Linewidth narrowing in freespacerunning diamond Brillouin lasers
1 Introduction
Highpower narrow linewidth singlefrequency lasers have attracted extensive interest due to their superior characteristics in some fields, such as gravitational wave detection^{[}1^{]}, quantum information optics^{[}2^{]}, atomic physics^{[}3^{]} and nonlinear frequency conversion^{[}4^{–}6^{]}. The most widely used approaches to obtain highpower narrow linewidth lasers are master oscillator power amplification (MOPA) structures based on lowpower narrow linewidth seeds^{[}7^{,}8^{]} and singlecavity structures with direct highpower output^{[}9^{,}10^{]}. However, in MOPA systems, aside from the problems of beam quality deterioration and mode instability due to thermal accumulation^{[}11^{]}, the linewidth broadening introduced by spontaneous radiation during the amplification process cannot be ignored. Using a single cavity to obtain a highpower singlefrequency laser overcomes the complexity of laser structures but faces the problem that the linewidth cannot be further narrowed due to strong pumping intensity noise^{[}9^{]}. In addition, the output wavelength in the abovementioned solutions is critically dependent on the energy level structure of the gain medium, which limits the applicability of conventional highpower singlefrequency lasers in some special wavelength laser applications (e.g., sodiumguided star research)^{[}12^{]}. The emergence of optical nonlinear frequency conversion techniques accompanies the development of highpower laser technology to provide a new perspective for achieving highpower narrow linewidth laser radiation at specific wavelengths. Of these, stimulated Brillouin scattering (SBS), a thirdorder nonlinear effect, offers excellent advantages in applications of realizing ultranarrow linewidth lasers^{[}13^{–}15^{]}, microwave photonics^{[}16^{–}18^{]} and optical storage^{[}19^{]}. The heavy attenuation mechanism of acoustic phonons combined with the strong feedback provided by the cavity enables microwaveguide Brillouin lasers (BLs) to produce laser outputs far below the linewidth of conventional singlefrequency lasers and close to the quantum noise limit^{[}20^{,}21^{]}. Similarly, strong linewidth narrowing has been reported in fiber BLs^{[}22^{]} and laser outputs of subkHz magnitude subject to technical noise^{[}23^{]}. These demonstrations based on waveguide structures also face some insurmountable problems in power boosting. The high Q cavity achieves a narrow linewidth with a concomitant lowering of the threshold of the higherorder Stokes frequency component, resulting in a limited singlefrequency BL power scale^{[}24^{]}. The phasematching condition of highorder Stokes light can be broken using intermode scattering through a tailored cavity structure^{[}25^{]}, but the thermal effects accumulated in the cavity at power scaling detune the phasematching condition of the singlefrequency BL, leading to powerclamping phenomena^{[}20^{]}. The mechanism of Stokes linewidth compression in fiber BLs relies on the employment of long fiber cavities^{[}26^{]}, whereas their phasematching conditions require a free spectral range (FSR) comparable to the Brillouin gain linewidth, which thereby limits further linewidth compression and power boosting. Increasing the pump power in anticipation of achieving a singlefrequency power increase will inevitably produce spontaneous radiation of higherorder Stokes frequency components^{[}22^{,}27^{]}.
Addressing the power improvement constraint faced by waveguide BLs, freespacerunning BLs offer a solution to achieve high power and narrow linewidth output. Separating the cavity and gaining medium in the spatial cavity facilitates thermal management and control of phasematching conditions at high power. In addition, the diversity of Brillouin gain materials makes it possible to realize highpower lasers with narrow linewidths at special wavelengths. The power of the freespacerunning BL was increased from the initial 20 mW restricted by the gain level^{[}28^{]} to a level exceeding 20 W without saturation of the power curve^{[}29^{]}. The output wavelength also extends from the visible to the nearinfrared band^{[}28^{–}30^{]}. However, even with these impressive demonstrations, the characteristics of Stokes linewidth narrowing in spatially structured BLs have not been reported so far, which has been the most crucial basis for extensive research and development of BLs. In our previous study, we reported that the singlefrequency BL output at 1064 nm and 46.9 kHz exhibited severe broadening compared with the linewidth of the pump (~7 kHz), which was attributed to the effect of technical noise at that time^{[}29^{]}.
This study theoretically and experimentally analyzes the linewidth narrowing characteristics of freespacerunning BLs. Targeting ultranarrow laser linewidth measurements, we analyzed the influence of fiber length on linewidth measurement results from the mechanism of delayed selfheterodyne interferometry. Subsequently, the corresponding structures are used for BL linewidth measurements at different coupling rates. Based on the analysis of the experimental results, we discuss a solution for simultaneous power enhancement and linewidth compression in spatially structured BLs.
