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Abstract
The main drawback of fiber lasers is their high sensitivity to fluctuation in the properties of their surroundings, where even a minuscule fluctuation in the ambient parameters can destabilize them. In this paper, a new passive feedback mechanism inherent to Brillouin fiber lasers (BFLs) is presented and studied. This mechanism, stemming from the interplay between thermal optical-length variations and the gain-line induced frequency dependent lasing power, triggers unexpected and counter-intuitive phenomena such as self-frequency-stabilization, multi-stability, and memory effects. This feedback sheds light on the dynamic behavior of BFLs and can be controlled and modified by engineered the gain lineshape.
© 2017 Optical Society of America
1. Introduction
Fiber lasers exhibit highly attractive properties: extraordinarily low losses [1], great scalability [2–4] and the capacity to retain high power with negligible structural damage [5,6]. On the other hand, they are strongly susceptible to a variety of nonlinear processes such as Brillouin [7] and Raman [8] scatterings, Kerr [9] effect and SPM [10]. These effects trigger at relatively low power thresholds and can generate unexpectedly complex and even chaotic dynamics such as frequency variations [11], back-reflections and other phenomena which may be advantageously utilized for specific laser types or the study of nonlinear systems [12–14] but are often undesirable [15,16]. Furthermore, even below the nonlinear threshold, fiber lasers are affected by a multitude of environmental parameters such as temperate variations, air pressure and acoustic noise which manifest as constant phase noise and necessitate tight control over the pump frequency and/or the optical length of the laser cavity in order to maintain the lasing parameters within a desirable range [12]; nevertheless, fiber lasers are an attractive foundation for the study of a variety of physical phenomena such as rouge waves [13], wave turbulence [17], dissipative solitons [14], classical thermodynamic systems [18] and random lasers [19–21].
Stimulated Brillouin scattering (SBS) [22–24] is a nonlinear stokes process which diffracts lights at a lower, Doppler shifted, frequency through the generation of a moving acoustic grating [11]. Through SBS, a probe wave can be amplified at the expense of the energy of the pump while preserving its phase. In silica fibers, backscattered SBS has the lowest power threshold of all nonlinear mechanisms, and hence it is first to occur, even at pump levels below a single milliwatt [25]. SBS is a major problem in optical communications as it restricts the maximum transmitted power into optical fibers [15]; however, the dependence of SBS on various physical parameters (e.g. strain, temperature, etc.) facilitates utilizing it for sensing applications [26–29] and as an optical gain mechanism [30].
Brillouin Fiber Lasers (BFLs) are a specific type of lasers which utilize SBS in silica fibers as their gain mechanism [14,25,31–33]. At telecom wavelengths (~1550nm), SBS is characterized by efficient scattering of photons with a frequency shift of ~11GHz over narrow bandwidth (<50MHz). This mechanism has been utilized (among many additional applications) for obtaining a spectrally narrow gain profile which enables single mode lasing even for 10-20m long fiber cavities [11]. BFLs are famous for their intrinsically narrow linewidths [34,35] even when pumped using highly jittery sources [32,36]. In addition, they have been utilized for microwave and terahertz wave sources [37] and frequency comb generation [38].
Dielectric materials are susceptible to thermally induced strain and changes in their refractive indices (the thermo-optic effect) [39,40]. These effects are highly important in fiber lasers due to their relatively large cavity length. The temperature dependent phase being accumulated at a single roundtrip inside the cavity can be expressed as:
where k0 is the wavenumber in free-space, dn/dT is the thermo-optic coefficient, α is the thermal expansion coefficient and ΔT is the temperature difference. In silica fibers, α is approximately 5∙10−7K−1 [41] and dn/dT at telecom wavelengths is ~10−5K- [42]. For 10m long cavities, the influence of the thermo-optic effect on the phase is an order of magnitude stronger than that of the thermal expansion; however, both effects cause an increase in the effective optical length with temperature and hence contribute in a similar manner. These effects are generally manifested as low bandwidth phase noise which affects signals propagating in dielectric waveguides due to thermal fluctuations in the external environment. Nevertheless, thermal fluctuations can also be generated by the dissipation of optical power inside the waveguides, leading to similar effects [43–45].In this paper, we describe a series of highly consistent and repeatable phenomena which indicate the existence of an inherent thermal feedback process controlling the lasing frequency and power of BFLs through an interaction between the frequency dependent lasing power and its correspondingly induced thermo-optic change in refractive index and thermal strain. This feedback mechanism self-stabilizes the lasing parameters of BFLs in the lower frequency half of the gain line and prevents stable lasing in the upper part of the gain line. The experimental apparatus required for utilizing this feedback is uncomplicated and the measured results are simple to view as they are characterized by a strong, stable and narrow lasing line and dynamic transient times in the order of the fiber thermal time constant (~1s). This fundamental mechanism enables the introduction of additional degrees of freedom and may prove attractive for various studies such as the dynamics of nonlinear systems as well as for many applications. Although thermal dynamics, line broadening and hysteresis have been studied and demonstrated in micro-cavities [43,44] and DFB lasers [46,47], to the best of our knowledge, this is the first report on a self-stabilizing feedback mechanism and multi-stability phenomena in large fiber lasers.
