1. Introduction

The first observation of supercontinuum (SC) generation in microstructured fiber (MSF) by Ranka et al. [1] triggered numerous investigations, especially on the characteristics of the supercontinuum on depending initial parameters like pulse duration, peak power, wavelength and chirp [2–5]. Additional efforts have been invested to understand the mechanisms of supercontinuum generation in MSF [2, 6]. Different nonlinear effects like four-wave mixing, self-phase modulation and Raman scattering contribute to the SC generation process by generating new frequencies or redistributing the spectral content. Also, numerical simulations of pulse propagation in MSF have been performed, for the first time by Housakou and Herrmann [7], who used the integration of a set of reduced Maxwell equations, but did not take into account the Raman effect. However, a better agreement between simulation and experiment can be achieved by solving the nonlinear Schrödinger equation including the Raman effect as well [8].

Following the theoretical understanding and the related experimental improvements SC sources were applied, e.g., to coherent anti-Stokes Raman scattering microscopy [9], optical coherence tomography [10, 11] and precision frequency metrology [12, 13]. These applications require tailored supercontinua, featuring for instance a special spectral shape or a very high pulse-to-pulse coherence. As a consequence, the interest recently has focused on the improvement of specific SC properties. For instance, it has been reported that the spectral shape of supercontinua generated in MSFs, the phase stability and hence the coherence could be influenced to some degree by changing the pump pulse length [14], the input wavelength [16] or modulating the pump pulse shape [17]. However, it has turned out that the strong dependence of the SC generation process on initial conditions results in significant temporal and spectral fluctuations within the SC light distribution depending on pump pulse fluctuations in amplitude and phase [14, 15] and until today, a full control of either SC spectral or temporal shape has not been demonstrated.

An additional parameter to influence the SC generation could be introduced by feedback within the generation process. In earlier work, the concept of using feedback in SC generation has been investigated by Moselund et al. [18] for the case of picosecond pulses. Tlidi and colleagues [19] have been engaged in the investigations on modulational instabilities (MI) in microstructured fiber feedback resonators and have predicted frequency shifts by MI depending on the fourth order dispersion coefficient. Steinmeyer and colleagues have studied a feedback system, based only on self-phase modulation (SPM) in a single mode fiber and pumped with picosecond pulses, in which they have found nonlinear dynamics like period doubling [20].

In this work, we numerically investigated the influence of feedback on SC generation in a MSF and in the femtosecond regime, where not only SPM but additional effects have to be taken into account. With this, we want to contribute to the goal of creating tailored SC for sophisticated applications. As in many other optical systems like injection locked lasers [21] or self-injection locked OPOs [22], the inclusion of feedback has been shown to lead to a significant effect on the system behavior as for instance self-stabilization, and we expect a considerable impact from the implementation of feedback in our case as well.

In the following we first introduce the numerical implementation for a single shot SC generation (section 2.1), its verification via a comparison with our experimental data (section 2.2), and the extension of the numerical code by a feedback loop (section 2.3). In section 3 the results of the simulations with feedback are presented first with a simplified (section 3.2), then with the full model (section 3.3), and finally including noise (section 3.4) in order to estimate the experimental realizability. Following in section 4, a first experimental verification of the numerically observed effects is shown.

2. Numerical model

2.1. Propagation equation

In order to model the pulse propagation inside the fiber, we used the generalized scalar nonlinear Schrödinger equation (GNLSE) [2]:

As the slowly varying envelope approximation was considered, here, E=E(z, t) is the pulse envelope, where t is the time in a frame of reference moving with the group velocity β ^-1 ₁. The values β_k are the dispersion coefficients at the center frequency ω ₀, α is a measure of linear fiber loss and γ is the usual nonlinear parameter of the fiber defined as γ=(ω ₀ n ₂)/(cA _eff), where n ₂ is the nonlinear refractive index, c is the vacuum speed of light, and A _eff is the effective mode area. The response function R(t), which includes the instantaneous electronic and the delayed Raman contribution, can be expressed by R(t′)=(1-f_R)δ(t′+f_Rh_R(t′), with f_R=0.18 representing the fractional part of the delayed Raman effect. For h_R(t′) we used the analytic form of the Raman response function [23]: h_R(t′)=[(τ² ₁+τ² ₂)/(τ₁τ² ₂)]exp(-t′/τ₂) sin(t′/τ₁) with τ₁=12.2 fs and τ₂=32 fs. Equation (1) was solved numerically and stepwise along the fiber by using a split-step Fourier method [24]. In order to resolve the temporal and spectral features, which occur during the SC generation process, we used an array size of 2¹⁶ points and the temporal resolution was set to 0.5 fs, corresponding to a spectral resolution of approximately 30GHz. Due to the usually large nonlinear parameter of MSF and resulting rapid spectral changes with propagation, a small step size in propagation direction of 10 µm was required, which was kept fixed after it had been verified to be sufficiently small.

2.2. Verification of algorithm for single shot SC generation

The numerical implementation was tested by performing basic simulations for which well-known results from literature [2] are available and complete agreement was obtained. As a second test, the results of the numerical model were compared with our own experimental results obtained with the following setup: Experimentally, we generated a SC in a microstructured fiber (45mm long, Crystal Fibre NL-PM-750), which was pumped with 60 fs pulses (measured with an autocorrelator, APE AutocorrelatorMini) at a repetition rate of 82MHz from a mode-locked Titanium:Sapphire laser (Spectra-Physics Tsunami) at a center wavelength of 775 nm. The generated spectra were recorded with an optical spectrum analyzer (OSA, ANDO AQ 1425). In order to accomplish a representative comparison between simulation and experiment, the parameters used for the numerical simulation were adapted from the experimental values. The nonlinear parameter γ=0.095 (Wm)^-1 was extracted out of the data sheet of the MSF [25]. By fitting an 8^th order polynomial to the dispersion curve displayed also on the data sheet, the dispersion coefficients β_k around 775 nm of the MSF were retrieved up to the 10^th order. The initially injected pump pulse was numerically implemented as a hyperbolic secant field profile E(0, t)=√P ₀·sech((2log(1+√2)t)/Δt)=E _pump, matching the temporal profile of the experimentally available, Fourier-limited pulses of the Titanium:Sapphire laser. The full width of the pulses at half maximum intensity (FWHM) was Δt=60 fs, and the peak power P0 was used as the variable parameter in our numerical investigations. With these parameters, a good agreement between experiment and simulation was observed. With the parameters given above, the GNLSE was numerically integrated for a number of different input pulse powers between 1mW and 18mW, and the results are shown in Fig. 1, where a) numerically calculated and b) experimentally measured spectra are displayed for varying effective average pump power. In both graphs, the spectral power is shown on a color-coded logarithmic density scale, and the effective average pump power increases from bottom to top, such that each single horizontal line within the graphs is representing a complete spectrum. We would like to note, that the effective average pump power inside the fiber was deduced from the fiber coupling efficiency, which was approximately 50%. As can be seen from Fig. 1, there is a very good agreement between simulation and experiment as all spectral features can be identified in both graphs. For example, the spectral feature evolving on the short-wavelength side (see the feature labeled ”1” in Fig. 1) as well as the Raman-shifted peak on the long-wavelength side (feature 2), or even the finer spectral structure in the middle part of both spectra (feature 3) show strong similarity. From this verification we concluded, that our numerical model and its implementation with the experimental parameters accomplish realistic predictions, such that we could proceed with first numerical investigations on SC generation with feedback and femtosecond pump pulses.

 

Fig. 1. Relative spectral power as function of effective average fiber input power: (a) simulation and (b) experimental results. In both graphs, the relative spectral power is displayed on color-coded logarithmic density scale (see color bar on the right), with the wavelength given on the x-axis, and the effective average input power increasing from bottom to top (y-axis). The numbers 1–3 are indicating spectral features, which are explained in the text.

