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The delay and phase dependent behavior of a system for supercontinuum generation by using a microstructured fiber within a synchronously pumped ring resonator is presented numerically. The feedback introduced by the resonator led to an interaction of the supercontinuum with the following femtosecond laser pulses and thus to the formation of a nonlinear oscillator. Via the feedback phase different regimes of nonlinear dynamics, such as steady state, period multiplication, limit cycle and chaos can be adjusted systematically. The spectrum within one regime of nonlinear dynamics can additionally be modified independently from the regime of nonlinear dynamics.
©2010 Optical Society of America
1. Introduction
Ultrabroadband supercontinua [1,2] with relatively low input power are generated commonly by injecting laser pulses into microstructured fibers (MSF) [3] and can be modified by input pulse and fiber parameters [4]. The influence of optical feedback on the spectral SC composition was investigated in picosecond systems. In these feedback systems SC generation relies on seeded four-wave mixing [5,6]. Furthermore, in the case of feedback systems with weaker nonlinearities of a single-mode fiber, the occurrence of period doubling and chaos was investigated [7,8]. This system behavior is caused by self-phase modulation and multiple-beam interference and depends on the power in the feedback system.
In our previous work on the highly nonlinear SC feedback system pumped with femtosecond laser pulses, four regimes of nonlinear dynamics were found numerically and experimentally [9,10], namely steady state, period multiplication, limit cycle, and chaos.
While in our previous work the existence of the different nonlinear dynamics was proven, the subject of this contribution is the dependence of the system behavior and of the emerging spectra on the feedback delay. We found a dependence on the phase of the feedback loop, similar to other nonlinear oscillators [7,8]. In our system different optical spectra and different regimes of nonlinear dynamics can be scanned within a phase detuning of 2π, which corresponds to a delay detuning of one single wavelength. For larger delay detuning the bifurcation structure can be changed. The numerical investigations presented in the following are fundamental for the understanding of the system’s behavior and to enable future experimental realizations. Our investigations are a prerequisite for the possibility of rapidly switching between different kinds of nonlinear dynamics, of controlling the system such that it remains in a certain state, and of adjusting individual SC spectra. Therefore, this may be a further step towards the full control of a SC generating system.
2. Numerical results
For the simulations, we used a numerical model of the SC feedback system, which was developed and presented earlier [9]. A schematic diagram of the SC feedback system is shown in Fig. 1 . The electric field per resonator round trip (Einput) was calculated within four main steps indicated by the gray boxes in the diagram: firstly, the pulse propagation within the nonlinear fiber was described by the generalized scalar nonlinear Schrödinger equation (GNLSE) and was solved numerically using a split-step Fourier method. The resulting electric field behind the MSF is called Efiber in Fig. 1. Secondly, for simulating a delay detuning τ, a term iωτ linear with the light frequency ω was added to the spectral phase after the fiber propagation (resulting in Edelay in Fig. 1). Note, that the magnitudes marked with tildes in Fig. 1 are the spectral electric fields, i.e., the Fourier transformed temporal electric fields. Thirdly, the pulses were formed, to simulate realistic experimental conditions. Specifically, in order to emulate material dispersion, which would be introduced by microscope objectives in an experimental setup, used to couple into and out of the MSF, additional higher order spectral phase terms up to the 6th order were added [9.125·10−28 fs2(ω-ω0)2 + 1.044·10−42 fs3(ω-ω0)3 - 1.971·10−57 fs4(ω-ω0)4 – 2.226·10−72 fs5(ω-ω0)5 + 8.085·10−87 fs6(ω-ω0)6]. This resulted in the chirped SC pulse Eresonator. Fourthly, one resonator round trip was finished by superimposing a fraction of the SC within the resonator (ε·Eresonator) with the next incoming pump pulse (Epump), thus providing the initial conditions for the next resonator round trip.
Fig. 1 Schematic diagram of the numerical SC feedback system: pulse propagating through the MSF (by solving the generalized nonlinear Schrödinger equation GNLSE), introduction of a variable delay and dispersion, superposition of a fraction (ε) of the SC and the next pump pulse. The magnitudes marked with tildes are the spectral electric fields, i.e., the Fourier transformed temporal electric fields. For details, see text.
