The manipulation of a supercontinuum via delayed optical feedback is investigated experimentally. The supercontinuum is generated in a microstructured fiber and a feedback ring resonator introduces the optical feedback and leads to the formation of different regimes of nonlinear dynamics. Via the feedback phase the optical spectrum and the regimes of nonlinear dynamics can be adjusted systematically. The impact of delay detuning on two different length scales, namely on a sub-wavelength scale and on a larger scale in the order of 10 μm are discussed. Additionally, the adjustment of the optical spectrum without changing the regime of nonlinear dynamics is demonstrated.
© 2010 OSA
The generation of supercontinua is the result of spectral broadening of light due to nonlinear effects . A common technique for ultrabroadband supercontinuum (SC) generation with relatively low input power is the injecting of laser pulses into microstructured fibers (MSF) . The SC generation can be modified by input pulse and fiber parameters, such as the input wavelength, power, pulse duration , chirp  and polarization , as well as the dispersion properties, that can be changed via the structure of the MSF, the material of the fiber  or by tapering the fiber [7,8]. Today SC generation can be influenced such that the SC is usable for applications, such as, i.e., optical coherence tomography [9,10], fluorescence microscopy  or optical frequency metrology [12,13]. Beside optimizing the usual input pulse and fiber parameters to form specific supercontinua, also new approaches for improving the coherence properties for example by modulating the input pulse  and for shaping the spectrum of the SC by two-color pumping , seeding with a pulse at another central wavelength , fiber cascades , fiber Bragg gratings  or long period gratings  were demonstrated. Another possibility to influence the spectral supercontinuum composition is the introduction of an optical feedback. This was investigated in picosecond systems, where SC generation relies on seeded four-wave mixing [20,21]. The introduction of an optical feedback can lead to a number of different effects: for example, the spectral composition or the pulse shape could be modified. In the case of feedback systems with the weaker nonlinearities of a single-mode fiber, however, the occurring effects can be much more complex, such as period multiplication or chaotic behavior [22,23]. Such system behavior has been shown to be caused by self-phase modulation and multiple-beam interference and to depend on the power in the feedback system.
In our previous work on a SC feedback system pumped with femtosecond laser pulses, steady state, period multiplication, limit cycle and chaos were found numerically and experimentally [24,25]. Experimentally, these different regimes of nonlinear dynamics could be identified via the temporal pulse energy evolution, which corresponds to characteristic radio frequency spectra: displaying only one frequency at the pump laser’s repetition frequency in the case of steady state, two or more equally spaced frequencies in the case of period multiplication, non-equally spaced frequencies in the case of limit cycle, and an evenly increased noise floor in the case of chaos (for detailed explanation and examples, see ).
In our recent numerical investigations, we predicted a phase dependence of the system behavior . The numerical simulations showed that effects on two different scales of the delay detuning have to be considered to understand the system’s nonlinear behavior: For small (sub-wavelength) delay changes interferometric effects are dominant, resulting in different optical spectra and different regimes of nonlinear dynamics within a phase detuning of 2π. With large resonator length detuning, the bifurcation structure itself changed gradually, resulting in different compositions of nonlinear dynamics. Those numerical investigations showed that both, namely the manipulation of nonlinear dynamics as well as of the optical spectra are connected in a SC generation feedback system pumped with femtosecond laser pulses.
In the present work our numerical investigations are experimentally verified. The manipulation especially of the optical spectrum via the delay, and considerations of the system stability, which would be interesting for applications, are discussed. The 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.
2. Experimental setup and overview of the system behavior
The experimental setup of our system is shown in Fig. 1a . It consisted of a ring resonator with 45 ± 1 mm of MSF, synchronously pumped by a mode-locked Titanium-Sapphire laser. Two pulse trains of perpendicular polarizations were coupled into the resonator. The p-polarized pulse train (indicated by orange arrows) was used to generate the SC, where the s-polarized pulse train (indicated by blue dots) provided a reference for interferometric measurement of the phase detuning.
