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Abstract
Free-space Brillouin lasers (BLs) are capable of generating high-power, narrow-linewidth laser outputs at specific wavelengths. Although there have been impressive experimental demonstrations of these lasers, there is an absence of a corresponding theory that describes the dynamic processes that occur within them. This paper presents a time-independent analytical model that describes the generation of the first-order Stokes field within free-space BLs. This model is based on the cavity resonance enhancement theory and coupled wave equations that govern the processes of stimulated Brillouin scattering (SBS). This model is validated using an experimental diamond BL to numerically simulate the influence of the cavity design parameters on the SBS threshold, pump enhancement characteristics, and power of the generated Stokes field. Specifically, the model is used to determine the SBS cavity coupler reflectance to yield the maximum Stokes field output power and efficiency, which is also a function of the pump power and other cavity design parameters. This analysis shows that the appropriate choice of Brillouin cavity coupler reflectance maximizes the Stokes field output power for a given pump power. Furthermore, the onset of higher-order Stokes fields that are undesirable in the context of single-frequency laser operation were inhibited. This study aids in understanding the relationship between the cavity parameters and resultant laser characteristics for the design and optimization of laser systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Stimulated Brillouin scattering (SBS) has enabled numerous applications in the fields of microwave photonics [1,2], sensing [3–5], large-scale and high-energy laser systems [6–8], and ultra-narrow linewidth lasers [9,10]. As a third-order nonlinear effect, a higher pump intensity is required to achieve the threshold for the process. This has posed a great challenge to the development of continuous-wave Brillouin lasers (BLs). Extensive research has been carried out in past decades focusing on improving the Stokes gain to achieve high-efficiency systems, such as seeking media with inherently large gain coefficients [11] and developing structures with enhanced photo-phonon interactions [1,12,13]. Significant progress has also been made in the study of maximizing the Stokes gain by optimizing the laser design, including the employment of long fiber ring cavities to increase the interaction length [14,15], or resonant pumping structures to produce high circulating pump power. The combination of optimized laser structures and materials with large gain coefficients reduces the threshold of the BL to tens of microwatts [16,17]. However, the development of these systems introduces new challenges and barriers. For example, the long interaction length in fiber BLs simultaneously reduces the Stokes threshold for any order of Stokes components, limiting the power boost of single-frequency BLs to a certain extent [18]. In addition, the wide Brillouin gain bandwidth of the integrated cavity material (i.e., Si3N4 approximately 300 MHz), assisted by the high Q characteristics (approximately 108) more easily satisfies the phase matching and threshold for higher-order Stokes fields [9]. It is encouraging to observe the emergence of studies that achieve power scaling of single-frequency BLs by suppressing the generation of higher-order Stokes fields. In addition, large mode area fibers in the fiber BLs are used to alleviate the influence of the effective threshold intensity of high-order Stokes fields on single-frequency power boosting. Systems using this approach have achieved a Stokes power up to 5 W [14], which is nearly five times higher than those previously reported [19]. However, the generation of second-order Stokes components remains the main factor limiting single-frequency power scaling [18]. BLs using intermodal scattering in waveguide cavities provide opportunities for manipulating phase-matching conditions [12,20], which was successfully used in Ref. [21] to break the phase matching of high-order Stokes fields while increasing the single-frequency power to the level of hundreds of milliwatts. Nevertheless, thermal effects within these microstructures led to a detuning of the Stokes gain that resulted in the clamping of the Stokes power, similar to that observed when high-order Stokes fields were generated [22,23].
