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Abstract
We report a high efficiency Brillouin random fiber laser (BRFL) enabled by a random fiber grating (RFG) with demonstration of replica symmetry breaking (RSB). The RFG was characterized by optical coherence tomography (OCT) method, which measured the spatially resolved reflectivity of RFG by a tunable delay line. Multiple narrow linewidth peaks appeared in reflection spectrum of RFG, created by frozen scattering centers acting as narrow linewidth filters to select random modes in random fiber lasers based on Brillouin gain. With the scattering from RFG as disordered feedback, a BRFL with slope efficiency of 29.3% and lasing threshold of 10.2 mW was demonstrated with 1 kHz linewidth. Intensity dynamics show that RFG can reduce the noise of BRFL with a symmetric phase portrait, indicating the increased mean path length and coherence time of the Stokes photons. The probability distribution of the Parisi overlap parameter of intensity fluctuation spectra from trace to trace reveal a photonic spin-glass phase with RSB in the RFG enabled BRFL, providing a photonic platform to study the photon glassy behavior of random fiber lasers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Random fiber lasers are a new breed of lasers in which the conventional fixed cavity configuration is replaced by random feedback of scattering in disordered medium [1–3]. Multiple scattering of photons in random medium increase mean free path of light in random fiber laser system, leading to unique laser properties which raised great interest in research community to study the underlying physics of laser radiation [4–6], as well as potential applications in fiber sensing [7–9] and fiber-optic communication [10]. There are two typical types of random medium in constructing the random fiber lasers: one is the Rayleigh scattering fiber with the naturally inhomogeneous refractive index variation [11,12], and the other is the artificial controlled random medium such as the nanoparticles scattering in colloidal suspension inserted in the microstructure fibers [13,14], or random fiber grating (RFG) that has intentionally induced refractive index modulation [15,16]. Compared to the Rayleigh scattering from fiber’s intrinsic inhomogeneity, scattering from the RFG owns stronger random refractive index variations, which provide both stronger reflection as the distributed feedback for random fiber lasers and localized Fabry-Perot (FP) period interferometers at varied periods to reject laser noise, leading to a lower lasing threshold, narrower linewidth and lower laser frequency noise [17]. The RFG can be inscribed by a femtosecond laser by either using random repetition rate of the pulse or dithering the motion of translation stage randomly as described in [18]. The enhanced distributed scattering from RFGs were used to enable a low relative intensity noise random fiber laser [16], as the enhanced random reflections from the RFG could tolerate environmental perturbations and reduce number of resonating modes [17,19]. At the same time, the broad bandwidth reflection spectrum of the RFG make it a perfect random medium to generate the multi-wavelength random fiber lasers [20,21] for the applications in optical communications and microwave photonics. Because of the light localization in the random medium, it provides a good platform to study the intensity statistics of random fiber lasers based on Raman gain and Erbium-doped gain. The intensity noise and multimode structures of a Raman random fiber laser enabled by a long RFG as a random coherent feedback mechanism was studied for the applications in chaotic communication and random number generation [22]. Replica symmetry breaking (RSB) was recently found in a RFG based Erbium-doped random fiber laser [23], which demonstrated the transition from a photonic paramagnetic to a photonic spin-glass state. However, the RSB is only demonstrated in random fiber lasers based on Erbium-doped formed stimulated emission of molecules in microscopic state, it is vital to examine if RSB holds for the superposition of local wavelength scale structural changes under macroscopic state in random fiber lasers based on Brillouin gain. Besides, good stability of the random scatters is vital to generate the phenomenon of RSB [24], it is important to characterize the distributed scattering of the RFG as relative strong scattering strength make random scatters as “frozen” scattering centers that can tolerate the environmental perturbation. Optical frequency domain reflectometry (OFDR) was employed to characterize the property of random fiber grating with coherent light [25], which has strong coherent scattering and reflection noise to prevent detection of local spatial property of the RFG. The light reflection in the OFDR is a coherent superposition of all scattering sources, especially in the Fourier transform demodulation method that ignores multiple scattering processes of Rayleigh scattering. However, this is not the case for random grating, where multiple scattering processes can be the dominating process due to strong reflective index modulation in optical fiber, which is 3 orders stronger than Rayleigh scattering in optical fibers. Optical coherence tomography (OCT) [26–28] can form images of the random scattering medium with micrometers resolution using incoherent light, which is a good method to characterize the local optical property without the interference from other scattering sources. The spectral response of the RFG could also be retrieved by taking Fourier transform of the measured interference pattern.