Download Full Size | PDF

2.3. Implementation of the feedback loop

In order to investigate the effect of feedback on SC generation, the model, which was presented in sections 2.1 and 2.2, was extended by a feedback loop. A schematic diagram of the implementation can be seen in Fig. 2. Here, only the solid line (path A) was considered, whereas the alternative path B (dashed line) will be explained in section 3.1., and E _pump denotes the electric pump pulse envelope which was adopted to a hyperbolic secant field profile, as before. The approach was straight forward: After integrating the GNLSE with the pump pulse field profile as the initial condition E_N(z=0, t)|N=0=E ₀(0)=E _pump (N=number of feedback iterations, and t was omitted for simplified presentation), the SC pulse envelope E ₀(L)=GNLSE(E ₀(0)) (L=fiber length) at the end of the fiber could be obtained. Then a certain fraction e of a few percent of the generated SC was superimposed onto the following pump pulse, by simply adding up the two arrays, resulting in E ₁(0)=(ε·E ₀(L)+E _pump). The GNLSE was integrated along the MSF length again, now with the modified input pulse array values, in order to get the next SC pulse E ₁(L)=GNLSE(E ₁(0)). This procedure was repeated a few hundred times, by calculating the resulting recursive map

thus modeling a number of round trips along the MSF with feedback. For high feedback efficiencies (ε>45%), i.e. strong coupling, a reduced stability of the SC evolution and only a slow convergence was observed. The optimum fraction of the SC, to be fed back and superimposed onto the next pump pulse in order to achieve fast convergence, was identified to be ε=30% of the electric field (or 9% of the power). In addition, from a practical point of view, a power efficiency of 9% can be realized easily in an experiment using standard broadband beam splitters for combining the SC pulse inside the feedback loop with a new undepleted pump pulse. All other parameters of the numerical simulation remained unchanged. In order to reach a steady state within the feedback loop, or to be able to identify patterns in the temporal spectral evolution, the numerical integration has to be repeated for typically several hundred times. Such an iterative numerical calculation would, in a simple case, take about two days on a normal personal computer. The results presented in this contribution correspond to 1.5 years of CPU time and were accomplished within a manageable time frame of a few weeks by grid-computing using the Condor software [26].

 

Fig. 2. Schematic diagram of the SC feedback system. The path A (solid line) describes the scheme with simple feedback included, whereas the path B (dashed line) shows the scheme with an inclusion of a dynamical dispersion compensation (DDC). For details see text.

Download Full Size | PDF

3. Results

3.1. Numerical analysis of the effect of feedback on supercontinuum generation

Supercontinuum generation depends on numerous factors, namely the input pulse parameters and the fiber parameters. When applying feedback the supercontinuum is additionally influenced by the feedback efficiency, resonator length (see previous work with picosecond pulses [18]) and by dispersion introduced within the feedback resonator. Specifically, small variation of the resonator length (within one wavelength) changes the relative phase of the two fields, leading to either constructive or destructive interference. In order to simplify the analysis of the effect of feedback, in a first step we kept these latter parameters fixed and used the pump pulse power as the only variable parameter, which was accomplished by including a dynamic dispersion compensation (DDC) in the model. In addition to this first special case from which we expect to gain a fundamental understanding by varying an isolated parameter, we investigated as a second special case, the influence of feedback without any dispersion compensation, which should be easily realizable in an experiment (see section 3.3).

The operation principle of the DDC is illustrated in Fig. 2 as path B. After passing through the MSF, the generated SC had a nontrivial temporal pulse shape. This pulse shape varied concerning the number, the power, as well as the position of temporal intensity maxima, depending on the peak power of the input pulse. As a result, the above named input, fiber and resonator parameters cannot be changed independently from each other. In order to avoid this interdependency, we introduced the DDC, which means that the spectral phase of each generated SC was set to zero after every iteration, resulting in a compressed, Fourier-limited SC pulse. With the DDC the fed back SC was a symmetric pulse in the time domain with a clearly defined center position, which could precisely be adjusted to the following pump pulse for superposition to start the next iteration. With both pulses having constant, zero spectral phase, it was guarantied that the optical length of the feedback ring was constant for all wavelengths, and that superposition always led to constructive interference.