Free-space BLs can be designed using various gain media and cavity structures, leading to a wide diversity of Stokes wavelengths and control over the generation of higher-order Stokes components. Free-space BLs can achieve pump-limited outputs of up to 11 W in the visible band [24] and 22.5 W in the near-infrared band [25] using media with high thermal conductivity and Brillouin gain coefficient. Moreover, the power transfer characteristics of these studies do not exhibit clamping behavior and are characterized by a high signal-to-noise spectral purity. However, there are limitations because the cavity parameters are fixed and the theoretical description of the laser is incomplete, even though these reports have demonstrated impressive output characteristics. A beam waist inside the free-running cavity, and the size of the beam waist determines the gain level of Stokes light. This is unlike guided-wave structured BLs, which have a uniform distribution of the optical field inside the cavity. However, studies on resonantly pumped BLs do not specify the dependence of the Stokes output power on the cavity parameters or further optimize solutions [23,26]. In addition, free-space resonators offer far greater flexibility for examining the influence of laser design parameters on BL behavior. Understanding the relationship between the cavity parameters and resultant laser characteristics is vital for the design and optimization of these laser systems.
This study proposes and develops a steady-state model of the first-order Stokes field inside a free-space resonant pumped BL. The spatially distributed Stokes gain equations are derived from the SBS-coupled wave equations combined with the cavity resonance enhancement theory, yielding a time-independent analytical power model of the first-order Stokes field. The validity of the model was experimentally verified using a diamond BL. The influence of the cavity design parameters on the Stokes threshold, cavity enhancement characteristics of the pump, and Stokes power were investigated. Finally, the model was employed to explore the optimal cavity parameters that maximize the output power and efficiency. Furthermore, it was demonstrated that the second-order Stokes threshold could always be higher than the given pump power using an optimized coupling mirror reflectivity. This feature enhanced the single-frequency BL power independent of the generation of high-order Stokes fields, providing a new technical route for the realization of high-power single-frequency lasers.
2. Single-frequency BL model
The intensity distribution of the pump and Stokes fields within the cavity plays a significant role in the energy transfer between the fields [27]., The change in the intensity of the intracavity Stokes and pump fields in the paraxial approximation and under steady-state conditions can be respectively expressed as [28],
Additionally, the power in each beam is expressed as:
The treatment of wP(z) = wS(z) = w(z) with $\textrm{M}_{\textrm{P,S}}^\textrm{2}$ equal to 1 and bP = bS = b is valid beacause the quantum conversion efficiency of the pump field to the Stokes field is close to unity when considering the TEM00 intensity distribution. Integration of Eq. (1) in the polar coordinates (r from 0 to ∞ and θ from 0 to 2π) yields the power distributions of the intracavity pump and Stokes fields, which are respectively expressed as:
The solutions of Eqs. (6a) and (6b) are straightforward because the variables are separated and respectively expressed as:
The corresponding pump nonlinear loss GP and Stokes gain GS are respectively expressed as:
Arctan (l⁄ b) approximates to l/b, and leff is equal to the crystal length l when l << b. The expression for the Stokes gain is then simplified to gSIPl., leff approaches its limit, bπ/2 when l >> b. A cavity design that utilizes tight focusing is beneficial for achieving a high Stokes gain. However, such a structure requires the use of large-aperture mirrors. Furthermore, this design accompanies a correspondingly short Rayleigh length that limits the spatial mode matching.
The most critical characteristic of a resonantly pumped laser is power enhancement. The power enhancement factor of an intracavity pump field when the cavity meets the pump resonance condition is expressed as [29]
Here, the corresponding ratio of the optical power reflected by the cavity to the incident pump power is expressed as:
The passive losses in the cavity apart from the coupler are attributed to L, which includes characteristics such as the incomplete reflection of the cavity element and absorption losses. While the Stokes field operates in a steady state, the net Stokes gain is expressed as:
The residual pump power after one round trip of the cavity is 1-Rm, except for the loss caused by the coupler mirror at a given Stokes optical power, where Rm contains both nonlinear loss GP and passive loss L. This is expressed as:
The relationship between the Stokes output power and circulating Stokes power in the cavity is expressed as:
Substituting Eqs. (10) and (14) into Eq. (13) yields:
The pump enhancement factor from Eq. (16) is inversely proportional to the externally injected pump power when the Stokes field is in a steady state. The intracavity pump power remained constant after it exceeded the Stokes threshold. Using Eqs. (10), (13), and (16) to eliminate αEn and Rm, the first-order Stokes power can be derived as:
The Stokes power model can be further simplified to obtain a formally consistent expression that relates to the actual power curve parameters. This is accomplished using the approximations ($\textrm{x} \to {1,\; \textrm{ln}}({1/x} )\approx ({\textrm{1 - x}} ),\,$), ($\textrm{x} \to {1,\; 1 - }\sqrt {\rm x} \approx 0.5({{1 - {\rm x}}} )$), and ($\textrm{x} \to {0,\; }{\textrm{e}^\textrm{x}} \approx \textrm{1 + x}$). As a result, the updated calculations for PSout, Pth, and σS can be respectively expressed as:
The first-order Stokes power model of the BL is formally consistent with that of the Raman model, but with a more concise form owing to the minor difference in coupler reflectivity between the pump and Stokes fields [30]. It is worth noting that this particular case of the BL makes it impossible to separately optimize the coating reflectivity, as in the case of Raman lasers, which will be discussed in detail in Section 4.
