## Abstract

A coherent random fiber laser based on stimulated Brillouin scattering as gain and Rayleigh scattering as distributed feedback mirror was constructed. Its frequency is stabilized by a high finesse narrow-band Fabry-Perot interferometer (FPI) to select lasing frequency within the gain bandwidth. The light confinement within single-mode fiber enhances largely the random lasing directionality, which enables a high-quality coherent random lasing in the weak scattering region by using a milliwatt continuous-wave pump source. The FPI in the laser configuration acts as a frequency selection on the Rayleigh feedback light, and thus the random lasing frequency was locked at one of its transmission peaks giving a relative frequency fluctuation of ~2.5 × 10^{−11} at 100 s. The measured frequency jitter is within ~~ ± 20 kHz over 3 hours, 3 dB linewidth is ~50 Hz and frequency noise is ~20 mHz/Hz^{1/2} at 10 kHz.

© 2013 Optical Society of America

## 1. Introduction

Since Ambartsumyan, Basov and Letokhov proposed the concept of using multiple scattering in gain medium to generate coherent laser-like emission in 1966 [1, 2], random lasers (RLs) attract continuous interests due to their unique lasing mechanism and potential applications [3–6]. The multiple-scattering centers in the RL increase the dwell time of photons in its gain medium, and when the dwell time exceeds the mean generation time of photons, a chain-like photon emission similar to the neutron generation in an atomic bomb is triggered [3]. Over the past decade, different types of RLs have been demonstrated in various bulk materials including crystal powder material [7], laser dye with nanoparticles [8], rare-earth powders [9], semiconductor powder [10], polymer films with silver nanoparticles [11] and even dye-treated human tissues [12]. Based on the feedback mechanisms, RLs can be divided into two categories: RLs with incoherent feedback and RLs with coherent feedback [3]. In RLs with incoherent feedback (also called incoherent RLs), multiple-scattering in gain medium only returns a portion of energy or intensity of light, and the lifetime of emission photons is extended by the stimulated amplification process [4]. In coherent RLs, higher scattering density results in higher possibility of light recurrence [3], which means after multiple scattering, light returns to its original position. The spatial resonance lying among the inter-scattering centers provides the field or amplitude feedback of light, and constructive interference between the scattered light supports coherent lasing spikes at certain resonant frequencies.

Because of the lack of lasing directionality, RLs in bulk materials generally need high scattering density and high-power pulse pump to achieve lasing, and their lasing qualities were also limited [7–13]. In order to improve the random lasing directionality, several types of random fiber lasers (RFLs) were demonstrated recently [13–20], and both incoherent [13–18] and coherent [19–23] random emissions were observed using optical fiber. Matos et al used a section of hollow-core photonic crystal fiber, and laser dye material with nanoparticles was filled into the hollow core as the gain medium with multiple scattering [13]. A Raman RFL based on Raman gain and Rayleigh scattering feedback was reported in an open fiber cavity with a high lasing efficiency of ~15% [14, 15], and modeling of this incoherent Raman RFL was also presented [16–18]. Er-doped fiber was also used as gain medium to construct coherent RFLs, while randomly distributed Bragg gratings were fabricated to provide random feedback [19, 20]. In [21], a coherent random laser emission within a liquid core optical fiber with nanoparticles was reported in the extremely weak scattering regime.

Recently, we reported the observation of narrow-linewidth lasing spikes in high-quality coherent Brillouin RFLs [22, 23], in which stimulated Brillouin scattering (SBS) in optical fiber was used as the gain mechanism, while distributed Rayleigh scattering (RS) provides random feedback. Due to the weak spontaneous Brillouin scattering in optical fiber and efficient Rayleigh feedback configuration [23], a high-quality resonance of the Stokes light was achieved by using a milliwatt continuous-wave pump source. Coherent lasing spikes with linewidth as narrow as ~10 Hz were observed on the top of the Brillouin gain spectrum. However, high frequency instabilities of this coherent Brillouin RFL including frequency jitters and multi-mode emissions, were also observed, and they were mainly induced by 1) the inhomogeneous broadening of Brillouin gain spectrum due to the fiber non-uniformity and 2) the sensitivity of the open laser cavity to external perturbations [23].