With the inclusion of this DDC we have performed a series of simulations with stepwise increasing the effective average pump power from 0.25mW to 12mW, by iteratively integrating the GNLSE, as described above in section 2.3. After each iteration, the generated spectrum was recorded, until the simulations were stopped after 100 iterations, as after usually 30 iterations the system reached its specific final system state. From these simulations we obtained at different average pump powers different system dynamics, like steady state, period doubling, and period quadrupling up to chaos. These different system dynamics are illustrated in Fig. 3 where the spectral phase at a wavelength of 775 nm (black dots) is displayed as a function of average pump power. This bifurcation diagram shows the steady-state operation as the values of the first 50 iterations are not shown. The spectral phase of the SC components are given with respect to the initial pump pulse phase, which was set to zero. On a first view, one can divide the power axis up into four main intervals: In the first interval from 0mW to 9.122mW, the spectral phase converged to a fixed point [27], indicating a steady state. With increasing average power up to 9.122mW the phase convergence was slower, until a bifurcation occurred around 9.123mW. Here, the system transformed into a period-2 cycle (period-doubling). This system behavior consisted of two fixed points which alternated with every further iteration. With further increase of the average power, the period-2 cycle was maintained up to an average power of 10.1mW. Within the third interval from 10.1mW up to 10.3mW, each of the two fixed points again split into two fixed points, thus transforming into a period-4 cycle, indicating a second bifurcation point. For a better visualization of this behavior, the second and third interval are shown magnified in Fig. 3 b. The last interval from 10.3mW up to the highest investigated average pump power of 12mW was a chaotic regime, in which only very narrow windows of periodic behavior could be found, e.g., there were windows of period-4 cycle at 11mW and 11.05mW of pump power. Note, that the phase states observed in this chaotic regime did not cover the whole phase space cross section, but a connected region of it, which implies that the system showed a fractal dimension <2n [27].

 

Fig. 3. Bifurcation diagram with the spectral phase of a single spectral component as a function of average pump power: (a) for a pump power up to 12mW at a wavelength of 775 nm and (b) in more detail within the range from 8.8mW to 10.5mW at a wavelength of 775 nm (black dots) and 750 nm (gray dots).

Download Full Size | PDF

We would like to note, that for a complete description, the investigated system has to be observed in a 2n phase space, with the used array size n, as from each complex-valued array element a phase as well as an amplitude can be derived. The results presented here, however, did not depend on the individual number of array element considered. For the results presented in the following, the spectral phase at the wavelength of 775 nm was considered only as an example. In order to show that for a different wavelength the bifurcation points occurred at the same pump power and that the same system behavior within the same intervals was developed, in Fig. 3 b the spectral phase at a wavelength of 750 nm (gray dots) is included. We have verified, that the system displayed the same qualitative behavior within all the other spectral components by varying the bifurcation parameter (effective average input power), no matter which phase, or even amplitude or spectral intensity was regarded. The system behavior could also be deduced from the total pulse energy, which represents a measurable observable for an experimental verification.

3.2. Nonlinear dynamics with dynamical dispersion compensation

The above presented simulations with included feedback showed different system dynamics depending on the input pump power. In the following we analyze the spectral evolution by extracting one example of each periodic regime and having a closer look at the vicinity of the bifurcation point from steady state to period-2 cycle. For a convenient visualization of the spectral evolution, the spectra obtained after every iteration are plotted on top of each other with the spectral energy density being color-coded on logarithmic scale as shown in Fig. 4 a. There, the spectra obtained during the first 50 iterations are displayed for an average pump power of 6mW. As can be seen, the spectral shape changed substantially during a transition phase, which took about ten iterations. Afterwards the spectral shape was constant and the spectral width was maintained, indicating a steady-state operation.

 

Fig. 4. Left: Spectral evolution of SC pumped with 6mWaverage power, the spectral power density is displayed on logarithmic density scale (see color bar). Right: Variability V ¹ _N of the whole spectrum as a function of the number of iterations, on linear (b) and on logarithmic scale (c). For definition of V ¹ _N see text.