2. Experimental setup
Figure 1 depicts a schematic of the experimental BFL setup. A stable, 1.7kHz-wide DFB laser source at 1550nm is amplified to 750mW and employed as a pump source for the BFL. The pump is injected into a 7m long (FSR of ~30MHz) single mode fiber ring resonator using an optical circulator which prevents the pump from resonating in the fiber cavity as it absorbs counter-clockwise travelling waves. The resulting back scattered Brillouin signal, on the other hand, can resonate in the cavity as waves travelling in a clockwise direction can pass through the circulator. The cavity contains roundtrip losses of ~20% (finesse of ~28), mostly emanating from losses inside the circulator. A piezoelectric fiber stretcher is used for introducing variation in the optical length of the cavity and a 1:99 fiber coupler taps the Brillion lasing signal. A 1:1 coupler beats the tapped Brillouin signal with the pump and the combined signal is inserted into a heterodyne detection scheme consisting of a fast optical detector connected to a high resolution RF spectrum analyzer. In the counter-clockwise direction, the 1% tap also samples the depleted pump before it is absorbed by the circulator and the depleted pump level is used in the calibration process of the system and as an indirect measure for the lasing power. The resonator is covered using a polystyrene box for isolation which reduces the magnitude and primarily the rate of the environmental perturbation below the bandwidth of the described feedback mechanism.
Fig. 1 A schematic of the experimental setup. Inset: A typical down-converted laser spectrum as measured using heterodyne detection in relation to the location of the gain line center.
It should be emphasized that this specific BFL setup is characterized by its ability to lase across an entire cavity FSR around the peak of the gain spectrum. A slightly different BFL configuration, requiring mutual resonances of both pump and Brillouin (e.g., a doubly resonant BFL [47–49]), is commonly used and although it is characterized by lower pump thresholds, its lasing frequencies are limited to nearly discreet points where the cavity resonates at both the pump and Brillouin signals. We note that this thermal feedback mechanism also exists in the more commonly used doubly resonant configuration.
3. Feedback mechanism
Surprisingly, the BFL configuration studied here exhibits remarkably stable lasing, even when actively perturbed with frequency variations of tens of MHzs. Specifically, stable lasing only occurs in the lower half of the gain spectrum (i.e., at frequencies lower than the peak of the gain spectrum). On the other hand, lasing in the higher half of the gain spectrum is highly unstable and quickly drifts away until it settles in the lower frequencies half of the gain.
Figure 2 shows the measured dynamics of the lasing frequency (relative to the gain spectrum peak marked by a red horizontal line) while undergoing a series of external perturbations. The times at which the perturbations were induced are marked by Roman numbering. A real-time video recording of the laser dynamics under a set of perturbations similar to those shown in Fig. 2 is provided as Visualization 1 in the supplementary material. It also shows that when not perturbed intentionally, the laser can exhibit stable lasing for long periods. As mentioned before, the lasing in the lower part of the gain spectrum is extremely stable. In fact, even minor isolation was capable of lowering the impact of the environmental perturbations to a level which was entirely compensated by the intrinsic thermal feedback (see Visualization 1 in the supplementary material).
Fig. 2 The experimentally measured shift in lasing frequency in response to successive perturbations (marked using numbered arrows).