For the numerical results presented in the following, the same fiber parameters were used as in our previous work (derived from the datasheet of the experimentally used MSF [11]), and the pulse parameters were kept constant at an average input power of 3 mW of pulses with a hyperbolic secant field profile and a duration of 60 fs (full width at half maximum) at a central wavelength of λ0 = 2πc/ω0 = 775 nm. The simulations were calculated in a frame of reference moving with the group velocity β1 −1(ω0). The feedback power efficiency was set to ε2 = 36%. Only the delay between the fed back SC and the next laser pump pulses was used as a variable parameter. Simulations were carried out for a delay position from −170 fs to 165 fs in steps of 0.2 fs. The complete parameter scan shown in Fig. 2 required more than 17000 hours (2 years) of computing time, which could only be realized within a tolerable time frame (2-4 weeks) by grid-computing using the Condor software [12].
Fig. 2 Numerical simulation a) of a delay scan from −170 fs to 165 fs; blue: regimes of nonlinear dynamics, red: convolution function of the intensity profiles of the SC pulse after the first resonator roundtrip and the next laser pulse (pulse overlap), black: linear cross-correlation function of the two fields (energy of overlapped pulses), green: central wavelength of the overlapping SC interval; b) and c) zoom into the delay range from −5 fs to 45 fs; b) blue: regimes of nonlinear dynamics, black: linear cross-correlation function; c) corresponding optical spectra.
For each delay position the observed regime of nonlinear dynamics was recorded (blue filled cycles in Fig. 2 a). Several regimes of nonlinear dynamics could be observed, namely steady state (P1), period multiplication of different orders (P2, P3, P4, P7, and P8), and limit cycle (LC). For the regime of steady state one system state is exactly reproduced after each cavity round trip. For period multiplication of the order N there are N system states that are assumed alternatingly. This means, that the same system state is not repeated after every round trip, but after the Nth round trip. Limit cycle are self-sustained, periodic oscillations of the system state. An infinite number of system states is assumed, which form a closed orbit in phase space representation. Details to the different regimes of system dynamics and their experimental observation are described in our previous work [9,10].
In order to obtain an estimate for the influence of the delay detuning on the system, we considered two quantities of the electric field in front of the MSF after the first round trip, namely the convolution of the intensity profile of the incoming laser pulse |Epump|2 with the intensity profile of the SC |Eresonator|2 generated after a single pass through the resonator, which illustrates the overlap of the two pulses, and the cross-correlation of the same two pulses in order to understand the phase dependent effects.
The convolution C(τ) of the laser pulses with the single-pass SC was calculated by:
The result of Eq. (1) is shown as the red curve in Fig. 2 a. The convolution function is a measure for the delay range, where the pulses overlap, namely from about −170 fs to 150 fs. As the interval, where nonlinear dynamics were observed, reaches only from −52 fs to 132 fs, the existence of certain nonlinear dynamics cannot completely be explained with the convolution function, obviously. Besides the convolution function additional effects, mainly the influence of the overlap of different wavelength components has to be taken into account. And due to the chirp introduced by material dispersion, the SC pulse is much longer than the pump laser pulse so that the convolution roughly reflects the SC temporal pulse shape. The SC pulse chirp is illustrated by the dotted green curve in Fig. 2 a, which shows the central wavelength of the overlapping interval of the SC for each delay position: the central wavelength increases with increasing delay. This means that for the highest maximum of the convolution function (around a delay of 74 fs), where several regimes of nonlinear dynamics were observed, longer SC wavelength components (around 784 nm) interact with the pump pulse, for the second lower maximum (around a delay of −87 fs), where only the steady state was observed, shorter wavelength components (around 768 nm) interact. We have observed that the interaction with short wavelength components has less impact on the nonlinear dynamics than with longer wavelength components, which is important, when the system behavior on a long-scale delay detuning is investigated. But, for a delay detuning on a sub-wavelength scale, which is discussed in this paper, the overlapping wavelength regions can be assumed to be constant, and the phase dependent effects do not result from the overlap of different wavelength components.In order to identify the phase relations between two pulses commonly interferometric techniques are used. Therefore, we calculated the linear interferometric cross-correlation Icc(t) of the fields of the incoming laser pulse and the SC generated in single pass:
