1. Introduction

Optical fibers with small core diameter allow high peak intensities in combination with relatively long interaction lengths, resulting in strong nonlinear effects. Thus, the propagation of ultrashort laser pulses in fibers near the zero-dispersion wavelength is suitable for broadband supercontinuum (SC) generation [1].

Supercontinua are usable for many applications, e.g., optical coherence tomography [2, 3], fluorescence microscopy [4] or optical frequency metrology [5, 6]. These applications require special SC properties, for instance a special spectral composition is required for optical coherence tomography, or a high degree of pulse-to-pulse stability for frequency metrology. To meet these demands, on the one hand SC can be modified by input pulse [1, 7, 8] and fiber parameters [9–11]. On the other hand the coherence properties were for example improved by modulating the input pulse [12], and the spectrum was shaped by two-color pumping [13], by using fiber cascades [14], or by fiber Bragg gratings [15].

While these approaches mostly improve one specific feature of the SC, the introduction of an optical feedback was demonstrated to lead to a number of different effects, such as the manipulation of the optical spectrum or the manipulation of the nonlinear dynamics with its characteristic temporal evolution of the SC pulse train. The influence on the SC spectral shape was demonstrated for optical feedback systems pumped with picosecond pulses [16,17], where SC generation relies on seeded four-wave mixing, as well as for femtosecond systems [18, 19], where self-phase modulation and higher order dispersion are the dominant effects leading to the spectral broadening. Furthermore, the occurrence of nonlinear dynamics due to optical feedback was shown for femtosecond SC feedback systems [20, 21], as well as for the case of feedback systems with the weaker effective nonlinearity of a conventional single mode fiber [22, 23]. By adjusting different regimes of nonlinear dynamics it is possible to change the pulse train evolution, e.g., to generate SC pulse trains with alternating spectral shapes, as fast as the laser’s repetition rate [18, 19].

Beside the spectral composition and the pulse train evolution, the stability of SC light sources especially for frequency metrology is an important topic, because fluctuations of the SC lead to a reduced signal-to-noise ratio and thus to a limited applicability of supercontinua. SC stability in dependence on the input pulse parameters has extensively been investigated, considering technical, i.e., low-frequency amplitude noise [24] and quantum shot noise [25, 26] as well as coherence properties [27]. As a result a rule of thumb was proposed: for high-stability SC generation, short input pulses (≈ 50 fs) with relatively low peak power (a few kW) should be used in combination with a relatively short (some cm), anomalously dispersive fiber.

In this work, the suitability of using the optical feedback to improve the SC noise properties is analyzed. The impact of an optical feedback on the technical SC noise properties is investigated numerically and experimentally by comparing the feedback system’s characteristics with those of a single-pass SC generating system, i.e., without feedback. Technical noise is typically the dominant noise source in the low frequency range of up to a few 100 kHz. Especially residual relaxation oscillations, that can hardly be avoided even with highly stable pumping conditions, lead to slow amplitude modulations of the pulse train of a mode-locked laser which can result in Q-switched mode-locked operation. Thus, in order to demonstrate the noise reduction method presented in this work, the laser was adjusted to the Q-switched mode locking regime, resulting in a pulse train modulated with a frequency of 450 kHz. In a first step, noise reduction of relatively high amplitude noise in the order of 1% to 10% of the pulse amplitude for this specific noise frequency is demonstrated numerically and experimentally. In a second step, the suitability of the presented noise reduction technique for lower amplitude noise and for other noise frequencies is discussed. In the following, the noise reduction of the power fluctuations as well as the noise reduction of the spectral amplitude fluctuations of individual frequency components will be presented.

In addition to our previous work [18, 19], which showed the possibility of adjusting the optical spectrum and the nonlinear dynamics with its characteristic pulse train evolution via optical feedback, the work presented here demonstrates that also the SC noise properties can be controlled via optical feedback.

2. Experimental setup

The experimental setup is sketched in Fig. 1. The laser system together with the SC generating setup including the feedback cavity are illustrated in part A. The analysis setup of the power noise is illustrated in part B and the spectral amplitude noise measurement setup in part C.

 

Fig. 1 Experimental setup; part A: laser system and feedback cavity, part B: power noise measurements, part C: spectral amplitude noise measurements. BS: beam splitter, MSF: microstructured fiber, MO: 40x microscope objective, GS: uncoated glass substrate, F: spectral bandpass filter, PD: photodiode, M: mirror only used for option B, for more details see text.