When the fiber cavity is perturbed (using a piezoelectric fiber stretcher) it may exhibit a variety of dynamical behaviors depending on the magnitude and the direction of the perturbation. Perturbations I and II in Fig. 2 correspond to small blue-shift (I) and red-shift (II) of the cavity resonance frequency. Following these perturbations, the lasing frequency shifts to a new value but then recoils and returns towards the original frequency in an exponentially decaying trajectory. Nevertheless, note that the end value of the lasing frequency following the perturbation is slightly shifted towards the direction of the perturbation. Thus, the laser dynamics exhibit a “memory” effect and successive or continuous perturbations can lead to significant shifts in the stable lasing frequency across the lower half of the gain spectrum.
By altering the optical length of the cavity more aggressively, it is possible to shift the lasing frequency momentarily to the higher half of the gain spectrum. Consequently, the feedback process either pushes the lasing frequency upwards until mode hopping occurs and the frequency stabilizes in the lower half of the gain spectrum (III) or gradually pushes the lasing frequency downwards until it stabilizes, again, in the lower half of the gain spectrum (IV). Note, again, the memory effect which causes the lasing to stabilize at a frequency which is slightly different than the original one (prior to the perturbation). This effect can be utilized for modifying the lasing frequency by applying successive or continuous perturbations. In this way the lasing frequency can be tuned across the lower half of the gain spectrum.
The typical time constant of the feedback mechanism is ~1s. In addition, there is a narrow region around the peak of the gain where the feedback does not occur (see Visualization 2 in the supplementary material). While reaching this region is possible by applying multiple successive perturbations, lasing in this region is unstable and tends to drift away towards the lower half of the gain spectrum.
The long time constant (~1s) implies that the origin of the feedback mechanism is a thermal process. The observed dynamics can be understood by considering the interplay between the frequency dependent lasing power and the thermal change of the effective optical length of the fiber at different lasing powers.
The BFL lasing power depends on the frequency detuning from the gain center - maximal at the center and drops as the off-center frequency detuning increases [50]. Figure 3 shows a measurement of the depleted pump after a single roundtrip inside the cavity as a function of the lasing frequency detuning where a ~40% difference is shown. This sweep of the depleted pump was measured by ramping the PZT voltage to produce a scanning perturbation across a single cavity FSR and sampling the depleted pump as described previously (see Fig. 1). It should be noted that an increase in a laser power due to a shift in the resonance frequency of the cavity is accompanied by a corresponding decrease in the pump power. However, as the lasing signal resonates in the cavity (while the pump does not), the extent by which the intracavity lasing power changes in response to detuning from the gain peak is several times greater than that of the change in pump. In our system, the depleted pump varies by ~120mW while the lasing power varies by more than 600mW over a 15MHz detuning from the gain peak. Consequently, the changes in the pump power partially counteract the feedback mechanism induced by the lasing power and, therefore, it is the latter which dominates the thermal processes in the cavity.
Fig. 3 A measurement of the depleted pump following a single round trip inside the cavity as a function of the frequency detuning between the lasing mode and the peak of the gain line.
Such changes in the lasing power also modify the heat absorbed by the fiber material, resulting in changes of the optical length of the cavity due to thermally induced strain and the thermo-optic effect. Consequently, the cavity is perturbed again, yielding an additional shift of the resonance frequency.
This dynamical process is illustrated in Fig. 4. The blue curve represents the lasing intensity as a function of the cavity detuning over a span of a single cavity FSR. Assume that the lasing frequency is at the lower (left) half of the gain spectrum (e.g. at frequency “1L”). Now, assume that a perturbation reduces the optical length of the cavity shifting it to a higher lasing frequency “2L”. This shift is accompanied by an increase in lasing power which subsequently (through the thermo-optic effect and thermal expansion) increases the optical length of the cavity resulting in a shift backward to a lower frequency, e.g. “3L” (this scenario corresponds to perturbation I in Fig. 2). Opposite perturbations would result in shifts to lower frequencies followed by a reduction of cavity optical length which shifts the lasing frequency back to a higher frequency (corresponding to perturbation II in Fig. 2). Evidently, this process confines the lasing frequency in the lower half of the gain line and stabilizes it. Let us now consider the complementary scenario where the lasing frequency is at the higher half of the gain spectrum. A perturbation that reduces the optical length of the cavity shifts the frequency from “1H” to a higher frequency “2H”. This shift reduces the lasing power and thus further decreases the length of the cavity resulting in an additional, self-feeding shift towards higher frequencies (“3H”). This process ends with the lasing mode hopping to the lower half of the gain line where lasing is stable (perturbation III in Fig. 2). An opposite perturbation would result in a shift to a lower frequency and a subsequent increase in cavity length (due to the higher lasing intensity) which would further shift the lasing frequency lower and lower until the mode enters the lower half of the gain line where equilibrium can be reached (perturbation IV in Fig. 2). Evidently, stable lasing is not possible in the higher frequencies half of the gain spectrum as noise constantly triggers these self-feeding feedbacks. In the vicinity of the gain peak, the variance in lasing power is small and minor changes in the lasing frequency are not accompanied by significant changes to the lasing power. Consequently, this feedback process has little influence in this region. Nevertheless, obtaining stable lasing in this region is difficult as environmental perturbations constantly push the lasing frequency away from this region in a random walk process.