In this paper, we introduce a high-finesse FPI into the coherent Brillouin RFL configuration and achieve the first frequency stabilized random laser to the best of our knowledge. The FPI used in experiments has a free spectral range (FSR) of ~21 MHz and a narrow bandwidth of ~30 kHz. The emission frequency of the coherent Brillouin RFL was locked to one peak of the FPI transmission spectrum, and thus the frequency jitters of the random lasing were decreased to tens kHz comparing with MHz of unlocked condition [23], and the multi-wavelength emission were also suppressed significantly. A theoretical model of this coherent Brillouin RFL was constructed to demonstrate its lasing mechanism as well as the function of narrow-bandwidth FPI to its frequency stabilization. The power evolutions of pump, Brillouin Stokes and Rayleigh scattered Stokes light in both Brillouin gain and Rayleigh feedback fiber were simulated, and the random lasing thresholds and efficiencies were also predicted under different gain fiber lengths. In experiments, three coherent Brillouin RFLs with gain fiber lengths of 2 km, 10 km and 25 km were built, while the Rayleigh feedback fiber was a section of 4.3 km non-uniform fiber. Their lasing thresholds and efficiencies were directly measured by a power meter, when their lasing spectra were measured by a heterodyne method. The frequency noise spectrum of the RFL with 25 km gain fiber was measured by a 3 by 3 fiber interferometer, and the results were compared with that of a commercial 3.5 kHz-linewidth fiber laser. In order to measure the coherence performance and frequency stability of this Brillouin RFL, two coherent Brillouin RFLs with same coherence properties were generated simultaneously from two gain fibers. The beat signal of those two RFLs was recorded and used to illustrate their coherence performance, linewidth and frequency stability. To the best of our knowledge, this is the first time that a frequency stabilized single-mode coherent random fiber laser is achieved by using an F-P interferometer, which selects one resonant frequency from coherent random feedback modes.

## 2. Theory and simulations

#### 2.1 Coherent Brillouin RFL configuration

The configuration of the coherent Brillouin RFL is shown in Fig. 1. When the pump light (*E _{P}*) is launched in the SBS gain fiber through optical circulator 1, the spontaneous Brillouin Stokes light in gain fiber is initiated from thermal noise [24], propagates in the opposite direction and has a frequency downshift of ~10.9 GHz with respect to the pump light [24, 25]. The spontaneous Stokes light has weak power at room temperature and its coherent time is determined by the phonons lifetime of the fiber material with a value of ~10 ns [25]. As propagating along the SBS gain fiber with length of

*L*, the Stokes light (

_{B}*E*) is amplified through the nonlinear electrostriction process, and simultaneously its coherent time is extended by the stimulated amplification process [26].

_{S}The stimulated Brillouin Stokes light out of the gain fiber is then sent into the RS fiber through circulator 2, where it works as the pump light of Rayleigh backscattering. The length of RS fiber is *L _{R}*, and the remaining Stokes light out of the RS fiber at

*z*works as the random laser output. The Rayleigh backscattered Stokes light (

_{2}= L_{R}*E*) accumulates in amplitude along the RS fiber and then it is sent to a FPI through port 3 of the circulator 2. The FPI works as the frequency selection component for the Rayleigh scattered Stokes light, and its free spectral range of the FPI is ~21 MHz to ensure one transmission peak within the ~30 MHz Brillouin gain spectrum. At the transmission peak, the Rayleigh backscattered light obtains the minimum cavity loss. Then an Er-doped fiber amplifier (EDFA) is used to compensate the optical loss induced by the FPI, and an optical filter with bandwidth of 0.1 nm is used to eliminate amplified spontaneous emission (ASE) noise of the EDFA. The Stokes light out of the optical filter passes through an optical isolator and is then sent back to the initiation end of the SBS gain fiber at

_{R}*z*, where together with the spontaneous Stokes light newly generated,

_{1}= L_{B}*E*works as the seed of new SBS amplification process.

_{R}#### 2.2 Modeling and simulations of Rayleigh random feedback

As shown in Fig. 2, the distributed Rayleigh scattering in RS fiber is induced by the non-uniformity of the fiber material and can be considered as frozen scattering centers which are randomly distributed along the RS fiber [14, 22, 23, 27]. Those scattering centers are fully deterministic in time, but their amplitudes and locations are randomly distributed. In this one-dimension random fiber laser, the Stokes light from SBS gain fiber is backscattered by those scattering centers and the backscattered light wave have the same frequency with the Stokes light. The backscattered Stokes light at the input end of RS fiber (*E _{R}|z_{2} = 0*) is the summation of all the scattered light waves from different scattering centers. In simulations, we assume that the total number of scattering centers in RS fiber is

*T*. For each scattering center

*j*(

*j*is from 1 to

*T*), its backscattering amplitude (

*A*) is a Gaussian random value with a zero mean and a standard deviation related to the Rayleigh scattering coefficient of the RS fiber, while the position of each scattering center (

_{j}*z*) can be considered as uniformly distributed over the whole RS fiber length. When we neglect the optical loss and polarization effect, the summation of backscattered light waves in RS fiber can be written as Eq. (1), where

_{j}*n*is the effective refractive index of the fundamental mode in RS fiber,

_{2}*c*is the speed of light in vacuum and

*f*is the frequency of the Stokes light.

