Experimental control over soliton interaction in optical fiber by pre-shaped input field

Experimental control over soliton interaction in optical fiber by pre-shaped input field Esben R. Andresen a, John M. Dudley b, Dan Oron c, Christophe Finot d, and Hervé Rigneault a a Institut Fresnel, Aix-Marseille Université, CNRS, Domaine Universitaire de St. Jérôme, UMR 633, 3397 Marseille Cedex 2, France; b Laboratoire d Optique P.M. Duffieux, Institut FEMTO-ST CNRS, 6 Route de Gray, 253 Besançon Cedex, France; c Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76, Israel; d Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne, 278 Dijon, France ABSTRACT Interactions between femtosecond solitons in a nonlinear photonic-crystal fiber are of fundamental interest. But many practical applications would abound if solitons could be arbitrarily superposed into multiples in the fiber. Here, we numerically and experimentally demonstrate a first step towards this aim, the creation of a soliton pair with arbitrary relative phase, delay, and frequency throughout almost the entire output parameter space with the aid of a pre-shaped fiber input field. Keywords: Fiber, CARS, SRS, soliton, pulse shaping, coherent control, microscopy, endoscopy, frequency conversion, interference. INTRODUCTION Nonlinearity in photonic-crystal fiber (PCF) is an often-used means to generate new wavelengths from lowintensity (nj-) femtosecond (fs-) pulses, as the frequency conversion benefits from the high nonlinearity as well as the novel dispersion properties. 2,3 Lately, there have been several reports of attempts to use amplitudeand phase shaped input fields to control the nonlinear interaction in the PCF, often with the aim of guiding the output wavelengths into a particular spectral region. 4 4 These studies are of interest for applications that would benefit from rapid control, low cost, and fiber delivery, like nonlinear microspectroscopy, microscopy, and endoscopy. 6,5 2 Here, we will aim to control the nonlinear interaction in the anomalous dispersion region of a PCF, which dictates that soliton dynamics is the predominant nonlinear effect; 2 an input pulse converges into a soliton, provided the durations and energies of the initial pulse and the formed soliton are roughly matched; the formed soliton redshifts due to intrapulse stimulated Raman scattering. 22,23 Therefore, attempts to control the PCF output must make use of solitons as building blocks. Our approach is to send one pulse into the fiber for every soliton we wish to create. In our previous paper, 4 we used an input pulse train of two degenerate pulses, in that case, the output parameters (output soliton relative delay T sol, output soliton relative phase Φ sol, and output soliton frequency shifts ω and ω 2 ) depended on the input parameters in a non-trivial way, in particular, all the output parameters depended on the input relative phase φ, limiting the accessible output parameter space. In the present paper, we lift the degeneracy of the input pulses, and we demonstrate how this can lead to a decoupling of the output parameters from φ. Further author information: (Send correspondence to H.R.) E.R.A.: E-mail: esben.andresen@fresnel.fr, Telephone: +33 ()4 9 28 8 84 J.M.D.: E-mail: john.dudley@univ-fcomte.fr, Telephone: +33 ()3 8 66 64 94 D.O.: E-mail: Dan.Oron@weizmann.ac.il, Telephone: +972 8 9346282 H.R.: E-mail: herve.rigneault@fresnel.fr, Telephone: +33 ()4 9 28 8 49 C.F.: E-mail: Christophe.Finot@u-bourgogne.fr

Figure. Sketch of the experimental setup. fs-laser, femtosecond laser; Mich, Michelson interferometer; PCF, photoniccrystal fiber; OSA, optical spectrum analyzer. 2. Numerical code 2. MATERIALS AND METHODS The numerical simululations used a standard generalized nonlinear Schrödinger equation model. 2,2 Our aim was to describe the experiments performed using the setup in Fig., thus we used the realistic Raman response and the full dispersion curve of the PCF used as detailed in Table. The PCF has a single zero-dispersion wavelength at 745 nm. We modelled the Michelson interferometer as a linear transfer function. The transfer function H acts on the laser field E laser to give E shaped (ω ω ) = H(ω ω )E laser (ω ω ) () where ω is the central frequency of E laser. E shaped is used as initial condition for the numerical simulation of the nonlinear propagation in the PCF. The laser field is taken as E laser (t) = (.763t ) P sech, (2) τ with τ the FWHM of E laser (t) 2. Table. The PCF parameters used at λ = 8 nm. β 2 -.78233377 4 fs 2 /m β 3 9.257249 4 fs 3 /m β 4-3.4363836 4 fs 4 /m β 5 4.94298465 4 fs 5 /m β 6 5.36273 3 fs 6 /m β 7-3.862889 3 fs 7 /m β 8.248442795 3 fs 8 /m β 9 2.8835734 3 fs 9 /m β -.695786 2 fs /m β -2.2886368 8 fs /m γ. (W m ) In the simulations, we consider a transfer function H (nondeg) that mimicks an unbalanced Michelson interferometer with a phase difference between the two arms: H (nondeg) (ω ω ) = (A+δA )e i(ω ω) t/2+i φ/2 +Ae i(ω ω) t/2 i φ/2. (3) It results in two replica pulses of amplitude (A+δA ) (leading pulse) and A with relative delay t and relative phase φ cf. Fig.. We also consider a transfer function H (deg), obtained from Eq. 3 by setting δa = which

