 (2004). 7. O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Grating-mediated waveguiding,” Phys. Rev. Lett. 93, 103902 (2004). 8. 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). 9. 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 (2003). 10. 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 (2003). 11. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings” Opt. Lett. 28, 710 (2003). 12. 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, 123902 (2004). 13. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature 433, 500 (2005). 14. Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, “Spatial soliton switching in quasicontinuous optical arrays” Opt. Lett. 29, 766 (2004). 15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12, 2831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831 16. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882 (2000). (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4244 #7339 $15.00 US Received 2 May 2005; revised 20 May 2005; accepted 21 May 2005 17. R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, “Nonlinear pulse reflection from chirped fiber gratings,” Opt. Express 3, 465 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-11-465 Diffraction inevitably broadens all finite light beams propagating in the homogeneous linear medium, but a transverse modulation of the refractive index can drastically affect this process. When transverse refractive index modulation is periodic and strong enough the homogeneous diffraction is replaced with the discrete diffraction arising because of the coupling between nearest neighbors in the array [1]. Because the diffraction coefficient for tilted waves depends on their propagation angles, launching of broad laser beam at suitable angles relative to the array axis allows engineering its diffraction broadening [2,3]. This phenomenon was predicted theoretically and experimentally observed in both oneand two-dimensional waveguide arrays [2,4,5], as well as in harmonic refractive index gratings [6]. The interesting phenomenon of diffraction suppression, or grating-mediated waveguiding, was recently reported in Ref. [7]. It is worth noticing that diffraction control of relatively broad laser beams does not require deep refractive index modulations. This fact offers the unique opportunity to use optically induced periodic waveguide arrays, or optical lattices, with the flexibly controlled parameters [8-13]. Besides modification of diffraction properties of light beams, the transverse refractive index modulations drastically affect properties of guided modes supported in the nonlinear regime [8-15]. Thus, the existence domain of lattice solitons is dictated by the Bloch wave spectrum for the lattice. However the challenging problem of diffraction management is still far from its complete solution. In particular, it is quite important to develop an effective approach to the broadband diffraction control that also allows suppression of diffraction for narrow beams, thus having a broad spatial spectrum. In this work we explore the prospects offered by lattice with quadratic frequency modulation for diffraction control and show that by a properly changing the lattice depth it is possible to achieve broadband diffraction suppression or reversal of diffraction sign for narrow beams launched in the central channel of the lattice parallel to the guiding lattice channels. We also have found that such diffractive properties alter qualitatively the properties of lattice solitons, especially those supported by defocusing media. We consider propagation of optical radiation along the ξ axis in cubic nonlinear medium with modulation of linear refractive index along transverse η axis, described by the nonlinear Schrödinger equation for dimensionless complex field amplitude q :


