Influence of a longitudinal-mode on stimulated Brillouin scattering characteristics in fused silica

Bin Chen; Bin Chen; Zhenxu Bai; Zhenxu Bai; Yifu Chen; Yifu Chen; Kun Wang; Can Cui; Can Cui; Yaoyao Qi; Yaoyao Qi; Jie Ding; Jie Ding; Bingzheng Yan; Bingzheng Yan; Yulei Wang; Yulei Wang; Zhiwei Lu; Zhiwei Lu; Zhiwei Lu

doi:10.1364/OE.498489

1. Introduction

High-power pump sources, along with multistage amplifiers, are extensively employed to enhance laser output power and optimize overall performance [1–3]. The quality of a high-power -laser beam is greatly influenced by wavefront distortion resulting from thermal lensing and thermally induced birefringence [4,5]. Research has shown that the wavefront distortion can be corrected by constructing a stimulated Brillouin scattering phase conjugation mirror (SBS-PCM) within a laser system [6–8]. SBS also exhibits minimal frequency shift, high energy-conversion efficiency, beam cleanup and linewidth narrowing effects [9–13]. To enhance the load capacity of SBS-PCM, Wang et al. [14] confirmed that a liquid medium has a higher optical breakdown threshold after being purified. Kang et al. [15] demonstrated the use of an ultra-clean closed SBS-PCM to maintain impurity particles in the FC-770 medium below 40 nm. Their findings indicated that even at an output power of 550 W (approximately 1.1 J pulse energy) and a pulse repetition rate of 500 Hz, there were no noticeable optical faults or significant thermal effects. To mitigate thermal effects and increase the repetition rate of SBS operations, Wang et al. [16] used a rotating eccentric focusing lens to achieve high pulse repetition rates. This lens has less coma, enabling the realization of SBS-PCM at the kilohertz level. Subsequently, an SBS pulse compression experiment was conducted using fused quartz in the 100-1000 Hz frequency range to obtain a sub-nanosecond output [17]. The above research indicates that solid optical media with high thermal conductivity and purity can effectively reduce the negative effects of thermal impact. Conducting high-power and high-repetition-rate SBS experiments is advantageous.

Numerous studies have examined the influence of various parameters on SBS in a medium, including the gain coefficient and phonon lifetime. In addition, the effect of the experimental structure on the SBS was studied. Previous studies have reported that a high compression ratio is facilitated by shorter phonon lifetimes and longer phase interaction lengths. Achieving high phase conjugation fidelity is advantageous when using a high gain coefficient and short interaction length [18,19]. However, their experiments typically focused on single-mode pumps, and there are relatively few studies on the SBS output when using multi-longitudinal-mode pumps [20–22].

The multi-longitudinal-mode can affect both the time- and frequency-domain characteristics of the laser pulses. First, in the time domain, the energy reflectivity of the SBS and other output results can be affected by the multi-longitudinal-mode. Additionally, the beat waveform can generate pulse spikes with considerably high peak power, which may damage the amplifier and other optical components [23,24]. Lee et al. [25] reported that multi-longitudinal-mode pulses exhibited enhanced SBS due to self-focusing. However, the nonlinear effect caused by the intensity spikes at a high pump power can suppress the SBS. Jaberi et al. [26] studied the influence of the longitudinal-mode structure on the energy reflectivity and time behavior of the reflected pulses in the SBS process. They observed that the beat frequency of the reverse-amplified Stokes pulse mode decreased significantly, and the fluctuations of the Stokes pulse were reduced in the SBS amplifier. Relatively few studies have investigated the impact of the frequency-domain characteristics on the SBS output, such as the linewidth of the multi-longitudinal-mode. Wang et al. compared Stokes linewidths in various liquid media. The study revealed that the primary factors influencing the linewidth of Stokes pulses are the Brillouin gain and pump power density. A medium with a wider Brillouin gain linewidth yields a laser with a broader linewidth [27]. However, the laws and characteristics of solid media for the SBS linewidth output are yet to be reported. Therefore, analyzing the effect of multi-longitudinal-mode pumps on SBS is essential for researching high-power solid-state SBS systems.

