Optical Beam Instability and Coherent Spatial Soliton Experiments George Stegeman, School of Optics/CREOL, University of Central Florida 1D Kerr Systems Homogeneous Waveguides Discrete Kerr Arrays Joachim Maier & Patrick Laycock Stewart Aitchison s Group (Un. Toronto) Yaron Silberberg s Group (Weizmann) Demetri Christodoulides Group (CREOL) 1D Quadratic Systems Robert Iwanow & Roland Schiek * Homogeneous QPM Waveguides Discrete Quadratic Arrays D Quadratic Systems Wolfgang Sohler s Group (Un. Paderborn) Falk Lederer s Group Ladislav Jankovic, Sergey Polyakov, Hongki Kim Homogeneous QPM KTP (PPKTP) Lluis Torner s Group (Un. Barcelona) Moti Katz (Soreq) D Semiconductor Amplifiers Erdem Ultanir & Stanley Chen Chris Lange s Group (Frederich Schiller Un.) Falk Lederer s Group (Frederich Schiller Un.) * Technical University of Munich
Interplay Between Self-Focusing and Diffraction: Spatial Solitons & Modulational Instability Diffraction Self-focusing (NLO) + Plane Waves (Very Wide Beams) Modulational Instability (Filaments) Narrow Beams Solitons
Spatial Solitons Spatial Solitons (1+1)D Spatial Solitons (+1)D (1+1)D - in a slab waveguide - diffraction in one D (+1)D - in a bulk material - diffraction in D Soliton Properties: 1. Robust balance between diffraction and a nonlinear beam narrowing process. Stationary solution to a nonlinear wave equation 3. Stable against perturbations Experimentally: 1. Must be stationary over multiple diffraction lengths. Must be stable against perturbations 3. Must evolve into a stationary soliton for non-solitonic excitation conditions
Material Nonlinear Mechanisms Discussed here Kerr P NL = ε c (3) E E n = n I Quadratic P NL = ε c () {E(ω) E(ω) + E * (ω)e(ω)} ω + ω = ω ω = ω - ω Not Discussed here Saturating Kerr P NL = ε {c (3) E E + c (5) E 4 E + } n = n sat I/[I + I sat ] Photorefractive n = - ½ n 3 r eff E DC - ½ n 3 s eff E DC (Semiconductor) Gain Medium P NL f(n, α, π, E) N carrier density (complex dynamics) α - loss π - electron pumping rate (determines gain) Reorientational (liquid crystals) n = (n - n )(sin θ[ E ] sin θ [])
Spatial Soliton Systems Soliton Type Material # Soliton Param. Soliton Size Power Dissipative (SOAs) AlGaAs 15 µm 1 s mws 1D Quadratic QPM LiNbO 3 x 5 µm 1 W D Quadratic PPKTP x µm 1 KW 1D Kerr AlGaAs (E g /) 1 x 4 µm 1 s W ALL spatial soliton generating equations are CW in time BUT, most spatial soliton experiments use PULSED lasers!! Temporal Pulse complicates situation Spatial Output I(t) Soliton-forming beam component t Diffracting background
Generic Experimental Layouts Laser Wavelength Peak power Pulse width Bandwidth Elliptical (1:1) or Circular Beam dimensions M (gaussian quality factor) Peak power & Pulse width Polarization at sample Flat phase at sample interface Frequency spectrum Beam shape & dimensions Beam energy distribution Beam frequency spectrum Beam pulse width Beam transmission (losses) Beam Shaping Beam Characterization Sample Output Beam Characterization NLO mechanisms # Diffraction lengths Optical quality
Nonlinear Wave Equation: Kerr Nonlinearity E exp[ i{ ωt kz}] Slowly varying phase and amplitude approximation (1st order perturbation theory) n E + ω E = ω µ P c ω ik E + E = 3 χ z c diffraction 3ε χ (3) E E (3) NL E nonlinearity E Plane Wave Stationary Plane Wave Solution E = Stationary E = z n = n,e E E = 1 ik n E E z i( ωt kz) Ee +, e c. c.