Fig. 4 A Schematic of the thermal feedback process. ΔLext indicates an external perturbation, ΔLtherm indicates a thermal feedback response while the designation above each lineshape indicates its sequence within the process and location with respect to the peak of the gain line. Brighter Lorentzian lineshapes indicate higher temperatures.
4. Simulation model
To further investigate and verify the origin of the feedback mechanism, the laser dynamic was modeled using a master equation approach. We assume that the thermal time constant of the fiber cavity is very large compared to that of cavity roundtrip. This assumption is justified because the cavity roundtrip is ~50ns while the thermal time constant (as observed experimentally) is ~0.8sec (see also appendix 6.2). Thus, we can model the system dynamics as quasi-steady-state which is adiabatically modified by the temperature variations. Consequently, the calculation time step (each iteration) consists of many roundtrips allowing the laser to reach lasing quasi-steady-state (not thermal steady-state) within that iteration. To model the thermal dynamics, we first calculate the lasing power, frequency and temperature at each iteration. The time dependent cavity length (obtained from the temperature shift) and the saturated gain (obtained from the current intensity) are used for obtaining the evolution of the laser intensity which serves as the driving force for the heat equation in the fiber laser. The BFL model consists of two basic elements:
where Δυ is as assumed to be 50MHz, which corresponds to the typical properties of Brillouin gain in silica fibers at telecom wavelengths, υ is the optical frequency, I is the power inside the cavity, and Isat is the saturation power.
where t is the cavity transmission coefficient (related to the roundtrip power losses), k is the wavenumber and L is the roundtrip length of the BFL cavity. The impact of the cavity can be modeled in this way because of the substantially smaller cavity roundtrip which allows the light to circulate many times in the cavity (reaching quasi-steady-state) during each iteration of the simulation.
We calculate the field in the cavity over a bandwidth which is roughly 20% greater than its FSR. The initial conditions are random white noise (corresponding to the spontaneous emissions of the amplifier) and a cold cavity length of L = L0.
At each iteration, the electric field is calculated by multiplying the field at the previous iteration with the saturated gain line (G(ν)) and the cavity response line (Rc(ν)) and adding a small amplitude white noise term. This calculation is used straightforwardly in order to obtain the (saturated) gain line at the following iteration and it is also used for calculating the effective temperature shift of the cavity which in turn, affects the cavity optical length. In view of the very different time scales of the lasing process and the thermal process, we model the temperature shift induced by the lasing intensity by introducing a parameter which indicates a linear relation between the temperature shift and the lasing power at thermal steady–state:
where Ilaser is the time dependent lasing power, ΔT∞ is the steady-state temperature shift of stable lasing at Ilaser and β is a positive linear coefficient linking the two parameters. This relation is assumed to be linear in view of the small changes in the effective cavity length (corresponding to ~1µm for a 10m long fiber cavity).The lumped capacitance model [51] describes the thermal dynamics of systems where the temperature gradients inside the region of interest are negligible. In the case of single mode fiber lasers, the region of interest is the mode field diameter inside the fiber (~10um according to the ITU G.652 specifications) where the thermo-optic interactions take place. Heat is absorbed spatially, along the fiber cavity in a manner corresponding to the nearly Gaussian beam profile of the mode and it then diffuses radially along the fiber cladding and outside the region of interest. In order to verify the validity of the lumped capacitance model in our system, it is necessary to calculate the Biot number:
where h is the heat transfer coefficient, Lc is the characteristic length of heat flow along the body and k is the thermal conductivity of the body. For silica, k≈1.4W/m∙K and the characteristic length can be taken as the mode field radius: Lc≈5µm. The heat transfer coefficient of the glass-glass interface which characterizes the heated region of interest and the outside region is not readily available; however, given that heat is only diffused and cannot be convected, we can safely estimate that h<<1000W/m2K, a value which represent a upper bound of heat transfer in free convection of liquids [52]. For the lumped capacitance model to be valid, it is necessary that Bi<0.1 which translates to h<28,000W/m2K in the described case. Generally speaking, heat transfer coefficients which exceed this value are only