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The laser system consisted of a Titanium:Sapphire laser (Tsunami by Spectra-Physics) followed by a Faraday isolator to avoid reflections back into the laser resonator, and a prism compressor to compensate the material dispersion especially of the Faraday isolator. After passing this prism compressor, a pulse duration of 54±6 fs (full width at half maximum, FWHM, and assuming a sech² intensity profile) was measured with an autocorrelator (model Mini by APE) at 775 nm central wavelength and 82 MHz repetition rate. In order to create a signal with a defined modulation frequency the laser was adjusted to operate in the Q-switched mode-locked regime [28], and with this technique an amplitude modulation of the laser’s 82 MHz output pulse train with a frequency of 450 kHz was achieved.

The pulses of the pump laser system were coupled into the ring cavity with a pellicle beam splitter (BS) with a splitting ratio of 12:88. This low coupling efficiency of the pump power (12%) was accepted in order to enhance the feedback efficiency. Within the ring cavity the SC was generated by focusing the pump pulses with a 40x microscope objective (MO) into a 45 mm long commercially available polarization maintaining microstructured fiber (MSF) (NKT Photonics, NL-PM-750 [29]) with a nonlinear parameter of γ = 0.095 W⁻¹m⁻¹. The input power values in the measurements mentioned within the following sections refer to the effective power within the fiber, thus considering the coupling losses. The fiber was rotated such that the linearly polarized pump light was coupled into the fast axis, where the widest SC was generated, and the polarization was maintained for all spectral components [8]. Behind the fiber the generated SC was collimated with a second 40x MO. The ring cavity round trip was completed by the SC passing the pellicle beamsplitter (transmission efficiency of 88%), to be coupled back into the MSF. In order to arrange a superposition of the back-coupled SC with the next laser pulses, the cavity length was matched to the repetition rate of the driving laser system. Since the effects that are presented in the following crucially depended on the feedback phase, a piezo actuator was used to accomplish a delay scan on the sub-wavelength scale. The phase shift could be measured with the help of a second, less intensive, perpendicularly polarized reference beam, as was explained in detail in [19]. This beam was additionally coupled into the feedback system and could be estimated to propagate linearly and independently from the highly intensive beam for SC generation. Thus, the interference of the reference beam within the resonator could be used to extract changes of the feedback phase.

The feedback efficiency was measured by recording the power at the output port with and without feedback and yielded values between 15% and 20%. Due to the material dispersion, the feedback resonator was normally dispersive for the SC wavelength range with a dominant quadratic phase term of 0.9 · 10⁻³ps² · (ω – ω ₀)² for the laser’s central frequency ω ₀ = 2π · c/775 nm. Thus, the feedback resonator was not synchronously resonant for all SC wavelengths, and by tuning the resonator length, i.e., the delay, different wavelengths could be chosen to be resonant (this was investigated in detail in [18, 19]).

For analysis of the system characteristics the partially reflected beam from an uncoated glass substrate (GS) was used. For the results presented in section 4.2 concerning the power noise reduction, the signal was detected with a silicon photodiode (PD 0) connected to a radio frequency spectrum analyzer (Advantest, TR 4131/E), as is shown in Fig. 1 part B. In order to measure the effect of spectral amplitude noise reduction presented in section 5.2, a spectral bandpass filter (F) was incorporated into the feedback cavity, so that only spectral components between 750 nm and 800 nm were fed back. Furthermore, the beam was split into four partial beams, as is shown in part C of Fig. 1, in order to enable spectrally resolved measurements of fast fluctuations: First, the beam was divided into two parts with a 50:50 beam splitter (BS), where one part was directly detected with a photodiode (PD 1) as a measure of the average SC power. The second part was dispersed using a prism, and three spectral ranges of 10 nm bandwidth (FWHM) with central wavelengths at 730 nm, 775 nm, and 830 nm were filtered with slits, and the according power values were separately measured with three photodiodes (PD 2, PD 3, PD 4). The photodiodes were chosen to have a response time between 0.1 μs and 1 μs, which was much longer then the laser’s repetition rate (12 ns) to average over the laser pulses, but which was fast enough to resolve the amplitude modulations of interest at a frequency around 450 kHz (≙ 2.2 μs). Time series of all four generated signals were measured simultaneously with a four-channel oscilloscope (Tektronix, DPO7000).