Simplest Case: Plane Waves in 1D Slab Waveguides Slab Waveguide y z x? (1+1)D - in a slab waveguide - diffraction in x-dimension χ (3) n = n,e E [ ] γz ikn E z i( ωt kz) E = E 1+ δ ( κ ) cos( κx) e perturbation plane wave solution to nonlinear wave equation e e Period (Λ) = π/κ δ << 1 perturbation amplitude γ = exponential gain coefficient
Modulational Instability in c (3) Slab Waveguides Insert trial solution into NLWE: Assume E satisfies linear WE Assume δ <<1 ω ik E + E = 3 χ z x c (3) E E 6 ------ - - - 75 75 KW kw 5 kw 5 KW at at γ κ = kn E k k κ peak : For g real E threshold : = E k = κ n n 4k, E κ n n, E MI Gain, cm -1 γ (cm -1 ) 5 4 3 4 6 8 1 1 14 6 1 14 Period Period, (µm)
Normalized Intensity Norm. Intensity, % (1+1)D Kerr MI Instability: AlGaAs 1 Below Half Bandgap λ = 1.55 µm 1 1 1 1 x [11] -1 y [1] z KW kw 3 KW kw 9 kw KW 1 8 kw Position (µm) Norm. Intensity, % cm 1 Al.4 Ga.76 As Al.18 Ga.8 As Al.4 Ga.76 As 1 GaAs substrate n x1-13 cm /W 1 1-1 -3 mm Position, (µm) µm kw 3 kw 9 kw 8 KW kw 55 KW 55 kw 1-1 -5 5 1 y n
Fourier Analysis of Intensity Pattern F.T. of Field 1 Intensity, % 1 nd harmonic Input 5 Position, µm 3 rd harmonic? Harmonics growth saturation Fourier spectrum of small scale noise on profile..1..3.4.5.6 Frequency, 1/µm
Noise Generated c (3) MI: Period Versus Power 4 35 3 Experiment --- Theory Period (µm) Period, µm 5 15 1 5 1 3 4 5 6 Peak Power Power, (KW) kw
Nonlinear Wave Equation: Kerr Nonlinearity E exp[ i{ ωt kz}] Slowly varying phase and amplitude approximation (1st order perturbation theory) n E + ω E = ω µ P c ω ik E + E = 3 χ z x c diffraction 3ε χ (3) E E (3) NL E E nonlinearity Stationary NLS Solution z E = Nonlinear Eigenmode Spatial soliton
(1+1)D Scalar Kerr Solitons y x Output Input Low Power High Power Dn = n,e E E( r) = n n, E n 1 k vac w y sec h{ w }exp[ i n k z vac w ] 1 parameter family Power x Width = Constant
199 Kerr Solitons in AlGaAs Waveguides
Connection Between MI and Spatial Solitons Λ Λ =, n k E n k n k E π π γ Peak γ, k n n E E Λ = π MI E w Same intensity 1 8 ] [ = Λ π w Spatial Soliton Peak field, 1 w n k n E E =
AlGaAs Waveguide Arrays 4.µm 8.µm n = 1.5x1-13 cm /W @ 155 nm 1.5µm Al.4 Ga.76 As Al.18 Ga.8 As Al.4 Ga.76 As 1.5µm 4.8mm @.5 coupling length 41 guides Bandgap core semiconductor: l gap = 736nm
Diffraction in Waveguide Arrays a n E n (x) Light is guided by individual channels Coupled mode equation: dan i + β an + c dz ( a + a ) n+ 1 n 1 = Neighboring channels coupled by evanescent tails of fields Light spreads (diffracts) through array by this coupling c a n is field at n-th channel center ß is propagation constant of single channel E n (x) is the channel waveguide field.