3. Comprehensive analysis of a free-space-running BL
3.1 Experiments
The layout of the free-space BL is illustrated in Fig. 1. The gain medium that was used to experimentally validate the BL model was a Brewster-cut diamond (4 × 1.2 × 5 mm3, 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 on the right side of Fig. 1. The pump laser had a maximum power of 60 W that was emitted from a fiber-based optical amplifier (OA) and used a single-frequency 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 of approximately 560 MHz. M1 denotes a plane mirror (R1 = 96%), M2 denotes a high-reflecting plane mirror, and M3 and M4 denote concave, high-reflecting mirrors with radii of curvature of 100 mm. 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 optimized to support the dual resonance of the pump and Stokes field frequencies. Frequency locking of the pump field to the cavity was achieved using the Pound-Drever-Hall frequency stabilization technique [31]. The frequency locking of the Stokes field was facilitated using the thermal feedback introduced by the Brillouin gain profile owing to the thermo-optical effect of diamond caused by the generation of the Stokes radiation [32,33]. 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 output from the system using an isolator (ISO).
Fig. 1. Schematic layout of a free-space, free-running BL based on a diamond. (EOM, electronic optical modulator; OA, optical amplifier; ISO, isolator; f1-2, lenses; HR1-2, high reflective mirrors; M1: partial reflectivity plane mirror; M2, high reflectivity plane mirror; M3-4, high reflectivity concave mirrors; PD, photo-detector).
In this system, the SBS gain coefficient gS was 14.9 cm/GW for a pump wavelength λ of 1064 nm, which was calculated and experimentally verified in a prior publication [25]. The passive loss L had a value of 1.76%, including the transmission loss of the cavity (except for the coupler mirror) and loss introduced by the gain medium (geometric refraction loss and absorption). The mode-matching factor γ was set to 0.96, and the equivalent beam size of the cavity waist w0 was approximately 76 µm. The specific parameters of the BL system parameters used in this study are listed in Table 1, except for the variables and ranges provided in the corresponding pictures. The values of the parameters outside the study subjects are not specifically indicated in the following sections.
Table 1. Parameters of the diamond BL system.
The power-transfer characteristics of the laser were investigated, and the experimentally determined Stokes and cavity-reflected powers as a function of the incident pump field power are plotted as filled red squares and black circles in Fig. 2, respectively. Additionally, the result of substituting the parameters from Table 1 into Eq. (19) is shown in Fig. 2. The predicted Stokes and cavity-reflected powers are denoted as red dash-dotted and black short-dashed lines, respectively. The model-predicted Stokes output power was consistent with the experimental results. In terms of the cavity reflected power (residual pump power), the experimental and model-predicted results diverged when the pump power exceeded the SBS threshold power (29 W). The reason for this discrepancy is that the power load across the optical path changes as the back-propagating Stokes power increased, which subsequently increased the thermal effect on the optical elements and reduced the mode-matching efficiency. Experimentally, the mode matching was optimized for a power load of 60 W. The generated Stokes power led to the best mode matching condition for a pump power of 45 W, corresponding to a Stokes output power of 15 W. Therefore, the total power load was the same as that of the pre-set. A higher pump power will lead to a further mismatch in mode matching, which exacerbated the discrepancy between the actual cavity reflection pump power and model prediction, resulting in a slightly lower Stokes output power than the prediction.