possible in cases where not only that heat convection is present but it is further accompanied by a change of phase [51] and therefore the lumped capacitance model can be used safely in our case. Using the lumped capacitance model, the heat dynamics of the system can be described using the following exponential relation [51]:where Ti is the initial temperature in the region of interest, T∞ is the final temperature in thermal steady state and τtherm denotes the thermal time constant. Accordingly, the temperature shift is calculated at each iteration of the numerical model using the following relation:where td is the iteration time step. This equation is based on the exponential dependence of the heat equation on time where the initial condition is the temperature shift at the previous iteration and the local “final” temperature shift, ΔT∞. Using the temperature shift, the shift in effective fiber length can be written as:where ΔT denotes the temperature shift above the ambient temperature and γ denotes the change in optical length due to the thermal expansion and thermo-optic effect of the silica fiber. This dependence is again assumed to be linear for the small levels of resonance shifts that were observed experimentally (corresponding to ~1µm for a 10m long fiber cavity). The overall effective cavity length can thus be expressed as the sum of the cold cavity length and the thermally induced shift:where L is the optical length of the cavity at time t, L0 is the length of the cold cavity and ΔLpert is the length difference induced by the external perturbations (if exists).Figure 5 presents the calculated lasing frequency under a set of perturbations similar to those that were used experimentally (see Fig. 2). The self-evident agreement between the measured (Fig. 2) and the calculated (Fig. 5) laser dynamics validates the model and supports the identification of the described thermal process as the origin of the observed phenomena. Note that the model is able to reproduce not only the general stability properties of the laser but also the more subtle effects such as the slight difference between the lasing frequencies before and after the perturbations (i.e. the “memory” effect) and again, this also reinforces the validity of the model. The minor differences between the graphs (mainly the steeper response of the numerical model) stem from a lower temporal resolution in the experimental setup, a small drift which exists in the experimental setup and the fact that the external perturbations were modeled as ideal step functions while in practice they are not. Also note that the starting point and perturbation strength are, naturally, not identical to the experimental ones.
Fig. 5 The simulated shift in lasing frequency in response to the same set of perturbations shown in Fig. 2.
Finally, we measured the magnitude of lasing frequency change in response to small cavity-length perturbations, as a function of the perturbation frequency. The results, shown in the appendix 6.2, indicate that the system behaves as a high-pass filter with a pole frequency of roughly 1.3Hz (characteristic time constant of ~0.78s).
As mentioned above, lasing in the lower part of the gain spectrum is surprisingly stable and minor isolation is sufficient for reducing the environmental noise to a level which can be compensated by the intrinsic thermal feedback. Visualization 1 in the supplementary material demonstrates this stability as well as the resilience and prominence of this mechanism in response to deliberate perturbations.
A related thermal feedback phenomenon has been observed in high-Q micro-cavities [43,44]; however, the underlying feedback mechanism in that case is different, stemming from the variation in the built-up intensity in the cavity due to thermally induced shifts of its resonance. This feedback is manifested in cavity line broadening and hysteretic wavelength response. Furthermore, this mechanism was explicitly shown to be irrelevant for short cavities whereas the phenomenon described here is demonstrated in a 10m long, laser cavity.
The feedback process described here may also have potential useful application such as counteracting the environmental drift and passively stabilizing dielectric laser cavities. It might also be possible to design fibers with stronger thermal response and reduce the time constant in order to allow for faster control or to lower the level of lasing power necessary for a significant feedback response. Additionally, Brillouin gain profiles greatly depend on the properties of the specific fiber being used where different fibers are often characterized by slightly different Brillouin shifts and gain widths. It is hence feasible to design custom Brillouin gain profiles by concatenating segments of different fibers to tailor more sophisticated gain profiles exhibiting, for example, gain wells that can serve as “frequency traps” and confine the lasing frequency to a specific spectral region.