2 Experiments and results
The operation mechanism and structure of the BL and the linewidth measurement system are shown in Figure 1. The gain medium used in the experiments was a Brewstercut diamond (4 mm × 1.2 mm × 5 mm, Element Six) with polarization parallel to <001>, which is consistent with the direction of maximum Brillouin gain through the crystal. In addition, the diamond was mounted on top of a copper heat sink that was actively temperature controlled to 20°C by a thermoelectric controller, as shown in Figure 1(b). The pump laser had a maximum power of 60 W that was emitted from a fiberbased optical amplifier (OA) and used a singlefrequency seed laser (seed) with a linewidth of less than 10 kHz at 1064 nm. The Brillouin cavity was a ring cavity with a free spectral range (FSR) of approximately 560 MHz. The pump beam was injected into the fundamental mode of the Brillouin cavity using focusing lenses (f1 and f2) and reflectors (HR1 and HR2). In addition, the cavity length was adjusted to support the dual resonance of the pump and Stokes field frequencies^{[}29^{]}. Frequency locking of the pump field to the cavity was achieved using the Pound–Drever–Hall frequency stabilization technique^{[}31^{]}. The generated Stokes field was emitted in the opposite direction of the incident pump beam when the pump field intensity exceeded the Stokes threshold and then separated from the system using an isolator (ISO).
Fig. 1. Illustration of the phase match, experimental setup and linewidth measurement system of a freespace, freerunning BL based on a diamond. (a) Demonstration of the Brillouin gain process. The grey curve refers to the cavity resonant mode with full width at half maximum (FWHM) γ. The red (blue) curves refer to pump (Stokes) frequency ω _{P} (ω _{S}) with the linewidth of Δν _{P} (Δν _{S}). The Brillouin gain spectrum linewidth (Γ ) and frequency shift (Ω _{B}) are also indicated. (b) Schematic layout of a freespace, freerunning BL based on a diamond (EOM, electronic optical modulator; OA, optical amplifier; ISO, isolator; f1 and f2, lenses; HR1 and HR2, high reflection mirrors; M1, coupler mirror; M2, high reflectivity plane mirror; M3 and M4, high reflectivity concave mirrors; PD, photodetector). (c) Linewidth measurement schematic. The beam is divided into two components using a 20%/80% coupler. The first component passes through an acoustic optical modulator (AOM) to generate a frequency shift of 100 MHz, and the second is delayed using a delay line (Nufern, 1060XP). The beat note with a center frequency of 100 MHz is generated in the PD after the two components are recombined using a 3 dB coupler and analyzed by an electric spectrum analyzer (ESA) (RSA5032, Rigol).
The two sets of critical parameters in the resonant cavity are the intrinsic loss rate (γ_{in}) and external coupling rate (γ_{ex}) as well as the intrinsic Q value (Q_{in} = ω/γ_{in}) and load Q value (Q_{L} = ω/γ=ω/(γ_{in}+γ_{ex})), where Q_{L} is the total Q value of the system^{[}32^{,}33^{]}. The intrinsic loss and external coupling loss in the spatial cavity can be expressed as γ_{in} = L_{0} × FSR and γ_{ex} = –ln R_{1} × FSR, respectively^{[}34^{]}. L_{0} is the passive loss caused by the rest of the optical elements in the cavity except for the coupling mirror, and R_{1} is the reflectivity of the coupling mirror. The fundamental linewidth of a resonant pumped BL can be expressed as follows^{[}35^{,}36^{]}:
The first part of the equation indicates the linewidth caused by the thermal phonon noise^{[}37^{,}38^{]}, and the second term indicates the Stokes linewidth caused by the pump phase diffusion^{[}26^{]}. Here, α is the linewidth broadening factor (zero when the Stokes frequency is not detuned), Q_{ex} is the Q value that describes the coupling loss, ℏ is the reduced Planck constant,
Table 1. Stokes linewidths corresponding to different coupling mirror reflectivities.