Fig. 2. Plot of the output Stokes and reflected pump powers from the experiment, calculated using the developed analytical model.
3.2 Dependence of Stokes threshold on cavity design parameters
The Stokes threshold was determined by a combination of the cavity design parameters, SBS gain medium, the pump field. Careful design of the cavity parameters can enhance the pump field to reduce the threshold power for the SBS process. The contour distribution of the Stokes threshold plotted using Eq. (19(b)) as a function of cavity design parameters, such as passive loss L, mirror reflectivity R1, confocal parameter b, and the SBS crystal length l is shown in Fig. 3.
Fig. 3. Plots showing the dependence of the pump power required to reach the SBS threshold as a function of cavity parameters: (a) dependence on the coupling mirror reflectivity R1 and the intracavity passive loss L; (b) dependence on the SBS crystal length l and confocal parameter b.
The threshold power was consistent with the variation of the passive loss, as shown in Fig. 3(a). In addition, the SBS threshold was minimized as the reflectance increased, but it increased rapidly when the reflectance exceeded a certain value. The pump loss in the cavity was dominated by the passive loss L from Eqs. (10) and (14) when the system operated close to the SBS threshold. The resonant enhancement factor of the pump field within the cavity at the threshold operation is expressed as:
The function produced from Eq. (20) achieves its maximum value when R1 is equal to 1- L, which is often referred to as the impedance matching condition. In addition, reducing the passive losses improves the enhancement factor when the corresponding impedance is matched. Thus, it is advantageous to design a Brillouin cavity with high reflectivity and minimize the passive loss of the cavity to reduce the SBS threshold. The influence of the SBS crystal length l and cavity confocal parameter b on the threshold value Pth is shown in Fig. 3(b). The variation in threshold power is inversely proportional to the crystal length for a given cavity confocal parameter. Moreover, tightly focused structures significantly reduce the threshold power for a given crystal length. According to Eq. (9), a small confocal parameter in conjunction with a considerable crystal length allows the effective interaction length to approach its limiting value bπ/2. At this point, the dependence of the threshold power on the crystal length and confocal parameter diminishes. In this case, the threshold power gradually decreases and tends towards the limiting value, which is expressed as:
The limitation of the cavity length is more often used to describe the theoretical limit of tightly focused fields. Although tightly focused fields can significantly reduce the threshold power, the beam waist size cannot be reduced indefinitely in a practical system. Moreover, it has become progressively more challenging to match the pump field to a cavity mode with a beam waist size that is too small.
3.3 Characteristics of pump enhancement and Stokes power output
The model produced from Eq. (19) can be used to predict the cavity pump power evolution and Stokes power characteristics. Plots of the intracavity pump and output Stokes powers as functions of these cavity design parameters are denoted as grey contours and colored lines in Fig. 4, respectively.
Fig. 4. Plots of the Stokes output power PSout and intracavity pump power $\textrm{P}_\textrm{P}^{\textrm{intra}}$ as functions of a range of cavity parameters with the pump power ranging from 0 to 100 W. Plots as a function of: (a) coupling mirror R1, (b) passive loss L, (c) effective cavity waist size w(0), and (d) crystal length l.
