1. INTRODUCTION
During the past decades, many attentions have been drawn to developing sub-nanosecond laser pulse generation techniques for applications, including high harmonic generation with high spectral purity of atomic and molecular spectroscopy (Ganeev et al., Reference Ganeev, Suzuki and Kuroda2014; Bertolotti, Reference Bertolotti2015) efficient pump source of parametric amplifier (Ishii et al., Reference Ishii, Turi, Yakovlev, Fuji, Krausz, Baltuška and Piskarskas2005; Popmintchev et al., Reference Popmintchev, Chen, Popmintchev, Arpin, Brown, Ališauskas and Baltuška2012) and high spatial resolution in the LIDAR Thomson scattering diagnostics. Stimulated Brillouin scattering (SBS) pulse compression is particularly compelling in sub-nanosecond laser pulse generation because of its easy realization, low expenditure, high-energy load (Guillaume et al., Reference Guillaume, Humphrey, Nakamura, Trines, Heathcote, Galimberti and Kar2014), and arbitrary wavelength operation (Gorbunov et al., Reference Gorbunov, Papernyĭ, Petrov and Startsev1983; Davydov et al., Reference Davydov, Shipilov and Shmaonov1986; Roy et al., Reference Roy and Rao1986; Yuan et al., Reference Yuan, Lu, Wang, Zheng and Chen2014; Hasi et al., Reference Hasi, Zhao, Lin, He and Lü2015). With the development of SBS pulse compression, the maximum Stokes pulse energy has been achieved from <1 mJ (Velchev et al., Reference Velchev, Neshev, Hogervorst and Ubachs1999) to more than 1 J (Feng et al., Reference Feng, Xu and Diels2014) with the minimum compressed pulse width from several nanoseconds (Damzen & Hutchinson, Reference Damzen and Hutchinson1983; Yoon et al., Reference Yoon, Shin, Kong and Lee2009) to hundreds picoseconds (Zhu et al., Reference Zhu, Lu and Wang2015). In these reports, however, laser pulses without tail modulation in the process of SBS pulse compressing are acquired only at low intensities (Schiemann et al., Reference Schiemann, Ubachs and Hogervorst1997).
With the increase of the pump energy, tail modulation of the Stokes pulse, which limits the output pulse power and width in the compressing as a result of the pump energy depletion and pulse width broadening, becomes more serious (Ottusch & Rockwell, Reference Ottusch and Rockwell1991; Kuwahara et al., Reference Kuwahara, Takahashi, Matsumoto, Kato and Owadano2000; Mitra et al., Reference Mitra, Yoshida, Fujita and Nakatsuka2006; Laroche et al., Reference Laroche, Gilles and Girard2011) Dane et al. (Reference Dane, Neuman, Hackel, Norton and Miller1992) proposed a two-cell structure and presented that the wave profile trailing edge would broaden and lengthen with the increasing of the pump energy. Also, Yoshida et al. (Reference Yoshida, Fujita, Nakatsuka and Ueda2007) found that the Stokes pulse broadening at higher pump energy in the quartz was attributed to the Stokes trailing-edge amplification caused by the residual pump. The tail modulation phenomenon still existed in Yoshida's (Reference Yoshida, Hatae, Fujita, Nakatsuka and Kitamura2009) later study that the pump pulse with the energy of 1 J was compressed from 13 ns to 160 ps. Xu et al. (Reference Xu, Feng and Diels2014) also noticed the phenomenon in high-energy SBS compression and indicated that tail modulation could decrease when the Stokes pulse duration was shorter than the phonon lifetime. In those reports, tail modulation in high-energy SBS compression is due to insufficient amplification of the Stokes leading edge and high pump energy simply. However, the mechanism of tail modulation generation and its suppression scheme were seldom investigated in SBS.
In this paper, we present an analysis of tail modulation generation as well as a numerical model for the Stokes waveform. In our experiment, a double-stage compression structure is conducted in order to validate numerical results. The first stage is used to control effective pump width within a rational range, and the second stage can achieve a steep leading-edge pulse without tail modulation. For the first time to our knowledge, we have demonstrated pulse compression from 8 ns to 102 ps accompanying no tail modulation and achieved a 90% maximum reflectivity with effective energy reflectivity of 42%.