The relationship between the output characteristics of the SBS and the longitudinal-mode of the pump laser was analyzed using the time domain, frequency domain, and energy. In this study, fused silica was used as the medium to produce SBS, and a stable pump source that generates output using either a single longitudinal or multi-longitudinal mode was built. The specific output parameters of the longitudinal-mode, including modulus, mode spacing, and linewidth, were determined. We demonstrated that the pulse spike resulting from the multi-longitudinal-mode is the primary factor responsible for an optical breakdown during SBS operation. The SBS generation threshold, energy reflectivity, linewidth, and other output parameters of the single and dual longitudinal-mode were compared. The study also quantified the effects of different intensities of the longitudinal-mode. This study establishes a foundation for a high-power Brillouin system based on a solid-state medium.

2. Theoretical model

Because of the long optical length of the cavity, the spacing between the longitudinal-mode in the free-space optical laser is small. Consequently, it typically operates in a multi-longitudinal-mode. The unidirectional traveling wave operation of the ring cavity can eliminate the spatial hole-burning effect, making it easier to achieve a single-longitudinal-mode output. The longitudinal-mode spacing of an annular cavity is calculated as follows:

(1)$$\Delta {\nu _q} = \frac{c}{{nl}}$$

where c is the speed of light in vacuum, n is the refractive index, and l is the physical length of the medium. More longitudinal-mode could be obtained by increasing the resonator length. A shorter resonant cavity can achieve larger longitudinal-mode spacing, thus reducing the number of longitudinal-mode capable of oscillation. The square ring cavity in the same area was more than 20% shorter than the 8-shaped ring cavity. Therefore, an optical length 66.5 cm square ring cavity had a theoretical longitudinal-mode spacing of 0.45 GHz.

A single-longitudinal-mode output can be achieved in a ring cavity operating in free space by inserting a Fabry-Pérot (F-P) etalon. The transmission spectrum linewidth of the F-P etalon is δν. To obtain a single-longitudinal-mode output, it must satisfy δν<Δν_q. The expression for δν is as follows:

(2)$$\delta \nu = \frac{{c({1 - R} )}}{{2\pi {n_1}\textrm{d}\sqrt R }}$$

where R is the reflectivity of the etalon, n₁ is the refractive index of the etalon, and d is the thickness of the etalon. In this experiment, a F-P etalon with a thickness of 0.5 mm and a reflectivity of 90% was used to satisfy the mode selection requirements.

SBS is a typical third-order nonlinear phenomenon that involves the pump field (E_P), the back-propagating Stokes pulse field (E_S), and the phonon field (ρ). In quantum mechanics, the scattering process can be viewed as the annihilation of a pump photon while simultaneously generating a Stokes photon and an acoustic phonon. According to previous research, the measured energy in a multi-longitudinal-mode structure is the sum of the energies of the longitudinal-mode [21]. Assuming that the longitudinal-mode are independent when the pump energy reaches the SBS threshold and that the acoustic field is established after a period, the Stokes pulse generated by the multi-longitudinal-mode is still a multi-longitudinal-mode pulse with the same frequency interval. The coupling-wave Eqs. (3) (4), and (5) can be used to simulate the SBS process as follows [28]:

(3)$$\frac{{\partial {E_\textrm{P}}}}{{\partial z}} + \frac{\alpha }{2}{E_\textrm{P}} + \frac{{{n_g}}}{c} \cdot \frac{{\partial {E_\textrm{P}}}}{{\partial t}} = \frac{{i{\omega _\textrm{P}}\gamma }}{{2{n_g}c{\rho _0}}}\rho {E_\textrm{S}}$$

(4)$$- \frac{{\partial {E_\textrm{S}}}}{{\partial z}} + \frac{\alpha }{2}{E_\textrm{S}} + \frac{{{n_g}}}{c} \cdot \frac{{\partial {E_\textrm{S}}}}{{\partial t}} = \frac{{i{\omega _\textrm{S}}\gamma }}{{2{n_g}c{\rho _0}}}{\rho ^\ast }{E_\textrm{P}}$$

(5)$$\frac{{{\partial ^2}\rho }}{{\partial {t^2}}} - ({2i\omega - {\varGamma _\textrm{B}}} )\frac{{\partial \rho }}{{\partial t}} - ({i\omega {\varGamma _\textrm{B}}} )\rho = \frac{{\gamma q_\textrm{B}^2}}{{4\pi }}{E_\textrm{P}}{E_\textrm{S}}^\ast $$

where z is the medium length, α is the absorption coefficient, n_g is the refractive index of the medium, t is time, ω is frequency, γ is the electrostrictive coefficient, ρ₀ is the average density of the medium, ρ is the change in the medium density caused by the photoelastic effect, Γ_B is the Brillouin linewidth, and q_B is the wave vector of the acoustic field.