which is shown as the black curve in Figs. 2 a and b. The cross-correlation function Icc(t) in Eq. (2) is a measure for the pulse energy that results from the interference of the two pulses at a certain delay position t. Via the cross-correlation function Icc(t), the absolute phase difference Δϕ between the two pulses is accessible, as Δϕ = ϕconst + cτ/λ, where ϕconst describes the phase offset introduced by the MSF and the resonator dispersion independent from the delay. A phase shift of 2π corresponds to the modulation period of the black curve in Fig. 2 a. The effects of the phase can be seen in Fig. 2 b, which is a zoom into the delay region from −5 fs to 45 fs. There, three different bifurcation structures can be identified: first, in the delay interval from −5 fs to 11 fs, a repeated switching between steady state and a period-2 cycle is observed. When the delay is increased, the bifurcation structure changes to show a repeated transition from steady state via a period-2 cycle to limit cycle, and back to steady state. For further increased delay, finally, the bifurcation structure consists of switching from steady state to limit cycle and back. Figure 2 b shows a clear correlation between the phase difference and the observed nonlinear dynamics: the more complex regimes (period-2 cycle and limit cycle) of nonlinear dynamics were only reached for constructive inference conditions resulting in a high pulse energy. For destructive inference conditions with lower pulse energy only the steady state was observed.Finally, plotting the average optical spectrum (averaged over 50 round trips after the transient time), a corresponding correlation can be observed, which is shown in Fig. 2 c: the spectrum broadened for high overlap energies and narrowed for low overlap energies. To investigate the dependence of the optical spectrum on the feedback phase in more detail, in Fig. 3 the phase dependent spectral evolution for one 2π-period for each of the three different bifurcation structures of Fig. 2 b are shown. For each delay position 50 successive spectra after the transient time are plotted on top of each other in one graph on a linear scale. For a better overview the spectra of the period-2 cycle after odd and even numbers of resonator round trips are plotted in black and red, respectively.
Fig. 3 Three examples of phase dependent spectral evolutions, each spanning a delay interval of 2.6 fs (~2π) and recorded at different delay positions. The occurring nonlinear dynamics are noted on the right axis; a) delay interval from −1.8 fs to 0.4 fs; b) delay interval from 15.6 fs to 18.2 fs; c) delay interval from 38.8 fs to 41.4 fs. For details, see text.
The first example (Fig. 3 a) shows the bifurcation from steady state to a period-2 cycle and back to steady state within the delay region from −2.2 fs to 0.4 fs. Within the regime of steady state from a delay position of −2.2 fs to −1.2 fs the spectral peak at a central wavelength of 775 nm with a bandwidth of 8 nm (full width at half maximum) is divided into two peaks separated by 42 nm, one at 756 nm and the other at 798 nm. This example clearly demonstrates that the spectrum can be modified in its structure within one single regime of nonlinear dynamics. After the spectral broadening within the steady state, a bifurcation to a period-2 cycle occurs at a delay detuning of −1.2 fs. Instead of the single, constant spectrum of steady state, now the system displays two different spectra (indicated with black and red), which are assumed alternatingly. The period-2 cycle can be observed up to the delay position of 0.2 fs, where a bifurcation back to steady state was observed. While, for delays close to the bifurcation points, the two spectra are similar, they become very different for delays in the center of the period-2 regime. At a delay position of −0.6 fs for example the red spectrum has one dominant peak at 775 nm, while the black spectrum consists out of several peaks and has very low spectral intensity at 775 nm. This 2π-period scenario repeats several 2π-periods in a very similar form.
The second example (Fig. 3 b) shows a 2π-period from 15.6 fs to 18.2 fs, i.e. the 7th cycle after the one shown as the first example of Fig. 3 a. Here, the system bifurcates from steady state to a period-2 cycle to limit cycle and back to steady state. The evolution of the steady-state spectrum is very similar to the first example but now the steady state is stable over a wider delay detuning. In the delay interval between 17.0 fs and 17.6 fs, the existence of two spectra is again an indication for period-2 cycle. In the regime of limit cycle from 17.8 fs to 18.0 fs the spectra of successive resonator round trips fill the whole area between a lower and upper limit spectrum. Note, that close to the bifurcation point from limit cycle to steady state at 18.0 fs, this area filled by limit cycle spectra is contracted, until it resembles the single-curve spectrum of steady state. This example demonstrates that the size of the filled area, which represents the difference between the spectra that can be assumed in the regime of limit cycle, can be changed by varying the feedback phase.
The third example (Fig. 3 c) shows the 2π-period around a delay position of 40 fs, i.e. the 9th cycle after the one presented in Fig. 3 b. Here the bifurcation structure does not longer contain a period-2 cycle, but only the direct transition from steady state to limit cycle and back. The spectral evolution of the steady state is again similar to the previous examples, except it prevails over an even wider delay detuning. Again one single spectral peak (at 38,8 fs) is divided into two dominant peaks (at 40 fs), with a separation varying from 52 nm (at 40 fs) to 42 nm (at 40.6 fs).