For SC generation, up to 300 mW of laser power (p-polarized) were injected into the feedback resonator, with 54 fs pulse duration (full width at half maximum, Fourier limited) at 775 nm center wavelength and 82 MHz repetition rate. A pellicle beam splitter (BS) was used to reflect 12% of the incident pump power into the ring resonator. The high pumping losses (88%) were accepted in order to enhance the feedback efficiency. The SC was generated by focusing the pump pulses with a 40x microscope objective (MO) into a commercially available polarization maintaining MSF (NKT Photonics, NL-PM-750 ). All following input power values refer to the effective power inside the fiber, thus considering over-all coupling losses of about 50%. The maximum power that could be coupled into the fiber with our setup was 18 mW (300 mW · 0.12 · 0.5). The results presented in the following were measured at a power of 5 mW inside the fiber. The fiber was rotated to couple the p-polarized signal beam into the fast axis to generate the widest SC and to maintain the polarization for all spectral components . The generated SC was collimated behind the fiber with a second 40x MO, before the resonator round trip of the SC was completed by passing the pellicle beam splitter (88% transmission). In order to arrange a superposition of the back-coupled SC with the next laser pump pulse, the resonator length was matched to the repetition rate of the driving laser system. The feedback efficiency was measured by recoding the power at the output port with and without feedback and yielded values between 15% to 20%.
Before the pump laser beam entered the feedback resonator, a fraction of about 40 mW was tapped with a half-wave plate (λ/2) and a polarizing beam splitter (PBS) to generate the reference signal. The reference pulses were chirped by dispersive material to around 600 fs and delayed relative to the signal pulses. The chirp reduced the peak power considerably to ensure that the reference beam was not affected by nonlinear effects within the fiber. Due to the time delay, nonlinear interactions like cross-phase modulation between the signal and the reference beam were avoided. The reference pulse was then spatially recombined with the p-polarized signal beam at a second polarizing beam splitter and injected into the ring resonator. To ensure that the polarization was maintained, the reference beam was coupled into the slow axis of the MSF.
To observe both the signal beam and the reference beam, a reflection of an uncoated glass substrate within the resonator was used, behind which the two pulse trains were easily separated using a polarizing beam splitter. The fraction of the signal beam was used to simultaneously record the optical spectrum (OSA, optical spectrum analyzer ANDO AQ 1425) and the radio frequency spectrum (silicon photo diode with rise time ~1 ns connected to a radio frequency spectrum analyzer Advantest TR 4131/E). The relative feedback phase was measured by recording the output power of the reference beam, which was sinusoidally modulated as a function of the delay resulting from varying interference conditions. This measurement device enabled an interferometric, real time monitoring of the delay position with the high, sub-wavelength precision required, where also (thermal) drift of the resonator length was taken into account. The optical length for the signal pulses was also slightly changed by nonlinear effects, which was not detected by the reference beam. Only mechanically induced length changes were monitored in order to enable a direct comparison with our numerical simulations. Note that because of the different refractive indices of the fiber for perpendicular polarizations of the signal and reference beam only relative length changes could be measured. The reference beam was spectrally filtered with a prism and a slit in front of the detector in order to expand the coherence length and to achieve a sufficiently high contrast for a delay detuning of up to 700 fs. In order to adjust the optical path length of the ring resonator coarsely, a stepping motor was employed, which moved a pair of retro-reflecting mirrors in steps of 8 μm corresponding to a temporal delay of 53 fs. For precise adjustment of the resonator length and phase dependent measurements, the delay was tuned on a sub-wavelength scale with a piezo actuator with a maximum elongation of 25 μm.
In order to experimentally verify the predictions found numerically in reference , the optical and radio frequency spectra at an average input power of 5 mW were recorded, while the resonator length was detuned. Figure 1b shows an overview of the regimes of nonlinear dynamics that were observed within the delay range from −210 fs to 450 fs. Specific delay positions were adjusted coarsely with the stepping motor, where a precise scan with the piezo actuator was performed to measure the phase dependent bifurcation structures, which were extracted from the radio frequency spectra. As was expected due to the normal dispersion introduced by the microscope objectives, the central wavelength of the spectral SC interval (~10 nm) that temporally overlapped with the pump pulses (overlapping SC wavelength interval) increased with increasing delay. This information was extracted from the optical spectra at different delay positions as is explained in the following: Consider that the spectrum S(ω) of two electric fields delayed by τ in the time domain E(t) = E0(t) + E0(t + τ) is the result of the square norm of the Fourier transform of the resulting electric field E(t):
One can see from Eq. (1), that the spectrum is periodically modulated with the modulation frequency τ, which represents the delay between the two pulses in the time domain. Due to the dispersion the SC experienced during one round trip, the delay in the SC feedback system was a function of wavelength τ (λ). The modulation frequency was high for spectral components with large delay, decreased as the spectral components approached each other, and was zero for temporal overlap. The overlapping SC wavelength interval as a function of delay was extracted from the optical spectra by localizing the area of vanishing modulation. At the delay positions of −210 fs and 450 fs the wavelength interval around 735 nm and 840 nm overlapped, respectively. The overlap with even shorter and longer wavelength components showed no impact on the system’s behavior and therefore was not depicted. The delay position, where the laser’s central wavelength (775 nm) overlapped, was defined as zero delay.