The resonant width γ, determined by the coupler reflectivity and inherent loss of the cavity, is of the same order of magnitude as the Brillouin gain width Γ (2π × 5.3 MHz) of the diamond at 1064 nm^{[}29^{]}. Therefore, the current system does not satisfy the adiabatic approximation (γ << Γ), that is, the Stokes linewidth behavior is dominated by the phase diffusion of the pump, which is similar to the linewidth behavior of most singlepass pumped fiber BLs^{[}22^{,}26^{]}. Consequently, the linewidth at the kHz level is much smaller than that of most conventional inversion lasers. Precisely measuring the Stokes linewidth experimentally is considerably significant for the study of BLs. Here we have adopted a short fiber delay method based on a delayed selfheterodyne structure to determine the linewidth of Stokes output, which is given in Figure 1(c). The photocurrent spectral density (PSD) function of the photodetector output after the laser passes through the measurement structure is expressed as follows^{[}39^{,}40^{]}:
Here, S_{1} is the Lorentz function, whose spectral profile is unaffected by the fiber length, S_{2} is the period modulation function, S_{3} is the δ function, Δf is the linewidth, and P_{0} and τ_{d} relate to the beat frequency signal power and delay time, respectively.
Figure 2 shows the PSD for different lengths of the delayed fiber for a laser linewidth of 1 kHz and the variation of the PSD with different linewidth values for a fiber length of 1 km. From Figure 2(a), the longer the fiber length, the smaller the amplitude and period of the function (the closer the first minima to the center frequency). When the fiber length increases to a certain degree, S_{2}≈1, the modulation of the PSD profile can be neglected, and the PSD shows a Lorentzian lineshape. The traditional delayed selfheterodyne linewidth measurement scheme uses that Lorentz spectrum. However, the traditional method of sampling Lorentzian spectral lines has some limitations. For example, when the laser linewidth is extremely narrow, the delayed fiber can reach hundreds to thousands of kilometers, and such a fiber length introduces additional 1/f noise that makes the measurement results unreliable^{[}39^{,}41^{]}. However, it can also be found in Figure 2(b) that the depth of the interferometric envelope is definitively related to the laser linewidth as the modulation period of the spectral lines introduced by S_{2} is a definite value, and the narrower the linewidth, the more distinct the envelope depth of the PSD.
Fig. 2. (a) PSD at different fiber lengths for a linewidth of 1 kHz. (b) PSD spectrum at different laser linewidth values for a fiber length of 1 km.
Therefore, the curve of PSD as a function of linewidth can be obtained under the determination of the fiber length and acoustic optical frequency shifter. Considering that the extreme points close to the central frequency are less sensitive to the electric spectrum analyzer (ESA) noise floor^{[}35^{]}, the difference between the vertical axis values at the two successive extreme points m = 0 and m = 1 on the righthand side of the PSD is taken here (see Figure 2(b)), and the relationship between the line width and the difference in magnitude is obtained as follows^{[}41^{]}:
Once ΔS is determined, the linewidth of the laser to be measured can be obtained by numerically solving Equation (3). In the experiments, subject to the influence of the instrument noise floor and 1/f noise introduced by long fibers, there should be an optimal value for the length of the fiber, that is, satisfying the extreme point near the center frequency and the envelope amplitude with significant contrast. The BL system with a highpower pumping source is seeded by a narrow linewidth semiconductor laser with a linewidth of 5.1 kHz using frequency noise integration. We use this laser as a light source to explore the optimal value of fiber length for the linewidth measurement system. The frequency offset f(m = 1) from the center of the first extreme point (m = 1) on the righthand side for different delay fiber lengths and the absolute value ΔS* of the difference in envelope amplitude corresponding to a laser linewidth of 5 kHz are given in Figure 3. From Figure 3(a), as the fiber length increases, ΔS* and f(m = 1) gradually decrease and tend to zero. After the fiber length exceeds the position where f(m = 1) = 0 (point 1), the PSD enters the transition region from the envelope spectrum to the Lorentz spectrum. The PSD in this region exhibits a weakly modulated Lorentzian type with S(m = 1) < S(m = 0). When the fiber length increases until the weak modulation disappears (point 3, l = c × 6τ_{c} = 600 km; τ_{c} represents the laser coherence time), the PSD shows a Lorentzian pattern at this point. For comparison purposes, the coherence length corresponding to the laser linewidth is also indicated in Figure 3(a) (point 2, l = c × τ_{c} = 100 km). The result shows that the Lorentz spectrum method requires the use of a delay much larger than the coherence time (τ_{d} > 6τ_{c}) to eliminate the influence of the interference envelope, while the introduction of long fibers will inevitably introduce noise into the laser to be measured, making the measurement results unreliable^{[}42^{]}. Linewidth measurement using the envelope becomes a new method to solve the linewidth calibration of ultranarrow linewidth lasers.
