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Abstract
The pulse duration of the near quarter-acoustic period (τa) is demonstrated in transient stimulated Brillouin scattering (SBS) pulse compression by the suppressing Stokes trailing-edge broadening at high intensities. A theoretical analysis reveals that the difficulty in attaining the transient compression limit is caused by the broadening of the Stokes trailing edge owing to insufficient pump depletion, and this undesirable phenomenon can be significantly suppressed by a high SBS gain coefficient. An average pulse duration of ∼1.05 τa was experimentally achieved in transient compression with a high-energy efficiency of over 30%. Benefiting from energy back conversion, compression below the transient SBS limit (< τa) also occurred when the pump peak power was increased to 150 MW.
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1. Introduction
Stimulated Brillouin scattering pulse compression technology [1,2] has been demonstrated to be an effective approach in achieving high peak power lasers with a single longitudinal mode (SLM), narrow linewidth, and arbitrary wavelength operation [3–5], while avoiding the defects of conventional mode-locked regimes, such as low-energy outputs (several nanojoules to hundreds of microjoules) and complex spectral compositions [6–8]. For peak power enhancement, the generation of the shortest compressed pulse duration τH (full width at half maximum, FWHM) has become a focus of research. Steady-state SBS has been widely employed to compress nanosecond laser pulses to a sub-nanosecond level; further increase of the compression encounters drastic difficulties because of the physical limitation of the phonon lifetime τB [9,10]. An emerging quasi-steady-state SBS is free from this issue and has been reported to achieve excellent compression. However, the compression rate and τB/τrising_time (τrising_time is the rising time of the step pump) are positively correlated [11]; increasing the phonon lifetime causes an extension of the actual pulse duration, while infinitely shortening the pump rising time is very difficult. In terms of physical limitations, the compression limit of transient SBS is a quarter acoustic period τa (on the order of picoseconds), which is clearly a more promising approach for achieving ultrashort pulses [12,13]. However, this suffers from low energy efficiency and Stokes duration broadening at high pump power. To date, compressed pulses near the quarter-acoustic period have not been achieved experimentally.
To push the shortest output pulse generation, it is desirable to suppress the scattered Stokes broadening, and the following approaches have been proposed. First, a special pump waveform, such as a step pulse, was shown to be more conducive in establishing the acoustic field in the gain medium, facilitating the compression of Stokes during the SBS process [14]. However, further shaping of the pump to sub-nanosecond durations is challenging. Second, the interaction length between the Stokes and pump pulses was demonstrated to be a key factor in achieving a short-pulse output. Yoshida et al. initially proposed that the spatial length of the pulse (c/2n) τπ is the optimal Stokes reflection, where τπ is the FWHM of the pump, c is the velocity of light, and n is the refractive index of the medium [15]. Feng et al. conclusively demonstrated that when the focusing length is approximately matched to the full length at half maximum (FLHM) of the incident Gaussian pulse, pulse compression is maximized [16]. However, an inherent characteristic of the transient SBS pump is its short duration (< τB), which only provides an extremely short interaction length. Meanwhile, transient SBS has an extremely high pump power density owing to the high generation threshold, which leading to an interaction length that is shorter than the theoretical setting length [17].
In this work, we focus on constructing a transient SBS compression system and achieving a laser output at the theoretical limit of the quarter-acoustic period. We demonstrate that the gain coefficient of the medium is a key factor in achieving the transient compression limit. As the pump power increases, the peak position of the Stokes shifts forward in the temporal domain owing to the threshold effect of SBS. This reduces the interaction length while weakening pump depletion, and the Stokes trailing edge is amplified by the residual pump and broadening. It has been theoretically demonstrated that a high gain coefficient effectively suppresses the Stokes forward shift with an increase in the pump. To suppress the broadening of Stokes signals in high-power pumps, it is necessary to use a medium with a high gain coefficient.
By choosing a high gain coefficient medium 3M Fluorinert electronic liquid PF-5060, pulses with average duration of 1.05 τa, which is near the theoretical limit, were achieved experimentally. The transient energy conversion efficiency was over 30% owing to the high gain coefficient of the SBS medium. Pulses with durations shorter than the quarter acoustic period were also obtained, which were attributed to the energy back-conversion from the trailing edge to the pump. This is the first time that transient compression has been demonstrated to achieve a theoretical limit with a high energy efficiency. The picosecond laser pulses can be potentially employed in the fields of fast ignition radiation [18], LIDAR Thomson scattering diagnostics [19], and precision laser micromachining [20].