Diffraction Via Nearest Neighbor Coupling Channel intensity distribution depends on: 1. Field amplitudes in neighboring channels. Relative phase between channels 3. Phase change during coupling process (usually π/)
Discrete Solitons in Kerr Waveguide Arrays Single channel input Moderately localized solitons High Power Eisenberg et al., Phys. Rev. Lett., 83, 716 (1998) Single channel output Strongly localized solitons
Beam Collapse in Waveguide Arrays Slice of output power distribution Input Power (a.u.) 1. Input Power [W] 1-6 1. 1-7 -4-4 -6-4 - 4 6 Position [um] Position (µm)
Soliton Type Material Soliton Param. Soliton Size Power Photorefractive 1 mws Dissipative (SOAs) AlGaAs 15 µm 1 s mws 1D Quadratic QPM LiNbO 3 x 5 mm 1 W D Quadratic PPKTP x µm 1 KW 1D Kerr AlGaAs (E g /) 1 x 4 µm 1 s W
c () : Type I Second Harmonic Generation P NL = ε c () {E(ω) E(ω) + E * (ω)e(ω)} Γ χ () ω - ω = ω Up-conversion Down-conversion ω + ω = ω ω + ω = ω ½ a 1 exp[i(ωt k 1 z)]+cc ½ a exp[i(ωt k z)] + cc Wavevector (momentum) conservation: k 1 = k Wavevector mismatch: k = k 1 -k Phase-mismatch: kl = (k 1 -k )L Diffraction Nonlinear Coupling ) ( ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( * 1 1 1 1 1 = Γ = Γ kz i kz i e z a z a z a y k z a z i e z a z a y k z a z i
Characteristic Processes and Lengths in Second Harmonic Generation On Phase-Match 1. Energy exchange between fundamental and harmonic. p/ phase difference k = Off Phase-Match 1. Perioidic energy exchange. Rotating phase difference k > k 1 I(ω) I(ω) z L pg Parametric Gain Length = ωd () eff cn E( ω, z = ) Coherence Length L c = π / k z
c () -Induced Beam Dynamics: 1D Beam Narrowing Due to Wave Mixing a 1 a i Γa 1 exp[ i kz] = z k y a1 1 a1 * i Γa1 a exp[ i kz] = z k y 1 a 1 (1) e.g. k= exp[±i kz] = 1 () ignore diffraction / y= (3) writing a/ dz as a/ z a a a 1 z a is narrower than a 1 along y-axis e.g. a 1 exp[-y /w ] a exp[-y /w ] z y a 1 a a 1 * z a 1 is narrowed along y-axis e.g. a exp[-y /w ] a 1 exp[-3y /w ] z
Recipe For Plane Wave Instability & Solitons 1. Find plane wave stationary solutions, i.e. solve nonlinear wave equations in absence of diffraction.. Add to plane wave solution: noise with spatial Fourier component κ, amplitude δ<<1 and gain coefficient γ 3. Solve for intensity regimes with γ real and >. If they exist, plane wave solutions are unstable over those parameter ranges. 4. If plane wave solutions are unstable at high intensity, nonlinear eigenmodes are solitons
1D Plane Wave Eigenmodes and Modulational Instability Nonlinear Wave Equations (No Diffraction) da 1 /dz = iγb a 1* exp[-i kz] db /dz = iγa 1 exp[i kz] k = k(ω) - k(ω) - there are unstable stationary eigenmodes, each consists of a fundamental and harmonic wave - fundamental and harmonic fields are either co-directional or counter-directional a 1 b - fundamental only at input and +ve phase-mismatch co-directional dominates - consider a perturbation with periodicity π/k, gain coefficient g and amplitudes F 1 and F a 1 = [ρ 1 + F 1 cos(kx)exp(gz)]exp[ik 1 z] b = [ρ + F cos(kx)exp(gz)]exp[ik z] For g real, periodic pattern grown exponentially with distance z!