As shown in Fig. 3(a), in a system of polar coordinates, the backscattered light wave (at *z _{2} = 0*) from each scattering center can be considered as an effective scattering vector with amplitude of

*A*and relative phase of

_{j}*φ*=

_{j}*4πn*. Thus the summation of those vectors corresponds to an effective Rayleigh factor (

_{2}fz_{j}/c*R*) with effective amplitude (

*A*) and phase (

_{eff}*φ*), which is shown as the red line in Fig. 3(a). The Stokes light after a round trip in this random laser cavity (

_{eff}*E*) can be written as Eq. (2), where

_{S}’|z_{2}= 0*L*is the traveling length of the Stokes light outside the RS fiber and

*n*is the effective refractive index in SMF28 fiber.

_{1}As shown in Eqs. (1) and (2), the overall amplitude and phase of *E _{S}*’

*|z*are determined by the frequency of the Stokes light, distribution of the scattering centers in RS fiber, and traveling length of the Stokes light in the open laser cavity. The constructive interference between

_{2}= 0*E*and

_{S}’*E*in this open laser cavity happens at the resonant frequencies (

_{S}*f*), where phase delay between

_{res}*E*and

_{S}’|z_{2}= 0*E*is an integral multiple of 2π.

_{S}|z_{2}= 0In simulations, a group of 100 scattering centers was randomly generated along the RS fiber length, and we simulated the vectors *E _{S}’|z_{2} = 0* and

*E*as the function of frequency of the Stokes light. The parameters used in the simulation are summarized in Table 1. The simulated vector product of <

_{S}|z_{2}= 0*E*> is plotted in Fig. 3(b), when the frequency span is 400 kHz and the central frequency was selected to be 193.55 THz which corresponds to wavelength of 1550 nm. As shown in Fig. 3(b), several maximum values of <

_{S}’·E_{S}*E*> are observed at the resonant frequencies, where the constructive interference between

_{S}’·E_{S}*E*and

_{S}*E*can support coherent random lasing spikes [23]. It can be seen in Fig. 3(b) that the product values for different resonant frequencies are different, which is induced by the multiple scattering in RS fiber. Different from the conventional laser with fixed cavity length, the feedback of this coherent random fiber laser results from the summation of backscattered waves from multiple scattering centers, and the random distribution of those scattering centers results in uneven amplitudes of the feedback maxima.

_{S}’As demonstrated in [23], due to non-uniformity of the gain fiber, the Brillouin gain spectrum is inhomogeneously broadened [28], and within several MHz range the Brillouin gain factor is relatively uniform. Thus in the Brillouin RFL without FPI [23], frequency jitters and multi-spikes emissions were observed on the top of the random lasing spectrum, which result in the frequency instability of the coherent random laser (similar to mode-hopping in conventional lasers) [29]. As shown in Fig. 3(b), the FPI used in the setup has a narrow bandwidth of ~30 kHz, which selects one resonant frequency within its bandwidth. Thus random lasing frequency is locked to that transmission peak of the FPI.

#### 2.3 Light power evolutions in SBS and RS fiber

The power evolutions of the pump light (*P _{P}*), Stokes light (

*P*) in SBS gain fiber, as well as Rayleigh pump light (

_{S}*P*) and Rayleigh backscattered Stokes light (

_{RP}*P*) in RS fiber are simulated. In SBS gain fiber, the equations for

_{R}*P*and

_{P}*P*can be written as Eq. (3), where

_{S}*α*is the linear loss coefficient and

_{B}*g*is the Brillouin gain factor in SBS gain fiber.

_{B}*μ*is the spontaneous Brillouin scattering coefficient per unit length related to thermal-induced density fluctuation of the fiber material [25].

_{spon}In the RS fiber, the equations for power evolutions of *P _{RP}* and

*P*can be written as Eq. (4), where

_{R}*α*and

_{R}*r*is the linear loss coefficient and Rayleigh backscattering coefficient of the RS fiber respectively [22, 27].

The boundary conditions for Eqs. (3) and (4) can be considered as the power continuities among Stokes light in the random laser cavity, which can be written as Eq. (5).