Fig. 3. (a) Amplitude of the envelope differences and the frequency shift of the first extreme value point (m = 1) compared with the center at different fiber lengths for a linewidth of 5 kHz. (b) Experimentally measured linewidth values for a semiconductor laser corresponding to different fiber lengths.
We experimentally measured the linewidth of the laser mentioned above by selecting five delay lengths from the envelope region (lefthand side of point 1) in Figure 3(a) to determine the optimal fiber length for the measurement system, and the results are shown in Figure 3(b). It can be seen that the difference in the results for different delay fiber lengths within the selected envelope is more than 1 kHz. In particular, the linewidth values obtained using 100 m delay fiber measurements reached 9 kHz. The corresponding extreme points at this delayed fiber length are far from the center frequency. This results in the PSD being disturbed by the ESA noise floor and the amplitude difference being less than the ideal level, resulting in a larger calculated linewidth value. When the delayed fiber is 3 km and above, the amplitude and period of the envelope start to become insignificant, considering the presence of inevitable 1/f noise, which is measured to be approximately 8.6 kHz. For the laser to be measured, the PSD at 1 km delay is independent of the noise floor and free of 1/f type frequency noise, with a distinct envelope and significant amplitude difference. The result is closer to the actual level, with a corresponding linewidth of 7.36 kHz. The laser linewidth measured here is greater than the integral linewidth because the integral linewidth is linearly fitted to the frequency noise within the β split line and subsequently integrated, which will inevitably ignore part of the frequency noise, resulting in the corresponding integral linewidth narrowing^{[}43^{]}. The BL experiments were conducted according to the three sets of coupling mirror reflectivities selected earlier, and the Stokes linewidth was measured using a delayed fiber of 1 km at maximum pump power operation. The results obtained are shown in Figure 4.
Fig. 4. Stokes linewidth corresponding to the reflectivity of the three sets of coupling mirrors at 60 W power pumping. (a) Power spectrum of the corresponding delayed selfheterodyne amplitude difference. (b) Linewidth and envelope amplitude difference curves calculated by the coherent envelope method.
As the reflectivity of the coupling mirror increases, the Stokes output linewidth gradually narrows. The Stokes linewidths corresponding to the three sets of coupled mirror reflectivities are 3.2, 2.43 and 1.77 kHz, respectively, which achieve linewidth compression compared with the pump. Even at 96% reflectivity of the coupling mirror, the linewidth is still compressed by a factor of nearly 2.5. This result is much smaller than the previously reported Stokes linewidth of 46.9 kHz measured with the same parameters^{[}29^{]} because the delayed fiber used for the linewidth measurements in the previous experiments was 100 m, which caused the envelope frequency deviation to be too large, causing a portion of the corresponding PSD to be swamped with instrumental noise and leading to large calculated results.
The experimental results presented here illustrate that increasing the coupling mirror reflectivity reduces the Stokes linewidth. It can also be seen that the corresponding linewidth compression trend is consistent with that predicted in Table 1, but the linewidth results are all higher than the theoretically calculated values. We attribute the difference between the experimental measurements and the theoretical values to the introduction of additional technical noise^{[}26^{]} (e.g., ambient noise and electronic noise of the locking system).
The pump–Stokes power curves corresponding to the three reflectivities are given in Figure 5. The increase in the coupling rate contributes to the lowering of the threshold. The threshold value is about 29 W at a coupling rate of 96% and decreases to 8.9 W when the coupling rate increases to 98.5%. The increased coupling rate decreases the Stokes output power and efficiency at highpower pumping. At the maximum pumping power, the Stokes output powers corresponding to the three sets of reflectivities are 22.5, 16.9 and 13.6 W, and the corresponding optical conversion efficiencies are 37.5%, 29.1% and 23.4%, respectively.
3 Discussion
Table 2 gives the system parameters and corresponding output results for the waveguide, fiber and spacestructured BLs. The waveguide structure results in a much smaller overall level of corresponding Stokes linewidth than the other two structures due to the low intrinsic and coupling losses combined with the wider Brillouin gain linewidth. It is worth noting, however, that the corresponding output power is limited to the milliwatt order due to the smaller coupling loss and size. The linewidth narrowing mechanism in optical fibers relies on the use of long fiber cavities. While the use of large mode field fibers boosts the corresponding power, further realization of power enhancement and linewidth compression is limited by the higherorder Stokes frequency components generated. The higher Stoke gain level in spatially structured BLs is determined by the high pump power, thus allowing higher conversion efficiency at higher power levels.