MI: Gain Versus Period 1W kl = 9π 56W 35W 1W kl = 1π 56W 35W
MI Evolution: 1D SH Eigenmodes High Intensity SH Eigenmode 15 1 Fundamental -4-4 Transverse Position (µm) 5 15 1 Propagation Distance (mm) Harmonic 5-4 - 4 Transverse Position (µm)
TM (ω) 155 nm 1D Modulational Instability 55-75 µm LiNbO 3 kl = 1π 5 cms TE (ω) kl = 9π 1D case energy trapped between peaks in waveguide
Power Dependence of Breakup Period: c () 5 Experiment Period [mm] 15 Theory kl=9π 1 5 4 6 8 1 Peak Power [kw]
Quadratic Solitons Beam narrowing mechanisms robustly balance diffraction For the quadratic nonlinearity: quadratic solitons Consist of both a fundamental and harmonic component, constant with distance, ratio depends on soliton intensity and phase-mismatch! The fundamental and harmonic are in phase! Recall: for SHG with k=, a 1 and a are π/ out of phase and a grows with distance! a a 1 y Solitons are excited by focusing 1 s micron diameter beams at entrance facet Fundamental w Soliton L L >> L L dif pg
Beam Dynamics in 1D QPM LiNbO 3 Slab Waveguides uniform periodicity (L) k = k ω - k ω + π/λ d 33 1.55 µm d 33 L >> L dif L pg Fundamental Wave (FW) Only Input Rapid energy change and relative phase rotation Intensity FW HW L pg 1 3 4 5 Distance (mm)
Narrow Beam Inputs NORM. INTENSITY, % 1 75 5 5 w 7µm Input beam 5 5 75 1 POSITION, µm Soliton.75 kw.4 kw 5 5 75 1 5 5 75 1 POSITION, µm POSITION, µm 4 kw 5 5 75 1 POSITION, µm 5. kw 5 5 75 1 POSITION, µm 7. kw 5 5 75 1 POSITION, µm 75 5 5 Norm. Intensity, %1 Onset of MI 5 5 75 1 Position µm 1 kw
Multi-Soliton Generation M.I. period at peak of gain at input intensity Noise on input beam Input intensity and width for single soliton generation M.I. period at peak of gain at input intensity Width for single soliton generation Intensity >> single soliton intensity
1 3 4 5 Distance (mm) QPM Engineered Waveguides 17.68 Period [µm] 17.64 17.6 17.56 17.5 17.48 1 3 4 5 Position [mm] 1 3 4 5 -. µm steps for Λ in fabrication -intermediate values of Λ by averaging over multiple discrete periods -e.g. 17.65 µm 4 periods 17.6 1 period 17.6 averaged period actual periods (too dense to separate out) Rapid energy change and relative phase rotation Fundamental Wave (FW)Only Input Peak Intensity FW SH Fields in Phase Fixed Amplitudes Soliton
FWHM [µm] 15 1 Input Diffracted FD 1.5C,.7π SH 1.5C FD 11C, π FD 11.5C, -.7π INPUT Diffracted 5 5 1 15 Peak Input Power [W] INTENSITY [%] 8 6 4 ϑ=19.5 o C kl P peak =19 W FW FD (65.6%) (66.6%) SH (34.4%) -3 - -1 1 3 POSITION [µm]
PPLN Waveguide Arrays z x y z crystal axes x ca c b width β a β b ca c b separation refractive index diffusion profile ω TM ω TM a n b n d Ψ b z-cut LiNbO 3 substrates 16.8 µm QPM structure λ PM = 1557nm @13 C 7 µm Ti stripes in-diffused 11 guides, 51mm long loss FH =. db/cm
Coupled FH Decoupled SH Fundamental Harmonic λ = 1557nm transverse energy transport discrete diffraction Second Harmonic λ = 778.5nm χ () nonlinear coupling no transverse energy transport
Qualitative Results FH = 185W Normalized power [au] 1..5 Separation 13.5µm Coupling length 13mm kl = 5π Temp = 15 C 1W 4W 185W. - -1 1 Position [µm]
Summary 1. Spatial solitons are a very rich and diverse field. Spatial solitons have been studied in homogeneous and discrete systems, in waveguides and bulk media 3. Many nonlinear mechanisms can be used for solitons 4., 1 and parameter families of solitons demonstrated 5. Most solitons require temporal pulses to reach soliton threshold 6. Many spurious factors complicate understanding results 7. Dissipative solitons exist at 1 s mws power with nanosecond response many applications (not discussed) 8. Interactions are a very rich field with novel applications
Dispersion Relation For Waveguide Arrays i da dz n + β a n + c ( a + a ) n+ 1 n 1 = plane waves : a n = E exp[ i{ ωt kz z nkxd}] k z = β + c[exp{ ik x d} + exp{ ik x d}] k z = β + c cos[ k d] x k z b -π π k x d θ = π
Diffraction in Arrays D = d kz x dk diffraction D = cd cos( k x d) Zero diffraction k z Output Beam Width Input Beam Width b Negative diffraction k x d/π -π k x d π