Using Eqs. (3)–(5) as well as fiber parameters shown in Table 1, we simulated the output power (*P _{RP}|z_{2} = L_{R}*) of this Brillouin RFL as the function of pump power (

*P*), when the length of SBS gain fiber is selected as 25 km, 10 km and 2 km respectively. The simulated results are plotted in Fig. 4(a). It is shown that those Brillouin RFLs show theoretical lasing thresholds of 3 mW, 5 mW and 17 mW respectively. In theory, the random lasing threshold is defined as the condition of the single-pass Brillouin gain compensates the loss of the Stokes light in a roundtrip within the random laser cavity. When pump power is higher than the lasing threshold, the accumulated backscattered Stokes light in RS fiber exceeds the spontaneous Brillouin Stokes light initiated from thermal noise and dominates the SBS process. Under this condition, the SBS process in gain fiber transforms from a SBS generator to a SBS amplifier [22, 25], and the Stokes light oscillates in the random laser open-cavity.

_{P}|z_{1}= 0The power distribution of *P _{P}* and

*P*along the SBS gain fiber, and

_{S}*P*and

_{RP}*P*along the RS fiber were also simulated respectively, when SBS gain fiber length was selected as 25 km and the pump power was set at 10 mW (higher than the lasing threshold). As shown in Fig. 4(b),

_{R}*P*was amplified by the pump light when traveling backward in the SBS gain fiber, while

_{S}*P*decreased due to pump depletion as propagating forward. As shown in Fig. 4(c),

_{P}*P*decreased along the RS fiber due to fiber loss while

_{RP}*P*accumulated when traveling backward and reached its maximum at

_{R}*z*.

_{2}= 0As shown in Fig. 4(c), the high Rayleigh coefficient of non-uniform fiber provided an efficient random feedback, and more than 0.1% of Stokes light was backscattered. The lasing quality (*Q*) of this random laser can be defined as the ratio of the feedback Stokes light to the spontaneous Stokes light at the initiation end of the gain fiber, which can be written as *Q = P _{R}|z_{2} = 0/P_{spon}|z_{1} = L_{B}*. For the 25 km gain fiber, when the pump power was 10 mW,

*Q*can be calculated to be as high as 1.94 × 10

^{3}, which is due to this efficient random feedback scheme and the weak spontaneous Brillouin Stokes light in optical fiber. The high lasing quality can extend largely the effective traveling length of the Stokes light in this random laser open-cavity, and the linewidth of its coherent random emission is narrowed by the high-quality light oscillation.

In sum, the one-dimensional coherent random fiber laser gives a unique feedback properties. Because the feedback light beams from Rayleigh mirrors are coherent, the interference between reflected light beams in Rayleigh fiber supports an uneven frequency-dependent feedback pattern as shown in Fig. 3, and some resonant frequencies appear at the frequencies where constructive interference happens. For the other aspect, due to the random distribution of those Rayleigh mirrors, in statistics the mean feedback light power in Rayleigh fiber increase linearly along the fiber length when the Rayleigh pump depletion is neglected. Thus when we simulate the random lasing threshold and efficiency of this coherent random fiber laser, a series of power evolution equations were used for simplification. Due to those unique feedback properties, the coherent electrical-field summation in section 2.2 gives a good explanation of the coherent random lasing spikes we observed in experiments, while for simulations of its lasing threshold and efficiency, the power model shows good agreement with the experimental results.

## 3. Experiments

#### 3.1 Lasing threshold and efficiency measurements

In the laboratory, both the SBS gain and RS fiber were put into an aluminous soundproof box to isolate the random laser setup from environmental perturbations. As shown in Fig. 5, a commercial frequency stabilized semiconductor laser (FSSL, 500 kHz linewidth) and an EDFA were used to provide a continuous-wave pump light. The RS fiber is fixed as a section of 5.4 km non-uniform single-mode fiber (Corning Inc.), while three sections of SMF28 fiber with lengths of 25 km, 10 km and 2 km were used as the SBS gain fiber respectively. In the fabrication process of the non-uniform fiber, a continuous refractive index changing in fiber core was induced by modulating the Ge-doped concentration of its core area [30], which results in both higher fiber loss (0.45 dB/km) and larger Rayleigh coefficient (−34 dB/km) than conventional SMF28 fiber. The laser output powers were measured by using a power meter. For three different gain fiber lengths, laser output powers as the function of pump power were plotted in Fig. 6(a). It can be seen that the measured lasing thresholds for those three random lasers are ~2.5 mW, ~5 mW and ~20 mW respectively, which agree well with the theoretical predictions shown in Fig. 4(a). After their lasing thresholds, the random lasers show lasing efficiencies of 13.8%, 13.6% and 5.75% respectively.