The intracavity pump power was obtained from Eq. (20) when the pump power is below the Stokes threshold (PSout = 0), as shown in Fig. 4 (a). Here, the intracavity pump power enhancement factor was related only to the coupling mirror reflectivity and passive loss. Furthermore, the variation in the intracavity pump power was proportional to the incident pump power. The cavity enhancement factor reached its maximum value when the coupling mirror reflectivity matches the passive loss of the cavity. Thus, minimized pump power was required to reach the Stokes threshold. The intracavity pump field loss included the passive loss and nonlinear losses introduced by Stokes generation when the pump power was above the Stokes threshold (PSout > 0). Most of the intracavity pump power was converted to Stokes output power as the pump power increased at a given coupler mirror reflectivity, resulting in an insignificant increase in the intracavity pump power. The maximum Stokes output power (inflection point of the colored lines) occurred when the coupling mirror reflectivity matched the intracavity loss (impedance matching) at a given pump power. An increase in pump power led to a more extensive nonlinear loss of the intracavity pump field because the pump power level determines the maximum gain of the Stokes field, which reduced the value of the coupling mirror reflectance corresponding to the maximum Stokes output power and increased the Stokes conversion efficiency.
The evolution of the intracavity pump power for a pump power less than the SBS threshold for variable passive loss was consistent with that observed in the case of the change in coupling mirror reflectivity, as shown in Fig. 4 (b). Reducing passive losses enhanced the cavity enhancement factor, leading to a corresponding reduction in the power required to reach the Stokes threshold. Reducing the passive loss in the cavity is critical for improving Stokes power and conversion efficiency because the passive loss determines the maximum pump-resonance enhancement factor that can be achieved. For example, the Stokes output power increased from 8 to 24 W at a pump power of 40 W, reducing of passive losses from 0.02 to 0.01 with a corresponding three-fold increase in the conversion efficiency.
The cavity enhancement factor was independent of the waist w(0) when the pump power was below the threshold for the SBS. Conversely, the intracavity pump power was not influenced by the waist size. A waist size existed that maximized the corresponding Stokes output power for a given pump power that was greater than the SBS threshold, as shown in Fig. 4(c). From Eq. (9), the beam waist size affects the confocal parameter, which in turn modifies the effective interaction length and intracavity nonlinear loss of the pump. The maximum Stokes output power at a given pump power was achieved by adjusting the Stokes gain to match the pump impedance after the coupling mirror reflectivity was determined. Therefore, the beam size corresponding to the maximum conversion efficiency increased when the pump power was increased. Similarly, limited by the confocal parameter, the Stokes gain reached its limit at crystal lengths that were much larger than the confocal parameter. The change in the Stokes power did not vary significantly with the crystal length, as shown in Fig. 4(d).
4. Optimization of single-frequency BL power and efficiency
It is necessary to customize the cavity parameters for specific power levels in the design of lasers that utilize nonlinear frequency conversion processes to optimize the optical conversion efficiency of the system. The cavity design parameters that are most easily varied and controlled are the coupling mirror reflectivity, crystal length, and confocal parameters. An optimal value of coupling mirror reflectivity can be found for a given pump power after the other cavity parameters are determined that maximizes the corresponding Stokes output power, as detailed in Section 3. This value can be mathematically determined as a solution to the partial derivative of the Stokes output power PSout for the coupler reflectivity R1. Using Eq. (19), the partial derivative of PSout with respect to R1 is derived as,
The optimal coupling mirror reflectivity as a function of pump power should be obtained at the extremum of Eq. (22). Here, the corresponding R1 = R1-opt was solved as an analytical expression with the pump power and cavity parameters. However, there was no difference in the reflectivity of the coupling mirror for the pump and Stokes owing to the very small frequency shift of the SBS process, which is unlike other third-order nonlinear frequency-shifted lasers, such as resonantly pumped continuous-wave Raman lasers [30]. This simplifies the mathematical description of a BL in contrast to that of a Raman laser. Nevertheless, solving for $\partial {\textrm{P}_{\textrm{Sout}}}/\partial {\textrm{R}_{1}}{ = \; 0}$ should theoretically lead to an optimal value for the Stokes output and resonant enhancement characteristics of the pump. This is because the coupling mirror reflectivity of the BL system affects simultaneously the characteristics of resonant enhancement of the pump field and coupled output of the Stokes field. However, this optimal value cannot be obtained using a single equation. $\partial {\textrm{P}_{\textrm{Sout}}}/\partial {\textrm{R}_\textrm{1}}{ = \; 0}$ yields a transcendental equation that cannot be derived from an analytical solution similar to that of RS-opt for a resonantly pumped Raman laser [30]. The corresponding R1-opt of a BL can only be derived using a numerical solution.