In this paper, a RFG with random phase shifts was first characterized based on the time domain OCT technique. By using a light source with 40 nm bandwidth, the spatial resolution of the OCT setup is around 60 µm. The average reflectivity of the RFG remains constant, while local frozen scattering centers were found in the detailed structure of the reflectivity in the spatial domain. The reflection spectra of the RFG shows fine structure which can be used as narrow linewidth filters to select random modes in random fiber lasers. Then, a Brillouin random fiber laser (BRFL) based on RFG as random distributed feedback was built to illustrate the difference between RFG based BRFL and Rayleigh scattering based BRFL. Compared to the Rayleigh scattering based BRFL, the BRFL enabled by RFG showed a reduced lasing threshold and high lasing efficiency. Parisi overlap parameter was calculated based on the output intensity fluctuation spectra of the BRFL, revealing a photonic spin-glass phase of the BRFL enabled by the RFG for symmetry breaking property.
2. RFG characterization
The spatial structure characterization method is based on time domain OCT technique, as shown in Fig. 1. The experimental setup consists of a Michelson interferometer with a low coherence broad bandwidth light source, which is an amplified spontaneous emission (ASE) noise source with 40 nm bandwidth at wavelength of 1530 nm to 1570 nm. Light is split into two arms through the 3 dB coupler and then recombined from reference and sample arm respectively. The reference arm is composed of a Faraday mirror and a variable delay line. The variable delay line has a resolution of less than 1 fs, corresponding to a 0.2 µm spatial resolution in the fiber. The polarization controller (PC) before the random fiber grating in the sample arm is used to optimize the polarization state of the light to get signal with highest signal to noise ratio (SNR). The combination of reflected light from the sample arm and reference arm gives rise to an interference pattern, but only if the travelling distance difference of light from the two arms are less than the coherence length of the light source. Due to the broad bandwidth of light source, the coherence length is shortened to 60 µm. By scanning the delay time of the delay line, a reflectivity profile of the sample is obtained. The intensity detected by the photodetector (PD) is proportional to the real part of the inverse Fourier transform of the product of the source spectrum and the complex reflection spectra of the RFG [27]:
The intensity detected on the PD at different delay time of the delay line is shown in Fig. 2(a). It is seen that the total length of the RFG is approximately 50 mm starting from the delayed length of 10 mm to 60 mm. The amplitude of the beat pattern remains almost constant at all positions, indicating a constant average reflectivity of the RFG in the spatial domain. Figure 2(b) is the detailed reflectivity at a local position of the RFG, which shows that localized frozen scattering centers are created due to the random grating period fluctuation of the RFG. Those localized frozen scattering centers can provide distributed feedback for generation of random fiber lasers, creating random modes in the multi-scattering cavity with increased mean free path of photons. The reflectivity of the RFG calculated from Fourier transform in the OFDR method in [25] showed noise tails after the grating part, which is caused by multiple resonant light scattering in the RFG. In comparison, the reflectivity of the RFG measured by OCT shows a clean noise floor after the grating part without disturbance from other noise interference pattern. The incoherent light used in the OCT method ensures the discrete frozen scattering centers characterization while those frozen scatterings are blurred in the OFDR measurement results due to the disturbance of the multiple resonant light scattering sources. The reflection spectra of the RFG can be calculated according to Eq. (2) in the OCT method. The calculated