3. Numerical implementation

In order to simulate the experimental setup, we used a numerical model of the SC feedback system, which was developed and presented earlier [20]. A schematic diagram of the SC feedback system is shown in Fig. 2, illustrating the procedure for feeding back the whole spectral bandwidth within path A, and, for feeding back the spectrally filtered SC within path B. For both cases, the electric field for each resonator round trip (E _input) was calculated within four main steps indicated by the gray boxes in the diagram: firstly, the pulse propagation within the nonlinear fiber was modeled 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 E _fiber in Fig. 2. Secondly, the pulse envelope overlap of the SC pulse with the next pump pulse was chosen by introducing a delay (resulting in E _delay). Thirdly, the pulses were shaped by emulating the material dispersion, which would be introduced by microscope objectives in the experimental setup in order to simulate realistic experimental conditions. For path A this resulted in the chirped SC pulse E _resonator. In path B a spectral bandpass filter with a FWHM of 18 nm centered at 775 nm was additionally considered by multiplying the electric field with a super-Gaussian filter function. Fourthly, in both cases the n-th resonator round trip (n = 1,2,...) was finished by superimposing a fraction (ε) of the SC within the resonator ( $E_{\text{resonator}}^n$) with the next incoming pump pulse (E _pump), thus providing the initial conditions for the next resonator round trip ( $E_{\text{input}}^{n+1}\hspace{0.17em}=\hspace{0.17em}{E}_{\text{pump}}\hspace{0.17em}+\hspace{0.17em}\varepsilon \hspace{0.17em}\cdot \hspace{0.17em}{E}_{\text{resonator}}^n$). To create the initial conditions for the first resonator round trip (n = 1), the input pulse $E_{\text{input}}^1$ was set to the pump pulse E _pump. The electric fields before passing the MSF (E _input(t)), after passing the MSF (E _fiber(t)), and before superposition with the next pump pulse (E _resonator(t)) as well as their Fourier transforms (Ẽ _input(ω), Ẽ _fiber(ω), and Ẽ _resonator(ω)) were saved after each cavity round trip for data evaluation purposes.

 

Fig. 2 Schematic sketch of the program blocks performed within the numerical simulations of the SC feedback system. Path A: procedure for feeding back the whole SC spectral bandwidth for analyzing the average power noise. Path B: procedure for feeding back the spectrally filtered SC for analyzing the spectral amplitude noise. For a detailed explanation see text.

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For the numerical results presented in the following, the same fiber parameters were used as in our previous work (derived from the data sheet of the MSF used in the experiments [20]), and the pulse parameters were chosen to represent a squared hyperbolic secant intensity profile with a pulse duration of 60 fs (FWHM) at a central wavelength of λ ₀ = 775 nm.

4. Power noise reduction

In the SC generating feedback cavity two main effects have to be considered, namely the nonlinearity of the MSF and interference effects caused by the optical feedback. The fiber nonlinearity leads to a strong intensity dependent system behavior, especially to an intensity dependent phase shift, i.e., the nonlinear phase shift caused by the fiber Kerr-nonlinearity increased with the intensity. The second effect, the linear interference effect leads to intensity changes of the pulses that are re-coupled into the nonlinear fiber. If the intensity is increased or decreased depends on constructive or destructive interference conditions and thus on the feedback phase. Considering both effects, the feedback system acts as follows: in dependence on the intensity and pulse shape of the input pulse a certain nonlinear phase shift is introduced by the fiber nonlinearity, and a specific optical spectrum is generated. This leads to changed interference conditions of the fed back SC with the pump pulse and thus to a modified pulse intensity and pulse shape for the next cavity round trip. The aim of this work was to use both effects to stabilize the input pulses in front of the MSF for fluctuating pump pulses. Thus, the effect of the fiber nonlinearity and the interference conditions of the pulses within the ring resonator had to be balanced, such that higher pump powers were reduced and lower pump powers were increased via feedback resulting in stable input pulses. Since the input pulses form the initial conditions for SC generation, stable input pulses resulted in stable SC pulses. The effect of power noise reduction is demonstrated in the following by comparing the feedback system behavior to the behavior of the single-pass system.

4.1. Numerical results

In order to illustrate the phase and intensity dependence of the SC generating feedback system and to find parameter settings, where noise reduction can be expected, we numerically calculated the feedback system intra-cavity power for different pump power values. The results are shown in Fig. 3(a), where the feedback system intra-cavity power was calculated for pump powers from 1 mW to 10 mW in dependence on the feedback delay from 0 fs to 5 fs, which corresponds to a phase shift from 0 to 3.88π (= 2πc· 5 fs/λ ₀, where λ ₀ = 775 nm is the central wavelength and c is the velocity of light in vacuum).