4.1. Optimized coupling mirror reflectivity R1-opt for different pump’s power
The optimized coupling mirror reflectivity values R1-opt for different pump powers are denoted as black dashed line in Fig. 5. The distributions of the Stokes power PSout and conversion efficiency that were determined using different pump powers and coupling mirror reflectivity R1 are plotted with gray contour lines and colored backgrounds, respectively, to illustrate the influence of the optimized coupling mirror reflectivity values. The point at which the Stokes output power and optical conversion efficiency reach their maximum values was consistent with the intersection of the optimized coupling mirror reflectivity R1-opt curve. In addition, the decreased coupling mirror reflectivity R1 increased the Stokes threshold and slope efficiency σS from Eq. (19), which also led to an increase in the optical conversion efficiency for higher pump power operation. In contrast, most of the increased power of the Stokes field was confined to the cavity. The increased nonlinear loss of the pump in the cavity result in a mismatch with the coupler reflectivity, which subsequently reduced the pump enhancement factor. Finally, the circulating power of the pump field in the cavity achieved a dynamic balance related to the gain and loss of the Stokes field. Impedance mismatch can be suppressed to a certain extent by actively reducing the coupler reflectivity. Therefore, the increased pump power was correspondingly converted into the Stokes power.
Fig. 5. Plot showing the contour distribution of the Stokes output power PSout and optical efficiency as functions of pump power and coupling mirror reflectivity R1. The R1-opt curve corresponding to the optimized Stokes output power for a given pump power is also plotted.
4.2 Dependence of optimized coupler reflectivity R1-opt on crystal length, beam waist, and gain coefficient for a given pump power
The optimal coupler reflectivity from Eq. (22) is related to the power, crystal length, cavity waist size, and gain coefficient. Here, the pump power was set to 60 W and the variations in the optimized coupler reflectivity and Stokes output power were plotted, as shown in Fig. 6. The contour distribution (colored lines) of the optimized coupler reflectivity R1-opt and corresponding Stokes output power (gray background) as a function of the crystal length l and cavity waist size w(0) are shown in Fig. 6(a). The contour distribution of the optimized coupler reflectance and corresponding Stokes output power as a function of the gain coefficient gS and crystal length l are shown in Fig. 6(b). The nonlinear loss of the pump field increased if a small beam waist was used (tight focusing structure) for a given gain medium, which required a corresponding reduction in coupler reflectivity to realize impedance matching and maximize the output efficiency of the Stokes field. In addition, increasing the crystal length and Stokes gain also required reducing the coupler reflectivity to maximize the output Stokes power. However, it is worth noting that the radii of curvature of cavity mirrors M3 and M4 used in these experiments were 100 mm. The length of the diamond crystal used in this experiment was 5 mm. These mirrors had a separation of 105 mm, resulting in an equivalent beam waist size w(0) of 75 µm. The corresponding optimized coupler reflectance and Stokes output power were calculated to be 0.96 and 22.5 W for a pump power of 60 W, respectively. Further reduction in the beam size by decreasing the distance between M3 and M4 led to an unstable cavity. Therefore, the use of a medium with a larger SBS gain is necessary to further increase the output Stokes power under the limitations of the beam waist size and cavity length. In such a case and based on the proposed modeling, this must be balanced by considering the length of the medium and coupler reflectance to maximize the Stokes output power.
Fig. 6. Plots showing the optimized coupler reflectance R1-opt and Stokes output power PSout as functions of cavity parameters at a fixed pump power of 60 W. Plots as a function of: (a) SBS medium length l and cavity waist size w(0), and (b) SBS medium length l and gain coefficient gS.