mimicks a balanced Michelson interferometer with a phase difference between the two arms. From the simulated PCF output E out all the output parameters can immediately be found. 2.2 Experimental setup The experiment is performed by the setup shown in Fig.. The laser is a Titanium:Sapphire modelocked laser (Coherent Micra, 8 MHz, 8 nm, 3 fs); a Michelson interferometer imposes the transfer function H (deg) or H (nondeg) on E laser, and E shaped is launched into the PCF (6 cm of NL-2.-745, Blaze Photonics). The PCF output E out is characterized by a spectrum analyzer (ANDO AQ-635A), allowing us to measure the soliton frequency shifts ω and ω 2 and the relative soliton delay T sol, the relative soliton phase Φ sol from the spectral interference fringes and hence locate E out in output parameter space. 3. CONCEPTS As long as E out is comprised of two solitons it can be described by (leaving out a minor difference in amplitude): E out (t) = sech( t τ sol )e i ωt+iφ +sech( t T sol τ sol )e i ω2t+iφ2 (4) where τ sol is the duration FWHM of the soliton, which gives the following output spectrum E out (ω) 2 = E(ω ω ) 2 + E(ω ω 2 ) 2 +2 E(ω ω ) E(ω ω 2 ) cos( T sol ω + Φ sol ), (5) where Φ sol = Φ Φ 2. Under most circomstances (as long as the solitons are not colliding, i.e they are not exchanging energy through stimulated-raman scattering), this expression describes the PCF output spectrum. However, when the input pulse train is a pair of degenerate pulses and when T sol is small compared to τ sol, each output parameter depends on every input parameter. This is a consequence of the solitons being close to one another over the full length of the PCF; since the two solitons are nearly identical they must be close at the PCF input if they are to be close at the PCF output. We investigated this in Ref. 4. This inability to control each output parameter independently limits the access to certain regions of the output parameter space, notably it disallows some combinations of ( T sol, Φ sol ). In order to decrease the number of parameters, on which each output parameter depends, we propose the following. First, we fix the input relative delay t at > 6τ. t = 5 fs fulfills this for our τ = 3 fs. This way, the initial nonlinear propagation of each pulse is indendependent of the other, due to the lack of pulse overlap. Second, we add a little amplitude bias to the leading pulse. This slightly increases the magnitude of ω and hence decreases T sol due to the decrease in group velocity with ω in the anomalous dispersion region. This renders the impact of the (phase-dependent) soliton force 23,24 insignificant; partly because the solitons start out far from one another and only come close towards the end of the PCF (the soliton force decreases exponentially with temporal separation); partly because the soliton force between solitons decreases with increased group velocity difference. 25 The only mechanism of interaction that remains between the two solitons is the interaction of the leading soliton with the dispersive wave shed by the trailing soliton. 26 3 This interaction, which leads to an energy increase of the leading pulse, is independent of φ. In essence, this approach leads to the output parameters ω, ω 2, and T sol being independent of the input parameter φ, only Φ sol depends on φ, and in a linear way. We note that this method does not provide possibility for output solitons of equal ω since the input pulses start out at the same frequency but leave the PCF with a group-velocity difference; this is the price that is paid for the decoupling of the output parameters from φ. 4. Degenerate pulse pair 4. NUMERICAL RESULTS First, we illustrate the behaviour of a degenerate pulse pair. In Fig. 2, the initial condition for the simulation was a degenerate pulse pair and t was chosen so short that the pulses interact. The spectral components around the laser wavelength 8 nm arise from dispersive waves, the component around 85 nm from the two Raman-shifted solitons. This fact, while difficult to observe in the PCF output spectra presented in Fig. 2a, is more apparent in Fig. 2b, where the inferred T sol and Φ sol are plotted versus φ. T sol is not constant in φ, and Φ sol exhibits deviations from a straight line.