 

Fig. 3 Numerical results: feedback system output power as a function of the feedback phase for pump powers from 1 mW to 10 mW in steps of 1 mW. The feedback system showed a phase and intensity dependent behavior. Regions with approached proximate curves (one example is marked with the blue circle) indicate noise reduction; regions with far separated proximate curves (one example is marked with the red ellipse) indicate noise amplification. a) exact calculations; b) calculations with adapted wavelength-dependent response function of the photodiode that was used in the experiments.

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Each of the curves in Fig. 3(a) shows the combination of a delay dependence and an intensity dependence. The delay dependence can be seen from the power not being constant for different delay positions. The intensity dependence is reflected by the fact that the curves are not only offset by a value corresponding to the pump power, but that they also differ in shape. The deformation of the curves led to a varying distance between proximate curves as a function of the delay and the pump power. In regions where proximate curves approached each other closely (see for example the region marked by the blue circle), the feedback system intra-cavity power is nearly the same for different values of the pump power. For the example marked with the blue circle the intra-cavity power variations were reduced to 8.9% of the pump power variations of 1 mW, indicating power noise reduction. However, the feedback did not necessarily lead to a noise reduction; with other parameter settings, the input variations could also be amplified. For example, with a slightly changed delay position (1.8 fs instead of 1.4 fs), the distance between the proximate curves increased drastically, and the intra-cavity power fluctuations were amplified to 197.9% of the pump power variations (see the red ellipse in Fig. 3(a)).

Figure 3(b) shows the same results as Fig. 3(a) but with considering the wavelength-dependent response function of the photodiode that was used in the experiments for signal detection. Thus it shows how the graph would be measured under realistic experimental conditions, which was used to estimate the measurement error. Up to a pump power of around 5 mW no significant differences were observed, and thus a relatively low error of ±1 dB could be estimated in this power range. For increased pump power the bandwidth of the SC optical spectrum further increased resulting in a more significant impact of the response function of the photodiode. The evolution of each curve did not significantly change, but the detected power decreased. As a result proximate curves approached, suggesting an improved measured noise reduction. Thus the measurement error depended on the pump power and was estimated for each particular experimental case from those numerical investigations.

Figure 3(a) shows the feedback system’s answer, when it is driven at different, but temporally constant pump powers. The system’s response to dynamical changes is not included. Strictly speaking, this consideration only describes the system response to infinitely slow perturbation frequencies. However, it can be used to identify parameter ranges, where noise reduction can be expected. Thus, in the next step the promising parameter range marked with the blue cycle is investigated in more detail. In order to consider the system’s inertia and dynamical response the pump power was modulated. The amplitude variations of the feedback system were analyzed for pump power modulations at a frequency of 450 kHz, which models the amplitude modulations of the experimentally used laser in the Q-switched mode-locked regime. Figure 4(a) shows the results of a simulation at a delay position of 1.4 fs, were the pump power was set to 6.8 mW for 50 cavity round trips until the steady state was reached. Then the pump power was increased and decreased sinusoidally from 6.8 mW to 6.9 mW with a frequency of 450 kHz considering a pulse repetition rate of 82 MHz (corresponding to a period of 2.2 μs). The pump power evolution and feedback system intra-cavity power evolution of the transient phase and the first three modulation periods are plotted in Fig. 4(a) and (b), respectively. While the pump power and so the corresponding power of the single-pass system varied by 0.1 mW, the feedback system power only varied by 0.01 mW. A reduction of the relative noise of 11.5 dB was found for the perturbation frequency of 450 kHz.

 

Fig. 4 Numerical results: a) evolution of the pump power varied in the range from 6.8 mW to 6.9 mW; b) according evolution of the resulting feedback system output power.

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In comparison to the presented dynamical perturbation, for the stationary case, i.e., without considering the system’s dynamical response as was illustrated in Fig. 3, a considerably better noise reduction of 20.44 dB was calculated. This difference results from the increased impact of the dynamical system response as a function of the perturbation frequency, which leads to changed system behavior. Thus, for perturbation frequencies of up to 450 kHz a noise reduction between 11.5 dB and 20.44 dB can be estimated for the investigated parameter setting.

Considering that the influence of the system’s dynamical response increases with increasing perturbation frequencies, the noise reduction of 11.5 dB for a perturbation frequency of 450 kHz can be estimated as lower noise reduction limit for perturbation frequencies of up to 450 kHz for the investigated parameter setting.