4.3 Suppression of higher-order Stokes fields through the selection of optimized coupling mirror reflectivity
The model developed in this study is applicable to single-frequency BLs, wherein only a first-order Stokes field is generated. Typically, this discussion is valid only when the pump power is below the threshold of second-order (and higher) Stokes fields [9,23], and when these higher-order Stokes fields are not phase-matched. Through an analysis of the model and solution of the optimized coupler reflectivity, it was generally observed that the corresponding optimal coupler reflectivity value decreased with increasing pump power. The SBS threshold is related to the coupling mirror reflectivity, thus using the optimized coupler reflectivity as the cavity design parameter was beneficial as a means of effectively suppressing the onset of higher-order Stokes fields while improving the single-frequency Brillouin laser power. Motivated by this, the pump power required for the onset of the second-order Stokes field (following the method described in Section 2) was derived as follows:
The power required to reach the threshold for a second-order Stokes field was four times higher than that of the first-Stokes field for the given set of cavity parameters listed in Table 1. Equations (22) and (23) were used to analyze the power required to reach the second-order Stokes threshold and associated optimized coupling mirror reflectivity. To this end, the optimal coupling mirror reflectivity, Stokes conversion efficiency, and second-order Stokes threshold power PS2th were plotted as functions of the pump power from 0 to 5000 W, as shown in Fig. 7. The conversion efficiency of the pump field compared to that of the Stokes field steadily increased and gradually plateaued in the case where the coupler reflectivity was at its optimal value, reaching 91.7% at a pump power of 5000 W, as depicted by the red dotted line in Fig. 7(a). this saturation of the conversion efficiency was attributed to passive losses within the cavity. The threshold for the onset of the second-order Stokes field for an optimized coupler reflectivity was always higher than that of the input pump power, as indicated by the blue curve in Fig. 7(b). The pump power is plotted as a short black line to highlight this comparison. Thus, higher-order Stokes fields cannot be generated by selecting the coupling mirror reflectivity, which optimizes the Stokes power output of the single-frequency BL.
Fig. 7. (a) The change in optimized coupling mirror reflectivity R1-opt and corresponding Stokes conversion efficiency are plotted as functions of pump power. (b) Plot of the pump power required to reach the threshold for the second-order Stokes field for an optimized value of R1-opt.
It is clear from the above analysis and discussion that the design and optimization of single-frequency BLs are non-trivial, and a complex interplay exists between the cavity design parameters, SBS gain medium, and pump field. Maximizing the Stokes output from such a laser system requires careful consideration of several factors. This is especially the case to avoid the onset of high-order Stokes fields [9,22]. It is hoped that the analytical model and work presented here will serve as a basis and foundation for the design of new, efficient, and high-output power single-frequency BLs.
5. Conclusion
This study developed an analytical model that described the power transfer characteristics of a resonantly pumped single-frequency BL. The proposed model was based on classical SBS coupled wave equations incorporated the cavity resonant enhancement theory. The validity of this model was demonstrated for determining the output Stokes power as a function of various cavity design parameters. In contrast to the quantum descriptions previously used to describe the Brillouin dynamics in waveguide cavities, the proposed model considered the relationship between the focusing characteristics of beams propagating in free space, cavity parameters (passive loss, coupling mirror reflectivity/coupling coefficient), and the Brillouin gain coefficient of the gain medium. This model was verified by an experimental system based on a diamond crystal. It was shown that through the modeling that the onset of higher-order Stokes fields could be completely avoided by optimizing the cavity parameters for maximum first-Stokes generation at a given pump power, a problem that has long since limited other SBS laser systems. It is anticipated that this model will be a valuable asset for the design of high-power single-frequency BLs and the prediction of their associated power transfer characteristics.
Funding
National Natural Science Foundation of China (61905061, 61927815); Natural Science Foundation of Tianjin City (20JCZDJC00430); Funds for Basic Scientific Research of Hebei University of Technology (JBKYTD2201).
Acknowledgments
Duo Jin acknowledges support from the Hebei Provincial Department of Education's Postgraduate Student Innovation Capacity Development Grant Program (CXZZBS2021030); RPM acknowledges support from the Asian Office of Aerospace Research and Development (AOARD).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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