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Gaetano Assanto, Katia Gallo, Alessia Pasquazi and Salvatore Stivala

Spatial Solitons in 2D Lattices of a Nonlinear Nature

These authors describe their observation of two-color optical solitons in a 2D parametric photonic lattice with hexagonal symmetry in lithium niobate. Their results have enabled wavelength-controlled angular steering and the displacement of self-trapped filaments of light.

patial solitons are one of the most fascinating phenomena that result from the interplay between diffraction and nonlinearity: They offer unique possibilities for all-optical reconfigurable circuits

and interconnects. Scientists have studied spatial solitons theoretically and have also demonstrated them experimentally in several physical systems with intensity-dependent refrac-tive indices, ranging from glasses to semiconductors, and from photorefractive crystals to liquid crystals.

Such solitons can also exist in quadratic crystals—in other words, media where photons can interact via three-wave mixing to generate new wavelengths. This enables parametric amplification that can counteract beam spreading and causes the spatial locking of multi-frequency components (i.e., multi-color solitons or simultons).

The self-guiding of light in quadratic crystals has been achieved in bulk waveguides and with one-dimensional gratings in the nonlinearity as well; researchers realized such quasi-phase-matched structures in periodically poled lithium niobate. Later, researchers obtained 2D parametric photonic lattices by periodi-cally inverting the sign of the second-order susceptibility χ(2) in the plane of propagation. The advent of these quadratic two-dimensional periodically-poled structures has introduced novel versatile geometries for light-matter interactions.

S Parametric photonic lattices (PPLs) behave as uniform and homogeneous dielectrics under weak excitation (linear regime), whereas, at high powers, they allow for the simulta-neous quasi-phase-matching of several χ(2) processes. Unlike previous configurations with prescribed quasi-phase-matching periodicities in only one direction, beam self-trapping and light localization can be explored and exploited in the entire plane of the PPL, permitting spatial soliton steering by acting on wavelength, power and the direction of propagation of the input beam.

These possibilities introduce entirely new approaches to the steering of these self-localized beams, which was previously per-formed by means of refraction or internal reflection at a dielec-tric interface between media with differing (linear - nonlinear) indexes and/or crystal orientations, boundary forces in finite size samples, or walk-off control in molecular (liquid) crystals.

A previous case of power-controlled steering of two-color spatial solitons in a quadratically nonlinear medium was reported in potassium titanyl phosphate (KTP) by means of cascading via type II second harmonic generation. The direc-tion of energy propagation (Poynting vector) resulted from the interplay of both ordinarily (“o”) and extraordinarily (“e”) polarized frequency components of the parametric soliton generated via three-wave mixing (ω + ω = 2ω).

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HexLN planar waveguidesWe performed experiments in hexagonally symmetric pho-tonic lattices in periodically poled lithium niobate (HexLN) planar waveguides designed for twin-beam second harmonic generation (SHG) around the wavelength λω=1.55 µm. The waveguides enhance the transverse confinement and the inten-sity, resulting in more efficient parametric mixing with lower excitations required for soliton generation as compared to bulk. The PPL and the interaction are shown in part (a) of the figure below right: A pump beam at the fundamental frequency (FF) ω propagates with wavevector βω in the waveguide plane (XY) at a small angle θω with respect to the HexLN symmetry axis (X). Part (b) of the figure is a micrograph of the sample.

At high intensities, FF photon pairs can efficiently generate second harmonic (SH) light when momentum conservation is ensured by the lattice periodicity—i.e., 2βω−β2ω+ G ≈0 for propagation vectors βω and β2ω at the two harmonics and G due to spatial modulation of the susceptibility. In the planar HexLN, the FF beam excites two non-collinear SHG proc-esses, quasi-phase-matched at either λω = λ(+) or λω = λ(−),

(a) Refraction or (b) total internal reflection at a dielectric interface separating media with different (linear and/or nonlinear) susceptibilities. (c) Interaction and bending of a spatial soliton with a boundary potential (due, for example, to temperature distribution or molecular anchoring. (d) Soliton steering in a birefringent medium by changes in walk-off (by re-orienting the optic axis n for a given wave vector k). (e) Diffracting beams (dashed lines) and quadratic soliton (red solid lines and red-blue transverse profile) obtained by Type II second-harmonic generation for eeo three-wave mixing (blue indicates the harmonic, red the fundamental frequency). The final angle of propagation of the two-color simulton depends on the balance between the walk-off angles of the three waves. The green arrows emphasize the invariant profile as compared to the spreading components at low power.