reflectivity of the RFG (∼−30 dB reflection) is calibrated by measuring the actual reflectivity of RFG using optical spectrum analyzer (OSA). The RFG has a broad bandwidth reflection spectrum with a flat envelope in the spectral domain from 1530 nm to 1570 nm, as shown in Fig. 2(c). This is because the grating period variations of RFG that are characterized in the setup randomly ranges from 0 to 2.5 µm [18], which is larger than the optical wavelength around 1.5 µm, thus lead to totally uncorrelated phase superposition of light from random scattering centers. The detailed reflection spectra of the RFG in Fig. 2(d) around 1550 nm shows that numerous narrow linewidth peaks are formed, which can act as narrow linewidth filters in the application of random fiber lasers. The minimum spectral peak width measured by OFDR in [25] is around 16 pm while ∼10 pm spectral width is found in Fig. 2(d). The enhanced spectral resolution is enabled by the broad bandwidth light source combined with the high spatial resolution of the tunable optical fiber delay line. The mechanism of those narrow linewidth filters is random localized modes are created by the superposition of many FP interferometers that provided by the local frozen scattering centers, which is the key for low frequency noise and narrow linewidth laser operation that will be illustrated in the following section.
Fig. 2. (a) Interference pattern detected on PD at different delay length of the delay line; (b) Detailed structure of interference pattern from position 35 mm to 36 mm; (c) Reflection spectra of the RFG calculated from the Fourier transform of the interference pattern; (d) Detailed reflection spectra of the RFG from 1550 nm to 1552 nm.
3. RFG based high efficiency BRFL
The experimental setup of RFG based BRFL is shown in Fig. 3. The pump light source is based on a 3.5 kHz linewidth fiber laser (Rock Module, NP Photonics) which is amplified by an erbium-doped fiber amplifier (EDFA). After the EDFA, a PC combined with a polarization beam splitter (PBS) are used to generate a linearly polarized pump light. Then, the pump laser is injected to the random fiber laser cavity through circulator 1. The Brillouin gain medium is a 2 km Panda-type polarization maintaining (PM) fiber with fiber loss of 0.296 dB/km and a mode field diameter of 6.48 µm at wavelength of 1550 nm. The pump light in the Brillouin gain medium could stimulate backward Stokes light which travel anticlockwise between circulator 1 and circulator 2. The utilization of the linear polarization of pump light and PM fiber as Brillouin gain medium can greatly reduce the lasing threshold of the BRFL. In addition, the PM fiber provides a higher Brillouin gain coefficient due to (1) a smaller mode field diameter of 6.48 µm versus that of single mode fiber (10.4 µm) and (2) polarization matched stimulated Brillouin scattering (SBS) between identical linearly polarized pump and Stokes light that enhances the Brillouin gain by a factor of 2. The enhanced Brillouin gain makes relative short fiber provides Brillouin gain, which further reduces the Rayleigh scattering in the Brillouin gain fiber (2km) to reduce the cavity loss and to reduce the feedback from Rayleigh scattering in 2 km fiber length as the noise floor for the Stokes signal. After port 2 of circulator 2, the backscattered Stokes light from SBS was reflected by the RFG, which can enable random distributed reflection of the light through the multi-reflection of various scattering centers. The RFG scattered Stokes light is then injected back to the Brillouin gain medium after port 3 of circulator 2, providing distributed feedback for random fiber laser. As the pump power increases, the distributed feedback is amplified by the SBS to compensate the round trip loss for lasing oscillation. The laser emits after an isolator that was placed after the RFG to block the Fresnel reflection from the fiber connectors.
Fig. 3. Experimental setup of the RFG based BRFL and its measurement method for power, intensity dynamics and linewidth.