A heterodyne method shown in Fig. 5 was used to measure the lasing spectrum of the random laser with 25 km SBS gain fiber at various pump powers. In the heterodyne method, a portion of pump light from 10% port of coupler 1 was downshifted in frequency by an electro-optic modulator (EOM) and then amplified by EDFA 2 as the reference light. A RF source (HP 83640A) was used to control the modulation frequency of the EOM, while a tunable optical fiber (5 GHz bandwidth) was used to eliminate the ASE noise from EDFA 2. The beat signal between this reference light and the random laser output was measured by an electrical spectrum analyzer (Agilent E4446A). The resolution of this heterodyne method is limited by the linewidth of the reference light [31], which has the same value as that of the pump laser (500 kHz). Thus, we can only observe the coherent random lasing spectrum, but cannot measure the precise linewidth of the coherent random lasing output by using this heterodyne method.

Figure 6(b) gave the measured lasing spectra of this coherent Brillouin RFL at different pump powers. It can be seen when the pump power was higher than the lasing threshold, a sharp lasing emission peak with narrow linewidth appeared on the top of the Brillouin gain spectrum, which agrees well with our theoretical predictions.

#### 3.2 Laser relative intensity noise measurement

In the experiments, we also measured the output power trace of this coherent Brillouin RFL in time domain, when the length of SBS gain fiber was selected as 25 km and the pump power was 20 mW. As shown in Fig. 5, a photodetector (Thorlabs PDB130C) with bandwidth from DC to 350 MHz was used to measure the random laser output directly. The experimental result was recorded with an oscilloscope (Agilent DSO81204B), and the result was plotted in Fig. 7(a). It can be seen that the output intensity of the coherent Brillouin RFL shows good stability, and the straight time-domain line in Fig. 7(a) confirmed the single-wavelength operation of this random fiber laser. The power spectrum of this time-domain trace gave the relative intensity noise (RIN) spectrum of this coherent Brillouin RFL, which was plotted as blue solid curve in Fig. 7(b). The noise power spectrum of the photodetector was measured to be ~-140 dB/Hz, which is more than 30 dB lower the RIN of the RFL. Thus the influence of detector noise to RIN results shown in Fig. 7(b) can be neglected.

For comparison, the RIN spectrum of the commercial fiber laser (NP Photonics, 3.5 kHz linewidth) [32] was also measured with the same setup and plotted as red dash curve in Fig. 7(b). As shown in Fig. 7(b), the RIN spectrum of the coherent Brillouin RFL had the similar level with that of the pump fiber laser, but shows some low-frequency noise which was mainly resulted from external perturbations.

#### 3.3 Laser frequency noise measurement

In laboratory, the frequency noise spectrum of this coherent Brillouin RFL with gain fiber length of 25 km was measured by using a 3 by 3 unbalanced Michelson fiber interferometer and a digital phase demodulation scheme [33]. The experimental setup for the frequency noise measurement is shown in Fig. 8, where a section of conventional SMF28 fiber was included into one arm as the delay fiber, while two Faraday rotation mirrors (FRM) were used to eliminate the polarization fluctuation in the fiber interferometer. Two interferometric signals from port 2 and 3 of the 3 by 3 coupler were detected by two photodetectors (DC-350 Hz bandwidth) and then sent into the digital demodulation scheme to get the phase information of the fiber interferometer.

For the coherent Brillouin RFL, the interferometric signal from port 2 of the 3 by 3 coupler was recorded and plotted in Fig. 9(a), when the delay fiber used in the setup was 50 km (corresponding to 100 km delay line in Michelson interferometer). For comparison, the reference 3.5 kHz-linewidth fiber laser was also used to illuminate the same 3 by 3 interferometer, when 500 m conventional SMF28 fiber was used as the delay fiber (corresponding to 1 km delay line). The interferometric signal for the reference fiber laser was recorded and plotted in Fig. 9(b). For the interferometric signal of Brillouin RFL shown in Fig. 9(a), clear interferometric fringe was observed at 100 km delay line, this demonstrated a long coherent length of this RFL.