Download Full Size | PDF

In order to analyze the resulting stability of spectral amplitude and phase during the process of convergence, e.g., towards a fixed point we monitored the change of amplitude and phase. As a quantitative measure we defined a variability as

which may look like a contrast function. However, this function does not only take into account the spectral energy but also the spectral phase, which results in a calculation of the spatial separation between the respective coordinates in phase space (defined by amplitude and phase). Here, E_N(λ) is the Fourier-transformed electric field after the N ^th iteration. The upper index p has to be set to 1, if the variability between consecutive spectra should be determined, and has to set to a value of two, if a spectrum is not to be compared to the one directly following, but to the one after that, and so on. If p equals 1 the value of V ¹ _N is an indication for the average spectral change of two consecutive spectra. The more consecutive spectra differ from each other, the closer V ¹ _N will be to the maximum value of one, and if the spectral evolution converged towards a steady state, the variability should reach zero.

The variability V ¹ _N has been calculated for the SC with feedback shown in Fig. 4 a, and the result is displayed in Fig. 4 b on a linear scale as a function of the number of iterations. From this graph, it can be seen, that V ¹ _N showed a rather high value during the transition period, but decreased rapidly. By replotting V ¹ _N on logarithmic scale (Fig. 4 c) one can see, that the variability decreased exponentially and that after approximately 115 iterations the variability V ¹ _N reached a value in the order of 10^-24, which corresponded to the round-off errors of the simulation. This indicated that the system reached steady state.

 

Fig. 5. Occupied phase space of the spectral component at 790 nm: on the right (a) for an average pump power in the range between 8.75mW and 9.35mW and on the left (b) a magnification for an average pump power from 8.75mW to 9.10mW. The color indicates the number of feedback iterations after which this coordinate was reached, representing the respective system state (see color-coding bar on the right side).

Download Full Size | PDF

In the following the stability of the system in the vicinity of the first bifurcation is investigated, which is done by observing the system behavior in phase space. In Fig. 5, a cross section through the 2n-dimensional phase space is shown, which displays amplitude and phase evolution of a single spectral component, namely at 790 nm, since for this single spectral component the behavior in the transition region could easily be analyzed on a first view. Additionally, this cross section is the maximum fraction of phase space, which can be displayed reasonably, and which at the same time contains all important information, as qualitatively the same behavior was observed and verified for all spectral components. In Fig. 5 a the spectral component at 790 nm is displayed with the spectral power density (|E|²) in radial direction and the spectral phase (φ) in azimuthal direction for a pump power range between 8.75mW and 9.35mW, including the transition from steady state to period-2 cycle. The color indicates, after which number of iterations the respective coordinate was reached. By increasing the pump power the system switched abruptly into a period-2 cycle, which is presented by two new fixed points with different amplitudes and phases (outer two fixed points in Fig. 5 a). Fig. 5 b shows the evolution of the spectral component at 790 nm in a magnification, from left to right with the pump power increasing from 8.75mW to 9.1mW in steps of 0.05mW. Here, the system was approaching the respective fixed point on a spiral trajectory and from the color gradient it can be seen that the convergence was the slower, the higher the power and the closer the system was to the bifurcation point. This observation agrees well with general analysis of nonlinear dynamics, which states that any periodic attractor looses its stability when approaching a bifurcation point [28].

