Nonlinearity-induced broadening of resonances in dynamically modulated couplers
A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, A. Tunnermann, V. A. Vysloukh, F. Lederer, L. Torner
NNonlinearity-induced broadening of resonances in dynamically modulated couplers
A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, A. Tün-nermann, V. A. Vysloukh, F. Lederer, and L. Torner Institute of Applied Physics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Medi-terranean Technology Park, 08860 Castelldefels (Barcelona), Spain Institute for Condensed Matter Theory and Optics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
We report the observation of nonlinearity-induced broadening of resonances in dynamically modulated directional couplers. When the refractive index of the guiding channels in the coupler is harmonically modulated along the propagation direction and out-of-phase in two channels, coupling can be completely inhibited at resonant modulation frequencies. We ob-serve that nonlinearity broadens such resonances and that localization can be achieved even in detuned systems at power levels well below those required in unmodulated couplers.
OCIS codes: 190.0190, 190.6135
The precise control of wave packet evolution in systems with inhomogeneous refractive index landscapes is of major importance in optics [1,2]. When refractive index varies in both transverse and longitudinal directions a number of tools for controlling the propagation dy-namics become available. Such structures can be used to manage diffraction properties in waveguide arrays [3,4], where diffraction-managed solitons can form [5,6]; it can be also used to drag solitons in the transverse plane [7,8], or to initiate soliton center oscillations and shape conversions [9,10]. Dynamic localization (DL) in waveguide arrays and coherent de-struction of tunneling (CDT) in directional couplers, are possible either when guiding chan-nels bend periodically in the propagation direction [11-16] or when the guiding channels re-main straight, but their refractive index or widths oscillate periodically [17,18]. It turns out q , which governs the propagation of light beams along the ξ -axis of two-channel coupler under the assumption of cw radiation: q qi q q p η ξξ η ∂ ∂= − − −∂ ∂ R q by w de (1) Here and are the transverse and longitudinal coordinates normalized to the character-istic transverse scale and diffraction length, respectively, while the parameter p describes refractive index modulation depth. The refractive index distribution in the coupler is given ] w , where a super-Gaussian refractive index profile of each waveguide is fitted to the shape of the real laser-written waveguides in our sample [20], μ describes longitudinal modulation pth, and Ω is the longitudinal modulation frequency. The parameter w η characterizes the waveguide width, while s w stands for the separation between the coupler channels. We set 0.3 w η = that is equivalent to μ width of the waveguides, and s w = that is equiva-lent to the separation of channels of
32 m μ . W et p = that corresponds to a real modulation depth of refract − × . In all simulations we used the input η ( , ξ [1 ) sin( )]exp[ ( / 2) / ] [1 sin( )]exp[ ( 2) / R w w η η η ξ μ ξ η μ ξ η = + Ω − + + − Ω − − e also sive index ∼ s s / eams ( ) w η , re ( ) w η describes the shape of the fundamental linear mode of a single waveguide, and A is the input ampli q A ξ = = whe tude. b In unmodulated linear couplers ( light periodically switches between both chan-nels if only one of them is excited at . We define the linear beating period as the distance at which light returns to the input waveguide after one complete switching cycle (accordingly, the beating frequency is given by ). For our set of parameters one has . A longitudinal out-of-phase modulation of the refractive index in the coupler channels results in the inhibition of coupling that takes place only for properly selected val-ues of the modulation frequency Ω [15-17]. To characterize localization, we use a distance-averaged power fraction trapped in the excited channel 0) μ = ξ = b T b T π Ω = b T = ) ) d d ξ η −∞ −∞ ∫ ∫ ∫ arge stance). d η L r Ω b / 2 is close to a L U L − = m ( ) U Ω μ ∼ b / Ω Ω L b / Ω at ( , q ξ η → di ( , 0 q r Ω η = , where is the final distance. The depend-ence is characterized by multiple parametric resonances. Here we only consider the principal resonance with the largest frequency (Fig. 1). The resonance frequency grows with increasing longitudinal modulation depth μ [Fig. 1(a)], and for moderate μ values . In the limit one has Ω → . The half-width of the principal resonance defined at the level is a monotonically decreasing function of the averag-ing distance [Fig. 1(b)]. To understand the impact of nonlinearity on resonance curve it is desirable to perform averaging over sufficiently long distance L . However, the value δ Ω L → ∞ is determined mostly by the amplitude A of the input beam when nonlinearity is taken into account since in this limit a vanishing resonance width would only occur for a vanishing amplitude of the propagating beam. In contrast, in our calculations the resonance width is always finite due to the nonzero power used in the initial conditions. In Fig. 1(c) it is shown how a variation of μ results in a shift of the principal resonance. Around the resonance the dependence m U r sinc[ ( )] L α Ω − Ω function super-imposed on a constant pedestal (this is a direct consequence of averaging of an