This set of coupled wave Eqs. was used as the basis for our numerical simulation and was modified to a form suitable for the focusing structure. After solving the coupled wave Eqs. and substituting the expression for the light intensity, they can be rewritten as follows:

(6)$$\frac{{\partial {A_\textrm{P}}}}{{\partial z}} + \frac{\alpha }{2}{A_\textrm{P}} + \frac{{{n_g}}}{c} \cdot \frac{{\partial {A_\textrm{P}}}}{{\partial t}} ={-} igl{A_\textrm{S}}$$

(7)$$- \frac{{\partial {A_\textrm{S}}}}{{\partial z}} + \frac{\alpha }{2}{A_\textrm{S}} + \frac{{{n_g}}}{c} \cdot \frac{{\partial {A_\textrm{S}}}}{{\partial t}} ={-} ig{l^\mathrm{\ast }}{A_\textrm{P}}$$

(8)$$\rho ({z,t} )= \frac{1}{{\sqrt {2\pi } }}\mathop \int \nolimits_{ - \infty }^t f({t - \tau } ){A_\textrm{P}}({z,\tau } ){A_\textrm{S}}^\mathrm{\ast }({z,\tau } )d\tau $$

where amplitude intensity A replaces electric field intensity E, g is the gain strength, which is the three-wave coupling strength in the SBS process, $g = {g_B}{\varGamma _B}\textrm{ / }2w$, g_B is the steady-state Brillouin gain coefficient, ${g_B} = ({{\gamma^2}{\omega_P}^2} )/({{n_g}{c^3}v{\rho_0}{\varGamma _B}} )$, w is the beam cross-sectional area, and τ is phonon lifetime. The coupled wave Eqs. (6), (7), and (8) were discretized using the time-implicit finite difference method and the space-backward difference method. The numerical model can simulate the energy reflectivity and time-domain waveform.

3. Experimental setup

The device used in the SBS experiment is shown in Fig. 1. A self-built laser diode side-pumped Q-switched ring-cavity laser was used as the pump laser oscillator. A Faraday rotator (FR) and half-wave plate (λ/2) were used to run the laser in one direction. The pulse-repetition rate was fixed at 1 Hz. The central wavelength was 1064 nm. The output pulse from the oscillator was amplified using a double pass. An attenuator was created using a polarizer (P) and half-wave plate (λ/2) to change the energy from 0-100 mJ.

Fig. 1. The device for the SBS experiment: M₁-M₆: mirror, PH: pinhole, FR: Faraday rotator, λ/2: half-wave plate, λ/ 4: quarter-wave plate, P₁-P₃: polarizer, L: lens.

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By controlling the insertion of the F-P etalon, the laser was able to operate in either a single-longitudinal-mode or dual longitudinal-modes, while maintaining an optical length of 66 cm. Pulse waveforms were detected using a fast photodetector (Thorlabs DET08C) and recorded using an oscilloscope (Tektronix MSO64) with a bandwidth of 1 GHz. As shown in Fig. 2, the pulse width in both cases is 11-13 ns (full width half maximum). After performing fast Fourier transform on the waveform, it was evident that the longitudinal-mode spacing was 0.9 GHz.

Fig. 2. (a) Typical single-longitudinal-mode waveform. (b) Typical dual longitudinal-modes waveform (inset: spectrum of waveform after Fast Fourier Transform).

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The F-P etalon can measure the linewidth and spatial distribution of a beam in conjunction with beam profiling cameras. The F-P etalon consists of two mirrors with a reflectivity of 99.5%. The distance between the two mirrors was 50.1 mm, the free spectral range was 3 GHz, and the finesse exceeded 600. The pixel size of the beam profiling cameras was 11 µm, and its resolution was 1024 × 1024, which corresponds to the linewidth measurement accuracy of less than 1 MHz [29,30]. As shown in Fig. 3, the pump laser linewidth in the single longitudinal and dual longitudinal-modes were 65.2 ± 4.32 MHz and 70.2 ± 3.75 MHz, respectively. The longitudinal-mode spacing was approximately 0.9 GHz in the case of dual longitudinal-modes. In both cases, the spatial beam exhibited a Gaussian distribution.