Table 2. Platform parameters and corresponding output linewidth and power in several BLs.

Since freespacerunning BLs are designed for high power and narrow linewidth, the phase diffusion model for Stokes linewidths and the previously reported power model^{[}48^{]} are used here to analyze the influence of intrinsic and coupling losses on the Stokes output linewidth and power. Taking the variation of the passive loss L_{0} in the cavity from 0.001 to 0.03 and the coupling mirror reflectivity R_{1} from 0.9 to 0.9999, the variation of the Stokes linewidth and power at a fixed pump of 60 W is plotted as shown in Figure 6.
Fig. 6. (a) Stokes output linewidth determined by passive loss and coupling mirror reflectivity. (b) Stokes output power determined by passive loss and coupling mirror reflectivity at a fixed pump power of 60 W.
With the pass loss determined, the linewidth compression effect becomes significant as the reflectivity of the coupling mirror increases. Since the passive loss is negligible compared with the coupling loss, further compression of the Stokes linewidth by reducing the passive loss is ineffective. It is worth noting that although this term does not affect the linewidth behavior, the smaller the passive loss, the higher the pumping enhancement in the cavity and the Stokes output power. Reducing the passive loss helps to improve the linewidth compression effect and conversion efficiency of the Stokes output. In the BL system mentioned in this paper, the passive losses in the cavity are primarily due to the introduction of the Brewster element. By replacing the flatcut medium, the corresponding loss can be reduced from 0.015 to 0.005. Although the linewidth is only reduced by almost 200 Hz to approximately 1.3 kHz, the corresponding power is increased by 20 W to almost 48 W, where the optical conversion efficiency reaches 80%.
In addition to the two parameters mentioned above, the gain bandwidth of the Brillouin medium also affects the compression of the Stokes linewidth. For example, when the gain linewidth is increased from 5 to 40 MHz, the Stokes linewidth decreases from 1.7 kHz to 90 Hz. However, as can be seen from the previous experimental results, the broadening introduced by the technical noise is not negligible when the Stokes linewidth is reduced to a certain level. The linewidth of the BL system is mainly determined by the technical noise (e.g., environmental vibration, circuit locking noise) when the Stokes optical fundamental linewidth determined by the system parameters is much smaller than the broadening caused by the system technical noise. Further narrowing of the Stokes linewidth requires the suppression of technical noise from technical aspects, such as frequency stabilization^{[}32^{,}49^{]}. Overall, it is apparent from the above study that using freespacerunning BLs can achieve narrower laser radiation compared with a pump. Combined with the inherent low quantum loss of the SBS process, it is expected that ultranarrow linewidth laser radiation close to the quantum conversion limit can be achieved.
4 Conclusion
This study theoretically and experimentally analyzed the linewidth narrowing characteristics of freespacerunning BLs and analyzed the methods to simultaneously achieve linewidth compression and power enhancement. Experimental results show that the Stokes linewidth behavior in freespacerunning Brillouin cavities is determined by a combination of phase diffusion of the pump light and technical noise. At a coupling mirror reflectivity of 96%, we obtained a Stokes output with a power of 22.5 W and a linewidth of 3.2 kHz, which is nearly 2.5 times compressed compared with the linewidth of the pump. In addition, a Stokes light output of 1.7 kHz and 13.6 W was achieved by increasing the coupling mirror reflectivity to 98.5%. Furthermore, we analyzed the solution of simultaneously improving the output efficiency and linewidth compression by replacing the Brewster element with a parallelcut element to reduce the insertion loss of the intracavity element, which can theoretically achieve a linewidth of 1.3 kHz and a Stokes optical power output of up to 80% optical conversion efficiency. It is worth noting that as the fundamental linewidth is further reduced, the technical noise of the system will be the major obstacle restricting the realization of the fundamental linewidth in freespacerunning BLs. Fortunately, the corresponding frequency stabilization or noise suppression techniques will help solve this problem. This study will provide a new technical route to achieve highpower, highefficiency ultranarrow linewidth laser radiation at special wavelengths.
[2]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[19]
[20]
[22]
[23]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
Duo Jin, Zhenxu Bai, Zhongan Zhao, Yifu Chen, Wenqiang Fan, Yulei Wang, Richard P. Mildren, Zhiwei Lü. Linewidth narrowing in freespacerunning diamond Brillouin lasers[J]. High Power Laser Science and Engineering, 2023, 11(4): 04000e47.