The existence regions of the different bifurcation structures of the nonlinear dynamics are illustrated in Fig. 1b: For wide delay regions only the steady state (P1) was observed. In the delay region around −20 fs and 150 fs a switching between steady state (P1) and a period-2 cycle (P2) was detected. Those delay regions corresponded to an overlapping SC wavelength interval centered around 770 nm and 805 nm, respectively. Furthermore, around the delay position of 210 fs and an overlapping SC wavelength interval around 820 nm a transition from the regimes of steady state to a period-2 cycle to limit cycle and back to steady state was observed. Around a delay position of 290 fs and an overlapping SC wavelength interval around 830 nm the direct transition from steady state to limit cycle and back to steady state was identified.
The above results confirm two of our numerical predictions : firstly, the interaction with longer wavelength components in the region from 800 nm to 835 nm had more impact on the nonlinear dynamics than with shorter wavelength components. Secondly, a large scale delay detuning, resulting in a detuning of the overlapping SC wavelength interval, changed the specific bifurcation structures of the observed nonlinear dynamics.
3. Phase dependent system behavior
In order to investigate phase dependent effects three detailed piezo scans at the positions marked with A (130 fs), B (215 fs) and C (270 fs) in Fig. 1b are presented in the following paragraphs. Each of the three measurements spanned a delay interval of 8π (~10 fs at 775 nm), during which around 1000 radio frequency spectra and (due to the longer recording time) 80 optical spectra were recorded. This resulted in an average phase difference of λ/250 for successive radio frequency spectra and λ/20 for the optical spectra. The overlapping SC wavelength interval could be assumed to be constant for each of the detailed piezo scans. The delay between the measurement interval A (Fig. 2 ) and B (Fig. 3 ) was 65π (~85 fs at 775 nm), and between interval B, and C (Fig. 4 ) the delay was 42π (~55 fs at 775 nm).
In Fig. 2 the first example of a phase-dependent measurement is shown. The evolution of the radio frequency spectra is plotted in Fig. 2a and the according evolution of the simultaneously measured optical spectra in Fig. 2b. Each line in the two graphs corresponds to a radio frequency spectrum (Fig. 2a) and an optical spectrum (Fig. 2b), where the spectral power density is color-coded on a logarithmic scale. A more elaborated description of this measurement approach can be found in . The phase dependent bifurcation structure of the nonlinear system dynamics can be deduced from the data shown in Fig. 2a. In each of the radio frequency spectra the frequency peak at 82 MHz indicated the laser’s repetition rate. In this first example an additional peak at half of the repetition rate appeared and disappeared when the delay was detuned, and the cycle was repeated after the delay was changed by 2π. Within a 2π delay shift the system changed from steady state (P1) to a period-2 cycle (P2) and back.
Phase-dependent effects could be found in the simultaneously recorded optical spectra (Fig. 2b) as well. Specifically, the spectra were narrowed and broadened in a repeating 2π-cycle. Comparing the bifurcation structure of Fig. 2a and the spectral changes of Fig. 2b, one observes, that with increasing feedback phase, there was mostly a gradual spectral narrowing followed by a abrupt transition to a broad spectrum. Furthermore, one can observe that the transition from P2 to P1 was within the interval of gradual spectral narrowing, while the transition from P1 back to P2 was close to the abrupt transition of the optical spectrum, but the two transition points did not coincide (see the horizontal white lines indicating the transitions between nonlinear dynamics). Instead, the transition of the optical spectrum took place at a slightly smaller feedback phase than the transition between the nonlinear dynamics, which is in good agreement with our numerical investigations , but a demonstrative explanation needs a deeper insight into the system behavior and is subject of our future investigations. In the regime of steady state (P1, Fig. 2c) the −30 dB-width of the broadest spectrum was 260 nm, and of the narrowest 120 nm, i.e., the spectral bandwidth differed by more than a factor of two. For comparison: the spectral width of the according spectrum without feedback was 200 nm. In the case of the period-2 cycle (P2, Fig. 2d) the −30 dB-widths of the broadest and narrowest spectra were 270 nm and 210 nm, respectively. In addition to the difference in widths, the spectra within both regimes differed by their spectral shapes, i.e., by the relative spectral intensities at different wavelengths. Note, that in the case of a period-2 cycle, each of the measured spectra was in fact the average over two spectra generated alternately, due to the acquisition of spectra being much slower than the 82 MHz repetition rate of the driving laser.