oscillating function over a l r Ω δ μ U Ω ( ) m Figure 2 illustrates how the resonance is affected by the nonlinearity. The important result is that the resonance broadens with increase of input amplitude A , which is shown in Fig. 2(a). The resonance width increases almost linearly with input power correspond-ing to the findings in Ref. [20]. As the width of resonance approaches a minimal value (corresponding to the resonance width in the linear medium) that decreases with in-creasing averaging distance L [see, e.g., Fig. 1(b)]. The resonance broadening is illustrated A ∼ A → n Fig. 2(c) where the curves 1 and 2 correspond to small and moderate values, respec-tively. The broadening of resonance can be considerable even at moderate amplitudes, which are well below the critical amplitude required for coupling suppression in un-modulated system. A ) ex = cr A ∼
115 mm × ∼ − / b δ Ω Ω -1 [mm ] To confirm these trends experimentally we fabricated a sequence of directional cou-plers using the femtosecond laser writing technique [4] with laser pulses at , a temporal width of 140 fs and a repetition rate of 100 kHz. The sample material of choice was fused silica of highest quality, the focusing was achieved by 20x objective (NA . For the experimental analysis of the predicted resonance broadening, we launched light at in one waveguide using a 2.5x objective (NA . The end facet of the sample was imaged on a CCD camera, using a 10x objective (NA . The induced in-dex change depends on the writing speed v approximately as δ α , where the constants are fitted numerically to the experimental results [20]. A sinu-soidal index change along the individual guides is then achieved by varying writing speed as . The sample length was and waveguide spacing was . The beating period is ∼ that corresponds to T . The guides exhibit an average refractive index of and a modulation amplitude of . For these parameters, the modulation frequency corresponding to principal resonance amounts to Ω ≈ (i.e.,
Ω ≈ ). To analyze localization in the vicinity of , we fabricated couplers with modulation frequencies
Ω = , 0.176, 0.173, 0.170, 0.167, 0.165, and .
800 nm λ = = p( / ) n v β ≈ −
100 ) +
800 nm λ = δ ln( /[ v p β α
32 m μ − ≈ × r Ω = =
105 mm b Ω b n = + γ − , , α β γ in( ) r s ]) z μ γ Ω m − r -1 m The width of the resonance curve was defined using those powers, where 70% of input power still remains in the excited channel. In the resonant coupler a minimal power of 50 kW was injected to achieve localization. The powers for the detuned couplers defined with the above mentioned criterion are: Ω [kW] P .167 340 0.165 370 0.162 400 as shown in Fig. 2(b). The circles represent measured points, including the error of about 15 kW. For very small detuning the dependency is nonlinear, as confirmed by the continuous model [see Fig. 2(a)]. However, dependence becomes almost linear as detuning in-creases, which is consistent with results of Ref. [20] obtained for discrete model. Representa-tive output patterns for couplers with different modulation frequencies are shown in Fig. 4. Note that in all cases the left waveguide was excited. In the left column [Fig. 4(a)] the cou-pler at the resonance frequency is shown. For all applied input powers, the light remains lo-calized. In contrast, in detuned coupler with [Fig. 4(b)] the light at 50 kW couples strongly into second guide (first row). At 250 kW input power, about 70 % of the injected light remains in the excited channel (second row), defining the width of the reso-nance curve. For higher peak powers (e.g., 340 kW in third row), light almost completely localizes in the excited waveguide. When frequency is detuned from resonance even farther, i.e. when , at 50 kW a large fraction of the light couples into the second waveguide [see first row of Fig. 4(c)]. Even at 250 kW the localization is just slightly larger (second row). More than 70 % of the light remains in the excited guide only above an input power of 340 kW (third row), which then again defines the width of the resonance curve. ( ) P δ Ω -1 Ω = -1 Ω =
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K. Staliunas and C. Masoller, Opt. Express , 10669 (2006). 19. X. Luo, Q. Xie, and B. Wu, Phys. Rev. A , 051802(R) (2007). 20. D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, Opt. Express , 2151 (2006). igure captions Figure 1. (a) Resonance frequency versus μ . (b) The half-width of resonance curve de-fined at the level versus coupler length at and . (c) versus at A , for modulation depth (curve 1) and (curve 2). m U = b / Ω Ω = μ = A = μ = m U = b L T = μ Figure 2. (a) Theoretically calculated value versus half-width of resonance curve de-fined at the level at and . (b) Experimentally obtained dependence of input peak power required for localization in the launching channel versus normalized detuning from resonance frequency. (c) versus at , for input amplitudes (curve 1) and A (curve 2). A m U = b μ = = μ = L = b L T = m U / Ω Ω b T A = Figure 3. Propagation dynamics in (a) unmodulated coupler at , , and modulated couplers at (b) , , (c) , , (d) , . Propagation distance is L , while modulation frequency in (b)-(d) is Ω = . In all cases left waveguide is excited at . 0.00 μ = μ = b T A = A = μ = b Ω A = μ = = A = = ξ Figure 4. Output intensity distributions for an excitation of left channel of modulated coupler when (a) , (b) , and (c) . The in-put peak power is in first row, in second row, in third row. b Ω = Ω kW b Ω = Ω kW b Ω = Ω
340 kW13