Intensity [a.u.].8.6.4.2 a) φ = φ = π/2 φ = π φ = 3π/2 T sol [fs] 78 8 82 84 86 88 8 7 b).5 6 5.5 4.5.5 φ [rad/π] Figure 2. Simulation with the mask H (deg) for the parameters τ = 3 fs; t = 6 fs; A =.5; δa = ; and P = 3796 W. (a) PCF output spectra for four input relative phases φ; (b) corresponding output soliton relative delay T sol and phase Φ sol. Intensity [a.u.] T sol [fs].8.6.4.2 a) φ = φ = π/2 φ = π φ = 3π/2 Φ sol [rad/π] 78 8 82 84 86 88 25 b).5 2.5 5.5.5 φ [rad/π] Figure 3. Simulation with the mask H (nondeg) for the parameters τ = 3 fs; t = 5 fs; A =.484; δa =.38; and P = 454 W. (a) PCF output spectra for four input relative phases φ; (b) corresponding output soliton relative delay T sol and phase Φ sol. Φ sol [rad/π] 4.2 Non-degenerate pulse pair The result of the simulation with a non-degenerate pulse pair are shown in Fig. 3a,b. The simulation was repeated several times with φ-values ranging from -π to π. Simulated PCF output spectra for four different φ separated by π/2 are presented in Fig. 3a. It is clear that the two solitons are not centered around a common frequency (the visibility of the fringes is not % throughout) due to the induced difference in group velocity or, equivalently, different frequency shifts; ω = -.62 fs and ω 2 = -.38 fs. We infer T sol and Φ sol and plot them versus φ in Fig. 3b. T sol is seen to be constant with φ, signalling the complete decoupling

of the two; and Φ sol is linear in φ, indicating that the input relative phase φ determines the output relative phase Φ sol up to an additive constant. 5. Degenerate pulse pair 5. EXPERIMENTAL RESULTS Intensity [a.u.] 78 8 82 84 86 88 Figure 4. Experiment, degenerate input pulse pair with t = 93 fs. PCF output spectra for two different φ separated by π. The first experiments is a demonstrative example of the restricted access to the output parameter space when launching a degenerate pulse pair into the PCF. The result of such an experiment with a small temporal separation t = 93 fs is presented in Fig. 4. The two spectra were measured with exactly the same parameters but for φ which differed by π. The two spectra are clearly not shifted by half a fringe period relative to one another, which shows that it is not exclusively Φ sol that changes upon changing φ, notably T sol is coupled to φ. Though we attempted to match the parameters of this experiment to the parameters used in Fig. 2, the φ-dependence is more significant in Fig. 4. This is most likely due to experimental imperfections that are not incorporated in the simulation, such as higher-order dispersion of the optics. 5.2 Non-degenerate pulse pair We now present experiments that demonstrate the decoupling of T sol from φ when the degeneracy of the input pulse pair is lifted. Using the same parameters as in the simulations above, we set t = 5 fs and increase the amplitude of the leading pulse so as to approach the solitons at the output to within about 2 fs of one another. The results are presented in Fig. 5. Figure 5a presents PCF output spectra for φ =, π/2, π, and 3π/2. One immediately notices that, contrary to Fig. 4, the four curves only differ by a shift of the fringe pattern which signifies a simple relative phase shift. From the spectral interference of the solitons, we infer T sol and Φ sol and plot them versus φ in Fig. 5b. Despite the scatter of the data points, it is evident that T sol is constant in φ and that Φ sol is equal to φ, up to an additive constant, unlike the case of the degenerate pulse pair Fig. 4. The observations are in accordance with the corresponding simulation (Fig. 3). We note that the visibility of the fringes is not %, which indicates that there are some pulse to pulse fluctuations in Φ sol. At present, we believe that these arise primarily from laser fluctuations although some small intrinsic fluctuations are expected even for a noiseless laser. 6. CONCLUSION We have demonstrated, numerically and experimentally, how a properly shaped pulse pair launched into a PCF can result in an output soliton pair whose relative parameters Φ sol, T sol, ω, and ω 2 can be controlled independently by the input pulse shaping throughout almost the entire output parameter space. The exceptions are for T sol < 2 fs, for which the solitons exchange energy through stimulated-raman scattering; and ω = ω 2, rendered unaccessible by the introduction of a group-velocity difference between the two solitons. Though not yet demonstrated experimentally, our numerical results show that similar results can be obtained by phase-only shaping of the PCF input field, allowing for greater throughput than when using a Michelson interferometer. These results could be of future value for the realization of compact fiber-based light sources for biophotonic applications such as nonlinear microspectroscopy and microscopy or in the emerging field of nonlinear endoscopy.

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