Although the mechanism described here was studied in BFL, it is also relevant for other fiber or dielectric laser cavities and must not be neglected. The unintuitive consequences as well as the potential benefits of this mechanism can shed light on the complex and often non-obvious dynamics of fiber lasers. We emphasize that this mechanism does not stem from the properties of the cavity components, as we also observed it in the more commonly used doubly resonant configuration.
5. Conclusions
We described and demonstrated an opto-thermal feedback mechanism, inherent to BFLs, which controls and stabilizes the lasing frequency of the laser. This feedback is positive in the upper frequency half of the gain spectrum and negative in the lower half, thus enabling stable lasing only at frequencies which are lower than the gain center. In this spectral region, multiple equilibrium solutions of the coupled lasing and thermal equations are possible, where transition between these solutions can be triggered by perturbing the system. Very good agreement was found between the experimentally measured and the numerically modeled dynamics of the laser. These results enhance our understanding of the lasing dynamics of dielectric cavity lasers and open up new avenues for manipulating and controlling the properties of fiber lasers; particularly, the ability to modify the shape of the gain profile (e.g. by modulating the pump) and control the dynamics of the lasing wavelength is a key element in the realization of a wide variety of all optical processing elements, optical memories, and more.
6 Appendices
6.1 Measurement of β
In order to measure the β coefficient in our system, we actively locked the laser to the peak of the spectral gain (using a previously developed technique – see reference [49] for full details). This allowed us to fix the lasing at a predetermined wavelength which is located at a region where the frequency dependent feedback is minimal and does not interfere with the measurement. In this locked state, slight changes to the pump power induce corresponding changes to the lasing power which also change the amount of absorbed heat in the fiber. By measuring the control signal, we were able to learn the amount by which the control system compensated for the temperature induced variation in the optical length of the cavity and extract the underlying change in temperature (using Eq. (2)). By comparing the measured change in lasing power to the measured change in temperature, we determined that in our system β≈9.3∙10−2K/W.
6.2 Fiber’s thermal time constant measurement
In order to identify the time constant associated with the thermal feedback mechanism, we measured the magnitude of lasing frequency change in response to small signal perturbations with fixed amplitude and as a function of the perturbation frequency. These measurements were around a stable lasing point in the lower half of the gain line. The frequency response of the system is depicted in Fig. 6. Clearly, the system behaves as a high-pass filter due to the fact that high frequency perturbations cannot be compensated by the slow acting thermal feedback. Consequently, at high perturbation frequencies, the lasing frequency follows the perturbation as expected for a passive cavity. At lower perturbation frequencies, the thermal feedback can significantly suppress the frequency response below the existing noise level. The pole frequency of the system is fitted as roughly 1.3Hz corresponding to a characteristic time constant of ~0.78s.
Fig. 6 The response of the system to a small signal saw-tooth perturbation of different frequencies.
6.3 Influence of the circulator on the feedback
When we first encountered this thermal feedback phenomenon, we believed it to be a parasitic effect stemming from one (or more) of the elements in the system. Consequently, we invested significant efforts attempting to identify the origin of this feedback (and eventually eliminate it). The center of our attention was the circulator as it was the only “non-trivial” device inside the cavity which is also responsible for the dissipation of the remaining power of the depleted pump - an obvious source of thermal variances. In that respect, we tried several things:
- 1. We changed the circulator (we effectively built several such resonators with entirely different components including different couplers and fibers).
- 2. We connected the circulator to a large heat sink.
- 3. We constructed a two-circulator scheme where the depleted pump is allowed to exit the resonator instead of being absorbed, which greatly diminished the thermal stress on the circulator.
In all cases, the described feedback remained unchanged. Finally, in order to further rule out the role of the circulator in this feedback, we built a doubly resonant BFL with no circulators at all and we have also observed a similar looking self-stabilization phenomenon in this scheme, leading us to conclude that this effect does not stem from components imperfections but rather constitutes a fundamental effect. A full discussion of the precise feedback in a doubly resonant configuration is beyond the scope of this paper and is somewhat less “interesting” (the lasing frequency range is very limited and the mutual contribution of the pump and lasing signals cannot be easily separated); however, these control experiments rule out the possibility that the observed phenomena are induced by the components and clearly indicate that the observed thermal variations in the optical length are indeed the result of the described feedback mechanism.
Acknowledgments
The authors thank Shlomo Ruschin and Tal Carmon for helpful discussions and comments.
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