In order to analyze the spectral evolution for observed higher periods, the results of the simulation with dynamic dispersion compensation for two further examples extracted from period-2 and period-4 regimes are shown in Fig. 6. On the left side of Fig. 6 the spectra obtained behind the fiber during the first 50 iterations are displayed for an average pump power of 9.5mW (Fig. 6 a) and of 10.3mW (Fig. 6 c). In Fig. 6 a one can see that after a short transition regime of about five iterations, the generated spectra did not repeat after each iteration, but after every second round trip. For a clear verification, V ¹ _N was calculated as well as V ² _N and V ² _N+1, and these values are plotted as a function of the feedback iteration number in Fig. 6 b. As expected, V ¹ _N did not decrease at all, but remained at a constant value, close to one. On the other hand, V ² _N and V ² _N+1, which are measures for the similarity of the spectra obtained after even and odd numbered iterations, respectively, both did decrease exponentially, reaching the numerical noise floor after approximately 100 iterations. This is a clear evidence, that the system was showing two different fixed spectra, which were alternating after each iteration. We have thus observed period doubling, in analogy to classic (mechanic or electric) nonlinear oscillators for the first time within femtosecond SC generation. At a pump power of 10.3mW we observed a quadrupled period as shown in Fig. 6 c. This spectral evolution exhibited once again a transient oscillation regime of about 15 iterations. But then, the generated spectra did repeat only after every fourth one. For a clear verification, V ² _N and V ² _N+1 were calculated, but also V ⁴ _N, V ⁴ _N+1, V ⁴ _N+2, and V ⁴ _N+3, which represent the similarity of spectra obtained after every fourth iteration, each with a different initial spectrum. In Fig. 6 d these curves are plotted as a function of the number of iterations. As can be seen, V ² _N and V ² _N+1 did not decrease, but remained close to one. However, V ⁴ _N, V ⁴ _N+1, V ⁴ _N+2, and V ⁴ _N+3 all decreased exponentially to the numerical noise level, which they reached after approximately 950 iterations. This time span was somewhat longer than the previously investigated period-2 cycle (see Fig. 6 b), however, the observation of V ⁴ _N reaching the noise level is a proof for period quadrupling in the SC feedback system.

 

Fig. 6. Inclusion of dynamic dispersion compensation: Spectral evolution of SC pumped with (a) 9.5mW and (c) 10.3mW average power, the spectral power density is displayed on logarithmic density scale (see color bar). Corresponding variability as a function of the number of iterations on logarithmic scale for (b) 9.5mW and (d) 10.3mW average power.

Download Full Size | PDF

3.3. Simulation of the feedback system without dispersion compensation

The inclusion of the above presented dynamical dispersion compensation in numerics, used for clearly analyzing the effect of feedback on nonlinear dynamics, would strongly limit the potential for future experimental realization, as such a dispersion compensation adapting dynamically every round trip cannot be accomplished with commonly known state-of-the-art approaches. In order to avoid at first these technical limitations and to retain the possibilities of identifying parameters for an experimental realization, we excluded this dispersion compensation within the following calculations. Again, simulations with different average effective pump powers have been performed. From these simulations we also obtained steady states as well as period doubling, higher orders of period-multiplication and chaos, similar to the results obtained with DDC. However, the course of consecutive system behaviors was much more complex due to the interdependency of feedback parameters and resulted in an extremely complex and non-intuitive dependence on the input parameters. The effective average pump power is not a single suitable bifurcation parameter any more. For example, the system showed a period-2 cycle for 14mW, while increasing the average pump power to 16mW a steady state solution was developed. As a complete characterization of the system by considering further feedback parameters would exceed the scope of this contribution, we instead present a few instructive examples of calculated system dynamics to demonstrate the different system behaviors displayed in the case without dispersion compensation as well.

 

Fig. 7. Exclusion of dynamic dispersion compensation: Spectral evolution of SC pumped with (a) 16mW and (c) 14mW average power. The spectral power density is displayed on logarithmic density scale (see color bar). Corresponding variability as a function of the number of iterations on logarithmic scale for (b) 16mW and (d) 14mW average power.

Download Full Size | PDF

In Figs. 7 a and 7 b the spectral evolution and the corresponding variability are shown for a pump power of 16mW. Despite neglecting dispersion compensation the spectral evolution reached a steady state, which can be seen from the decreasing variability down to the numerical noise level. In Figs. 7 c and 7 d, the spectral evolution and the variability for a pump power of 14mW are displayed. The variability V ¹ _N stayed at a high level, whereas the variability V ² _N and V ² _N+1 decreased to the numerical noise level close to zero, which is clear evidence that also without DDC period doubling was observed.