Figure 4(a) shows the measured output power of BRFL against the injected pump power. The threshold is measured to be 10.2 mW of injected pump power, and the slope efficiency is measured to be 0.293. Compared to the BRFL enabled by the distributed Rayleigh scattering in [29] that had a threshold power of 14.6 mW and slope efficiency of 0.254. The threshold power of the BRFL enabled by the RFG in our setup is decreased by 30.1%, and the slope efficiency is increased by 15.4%. Due to the intentional induced large refractive index variation (10−4−10−5) by femtosecond pulse laser, the reflection coefficient of the RFG is relatively high. The RFG had a reflection coefficient of around −30 dB while the reflection coefficient of the 500 m Rayleigh scattering fiber in [29] is around −50 dB. The relative high reflection coefficient leads to a low cavity loss and thus a high quality factor, which further improves the slope efficiency of BRFL. Therefore, the utilization of RFG is important to improve the lasing efficiency of BRFL. A delayed self-heterodyne (DSH) method consisting of a Mach-Zehnder interferometer is employed to measure the linewidth of RFG enabled BRFL. In the DSH method, the output of BRFL is split into two parts by a 95/5 coupler, with the lower power part modulated by an acousto-optic modulator (AOM) with a frequency down shift of 40 MHz and higher power part sent to a 200-km delay fiber to make decoherence light in the two arms. The optical beat signal is collected by a PD and measured by an electrical spectrum analyzer (ESA). The spectrum of the photocurrent on PD has a Lorentzian shape with linewidth twice of the linewidth of the laser under test. For a Lorentzian shape, the 20 dB linewidth is approximately 10 times of the 3 dB linewidth. Therefore, the 20 dB linewidth of the photocurrent is approximately 20 times of 3 dB linewidth of the laser under test. The 20 dB linewidth of the photocurrent enabled by BRFL is measured to be 20 kHz, corresponding to 3 dB laser linewidth of 1 kHz. In comparison, the 20 dB linewidth of the photocurrent enabled by pump laser is measured to be 78 kHz, corresponding to 3 dB laser linewidth of 3.9 kHz, as shown in Fig. 4(b). The linewidth reduction of the BRFL are attributed to the random localized modes created by frozen scattering centers in the RFG combined with the relative high reflectivity of RFG, which reduces the loss of the random distributed cavity, leading to a high quality factor and narrow spectral linewidth output of BRFL.
Fig. 4. (a) Laser output power versus pump power of the BRFL; (b) Beat signal measured on ESA based on the DSH method to characterize the linewidth of BRFL (blue curve) and NP laser (red curve).
The intensity dynamics and the statistical feature of the RFG enabled BRFL is characterized by measuring the temporal trace of the laser output through a PD and an oscilloscope. For comparison, SBS emission without distributed feedback from RFG is also characterized. Without distributed feedback, stochastics intensity fluctuation presents an exponential probability distribution as observed in Fig. 5(a) and Fig. 5(b). The possibility of extreme events arises in the stochastics intensity fluctuation behavior with long tails in the probability density function. The SBS stochastics behavior was predicted in [30] by adding a Langevin noise source in the wave coupled equations among pump wave, Stokes wave and acoustic wave. In statistical optics, the negative exponential random variables are attributed by thermal process of acoustic phonon with totally uncorrelated phase relation [31]. The phase portrait is constructed by the two dimensional plot of temporal trace data points ${I_N}$ vs ${I_{N + k}}$ for delay $k = 1$, which exhibits motion on an outward spiraling and folding trajectory. The intensity in Fig. 5(c) and Fig. 5(f) are normalized with respect to the average output intensity. Figure 5(c) shows that little phase relation between the successive temporal points and reveals the chaotic nature of the amplified spontaneous Brillouin scattering process without lasing. In contrast, the statistical property of the temporal intensity traces is significantly modified as random lasing occurs that are enabled by the distributed feedback from the RFG. The temporal output trace of the BRFL follows a Gaussian probability distribution in Fig. 5(e) instead of the exponential probability distribution owing to the stable establishment of the random laser emission. The Gaussian probability distribution