Using a digital phase demodulation scheme [33], the frequency noise spectra of both the coherent Brillouin RFL and reference fiber laser were computed from their interferometric signals respectively, and the results were plotted in Fig. 9(c). As expected, the frequency noise spectrum of the reference fiber laser (red dash curve) exhibits a *1/f ^{1/2}* dependence at low frequency and a logarithmic roll-off at higher frequency range, which is induced by the thermal-induced fluctuation of its laser cavity length [33–35]. From the frequency noise spectrum of the coherent Brillouin RFL shown as solid black curve in Fig. 10, the thermal-induced

*1/f*noise is largely suppressed due to its long cavity length and randomly distributed feedback scheme. A quick drop at frequency range below ~3 kHz is the flick noise [36] induced by its slow frequency shift, while at frequency range higher than ~3 kHz, its frequency noise spectrum shows the white noise limit, which is mainly induced by fluctuation of laser output intensity. As shown in Fig. 9(c), the measured frequency noise value of this coherent Brillouin RFL at 10 kHz is as low as ~20 mHz/Hz

^{1/2}^{1/2}, which is about 20 dB lower than that of the reference fiber laser at 10 kHz.

To our knowledge, the low frequency noise of this one-dimensional coherent random fiber laser as shown in Fig. 9(c), can be mainly attributed to two effects: the first one is the ultra-long cavity length. According to the theory of Cranch and Foster [33, 34], the flick frequency noise of fiber laser shows *1/f* shape in frequency noise spectrum which results from the thermal noise-induced cavity length perturbations. This part of flick frequency noise is inverse proportional to the length of laser cavity [33]. Thus the ultra-long cavity of this coherent random fiber laser significantly eliminates this part of flick noise. The other effect is that because one source of flick frequency noise in fiber lasers comes from the thermal-induced vibrations of the laser mirrors, which can influence the constructive interference of light in laser cavity and thus perturb its resonant frequencies. However, in this one-dimensional coherent random fiber laser, the ensemble of Rayleigh scattering from distributed Rayleigh mirrors as a whole gives an averaged phase shift to the lasing light in one round trip. The emission frequency of this random laser is supported by the constructive interference of scattered light from all the Rayleigh mirrors. Those mirrors distribute randomly in Rayleigh fiber and their thermal-induced vibrations can be considered as independent Brownian motions, which may compensate with each other in the interferometric summation of the random feedback process. Thus the coherent random laser gives a more stable resonance pattern comparing with conventional fiber lasers in which two laser mirrors give a specific phase shift in one round trip.

#### 3.4 Laser linewidth and frequency stability measurements

In order to verify the coherence performance of this coherent Brillouin RFL, two Brillouin RFLs with the same linewidth but a lasing frequency difference of ~42 MHz were constructed, and both of them have the SBS gain fiber length of 25 km. The beat signal of the two random laser outputs was measured to characterize the coherence performance and frequency stability of this coherent Brillouin RFL [23, 37]. The setup used in experiments was shown in Fig. 10. The output of pump FSSL was divided into two light beams. The optical frequency of the first beam was downshifted by an acousto-optic modulator (AOM), whose modulation frequency was controlled by a function generator with a fixed frequency of 40 MHz. The other beam traveled through a section of 25 km delay fiber (longer than the coherent length of the 500 kHz linewidth pump FSSL) to make it incoherent with the first light beam. The two light beams with frequency difference of 40 MHz were amplified by two EDFAs, and then launched into two sections of 25 km SBS gain fiber as pump sources respectively. When both of the pump light powers were 10 mW, the beat signal of the two RFLs out of RS fiber was detected by a photodetector and then mixed with a sine signal generated by a function generator. As shown in Fig. 11, the low frequency component of the mixed signal was filtered out by an electrical low-pass filter and then recorded by a digital oscilloscope. When the frequency of the function generator was set to 42.1 MHz, the frequency of beat signal was shifted to low frequency range, where the coherence property of the coherent Brillouin RFL can be observed easily in time domain. As shown in Fig. 11(a), the measured beat signal between two RFL outputs shows a constant phase in a time period of ~0.01 s demonstrating a coherent length as long as ~3 × 10^{6} m.

The linewidth of the beat signal between those two Brillouin RFLs was also measured by using an electrical spectrum analyzer. The beat spectrum normalized to 42.1 MHz (two times of the free spectrum range of the FPI), was shown in Fig. 11(b). In order to increase the measurement accuracy, a 20 dB linewidth of the beat signal was measured to be 1.1( ± 0.5) kHz, which corresponds to the RFL Lorentzian 3 dB linewidth of 55( ± 25) Hz. The measured RFL linewidth of ~50 Hz is in good agreement with the coherent time of ~0.01 s shown in Fig. 11(a).