Fig. 3. (a) Typical single-longitudinal-mode linewidth. (b) Typical dual longitudinal-modes linewidth.

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Before entering the SBS medium, the beam radius was approximately 2.6 mm, the lens's focal length was 30 cm. The experiment was conducted at room temperature (∼20°C), meanwhile, no active cooling was applied to the fused silica. The distance between the lens and the medium was 20 cm. The SBS medium used in this study was a 20 cm fused silica (JGS1) with the following parameters: refractive index of 1.45, density of 2.2 g/cm³, phonon lifetime of 0.98 ns, Brillouin frequency shift of 16.3 GHz, and the steady-state SBS gain coefficient of 2.9 cm/GW [27]. Because of the phase conjugation property of the SBS-reflected beam, it passed through a quarter-wave plate (λ/4) and acquired S-polarization characteristics. Subsequently, it was reflected by a polarizer in the measurement system.

4. Results and discussion

In this experiment, we investigated the effect of the pump laser energy on the energy reflectivity while pumping in single and dual-longitudinal-modes, as shown in Fig. 4. The solid line represents the theoretical simulation results, and the data points accompanied by an error bar represent the experimental results. The SBS threshold is 10.2 mJ for the single-longitudinal-mode pump and 11.9 mJ for the dual longitudinal-mode pump. The SBS reflection energy gradually increases with increasing pump energy. The single-longitudinal-mode pump ramp efficiency was approximately 80%, and the dual longitudinal-mode pump ramp efficiency was approximately 77%. The appearance of bright sparks in the fused silica indicates the optical breakdown of the medium. The breakdown threshold of fused silica is approximately 35 mJ when pumped in the multi-longitudinal-mode. With the waist radius at the focal point of 40 mm, the measured breakdown threshold of fused silica is approximately 35 mJ using the multi-longitudinal-mode pumping, corresponding to the power density of 57.1 GW/cm². After the breakdown, the reflected energy of the SBS gradually decreased until it could no longer be produced. In contrast, under limited pump energy, no breakdown even if the power density exceeds 150 GW/cm² for the single-longitudinal-mode pumping.

Fig. 4. Evolution of SBS reflected energy with pump energy.

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A pump laser with varying single-longitudinal-mode rates was achieved by replacing the F-P etalon. Repeated experiments have verified the breakdown threshold to be approximately 35 mJ at different single-longitudinal-mode rates. The breakdown probability in fused silica gradually decreased as the single-longitudinal-mode rate increased. When the pump operates in 100% single-longitudinal-mode, the threshold for dielectric breakdown is significantly improved, resulting in an enhanced energy load capacity.

The results of multi-longitudinal-mode pumping were classified into two groups based on the intensity of the interference ring. The typical output of the SBS under different longitudinal-mode strengths, when the pump energy is 20 mJ, is shown in Fig. 5. It can be seen from Figs. 5 (a) and (c) that the intensity of Stokes is positively correlated with the pumping intensity over time whether single- or multi-longitudinal-mode pumping. The longitudinal-mode spacing of the Stokes pulse pumped by the strong and weak multi-longitudinal-mode was the same as that of the pump light, and the relative mode amplitude was lower than that of the pump light. Figures 5 (b) and (d) show that the results of multi-longitudinal-mode linewidth with different intensities are similar, and the Stokes linewidth obtained by multi-longitudinal-mode pumping is 5-10 MHz narrower than that obtained by single-mode pumping.

Fig. 5. (a) (c) Pump and Stokes waveforms for strong and weak multi-longitudinal-mode, (b) (d) Stokes linewidth for strong and weak multi-longitudinal-mode.

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Waveform simulations under single-longitudinal-mode pumping have been studied and have shown good agreement [24,31,32]. The simulation of waveforms in cases with multi-longitudinal-mode is yet to be studied. We selected a typical multi-longitudinal-mode waveform for the simulation, and the results are shown in Fig. 6. Compared with the experimental images shown in Fig. 5, the simulated waveform morphology of the multi-longitudinal-mode has a high degree of similarity. Therefore, numerical models can be used to study changes in multi-longitudinal-mode waveforms, similar to a real experimental environment.