4.2. Experimental results

In order to experimentally verify the numerically predicted effects of pulse noise reduction qualitatively, we used the setup in Fig. 1 part A without the spectral bandpass filter. For analysis a photodiode connected to a radio frequency spectrum analyzer was used as shown in part B of Fig. 1. The parameters were not exactly matched to the ones used in the simulations. Without the exact knowledge of all parameters, quantitative predictions of the system behavior seem unrealistic for the complex system under consideration. However, the presented results showed good qualitative agreement between experiments and simulations. The radio frequency spectrum of the feedback system was measured up to a frequency of 1 MHz at a pump power of 8.5 mW, while the delay was detuned by up to four wavelengths (8π) of the laser’s central wavelength of 775 nm. The result of this phase dependent delay scan is shown in Fig. 5(a). Note, that the stated values for the feedback phase are only relative with respect to the start position of the measurement. Each line in the graph corresponds to one radio frequency spectrum, where the spectral power is color-coded on a logarithmic scale. The radio frequency spectra were normalized to the average power for each single measurement. This was necessary because the system output power varied in dependence on the delay due to interferometric effects as was predicted by the numerical simulations (see Fig. 3(b)).

 

Fig. 5 Power noise measurement at a pump power of 8.5 mW with perturbations at a frequency of 450 kHz; a) normalized phase dependent evolution of the radio frequency spectrum at the feedback system output; b) corresponding evolution of the noise peak amplitude, where the red curve indicates the peak amplitude of the single-pass system.

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Figure 5(a) shows that the frequency peak at 450 kHz, which was introduced by the amplitude modulation of the pump laser pulses, was clearly visible in the radio frequency spectrum of the feedback system during the whole delay scan, but that the magnitude of the frequency peak varied as a function of the feedback phase. For a better illustration of the quantity of the peak amplitude variation in the radio frequency spectra, the evolution of the peak amplitude is plotted as a function of the delay in Fig. 5(b). The normalized peak amplitude of the single-pass reference measurement is plotted as the red line in the same graph for comparison. Note, that a silicon photodiode was used for the measurements, which showed a wavelength-dependent response function for the broadband spectral range of some hundreds of nanometers which was typical for the measured supercontinua. With the help of the simulations and considering the wavelength-dependent response function of the photodiode (see Fig. 3), an error of ±5 dB for the noise reduction was evaluated for the presented power range. It was observed, that with considering the characteristics of the photodiode typically an improved noise reduction was suggested. Thus, in order to include this inaccuracy, the measured noise reduction values have to be reduced by 5 dB, which resulted in a demonstrated power noise reduction of 15 dB. However, the effect of power noise reduction crucially depended on the feedback phase: when the feedback phase was shifted by half a wavelength (π), the power noise reduction was transformed into power noise amplification. Hence, the experimental results verify the numerically predicted effects, that a power noise reduction can be realized by introducing an optical feedback, and that this effect is strongly phase dependent.

5. Spectral amplitude noise reduction

5.1. Numerical results

For investigations on the spectral amplitude stability not only the power has to be considered, but also the variations of each single spectral component of the generated SC. As a first step, the numerical data of the previously generated time series (see section 4.1) were analyzed regarding their spectral contents, and the results are plotted in Fig. 6. The optical spectra that were calculated when the pump power was varied between 6.8 mW and 6.9 mW, are plotted for the single-pass system in Fig. 6(a) and for the feedback system in Fig. 6(b).

 

Fig. 6 Numerical results without filter: optical spectra for varying pump power between 6.8 mW and 6.9 mW (180 spectra are plotted on top of each other) a) without and b) with feedback; c) corresponding spectral variations without (red dashed line) and with (black solid line) feedback; d) corresponding relative spectral variations without (red dashed line) and with (black solid line) feedback.

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One can see, that the spectra that were calculated with feedback were more strongly modulated than those without feedback, which is the result of spectral interference effects in combination with the cavity dispersion.

For both the single-pass and the feedback system the spectral variations were calculated from the difference of the minimum and the maximum spectral intensity for each spectral component that was reached when the pump power was varied. The results are plotted in Fig. 6(c), which clearly shows that the variations of the feedback system (black solid line) were higher than those of the single-pass system (red dashed line) for most spectral components. However, for some spectral regions (around 800 nm, 830 nm, and 850 nm) the variations were suppressed via feedback. Because of the big difference of the spectral shapes for the single-pass and the feedback system the spectral variations have to be normalized to the according spectral intensities for a significant comparison, as is shown in Fig. 6(d) (with feedback: black solid line, single-pass: red dashed line). Also the normalized spectral variations were amplified via optical feedback for almost all spectral components. In addition to the reduction of the power noise, as was reported in section 4, the spectral amplitude fluctuations could only be reduced in a small wavelength interval from 791 nm to 803 nm via optical feedback. Due to the increased spectral amplitude fluctuations a power noise reduction seems not intuitive. But despite of high spectral amplitude fluctuations the power fluctuations can be reduced, if different spectral components fluctuate out of phase, i.e., if the spectral amplitude is increased for some spectral components while it is simultaneously decreased for others. Integrated over the whole optical spectrum in order to calculate the power fluctuations, this led to power noise reduction.