In order to characterize the frequency stability of the coherent Brillouin RFL, the frequency jitter of the beat signal between two RFLs were recorded in 3 hours with a time step of 0.5 s. As shown in Fig. 12, in 3 hours the measured frequency jitter was within ~ ± 20 kHz, which shows a two order of magnitude improvement compared with the Brillouin RFL without FPI [23]. The FPI used in this RFL cavity provides a wavelength-dependent selection of the Rayleigh feedback light, and thus the random lasing frequency is locked within the ~30 kHz bandwidth of the FPI transmission peak. In long-term, the frequency stability of this coherent Brillouin RFL is limited by this white frequency jitter [36]. By using the measured results shown in Fig. 12, the Allen standard deviation [36, 37] of this coherent Brillouin RFL with FPI can be calculated to be ~2.5 × 10^{−11} at average time of 100 s.

## 4. Conclusion

A coherent Brillouin RFL was demonstrated theoretically and experimentally. A high-finesse FPI was used in this random laser cavity to suppress its frequency instability. According to the theoretical simulation, the FPI provides a wavelength selection to the Rayleigh feedback light, and thus the frequency of this coherent lasing emission can be locked to one transmission peak of the FPI. In experiments, the performances of this frequency stabilized coherent RFL were measured respectively, including its lasing threshold, lasing efficiency, random emission spectrum, relative intensity noise spectrum, frequency noise spectrum, time coherence and frequency stability. Two orders of magnitude improvement in laser frequency stability was obtained by using a FPI with bandwidth of ~30 kHz. Single-wavelength, narrow-linewidth (~50 Hz), high frequency stability (<40 kHz frequency jitter) coherent random fiber laser was achieved in weak scattering domain by using milliwatt continuous-wave pump source.

## Acknowledgments

The authors would like to acknowledge the financial support of NSERC Discovery Grants.

## References and links

**1. **R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and V. S. Letokhov, “A laser with a nonresonant feedback,” IEEE J. Quantum Electron. **2**(9), 442–446 (1966). [CrossRef]

**2. **V. S. Letokhov, “Stimulated emission of an ensemble of scattering particles with negative absorption,” JETP Lett. **5**, 212–215 (1967).

**3. **H. Cao, “Random lasers: development, features and applications,” Opt. Photonics News **16**(1), 24–29 (2005). [CrossRef]

**4. **H. Cao, “Lasing in random media,” Waves Random Media **13**(3), R1–R39 (2003). [CrossRef]

**5. **H. Cao, “Review on latest development in random lasers with coherent feedback,” J. Phys. Math. Gen. **38**(49), 10497–10535 (2005). [CrossRef]

**6. **R. C. Polson, M. E. Raikh, and Z. V. Vardeny, “Universal properties of random lasers,” IEEE J. Sel. Top. Quantum Electron. **9**(1), 120–123 (2003). [CrossRef]

**7. **S. Gottardo, O. Cavalieri, O. Yaroshchuk, and D. S. Wiersma, “Quasi-two-dimensional diffusive random laser action,” Phys. Rev. Lett. **93**(26), 263901 (2004). [CrossRef]

**8. **N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature **368**(6470), 436–438 (1994). [CrossRef]

**9. **H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. **82**(11), 2278–2281 (1999). [CrossRef]

**10. **G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A **65**(1), 013807 (2002). [CrossRef]

**11. **X. Meng, K. Fujita, Y. Zong, S. Murai, and K. Tanaka, “Random lasers with coherent feedback from highly transparent polymer films embedded with silver nanoparticles,” Appl. Phys. Lett. **92**(20), 201112 (2008). [CrossRef]

**12. **R. C. Polson and Z. V. Vardeny, “Random lasing in human tissues,” Appl. Phys. Lett. **85**(7), 1289–1291 (2004). [CrossRef]

**13. **C. J. S. Matos, L. S. Meneze, A. M. Brito-Silva, M. A. M. Gamez, A. S. L. Gomes, and C. B. Araujo, “Random fiber laser,” Phys. Rev. Lett. **99**, 153903 (2007). [CrossRef]

**14. **S. K. Turitsyn, S. A. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania Castanon, V. Karalekas, and E. V. Podivilov, “Random distributed feedback fibre laser,” Nat. Photonics **4**(4), 231–235 (2010). [CrossRef]

**15. **A. A. Fotiadi, “An incoherent fibre laser,” Nat. Photonics **4**(4), 204–205 (2010). [CrossRef]

**16. **S. V. Smirnov and D. V. Churkin, “Modeling of spectral and statistical properties of a random distributed feedback fiber laser,” Opt. Express **21**(18), 21236–21241 (2013). [CrossRef]