(a) (b)

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[ Various approaches to spatial soliton steering ]

(a) Two second harmonic generation (SHG) processes are excited in the hexagonally poled lithium niobate (HexLN) photonic lattice waveguide by a fundamental frequency (FF) pump beam propagating at a small angle θω with respect to the symmetry axis X. Quasi-phase-matching is achieved via two distinct wave-vectors (G(+) = G01 and G(−) = G10, at π/6 with respect to X and with amplitudes |G(±)| = 4p√3/3L) of the HexLN reciprocal lattice. (b) Optical micrograph of a HexLN with period L=16.4 µm. (c) Pump-wavelengths for resonant SHG versus θω: data (diamonds) and fits (dotted lines).

Y

Y

Z

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m]

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βv

[ ]

respectively, by two lowest-order reciprocal-lattice vectors G(+) and G(−) of the lattice. The wavelength offset ∆λ = λ(+)−λ(−) between the resonances can be adjusted by tuning the FF angle of propagation in XY.

The HexLN, mounted on a piezo-electrically controlled stage and kept at a stable temperature of about 85° C to prevent pho-torefractive damage, is pumped at wavelengths in the 1.1−1.6 µm range by narrow line-width (< 2 cm−1) 20 ps pulses. The FF input is shaped into a cylindrical spot (lateral and vertical beam waists w0=27.5 µm and v0=3.4 µm, respectively) and excites the TM0 mode of the 18 mm-long PPL waveguide, where it can propagate for roughly 5.4 diffraction lengths (LD). Using coupled-mode theory, one can write a straightforward set of equations for the envelopes A and B(±) of FF and SH(±) waves, respectively:

(1)

with normalized coordinates ζ = x/LD and ξ = y/w0, with x being the launch direction for the FF beam in the PPL waveguide and y x. The FF input parameters, namely the wavelength (λω), peak-power (Pω) and angle of propagation (θω), determine the coupling coefficients Γ(±) and the phase-mismatches ∆κ(±) = (2βω − β2ω

(±) + G(±))LD of the two SHG processes governing the

Hexagonal 2D photonic lattice in a periodically poled lithium niobate planar waveguide

G(+) G(–)

∆λ

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dynamics. Equations (1) account for the in-plane angular deviation ρ2ω

(±) ∼ ±1.5 of the SH, as well as diffraction σω = 1/4 at FF and σ2ω

(±) ∼ 1/8 at SH, respectively.

Nonlinear wave dynamicsThe figures to the the right illustrate the pertinent nonlinear wave dynamics by comparing simulated beam propagation at frequency ω in the plane XY (b-c in the figures), with the FF transverse profiles acquired at the output (d-e), in the case of symmetric (θω=0°) and asymmetric (θω=0.54°) twin-beam SHG, respectively.

At low powers, the FF input beam launched along x spreads but remains centered in y=0; as the excitation increases, self-focusing prevails on diffraction and leads to self-confinement. In the symmet-ric case, a spatial simulton forms close to phase-matching at ∆κ(+) = ∆κ(−) ≥ 0, i.e., for input at λω ~ λ0 + δλ with λ0 the SHG resonant wavelength at θω=0° and δλ ≥ 0. Both non-collinear SHG processes tend to spatially lock the harmonics with angular deviation (ρ2ω

(±)) toward the SH wave vec-tor β2ω; but they balance and trap the two-color soliton on the input axis x (y=0).

The figure on the right shows measured and calculated output FF waists versus input wavelength owing to twin-beam SHG in the symmetric and asymmetric cases. In the symmetric configuration (θω=0°), the simulton forms close to the resonance, consistently with the predicted curve (yellow dotted line) from eqs. (1).