2. Theoretical analysis
SBS pulse compression originates from the asymmetric amplification of the leading edge and the trailing edge of the counter-propagating Stokes. The Stokes broadening phenomenon at high pump intensities is attributed to the over-amplification of the trailing edge owing to the residual pump during asymmetric amplification; this occurs because of the insufficient energy-extraction gain of the leading edge. We focus on the classical SBS generator model with a focusing lens and a cell [21] to analyze the Stokes broadening at high intensity pump. The focal length of lens and cell length determine the focusing length and media length, respectively. Considering that SBS is an automatically phase-matched pure gain process, a Stokes wave generated from the medium at z = L experiences exponential growth as it propagates through the medium, expressed as ${I_2}(z )= {I_2}(L ){e^{g{I_1}(L - z)}}$, where g is the gain coefficient, z is length of the SBS media, I2 and I1 are intensities of the scattered Stokes and pump, respectively. In transient SBS, it is difficult to further increase the genuine Pump-Stokes interaction length L-z by structure optimizations such as increasing the focus length. Firstly, transient SBS has an inherently short interaction length owing to its pump duration of a few hundred picoseconds. Secondly, the position of Stokes shifts forward with the increasing the pump intensity [22], which also shortens the genuine interaction length. And the increased pump intensity I may lead to optical breakdown and other nonlinear effects. Accordingly, increasing the gain coefficient of the compressor medium is the most feasible approach to increase the SBS gain.
The effect of the gain coefficient on SBS compression was analyzed using numerical simulations. SBS is a typical nonlinear effect based on the coupling of three-wave positive feedback, which can be simulated by a generator model. It involves two counter-propagating optical fields, EL (laser) and ES (Stokes), which are coupled through electrostriction to an acoustic field. The optical fields EL and ES are governed by Maxwell’s equations, and the acoustic field $\overline \rho $ obeys the Navier-Stokes equation. The slow amplitude approximation was applied to the laser pump and Stokes fields, preserving the second-order time derivatives of the Navier-Stokes equations, and a set of coupled wave equations describing the SBS process was obtained as follows:
Fig. 1. The evolution of the compressed pulse duration and energy efficiency with gain coefficient gB.
Currently, fluorocarbon liquids, which have the advantage of a high-energy load, are the most widely used SBS medium [23]. However, for transient SBS, the common media face the problem of insufficient gain. Hence, a new high-gain medium, PF-5060, was explored through the precise calculation of the medium’s SBS parameters. The calculation method was optimized based on previous studies [24,25], and the fundamental equation for the gain coefficient is given in [26]. To facilitate the calculation, the relationship between the frequency of the Stokes component and the acoustic vector can be deduced from the energy conservation and momentum conservation relationship in Brillouin scattering [27]. Therefore, the SBS gain coefficient can be expressed as
The SBS parameters of several media with representative gain coefficients were recalculated as shown in Table 1, and compared with those of previous studies [15,28]. Furthermore, the new perfluorocarbon medium PF-5060 has the largest gain coefficient and has not been used in the field of SBS compression.
Table 1. Parameters of several medium
Furthermore, the simulation results of the Stokes radiation temporal waveforms from the SBS generator at pump intensities of 10 MW/cm2, 20 MW/cm2, 30 MW/cm2, and 40 MW/cm2 are shown in Fig. 2, where the moment when the pump entered the cell was set as the zero of the x-axis. A Gaussian pump with a wavelength of 1064 nm and a duration of 8 ns was focused into a 120-cm-long cell through a lens with a focusing length of 75 cm. The intensity of the Stokes pulse was proportional to the incident pump intensity. Meanwhile, as the intensity increased, the Stokes generation time moved forward in the three types of media. In the pump density range from 10 MW/cm2 to 40 MW/cm2, the positions of Stokes radiation produced by the interaction of the pump with the media PF-5060, FC-770, and FC-40 were shifted forward by Δz = 48 cm, 49.2 cm, and 57 cm, respectively. Moreover, the energy of the pump wavefront cannot be fully extracted by backward Stokes in the case of insufficient SBS gain. The residual energy of pump wavefront excites a second Stokes following the main one provided that the SBS generation threshold is satisfied [21]. At the same pump intensities, the weakest second Stokes was generated in PF-5060, which verifies that the high gain coefficient of medium is beneficial for pump dissipation and second Stokes suppression.
Fig. 2. Theoretical analysis of Stokes temporal domain waveform evolution with pump intensities in three different media (a) PF-5060, (b) FC-770, (c) FC-40.
Simulation results show that PF-5060 can reduce position shift by up to 9 cm compared to the other two media and only a small quantity of pump energy remains. This indicates that a high gain coefficient provides PF-5060 with a longer genuine Pump-Stokes interaction length and a stronger pump depletion capability for fast energy extraction and maximum suppression of Stokes trailing-edge broadening.