17.
R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, "Nonlinear pulse reflection from chirped fiber gratings," Opt.Express 3, 465 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX- 3-11-465 Diffraction inevitably broadens all finite light beams propagating in the homogeneous linear medium, but a transverse modulation of the refractive index can drastically affect this process.When transverse refractive index modulation is periodic and strong enough the homogeneous diffraction is replaced with the discrete diffraction arising because of the coupling between nearest neighbors in the array [1].Because the diffraction coefficient for tilted waves depends on their propagation angles, launching of broad laser beam at suitable angles relative to the array axis allows engineering its diffraction broadening [2,3].This phenomenon was predicted theoretically and experimentally observed in both one-and two-dimensional waveguide arrays [2,4,5], as well as in harmonic refractive index gratings [6].The interesting phenomenon of diffraction suppression, or grating-mediated waveguiding, was recently reported in Ref. [7].It is worth noticing that diffraction control of relatively broad laser beams does not require deep refractive index modulations.This fact offers the unique opportunity to use optically induced periodic waveguide arrays, or optical lattices, with the flexibly controlled parameters [8][9][10][11][12][13].Besides modification of diffraction properties of light beams, the transverse refractive index modulations drastically affect properties of guided modes supported in the nonlinear regime [8][9][10][11][12][13][14][15].Thus, the existence domain of lattice solitons is dictated by the Bloch wave spectrum for the lattice.
However the challenging problem of diffraction management is still far from its complete solution.In particular, it is quite important to develop an effective approach to the broadband diffraction control that also allows suppression of diffraction for narrow beams, thus having a broad spatial spectrum.In this work we explore the prospects offered by lattice with quadratic frequency modulation for diffraction control and show that by a properly changing the lattice depth it is possible to achieve broadband diffraction suppression or reversal of diffraction sign for narrow beams launched in the central channel of the lattice parallel to the guiding lattice channels.We also have found that such diffractive properties alter qualitatively the properties of lattice solitons, especially those supported by defocusing media.
We consider propagation of optical radiation along the ξ axis in cubic nonlinear medium with modulation of linear refractive index along transverse η axis, described by the nonlinear Schrödinger equation for dimensionless complex field amplitude q : Here the longitudinal ξ and transverse η coordinates are scaled to the diffraction length and the input beam width, respectively.The parameter p is proportional to the depth of refractive index modulation, while the function In the linear limit ( 0 ) σ = Eq.( 1) can be reduced to the following equation for the spatial Fourier spectrum of the beam, where Further we consider propagation of symmetric collimated beam launched along ξ -axis in the central channel of the lattice at 0 η = .The possibility of broadband diffraction control of such beams in the lattice with a quadratic frequency modulation (FM) follows from analysis of dispersion relations.The dispersion relations can be obtained explicitly upon substitution of spectrum for the symmetric pairs of plane waves ( , ) [exp( ) exp( )]exp( ) having frequencies k ± and propagation constants b into Eq.( 3): The first term in the right side of Eq. ( 4) stands for the "positive" diffraction (convex phasefront) experienced by waves propagating in uniform media, the second one is a phase shift independent on k , and the last one describes the impact of the lattice due to the Bragg-type scattering.The sign of diffraction and its strength is dictated by the quantity 2 2 / d b dk [2,6].Notice that Bragg-type scattering is well studied in the context of nonlinear pulse reflection in chirped fiber gratings [16,17].Thus, pulse reflection in chirped gratings assumes synchronous energy transfer from spectral components of the incident wave-packet to spectral components of Bragg-reflected wave-packet, while a key feature of diffraction suppression that we address here is the broadband phasing of spectral components without energy exchange between them.It is the Bragg-type interaction between counter-propagating waves with equal amplitudes that results in appearance of phase terms k k pR pR k + in Eq. ( 4).The spatial spectrum of the lattice with a quadratic FM is given by the expression where Ai is the Airy function.This expression is depicted in Fig. 1(a).At "negative" one.In the spatial domain this is accompanied by a qualitative change of the phase front shape from convex to concave.Notice, that the possibility to control diffraction sign in the broad frequency band [ , ]   −Ω Ω with FM lattices is advantageous in comparison with the diffraction management in harmonic lattices, where zero diffraction can be achieved only for  C is readily apparent.These analytical results are confirmed by the direct numerical integration of Eq. ( 1).The propagation of such beam in the harmonic lattice is

1 α≤ < characterizes the frequency modulation rate. Parameter 1 σ
index, where Ω is the carrying spatial frequency and parameter 0 = ± defines the nonlinearity sign (focusing/defocusing).Further we assume that the refractive index modulation depth is small compared to the unperturbed index and is of the order of the nonlinear contribution.It should be pointed out that one can use either technique of Fouriertransform synthesis or computer generated holograms for optical induction of the desired lattice profiles as well as their direct technological fabrication.Notice that Eq. (1) conserves the total energy flow U and Hamiltonian H (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4245 #7339 -$15.00US Received 2 May 2005; revised 20 May 2005; accepted 21 May 2005

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, and it shifts to the high-frequency region with increase of FM rate α .Notice almost parabolic behavior of ( ) k R k dependence in a frequency interval k Ω > > −Ω .The dispersion curves ( ) b k are depicted in Fig. 1(b) for different values of the refractive index modulation depth p .The central result of this paper is that tuning the depth of the lattice affords principal modifications of the diffraction coefficient 2 d b dk , including reversal of its sign, for a broad frequency band [ , ] k ∈ −Ω Ω .The critical lattice depth that corresponds to the case of zero diffraction, i.e., the "positive" diffraction is replaced by the (C) 2005 OSA 30 May 2005 / Vol. 13, No. 11 / OPTICS EXPRESS 4246