Besides steady state and period doubling, results from simulations of the feedback system without DDC showed additional features within certain pump power ranges. As a representative example the spectral evolution of the system at 6mW pump power is shown in Fig. 8. This spectral evolution changed from iteration step to iteration step and the spectral width was reduced, compared to the spectral width which should be typically possible with the applied pump power. From the cross section through the phase space for a spectral component at 790 nm presented in Fig. 8 b, it can be seen that the amplitude-to-phase relation followed a closed orbit. This new regime resembles a limit cycle, which means that the system did not converge to fixed points, but that a self-sustained oscillation took place with a definite frequency and a definite circuit orientation [27]. The system behavior thus consisted of an infinite number of different phase space coordinates having in common that they all were lying on this closed orbit.

 

Fig. 8. (a) Spectral evolution of SC pumped with 6mW average pump power without dispersion compensation. The spectral power density is displayed on logarithmic density scale (see color bar). (b) Cross section through phase space of the spectral component at 790 nm, color is indicating the number of feedback iterations to reach each individual coordinate which corresponds to a certain system state.

Download Full Size | PDF

In summary, the SC generation with feedback displays a wealth of different system dynamics depending on the average pump power, from fixed points via period multiplication up to limit cycles and chaos.

3.4. Stability of a fixed state in the presence of pump power noise and quantum noise

In order to check the feasibility of an experimental verification, the effect of noise was additionally included in the simulations. For this, the peak power of the input pump pulse E _pump was randomly varied within a Gaussian distribution with a standard deviation (rms) of 0.5%, realistically modeling technical amplitude noise of the available pump laser. In addition, the effect of input quantum noise was included, with the phenomenological approach of adding one photon with random phase to every spectral mode [2]. These two different classes of noise have different effect on SC generation, as for only including quantum noise, the boundary structure of a spectral component in phase space is a circle, whereas for only including amplitude noise the boundary structure is more complex and not circularly shaped.

With these modifications, the calculations were performed again with the same parameters used for the data shown in Fig. 7, and the results of these noise calculations are shown in Fig. 9 for an average pump power of 16mW and 14mW. It can be seen in Fig. 9 a and Fig. 9 c that the spectral SC evolution revealed a short transition oscillation before it condensed down to steady state or a period-2 cycle, respectively. The variability plotted on a linear scale in Fig. 9 b decreased to an essentially higher value compared to simulations without noise. But nevertheless, the system finally turned into a steady state. Despite the included noise, the period-2 cycle, observed before at 14mW was maintained as well, which can be seen in Fig. 9 c and from the corresponding decrease of the variabilities V ² _N and V ² _N+1 (see Fig. 9 d).

 

Fig. 9. Inclusion of 0.5% rms technical noise and quantum noise: Spectral evolution of SC pumped with (a) 16mW and (c) 14mW average pump power, the spectral power density is displayed on logarithmic density scale (see color bar). Corresponding variability of the whole spectrum as a function of the number of iterations on logarithmic scale for (b) 16mW and (d) 14mW average power.

Download Full Size | PDF

In order to estimate the effect of noise on the stability of the different regimes of operation, the system coordinates were plotted in phase space. Fig. 10 a shows the fixed point environment of a cross section through phase space for 16mW average pump power at a wavelength of 775 nm, where the dot and the crosses display the solution without and with noise, respectively. The coordinates calculated with noise are scattered in a wider area around the fixed point obtained from the calculations without noise. Fig. 10 b shows the results for an average pump power of 14mW at a wavelength of 775 nm, representing a period-2 cycle. From this figure it can be seen, that also for period doubling the coordinates remained in a limited area in the vicinity of the respective fixed points, such that the two areas were clearly separated, and the coordinate in phase space was alternating between these two areas with every iteration. The cross section through phase space of the limit cycle for a spectral component of 790 nm at 6mW can be seen in Fig. 10 c. The solutions calculated without perturbation are given by the dots, which form a closed cycle. The coordinates obtained from simulations with noise (crosses) were scattered around this cycle, such that a closed belt with slightly varying width was created.