3. Experimental setup
The schematic diagram of the experimental setup is shown in Fig. 3. The pump source was a customized Q-switched Nd:YAG laser, which delivered an SLM output with a wavelength of 1064 nm. The twisted-mode cavity and resonant-reflector-mediated operation were used to obtain spatial hole-burning suppression and select the cavity axial-mode [29]. A laser pump with an energy of 100 mJ and a pulse duration of 8 ns was obtained, with a root mean square (RMS) of < 2.1%. The combination of a Faraday rotator (FR), two polarizers P1 and P2, and a half-wave plate λ/2 formed a spatial isolator, which prevented powerful backward scattered pulses that may damage the laser oscillator.
Fig. 3. Schematic of the experimental setup of the SBS compressor. FR: Faraday rotator; P: polarizer; λ/2: half-wave plate; λ/4: quarter-wave plate; L: lens; M: mirror; PBS: polarizing beam splitter.
Compact two-cell structures, which have a high pump load and high energy efficiency capacity, were used. A half-wave plate and polarizer were combined to form an adjustable energy attenuator to experimentally control the pump intensity of the injected compression system. The location of Stokes seed generation was random in the generator; consequently, SBS cells had to satisfy the optimal interaction length. The lengths of the generator and amplifier were 80 cm and 100 cm, respectively. While passing through the focusing lenses L1 and L2 located in front of the generator, the pump intensity in the focusing region reached the SBS threshold and created backward SBS pulses owing to electrostriction excitation, which is known as Stokes. The weak phase-conjugated Stokes was amplified through an amplifier system. The polarization states of the injected pulses were altered as they passed twice through a quarter-wave plate λ/4, and the output of the compressed short-pulse was eventually reflected by the polarizer. Each group of compact dual-cells was filled with the same medium.
The condition for transient SBS compression is that the pump duration (τp) should be shorter than the medium phonon lifetime (τB). Two-stage compression was employed, where the first stage is a pre-compression of the initial pump from 8 ns to 500 ps which is shorter than τB of PF-5060, and the second one is for building a transient SBS compression system. The output energy and Stokes duration of the first stage were measured at the back of the PBS to ensure the quality of the output pulse. Owing to the efficiency loss of the SBS compression, an amplifier was inserted to provide a second-stage compressor with a sufficient pump intensity. The first stage compressor functioned as a SBS phase conjugate mirror (SBS-PCM) as well, to effectively compensate for aberrations and improve the beam quality. Moreover, the laser energy was recorded using a laser energy meter (Vega Pyroelectric PE50BB-DIF-C (s/n:917609), Ophir Optronics, Israel), and the pulse duration characteristics were measured using a fast phototube (UPD-40-UVIR-D, Alphalas GmbH, Germany; rise time < 40 ps) combined with a digital oscilloscope (DPO71254C, Tektronix, USA; bandwidth: 12.5 GHz; sampling rate: 100 Gsamples/s).
4. Results and discussion
The output characteristics of the pre-compressor were measured, where the input pump pulse duration was 8 ns. The evolution of the output pulse duration with respect to the input energy is shown in Fig. 4(a). The value of each point was the average of 50 pulses, and the arithmetic square root of the variance (standard deviation, Sdev) was calculated to clearly show the stability and volatility of the output pulse duration. As the injected energy increased, the duration of the pulses from the SBS compression first rapidly decreased (green region). Subsequently, the input energy continued to increase, and the evolution of the pulse duration stabilized (red region), saturating at approximately 500 ps. As shown in Fig. 4(b), the output pulse energy and conversion efficiency were synchronously recorded. The output energy steadily increased monotonically, and the corresponding energy efficiency first increased rapidly at low incident energies and then remained steady at high incident energies. Output Pulses with an average duration of 500 ps and an efficiency of over 60% were achieved in the pre-compressor. The experimental observations show that the pump of the secondary transient compressor was successfully generated with a duration shorter than the phonon lifetime of PF-5060 (τB =1.4 ns). Moreover, the stability of the output duration in the high incident energy range tentatively confirmed the suppression of Stokes broadening by the high gain coefficient.
Fig. 4. Output characteristics of the first-stage compressor. The error bars represent the Sdev of mean value. (a) Evolution of SBS compression pulse duration with respect to input energy. (b) Output energy (black square) and energy efficiency (blue rhombus) of Stokes pulses versus input pump energy.