Fig. 6. Waveform simulation of multi-longitudinal-mode pump.

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Forty multi-longitudinal-mode pump energies with varying intensities were collected, and the results are listed in Table 1. The SBS energy obtained by varying the longitudinal-mode strengths indicated that the SBS threshold was correlated with the intensity of the main longitudinal-mode. Therefore, the threshold increases with an increase in the number and strength of the multi-longitudinal-mode.

Table 1. Energy comparison of strong and weak multi-longitudinal-mode

View Table

As shown in Fig. 7, the waveforms under single longitudinal and multi-longitudinal-mode pumps broadened continuously with increasing pump energy. This is because when the interaction length is short, the higher-power-density pump light interacts with the Stokes trailing edge and the phonon field, gradually amplifying the Stokes trailing edge. The higher the power density of the pump, the greater the relative amplitude of the trailing-edge bulge and the more severe the broadening of the Stokes pulse.

Fig. 7. Evolution of waveform with energy reflectivity during single-longitudinal-mode pump and multi-longitudinal-mode pump.

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The changes in the pulse width and linewidth under a single-longitudinal-mode pump are shown in Fig. 8. When the SBS energy reflectivity was less than 14%, pulse width rapidly reduced from 7.3 ns to 5.9 ns. This is because the pump pulse energy was primarily extracted by the leading edge of the Stokes pulse. When the SBS energy reflectivity exceeded 14%, the extraction efficiency of the Stokes pulse saturated, and the pulse width started to broaden. Restricted by the time-bandwidth product, the Stokes linewidth and pulse width exhibited opposite trends. The linewidth fluctuations ranged from 75.5-93.6 MHz. Because of the limited pump energy, the linewidth measured 75.6 ± 2.42 MHz, corresponding to a maximum energy reflectivity of 69%. As the pump energy increases, the pulse width and linewidth approach the pump pulse width and linewidth. The regular variation in the linewidth provides a new method for obtaining a laser with an adjustable linewidth [33].

Fig. 8. Evolution of Stokes pulse width and linewidth with energy reflectivity.

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5. Conclusions

In this study, we investigated the variations in the generation threshold, energy reflectivity, linewidth, and pulse width under different longitudinal-mode and pump energies using fused silica. It has been observed that SBS can be generated by multi-longitudinal-mode. However, when compared to single-longitudinal-mode pumping, the threshold for SBS generation is higher, while the threshold for light breakdown is lower. The experimental results for energy reflectivity and pulse waveform of SBS generated by multi-longitudinal-mode pumping are consistent with theoretical simulations. The results indicate that the time-bandwidth product and the trends of the Stokes linewidth and pulse width change in opposite directions influence the SBS process. As the pump power increases, the SBS system approaches the pump parameters. A narrower pump linewidth assists in exciting a narrower Stokes linewidth. Therefore, it is necessary to adjust the pump laser and select an appropriate SBS medium and structure to attain the desired output parameters using SBS. Single-longitudinal-mode pumping can effectively enhance the SBS loading capacity of fused silica. This has significant guiding for developing more solid media for high repetition rate SBS pulse compression or phase conjugation research.

Funding

National Natural Science Foundation of China (61927815, 62075056); Natural Science Foundation of Tianjin City (20JCZDJC00430); Funds for Basic Scientific Research of Hebei University of Technology (JBKYTD2201).

Acknowledgments

Bin Chen acknowledges support from the Postgraduate Training Program for the Cross-discipline of Hebei University of Technology (HEBUT-Y-XKJC-2021101).

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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	Pump energy	Stokes energy	Residual pump energy	Energy reflectivity
Strong multi-longitudinal mode pump	20.2 mJ	1.5 mJ	18.7 mJ	7.4%
Weak multi-longitudinal mode pump	19.8 mJ	2.6 mJ	17.2 mJ	13.1%

Influence of a longitudinal-mode on stimulated Brillouin scattering characteristics in fused silica

Abstract

1. Introduction

2. Theoretical model

3. Experimental setup

4. Results and discussion

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (8)

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