Due to the nonlinear effects, the generated supercontinua strongly depended on the exact form and amplitude of the input pulses and thus, in order to further improve the spectral amplitude noise characteristics for more wavelength components, the influence of feedback on the input pulses has to be evaluated in more detail. Therefore, the impact of feedback on the input pulses E _input is illustrated in Fig. 7(a), where the input pulses of the feedback system after the transient phase are plotted for constructive (blue dotted line) and destructive (black solid line) interference conditions as well as the pump pulse (red dashed line) at a power of 7 mW. The amplitude of E _input was increased and reduced for constructive and destructive interference conditions, respectively, which can be used to compensate the pulse amplitude fluctuations of the pump pulse and, therefore, is highly welcome. But beside the amplitude fluctuations also the pulse shape especially in the rising edge of the pulse was changed considerably, which is unwanted, because fluctuations of the pulse form do not help to compensate pump pulse amplitude fluctuations, but they result in fluctuations of the spectrum of the generated supercontinua. Note, that the main interference effects that are responsible for pulse amplitude fluctuations are due to the feedback of spectral components of the SC in the same wavelength range as the pump pulse spectrum. All other wavelength components do not interfere with the pump pulse, but they still influence the shape of the input pulse. By considering an additional spectral bandpass filter in the feedback cavity with a transmittance range around the pump pulse central wavelength of 775 nm, it was possible to reduce the fluctuations of the input pulse shape, thereby creating better controlled experimental conditions (see Fig. 7(b)). On the one hand, the input pulse form nearly stayed unchanged and was not affected by the feedback. On the other hand, the interference effects were preserved, and thus this modified setup provided optimized conditions for the compensation of pump pulse fluctuations, such that the input pulse E _input in front of the fiber was effectively stabilized.

 

Fig. 7 Feedback system input pulses for constructive (blue dotted line) and destructive interference (black solid line) conditions and pump pulse (red dashed line) at a pump power of 7ṁW a) without any filter and b) with spectral bandpass filter. c) Numerical results with spectral filtering: resulting feedback system output power in dependence on the feedback phase, when the pump power was varied from 1 mW to 10 mW in steps of 1 mW. The feedback system showed a phase and intensity dependent behavior. Regions with approached proximate curves (one example is marked with the blue circle) indicate noise reduction.

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The impact of the additional spectral filter on the SC feedback system without considering the dynamical system response was investigated by numerically calculating the intra-cavity power in front of the filter of the modified feedback system for the same pump power values as in Fig. 3 (from 1 mW to 10 mW). The results are depicted in Fig. 7(c), where the response of the feedback system again shows a complex dependence on feedback phase and pump power. Between 5 mW and 6 mW at a delay position of 1.2 fs two curves approach each other, i.e., this parameter range is expected to show noise reduction via feedback, and detailed numerical calculations including the dynamical system response to analyze the reduction of power variations were performed within this parameter range. The feedback system power was calculated for a pump power variation between 5.7 mW and 5.8 mW following the same routine as in Fig. 4. While the pump power was varied by 0.1 mW (Fig. 8(a)) the feedback system output power variations (Fig. 8(b)) were reduced by more than a factor of two to 0.042 mW. For this example a reduction of the relative noise of 3.54 dB was found for the perturbation frequency of 450 kHz.

 

Fig. 8 Numerical results with introduced spectral filter: a) evolution of the pump power varied sinusoidally in the range from 5.7 mW to 5.8 mW; b) according evolution of the resulting feedback system power.

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The optical spectra as well as the spectral amplitude variations for the pump power region from 5.7 mW to 5.8 mW for the single-pass system and the feedback system are plotted in Fig. 9. For this case, the spectrum of the feedback system (Fig. 9(b)) was very similar to the spectrum of the single-pass system (Fig. 9(a)), because spectral interference effects were suppressed by the spectral filter. The spectral amplitude variations and the spectral amplitude variations normalized to the corresponding mean spectra are plotted in Fig. 9(c) and Fig. 9(d), respectively. In both figures the spectral variations for the single-pass system are plotted as a red dashed line and those for the feedback system as a black solid line. The shape of the curves for the single-pass system and the feedback system look very similar. However, the black solid curve is below the red dashed curve, which shows that the spectral amplitude variations could be reduced for almost all spectral components. Nevertheless, the noise reduction was a function of wavelength. In average the noise was reduced by 2.37 dB in the wavelength region between 640 nm and 900 nm. Note, that the normalized spectral power density around 763 nm and 780 nm was very low (≪ 1) (see Fig. 9(a) and (b)), and thus the variation function was divided by a value ≪ 1 at these spectral components in order to calculate the normalized spectral amplitude variations. This resulted in very high relative variations for those wavelength components (see Fig. 9(d)), that were not representative and, therefore, should not be taken into account for physical interpretations.