The second phase-dependent measurement around a delay position of 215 fs is shown in Fig. 3. In the radio frequency spectrum evolution (Fig. 3a) a semicircle structure was observed, which was formed by unequally spaced frequency peaks indicating a limit cycle (LC) . When increasing the delay, the characteristic frequencies of the limit cycle shifted until the system switched to a period-2 cycle with a defined peak at half the laser’s repetition rate. After that, the system state switched to steady state, and finally again to a limit cycle, and the whole bifurcation structure was repeated after a 2π-period. Corresponding to the more complex bifurcation structure, also the optical spectra (Fig. 3b) were more complex. Again, the spectral bandwidth varied, but also spectral features like peaks and dips were shifted within one 2π-interval. Within each of the regimes P1 and LC, the optical spectra could be changed considerably via the delay, as can be seen from the spectra enclosed by the white lines indicating the transition between nonlinear dynamics in Figs. 3a and 3b. To illustrate this for each case, two spectra of the regimes of P1 and LC are plotted together in Figs. 3c and 3d, respectively. Note, that the spectra in the limit cycle regime were averaged over all its different spectra. Since the period-2 cycle existed only within a very small delay region, the spectrum was not shifted much in this example and is, therefore, not illustrated separately.
The last example (Fig. 4) of measurements around a delay position of 270 fs shows a cross-like bifurcation structure (Fig. 4a). This structure was formed by unequally spaced frequency peaks of a limit cycle, which appeared alternately with steady state by varying the delay, indicating a direct bifurcation from steady state to limit cycle and back to steady state. The simultaneously measured optical spectrum evolution in Fig. 4b shows significant variations in width as well as shape within each of the regimes of nonlinear dynamics, very similar to the measurements plotted in Fig. 3.
For higher input powers more complex bifurcation structures could be observed including higher orders of period multiplication and chaotic behavior. However, all measurements showed a 2π-periodicity of the bifurcation structure and the optical spectrum evolution.
With these phase dependent measurements two further numerically found predictions  could be verified: firstly, phase dependent bifurcation structures in the regimes of the nonlinear dynamics and corresponding changes in the optical spectra with a periodicity of 2π were found experimentally. And secondly, the potential of the feedback system for spectral shaping within one regime of nonlinear dynamics was demonstrated.
In our experiments the measurement period to record one single optical spectrum was in the order of some seconds. In spite of this long averaging time and without using any active delay-stabilization devices, it was possible to observe the strongly phase-dependent modulations in the optical spectrum. For applications that require a stable spectrum for a longer duration of minutes or hours, where thermal drifts have to be taken into account, an active delay stabilization device will be required. Since the presented experiments show about 20 different spectra within a delay detuning of one wavelength with mostly gradual changes between successive spectra, a precision of about λ/20 will be needed for those applications.
4. Summary and conclusions
The reported experimental results verify the numerical predictions that changing the feedback delay, leads to effects on two different length scales: on a scale of small delay detuning in the order of a wavelength, the system behavior depended on the feedback phase. Specifically, a certain bifurcation structure was displayed, during which the system switched between multiple regimes of nonlinear dynamics, and which was repeated in similar form with a periodicity of 2π. Additionally, the optical spectrum was influenced in its shape and bandwidth by the feedback phase with the same periodicity and could be adjusted independently from the regimes of nonlinear dynamics. On a scale of larger length detuning of several tens of wavelengths the system behavior showed very different bifurcation structures as well as different evolutions of the optical spectra during 2π-periods at different delay positions.
Both our experimental and numerical investigations  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 independently, which can be an important prerequisite for the generation of tailored supercontinua.
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