For a comparison, the black dotted line in the figure shows that the behavior in the case of a conventional singly resonant SHG exhibits a narrower bandwidth. In the asymmetric case (θω=0.58°), as the FF wavelength is scanned at the input, the spectral response shows two dips in output waist as well as opposite lateral displace-ments owing to the dominant pulling action from one of the two concurrent SHG processes.

Output waists and PPL wavelength responseWhen symmetry is broken by launching the FF at an angle θω with respect to the axis X, twin-beam SHG is quasi-phase-

(a) Quasi-phase-matching diagram with θω ≡ 0°, i.e. λ0 = λ(+) ≡ λ(−). (b) Simulated propagation of a continuous wave fundamental frequency (FF) beam with input at λω = λ0 +1.5 nm = 1,551.5 nm for low excitation. (c) Same as b) for high excitation. (d) Experimentally acquired fundamental frequency (FF) output profile for a launched peak power Pω = 1 kW; (e) same as in (d) for a launched peak power Pω = 60 kW. Similar transverse profiles could be observed at the second harmonic wavelength. Here X and Y are the lattice axes, and x (^ y ) is the direction of propagation of the input beam at the fundamental frequency.

(a) Quasi-phase-matching diagram with θω ≡ 0.54° , i.e., ∆λ = λ(+) −λ(−) ≡ 8.6 nm. (b) Simulated propagation of a continuous wave fundamental frequency (FF) beam in the asymmetric case for high excitation and λω = λ(−) + 1.5 nm = 1,548.4 nm. (c) As in (b) but λω = λ(+) + 0.5 nm = 1,556.0 nm; (d) Acquired FF output profiles at Pω = 70 kW, corresponding to (b); (e) as in (d) but corresponding to (c). Similar transverse profiles could be observed for SH light. Here X and Y are the lattice axes, and x (^ y ) is the direction of propagation of the input beam at the fundamental frequency.

y [µm]

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[ Asymmetric twin-beam SHG configuration ]

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(a) Symmetric twin-beam second-harmonic generation (SHG) with θω ~ 0° versus fundamental frequency (FF) wavelength, measured FF output beam waist (circles) for Pω = 25 kW and fit (yellow dots, Γ2 = 16.3) along y. The dotted line refers to singly resonant SHG (same parameters) for comparison. The bandwidth in twin-beam SHG is twice as large (14 nm) as it is in conventional SHG (7 nm). (b) Asymmetric twin-beam SHG for θω=0.58°, output waist (top) and lateral displacement (bottom) versus FF wavelength. Data for Pω = 22 kW (circles) and simulations for Γ2 = 16.

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[ References and Resources ]

>> W. E. Torruellas et al. Appl. Phys. Lett. 68, 1449 (1996).>> V. Berger. Phys. Rev. Lett. 81, 4136 (1998).>> L. Friedrich et al. Opt. Lett. 23, 1438 (1998).>> B. Bourliaguet et al. Opt. Lett. 24, 1410 (1999).>> G.I. Stegeman and M. Segev. Science 286, 1518 (1999).>> N.G.R. Broderick et al. Phys. Rev. Lett. 84, 4345 (2000). >> G. Assanto and G.I. Stegeman. Opt. Express 10, 388 (2002).>> A.V. Buryak et al. Phys. Rep. 370, 63 (2002).>> L. Jankovic et al. Opt. Lett. 28, 2103 (2003).>> Y.S. Kivshar and G. P. Agrawal. Optical Solitons: From Fibres to

Photonic Crystals. Academic, N.Y., 2003.>> F. Baronio et al. Opt. Lett. 29, 986 (2004).>> C. Conti and G. Assanto. “Nonlinear Optics Applications: Bright

Spatial Solitons,” in Encyclopedia of Modern Optics, , R.D. Guenther, D.G. Steel and L. Bayvel, eds., Elsevier, Oxford, 2004, 5, 43-55.