can be interpreted by a Fokker-Plank statistical equation [31], which describes the intensity statistical probability distribution of coherent laser source. As random grating acts as mode selection elements by many variable local periods, it eliminates many thermal amplification modes with low optical power to compete finite Brillouin gain, which manifest as the large range of low power distribution as shown in Fig. 5(b). The contribution of random grating feedback leads to a significant less number of the mode numbers survived for amplification by 2 km SBS gain fiber to reach threshold. Hence, a stable and Gaussian power distribution is expected as observed in Fig. 5(e). The transition from the exponential probability distribution of the SBS noise to Gaussian distribution of the BRFL manifest that light with uncorrelated phase relation is transformed to highly coherent light after the distributed feedback being added in the experimental setup. The corresponding phase portrait in Fig. 5(f) shows a confined cycle signature, which means the successive temporal points are coherently related in phase with each other with the increasing mean path length and coherence time of the Stokes photons. The statistical feature of BRFL reveals that the thermal noise characteristics of lasing emission can be tamed by the distributed scattering from RFG, which shows promising prospect in generating the coherent narrow linewidth lasers.
Fig. 5. (a) temporal trace (b) power density distribution (c) phase portrait of intensity dynamics of SBS without distributed feedback from RFG; (d) temporal trace (e) power density distribution (f) phase portrait of intensity dynamics of Brillouin random lasing with distributed feedback from RFG;
The characteristic of the intensity fluctuations of the BRFL can be analyzed in a way comparable to RSB [32], which was used to quantify the phase transition from the paramagnetic to spin glass state in spin glass theory. To do the RSB analysis, the spectra of the intensity fluctuation is calculated by doing fast Fourier transform (FFT) of the temporal traces, as shown in Figs. 6(a)–6(c). In order to compare the different properties of the temporal traces generated by different mechanisms, three types of Stokes light are analyzed which are SBS noise, Rayleigh scattering based BRFL, and RFG based BRFL respectively. The Rayleigh scattering based BRFL is obtained by changing the RFG in the experiment setup to a 500 m long Rayleigh scattering fiber. The overlap of the intensity fluctuation spectra from trace to trace, which is known as Parisi parameter, can be calculated by the correlation between intensities of two traces a and b as [33]
Fig. 6. Intensity fluctuation spectra of (a) SBS noise (b) Rayleigh scattering based BRFL (c) RFG based BRFL; Probability distribution of q value of (d) SBS noise (e) Rayleigh scattering based BRFL (f) RFG based BRFL.
4. Conclusion
In summary, a high efficiency BRFL constructed by the RFG is built and experimentally realized. The RFG that used in the BRFL experiment is characterized by the time domain OCT method. Characterization results find constant average reflectivity of the RFG in the spatial domain. Frozen scattering centers are detected in the detailed structure of the RFG reflectivity in the spatial domain. The reflection spectra of RFG have a broad bandwidth due to the random phase shifts introduced in the RFG. A large number of narrow linewidth peaks are observed in the fine spectra structures of the RFG, which can act as narrow linewidth filters to select random lasing modes in random lasers. The slope efficiency of the RFG based BRFL is increased to 29.3%, demonstrating a high efficiency BRFL compared to the Rayleigh scattering based BRFL. Narrow linewidth of the random laser radiation on the order of kHz is characterized using DSH method. The RFG exhibits noise rejection effect in the output intensity dynamics of BRFL, which further show RSB in its spectra after doing the FFT analysis. The characterization of the RFG and performance of the RFG based BRFL gives us a new perspective to understand the fundamental physics of the random lasing process, paving the way for RFG applications in other types of random fiber lasers and in the field of communication, high precision metrology and sensing.
Funding
Canada Research Chairs (950231352); Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-06302).
Disclosures
The authors declare no conflict of interest.
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