The characteristics of transient SBS compression are shown in Fig. 5, where the amplified first-stage Stokes pulse serves as the pump for the second-stage compressor. The evolution of the measured pulse duration as a function of the input energy is illustrated in Fig. 5(a), where the value of each point is the average of 50 pulses. The pulse of the transient SBS was continuously compressed throughout the energy interval. The output pulse duration steadily decreased and eventually remained at approximately a quarter of the acoustic period, accompanied by increasingly high stability. However, further energy boosting resulted in a clear optical breakdown. The evolution of the waveform is shown in the inset of Fig. 5(a), where the Stokes leading edges were significantly amplified and increased vertically, benefiting from the high gain, while the trailing edges were not discernibly broadened. Stable output pulses near the physical limit of the transient SBS (∼1.05 τa) were obtained owing to the sufficient depletion of the pump by the leading edge. Notably, several output pulses with durations of < τa were recorded for input pulse energies of 70–80 mJ, as shown in Figs. 5(b), 5(c), and 5(d). The phenomenon described in [16], in which energy back-conversion occurs in transient SBS compression at high pump intensities, was experimentally demonstrated. Owing to the extreme pump depletion, the intensity at the Stokes trailing edge was considerably higher than that of the remaining pump. The energy back-conversion of Stokes-to-pump further reduced the intensity of the Stokes trailing edge, breaking the physical limit of the transient SBS compression.
Fig. 5. Output characteristics of the transient SBS compression. (a) Experimentally measured dependence of the compressed pulse duration on the input pulse energy under the transient condition; the three insets are representative waveforms in the experiment. The shapes of the output Stokes pulses with durations lower than the quarter acoustic period are illustrated in (b), (c) and (d).
SBS pulse compression is essentially an asymmetric amplification of the returning Stokes pulses. It contains simultaneous compression and broadening, which originate from the amplification of the leading and trailing edges of Stokes, respectively. The Stokes wavefront always encounters and dissipates the pump first, and the remaining pump intensity determines the magnitude of the amplification at the Stokes trailing edge. When the SBS threshold was reached, the Stokes pulse was rapidly compressed because of the extremely high amplification efficiency of the Stokes wavefront. Subsequently, when the SBS process reached the saturation gain region, the amplification efficiency of the wavefront decreased, and the trailing edge absorbed the remaining pump energy for amplification, in which the compression and broadening of Stokes constitute dynamic equilibrium. Finally, as the incident energy further increased, the trailing edge exhibited more dramatic amplification, resulting in broadening.
For PF-5060, the high transient gain substantially improved the efficiency of pump depletion, and the amplification of the trailing edge was weak owing to the insufficient residual intensity. Therefore, compression reached the theoretical limit owing to the suppression of Stokes trailing edge broadening. Moreover, pulses with durations below the compression limit were experimentally obtained because the energy back-conversion phenomenon was not considered in the transient physical limit model proposed by Velchev et al [12].
In addition, the energies and efficiencies of the output pulses in transient SBS compression achieved by the second stage compressor are illustrated in Fig. 6. Benefiting from the high Brillouin gain coefficient of PF-5060, the energy efficiency was over 30%, which demonstrates another significant advancement in transient SBS compression. Moreover, PF-5060 demonstrated excellent load performance because it contains only C-F, which is more stable than extra bonds such as C-N and C-O in other fluorocarbon media. Based on the results of this research, a reliable route for compression-limit pulse generation with high energy efficiency in transient SBS is provided in terms of gain coefficient. It should be noted that several optical elements used are uncoated, which limited the energy conversion efficiency and the injection pump intensity. By improving the experimental conditions, higher energy conversion efficiencies may be obtained. Additionally, further refinement of the medium will offer an effective solution for high-energy picosecond laser generation.
Fig. 6. Output energy (black square) and energy conversion efficiency (blue rhombus) of Stokes pulses versus input pump energy in transient conditions.
5. Conclusion
In summary, transient SBS compression pulses with durations near the compression limit and high energy efficiency were demonstrated. This was achieved by using the SBS medium having the higher Brillouin gain coefficient to suppress Stokes trailing broadening at high pump intensities and to improve the energy efficiency. Finally, a pulse duration of ∼1.05 τa in transient compression was achieved with an energy efficiency of over 30% in PF-5060. The phenomenon of energy back-conversion in transient SBS compression was experimentally demonstrated, and output pulses with durations < τa were obtained. By further optimizing the compressor parameters and improving the experimental conditions, picosecond ultrashort pulses with high intensities can be obtained using the transient SBS regime, which will be proven by further experimental studies.
Funding
National Natural Science Foundation of China (61905064, 61927815); Natural Science Foundation of Hebei Province (F2019202320); Research Projects of Higher Education Institutions of Hebei Province (QN2019201); Natural science research Foundation of Hebei University of Technology (JBKYXX2002); Hebei Province Graduate Innovation Funding Project (CXZZBS2021030, CXZZSS2021040).
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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