 

Fig. 10. Fixed point environment in an amplitude and phase diagram for simulations with and without noise (a) at 16mW and (b) at 14mW for a spectral component at 775 nm, and (c) at 6mW for a spectral component at 790 nm. The dots and crosses denote the coordinate in phase space without noise and with technical as well as quantum noise, respectively, whereas the color bar is indicating the number of feedback iterations after which the respective coordinate was reached.

Download Full Size | PDF

In summary, these results show that the system behavior without noise was maintained also in the presence of noise. This indicates, that the presented nonlinear dynamical evolution should be measurable in an experiment under normal laboratory conditions.

4. First experimental verification

A period-2 cycle, also called period doubling is, spoken in more technical terms, a complex amplitude modulation of the whole SC spectrum with half the frequency of the repetition rate. Therefore, this modulation should be measurable as a side-band within the radio frequency (RF) range with a RF spectrum analyzer connected to a silicon photo diode. For a comparison, the equivalent information has been extracted from the total SC pulse energy of the simulation displayed in Fig. 9 c, by calculating the Fourier-transform. The result is shown in Fig. 11 a, where a peak at 41MHz can be found, i.e., at half the repetition rate. In order to experimentally observe such a period doubling the experimental setup presented in section 2.2 and used for single shot SC generation was extended by a feedback loop. We used a beam splitter to synchronously superimpose the generated SC with the next following pump pulse. Within the ring cavity, an optical delay line was included, such that the temporal overlap of the pump pulse and the SC could be adjusted. Within the feedback loop a wedge-shaped glass plate was used to tap a fraction of the generated SC, in order to measure the RF spectrum with a photodiode connected to a RF spectrum analyzer (Advantest, TR 4131/E). The RF spectrum obtained at an effective average pump power of 14mW is shown in Fig. 11 b. One can clearly identify a peak at 41MHz, which was exactly half the repetition rate of the applied pump laser. Simultaneously to this measurement we have observed the RF spectrum of the pump laser using a second RF analyzer (Anritzu MS2721B Spectrum Master) and the whole time during the measurement only a peak at 82MHz occurred. From this we can conclude that the observed period-2 cycle can be attributed purely to feedback. We have thus experimentally observed period doubling in a femtosecond SC system with feedback, and have achieved a first experimental confirmation of the numerical investigations of this nonlinear system. A more detailed experimental research, that will include measurements of higher order periods as well as measurements of limit cycles, will be subject of our future work.

 

Fig. 11. (a): Simulated radio frequency spectrum with the inclusion of noise at 14mW effective average pump power. (b): Measured radio frequency spectrum at 14mW effective average pump power.

Download Full Size | PDF

5. Summary and conclusion

A supercontinuum generating feedback system consisting of a microstructured fiber incorporated in a ring resonator and synchronously pumped with femtosecond pulses was presented. For numerically investigating this feedback system a model was implemented, by solving the generalized nonlinear Schrödinger equation with the use of a split-step Fourier method and by superimposing the generated supercontinuum with the respective consecutive pump pulse, in order to simulate the feedback. The implementation has been verified by comparing results from simulation to experimental measurements and convincing agreements have been achieved. From results of performed simulations including feedback, nonlinear system dynamics like steady state, as well as period doubling, period quadrupling, limit cycles and chaos were observed for the first time in such a feedback system, pumped with femtosecond pulses. Including technical amplitude noise in the model as well as quantum noise, it could be shown, that these observed nonlinear system dynamics were maintained even with these perturbations. First experiments confirmed the measurability of the numerically predicted manifold of system dynamics by showing period doubling of the system’s repetition rate.

These results are giving a deep insight into the impact of feedback on SC generation and into the strong dependence on input parameters, like input pump power. In the future, we want to use the feedback to generate tailored supercontinua, with regard to the improvement of pulse-to-pulse amplitude stability as well as phase stability leading probably for instance to efficient pulse compression of parts of the supercontinuum. In order to achieve this goal, further numerical and experimental investigations on the impact of feedback are currently in progress in our research group by considering further parameters, and according results will be presented in an additional publication.