These three examples lead to important conclusions: firstly, for a future experimental realization the strong dependence of the spectra on the feedback phase indicates the necessity of a good resonator length control. In Fig. 3, where a step length of 60 nm (corresponding to a temporal delay of 0.2 fs) was used, only a gradual change of successive spectra was observed. In an experimental realization a length stability in the same order (< 60 nm = λ/13) is expected to be required. Secondly, these examples show, that the optical spectrum strongly depends on the interference conditions, resulting in the possibility of spectral modification within one single regime of nonlinear dynamics via the feedback phase.
3. Summary and conclusions
The numerical simulations show, that effects on two different scales of the delay detuning have to be considered to understand the system’s nonlinear behavior: on a scale of small (sub-wavelength) delay changes, interferometric effects are dominant. A certain nonlinear behavior is assumed by the system, depending on the cross-correlation of the incoming pump pulse and the fed back SC pulse. The periodically changing interference conditions as a function of the delay position lead to a bifurcation structure with a periodicity of 2π, and to corresponding changes of the optical spectrum. Considering the phase relation between the laser pulse and the SC pulse after the first resonator round trip, qualitative predictions about the complexity of the expected nonlinear dynamics and the spectral bandwidth become possible: constructive interference conditions lead to more complex nonlinear dynamics and broader spectra. These predictions can be used either to avoid or to use nonlinear dynamics within applications.
On a larger scale of resonator length detuning, the bifurcation structure itself changes gradually. This can partly be understood from the convolution pulse overlap, which is a function of the intensity profiles of both the pump and the SC pulse: different overlap conditions change the interaction between the pulses and therefore results in different bifurcation structures as a function of the delay.
It was also demonstrated, that the optical spectrum can be modified by the delay of the feedback without changing the regime of nonlinear dynamics. For steady state it was shown, that it is possible to transform a single-peak spectrum into a double-peak spectrum and to vary spectral distance between the two peaks. Furthermore, it was shown, that it is possible to change each of the spectra of a period-2 cycle. Within the regime of limit cycle the size of the area filled by spectra of successive resonator round trips can be changed, which means that the difference between the spectra that can be assumed can be changed by varying the feedback phase.
Our numerical simulations demonstrate the potential of the SC system with feedback to adjust the optical spectrum even within a single regime of nonlinear dynamics via the feedback phase. This implies that the system dynamics and the optical spectrum can be adjusted in principle independently, which is an important prerequisite for the generation of tailored supercontinua. Our next step will be to verify these flexible system features experimentally.
References and links
1. R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970). [CrossRef]
2. R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24(11), 592–594 (1970). [CrossRef]
3. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000). [CrossRef]
4. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006). [CrossRef]
5. P. M. Moselund, M. H. Frosz, C. L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express 16(16), 11954–11968 (2008). [CrossRef] [PubMed]
6. Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. H. Knox, “Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber,” Opt. Lett. 30(10), 1234–1236 (2005). [CrossRef] [PubMed]
7. G. Steinmeyer, D. Jaspert, and F. Mitschke, “Observation of a period-doubling sequence in a nonlinear optical fiber ring cavity near zero dispersion,” Opt. Commun. 104(4-6), 379–384 (1994). [CrossRef]
8. G. Sucha, D. S. Chemla, and S. R. Bolton, “Effects of cavity topology on the nonlinear dynamics of additive-pulse mode-locked lasers,” J. Opt. Soc. Am. B 15(12), 2847–2853 (1998). [CrossRef]
9. M. Kues, N. Brauckmann, T. Walbaum, P. Gross, and C. Fallnich, “Nonlinear dynamics of femtosecond supercontinuum generation with feedback,” Opt. Express 17(18), 15827–15841 (2009). [CrossRef] [PubMed]
10. N. Brauckmann, M. Kues, T. Walbaum, P. Gross, and C. Fallnich, “Experimental investigations on nonlinear dynamics in supercontinuum generation with feedback,” Opt. Express 18(7), 7190–7202 (2010). [CrossRef] [PubMed]
11. N. K. T. Photonics, “NL-PM-750 data sheet,” http://www.nktphotonics.com/files/files/NL-PM-750-090612.pdf.
12. M. J. Litzkow, M. Livny, and M. W. Mutka, “Condor - a hunter of idle workstations,” in Proc. 8th Int. Conf. On Distributed Computing Systems (IEEE Computer Society Press, 1988), pp. 104–111.