 

Fig. 9 Numerical results with introduced spectral filter: optical spectra for varying system input power between 5.7 mW and 5.8 mW (180 spectra are plotted on top of each other) a) without and b) with feedback; c) corresponding spectral variations without (red dashed line) and with (black solid line) feedback and d) normalized spectral variations.

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The bandwidth of the spectral filter was chosen to be in the range of the bandwidth of the pump pulse. If it was too narrow, the feedback efficiency was reduced, and interference effects were not used optimally. If the bandwidth was too wide, the input pulse shape started to get distorted by the feedback and perturbations were introduced to the SC generation.

Beside the fluctuations of the optical SC spectrum, we verified that also the fluctuations of the SC pulse form were simultaneously reduced, which indicated that not only the spectral amplitude but also the spectral phase was stabilized synchronously. Our investigations showed, that noise reduction of the spectral amplitude over the whole SC spectrum via optical feedback is possible in principle.

5.2. Experimental results

In order to qualitatively verify the numerically found predictions concerning the spectral amplitude stability, an additional spectral bandpass filter with a transmission window with a FWHM of approx. 20 nm around the central wavelength of the pump pulse was inserted into the experimental setup. An experimental verification measurement by directly extracting the spectral amplitude variations from the recorded optical spectra was not possible, because the variations were too fast (≈ 450 kHz) to be measured with a spectrometer. Instead, we used the analysis setup described in section 2 and shown in part C of Fig. 1, where the power of three filtered spectral regions as well as the power of the whole SC spectrum were simultaneously recorded with four photodiodes connected to the four ports of a digital oscilloscope.

A pump power of 5 mW was chosen, and the according SC spectrum for the single-pass system is plotted in Fig. 10(a). The bandwidth and spectral shape were comparable to those of the simulated spectra in Fig. 9. The three spectral regions with a FWHM of 10 nm at central wavelengths of 730 nm (PD 2), 775 nm (PD 3), and 830 nm (PD 4), that were chosen for analyzing the spectral amplitude fluctuations, are marked in the spectrum. The power fluctuations were analyzed with the photodiode PD 1 by recording the average power of the whole SC spectrum as indicated in Fig. 10(a). The signals were measured as a function of the feedback phase from 0 to 10π. For the single-pass system (as a reference) as well as for each phase position of the feedback system four time series were synchronously recorded with the four ports of the oscilloscope. Each time series was normalized to its average power, and its radio frequency spectrum was calculated via Fourier transform. In Figs. 10(b)–10(d) the phase dependent evolution of the noise peak amplitude at 450 kHz, which was extracted from the radio frequency spectra, is plotted for each port. The red dashed line indicates the noise peak of the reference measurement for the single-pass SC generation. A phase dependent modulation of the noise peak amplitudes was found for all four signals, and within the blue shaded phase regions the noise was reduced synchronously for all signals. Note, that the absolute minimum of the noise peak amplitude within a delay detuning of one wavelength (2π) did not coincide for all signals, e.g., for the wavelength region around 775 nm (PD 3) the minimal noise peak (less than −15 dB) was found within the delay regions around π, 3π, 5π and 7π, where the noise of the other signals was amplified. Within the bandwidth of 10 nm, which was measured with the photodiodes PD 2 to PD 4, the response functions of the photodiodes can be estimated to be constant. Corrections concerning the wavelength-dependent response functions of the photodiodes have only to be taken into account for photodiode PD 1, which was used to measure the whole SC bandwidth. An error of up to 1 dB was evaluated from the numerical simulations (see Fig. 3) for the presented power range. For this case the error is much smaller than in the previous example, because here less power was used resulting in a less broadband SC spectrum and thus in less impact of the wavelength-dependent response function of the photodiode.

 

Fig. 10 Spectrally resolved analysis of the SC noise properties; a) generated optical spectrum without feedback. The spectral ranges that were recorded are shown as bars and labeled with PD 1 to PD 4; b) phase dependent noise peak amplitude with feedback (black solid line) and reference noise peak amplitude for single-pass SC generation (red dashed line) integrated over the whole pulse; c–e) phase dependent noise peak amplitudes for the spectral regions around 730 nm, 775 nm, and 830 nm.