>> G. Leo et al. Opt. Lett. 29, 1778 (2004).>> R. Schiek et al. Opt. Lett. 29, 596 (2004).>> M. Peccianti et al. Nature 432, 733 (2004).>> K. Gallo et al. Opt. Lett. 31, 1232 (2006). >> M. Peccianti et al. Nature Phys. 2, 737 (2006).>> C. Rothschild et al. Nature Phys. 2, 769 (2006).>> A. Alberucci et al. Opt. Lett. 32, 2795 (2007).>> K. Gallo and G. Assanto. Opt. Lett. 32, 3149 (2007).>> K. Gallo et al. Phys. Rev. Lett. 100, 053901 (2008).

matched at two distinct λ(+) and λ(−), yielding self-confinement at λω~λ(−) + δλ(−) or λω~λ(+) + δλ(+), with δλ(+)>0. Conse-quently, the spatial soliton acquires negative or positive angular and lateral shifts, depending on the pulling from either SHG processes. The cascading signature of quadratic solitons is the oscillatory character of the fundamental frequency beam as FF and SH exchange energy in the early stages of generation.

The figure on the left above illustrates the PPL wavelength response for twin-beam SHG excited in the asymmetric config-uration, with θω = 0.58°, λ(−) = 1546.5 nm, λ(+) = 1555.7 nm. At low power (500 W), the input beam undergoes diffraction as the generated SH component is too low to induce cascad-ing. At a high peak excitation of 22 kW, conversely, spreading is nonlinearly balanced and a simulton forms, with a lateral shift depending on the input FF wavelength. The top diagram associates the FF wavelengths to the wave-vector mismatches of both SHG processes.

Finally, the figure on the right shows the power-dependence of both output beam waist and lateral shift in a representative case of asymmetric configuration with θω = 0.54° and λω = λ(+)

+ 0.5 nm.

ConclusionWe envision that spatial solitons in 2D lattices of a nonlinear nature could be used for wavelength-selective routers and inter-connects, whereby self-localized two- (or three-) color beams can connect the input port to several distinct outputs real-izing wavelength vs. angle demultiplexing schemes. Quadratic spatial solitons in planar nonlinear lattices unveil new scenarios for physics and many enable applications of self-trapped light beams, including the engineering of soliton states for analog sig-nal processing and all-optical switching/routing in aperiodic lat-tices and quasi-crystals, as well as spatio-temporal light bullets.

In conclusion, two-dimensional photonic lattices of a parametric nature support novel simulton space dynamics via

SHG for θω = 0.54°, i.e., ∆λ = λ(+) − λ(−) ≡ 8.6 nm. The fundamental frequency (FF) output beam waist and lateral displacement versus input FF peak power are shown for λω = λ(+) + 0.5nm = 1,556.0 nm.

Asymmetric twin-beam SHG at θω = 0.58°, λ(−) = 1,546.5 nm, λ(+) = 1,555.7 nm. Transverse beam profiles (FF) versus input wavelength for low (500 W) and high (22 kW) excitations. The bottom panels are images of the FF beams at the output.

1,547.2 nm 1,547.8 nmSHG(–) SHG(+)

[ Asymmetric twin-beam second harmonic generation ]

200

Wai

st [

µm]

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ft [

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twin-beam second-harmonic generation. In the symmetric configuration, solitons can form in an enhanced spectral range for confinement; in the asymmetric cases, they can be steered by varying the input (fundamental frequency) power and wave-length. These distinctive features, spectrally sharp owing to the resonant phase-matched interactions, are the signatures of self-confinement in the presence of multiple parametric processes. t

[ Gaetano Assanto (assanto@uniroma3.it), Alessia Pasquazi and Sal-vatore Stivala are with The Nonlinear Optics and Optoelectronics Lab (NooEL) at the University Roma Tre in Rome, Italy. Katia Gallo is with the Royal Institute of Technology in Stockholm, Sweden. ]

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0.5 kW 22 kW

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λωλ(+)=1555.7 nmλ(−)=1546.5 nm

y (µm) y (µm) y (µm) y (µm) y (µm)

∆κ(−)>0, ∆κ(+)>0∆κ(+)=0

1,555.8 nm 1,556.0 nm