**17. **D. V. Churkin, A. E. El-Taher, I. D. Vatnik, J. D. Ania-Castañón, P. Harper, E. V. Podivilov, S. A. Babin, and S. K. Turitsyn, “Experimental and theoretical study of longitudinal power distribution in a random DFB fiber laser,” Opt. Express **20**(10), 11178–11188 (2012). [CrossRef]

**18. **S. Sugavanam, N. Tarasov, X. Shu, and D. V. Churkin, “Narrow-band generation in random distributed feedback fiber laser,” Opt. Express **21**(14), 16466–16472 (2013). [CrossRef]

**19. **M. Gagne and R. Kashyap, “Demonstration of a 3 mW threshold Er-doped random fiber laser based on a unique fiber Bragg grating,” Opt. Express **17**(21), 19067–19074 (2009). [CrossRef]

**20. **N. Lizaraga, N. P. Puente, E. I. Chaikina, T. A. Leskova, and E. R. Mendez, “Single-mode Er-doped fiber random laser with distributed Bragg grating feedback,” Opt. Express **17**(2), 395–404 (2009). [CrossRef]

**21. **Z. Hu, Q. Zhang, B. Miao, Q. Fu, G. Zou, Y. Chen, Y. Luo, D. Zhang, P. Wang, H. Ming, and Q. Zhang, “Coherent random fiber laser based on nanoparticles scattering in the extremely weakly scattering regime,” Phys. Rev. Lett. **109**(25), 253901 (2012). [CrossRef]

**22. **M. Pang, S. Xie, X. Bao, Y. Da-Peng Zhou, Y. Lu, and L. Chen, “Rayleigh scattering-assisted narrow linewidth Brillouin lasing in cascaded fiber,” Opt. Lett. **37**(15), 3129–3131 (2012). [CrossRef]

**23. **M. Pang, X. Bao, and L. Chen, “Observation of narrow linewidth spikes in the coherent Brillouin random fiber laser,” Opt. Lett. **38**(11), 1866–1868 (2013). [CrossRef]

**24. **R. W. Boyd and K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A **42**(9), 5514–5521 (1990). [CrossRef]

**25. **R. W. Boyd, *Nonlinear Optics* (Academic, 2010).

**26. **A. Yeniay, J. M. Delavaux, and J. Toulouse, “Spontaneous and stimulated Brillouin scattering gain spectra in optical fibers,” J. Lightwave Technol. **20**(8), 1425–1432 (2002). [CrossRef]

**27. **A. H. Hartog and M. P. Gold, “On the theory of backscattering in single-mode optical fibers,” J. Lightwave Technol. **2**(2), 76–82 (1984). [CrossRef]

**28. **S. Xie, M. Pang, X. Bao, and L. Chen, “Polarization dependence of Brillouin linewidth and peak frequency due to fiber inhomogeneity in single mode fiber and its impact on distributed fiber Brillouin sensing,” Opt. Express **20**(6), 6385–6399 (2012). [CrossRef]

**29. **G. A. Ball and W. W. Morey, “Continuously tunable single-mode erbium fiber laser,” Opt. Lett. **17**(6), 420–422 (1992). [CrossRef]

**30. **M. Li, S. Li, and D. A. Nolan, “Nonlinear fibers for signal processing using optical Kerr effects,” J. Lightwave Technol. **23**(11), 3606–3614 (2005). [CrossRef]

**31. **D. Derickson, *Fiber Optic Test and Measurement* (Prentice Hall, 1998).

**32. **C. Spiegelberg, J. Geng, Y. Hu, Y. Kaneda, S. Jiang, and N. Peyghambarian, “Low-noise narrow-linewidth fiber laser at 1550 nm (June 2003),” J. Lightwave Technol. **22**(1), 57–64 (2004). [CrossRef]

**33. **G. A. Cranch and G. A. Miller, “Fundamental frequency noise properties of extended cavity erbium fiber lasers,” Opt. Lett. **36**(6), 906–908 (2011). [CrossRef]

**34. **S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron. **5**(5), 378–384 (2007). [CrossRef]

**35. **S. Foster, G. A. Cranch, and A. Tikhomirov, “Experimental evidence for the thermal origin of 1/f frequency noise in erbium-doped fiber lasers,” Phys. Rev. A **79**(5), 053802 (2009). [CrossRef]

**36. **D. W. Allan, “Time and frequency (time-domain) characterization, estimation and prediction of precision clocks and oscillators,” IEEE Trans. Ultr. Ferr. Contr. **34**(6), 647–654 (1987). [CrossRef]

**37. **T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics **6**(10), 687–692 (2012). [CrossRef]