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The best simultaneous noise reduction of all four signals was found at a delay position of 7.7π, where the noise peak for the total SC spectrum was reduced by 3.3 dB (considering the measurement error), while the noise level of the individual spectral regions were reduced by 26.2 dB (730 nm), 5.4 dB (775 nm), and by 11.4 dB (830 nm). The simultaneous noise reduction for all measured signals within certain delay regions was in good agreement with our numerical prediction (see Fig. 9(d)).

6. Perspectives of reducing low amplitude noise

In the presented examples the impact on relatively high input fluctuations in the order of 1% to 10% of the average pump power were discussed. These examples were chosen to realize a comparison to our experiments, where relatively high fluctuations were introduced for systematic analysis. For the simulations, however, we carefully checked that the principles of our noise reduction technique are still suitable for smaller fluctuations (< 1%) by reducing the fluctuations stepswise. The results for the static feedback response for the parameter range marked with the blue cycle in Fig. 3 are listed in the following table.

Table 1. Power Reduction Results for Different Amplitude Values of Input Noise

View Table

The variation amplitude was reduced stepwise from 1 mW via 0.1 mW and 0.01 mW finally to 0.001 mW. This corresponds to a relative input noise of 7.69%, 0.73%, 0.075%, and 0.0073%, respectively. In this example the relative noise reduction could even be continuously improved from 11.15 dB to 35.86 dB by reducing the input variations. The tendency of improved noise reduction as a function of the variation amplitude was not universal for all parameter ranges. However, these findings clearly demonstrate that beside relatively high fluctuations also very small fluctuations can considerably be suppressed. This shows that our noise reduction technique can in principle be used to reduce a wide amplitude range of power fluctuations and a combination with other active noise reduction techniques of the pump source should be possible in order to achieve highly stable SC generation.

7. Summary and conclusions

Power noise reduction as well as spectral amplitude noise reduction of supercontinua via optical feedback were demonstrated numerically and experimentally. The supercontinua were generated within a highly nonlinear, microstructured fiber, which was incorporated into a ring cavity. It was shown, that the SC noise characteristics could be influenced via optical feedback based on the interplay of the fiber nonlinearity and the interference of the pulses in the ring cavity. Both effects could be balanced by adjusting the pump power and the feedback phase with the result of noise reduction.

Two different setups of optical feedback were investigated. Firstly, the whole SC spectrum was fed back without any filter within the feedback loop. With this setup a considerable noise reduction of the power could be achieved numerically (of 11.5 dB) as well as experimentally (in the order of 15 dB). The spectral amplitude noise, however, was only reduced for a narrow spectral region around 800 nm. Secondly, in order to improve the spectral amplitude noise characteristics, the complexity of the SC pulse shape in front of the MSF was reduced by adding a spectral filter to the feedback cavity. With this setup the input pulses in front of the MSF were effectively stabilized and beside the noise reduction of the power, additionally spectral amplitude noise reduction over the whole optical spectrum could be demonstrated in our simulations and was experimentally verified.

This second noise reduction method would allow for a stabilization specific to an application: depending on the requirements, a SC with stabilized spectrum regarding all wavelength components could be realized, or at the cost of increased noise at unused wavelength regions, certain wavelength regions could be stabilized with an even better noise reduction. In order to find parameter ranges with low noise at a certain wavelength for low perturbation frequencies, a road map very similar to the ones shown in Figs. 3 and 7(c) could be prepared. From the intensity of the desired wavelength component at the system output measured as a function of the pump power and the feedback phase one could extract parameter ranges, within which the intensity of this wavelength component can be expected to be insensitive to pump power variations.

The presented results demonstrate that in principal noise reduction can effectively be achieved with the help of optical feedback. However, due to the high phase sensitivity, for long-term stability an active resonator length control would be required. A phase measurement, based on a linearly, independently propagating reference beam, has already been installed. For future investigations this phase measurement could be expanded to a phase stabilization approach. The related verification experiment is actually in progress and will be published elsewhere.

Additionally, we proved with our numerical simulations that noise reduction via optical feedback can be used for the reduction of a wide amplitude range of power fluctuations within SC generation. The advantage of the presented passive, all-optical stabilization technique in comparison to active methods is that in principle it is relatively fast. Furthermore, it would also allow for reducing perturbations, for example with very low amplitude fluctuations that would hardly be possible to measure, so that active methods would fail. Thus, a combination of the presented passive noise reduction method with active methods should provide SC generation with outstanding high stability.