2. ANALYSIS AND SIMULATIONS
We define the effective pulse width of the pump to explain the Stokes tail modulation resulted from the SBS pulse compression. Effective pump width is defined as the part of the pump whose intensity reaches and exceeds the SBS threshold. It means that Stokes can be amplified by the pump immediately in the duration of the effective pump width once they encounter each other. The energy of the main peak range of the compressed pulse is concerned as the effective Stokes energy, and we define the ratio of the energy to the incident energy as effective energy reflectivity whose value shows the degree of Stokes tail modulation and characterizes the energy transform after SBS compression accurately. Figure 1 is the schematic diagram of effective pulse width of the pump changing with the energy. The generation time of Stokes pulse depends on the pump power injecting into the cell. If the pump power exceeds the threshold at the peak point “f”, the generation time equals the distance between point “f” and the focal point “a”. In this condition, the effective pulse width of the pump is close to zero and the Stokes leading edge cannot be amplified. So the emitted Stokes exhibits almost the same waveform as the pump light. With increasing of the pump energy (power), the time exceeding SBS threshold moves forward from point “f” to point “b” and the effective pump width (Δt) is Δt1, Δt2, Δt3, Δt4 accordingly. Under these conditions, backscattering stokes pulse leading edge is amplified and the corresponding pulse width is narrowed. When the effective pulse width of pump is equal to the back-turning time in cell, Stokes pulse is compressed sufficiently. The pump energy increases further and the Stokes leading-edge has transmitted the cell before the pump enters the cell completely. The Stokes trailing edge is amplified by the residual pump, which results in the tail modulation. In the case of Gaussian pump, effective pulse width of the pump continues to grow with the pump intensity, as shown in Figure 2.
Effective pulse width of the pump at different energy.
(a) Gaussian pump. (b) The effective pulse width of Gaussian pump.
Therefore, the key of suppression of the tail modulation generation is to let the pump effective pulse width in a proper range. Pulse temporal compression of the pump can decrease its effective pulse width bellow the back-turning time. It is convenient to use a stage SBS compression with long focal distance lens to shorten the pump's effective pulse width and retain most of energy. It should be noted that the compression degree of the Gaussian pump has an important impact on the output result. Here, we studied two general and typical situations. When the leading edge of Gaussian pump light is cut partly, this condition is called unilateral semi-cut Gaussian pump. If the leading edge of Gaussian pump light is cut completely, this condition is called unilateral full-cut pump. In the experiment, the compression degree of Gaussian pump can be controlled by the focal length of lens. Because of the limited interaction length, short focal length lens cut the leading edge of Gaussian pump partly and long focal length lens can do it completely.
To validate our analysis, we simulate the process of SBS compression by classical model (Ottusch & Rockwell, Reference Ottusch and Rockwell1991). The SBS process involves counter-propagating two optical fields E
L (laser) and E
S (Stokes), which are coupled through the electrostriction with an acoustic field
$\overline {\rm \rho} $
.
$$E_{\rm L}(z,t) = E_{\rm L}(z,t)e^{i(k_{\rm L}z - {\rm \omega} _{\rm L}t)} + c.c.,$$
$$E_{\rm S}(z,t) = E_{\rm S}(z,t)e^{i( - k_{\rm S}z - {\rm \omega} _{\rm S}t)} + c.c.,$$
$$\overline {\rm \rho} (z,t) = {\rm \rho} _0 + {\rm \rho} (z,t)e^{i(q_{\rm B}z - {\rm \Omega} _{\rm B}t)} + c.c.$$
Optical field E
L and E
S are governed by Maxwell's equations, and acoustic field
$\overline {\rm \rho} $
obeys the Navier–Stokes equation. Wave equations can be written in the form as Eq. (2).We solved Eq. (2) numerically by a generalization of the split-step method.
$$\displaystyle{{\partial E_{\rm L}} \over {\partial z}} + \displaystyle{{\rm \alpha} \over 2}E_{\rm L} + \left( {\displaystyle{n \over c}} \right)\displaystyle{{\partial E_{\rm L}} \over {\partial t}} = \displaystyle{{i{\rm \omega} _{\rm L}{\rm \gamma} ^{\rm e}} \over {2nc{\rm \rho} _0}}{\rm \rho} E_{\rm S},$$
$$- \displaystyle{{\partial E_{\rm S}} \over {\partial z}} + \displaystyle{{\rm \alpha} \over 2}E_{\rm S} + \left( {\displaystyle{n \over c}} \right)\displaystyle{{\partial E_{\rm S}} \over {\partial t}} = \displaystyle{{i{\rm \omega} _{\rm S}{\rm \gamma} ^{\rm e}} \over {2nc{\rm \rho} _0}}{\rm \rho} ^*E_{\rm L},$$
$$\displaystyle{{\partial ^2{\rm \rho}} \over {\partial t^2}} - (2i{\rm \omega} - \Gamma _{\rm B})\displaystyle{{\partial {\rm \rho}} \over {\partial t}} - (i{\rm \omega} \Gamma _{\rm B}){\rm \rho} = \displaystyle{{{\rm \gamma} ^{\rm e}} \over {4{\rm \pi}}} q_{\rm B}^2 E_{\rm L}E_{\rm S}^*.$$
We numerically simulate the Stokes waveform, pulse width, energy reflectivity, and effective energy reflectivity of Gaussian pump, unilateral semi-cut Gaussian pump and unilateral full-cut pump, respectively. FC-72 is chosen as the active medium. The length of cell and the focal length of the lens are 100 and 60 cm in the simulation, respectively. Parameters of FC-72 are in Table 1.
Physical properties of FC-72
Figure 3 shows the dependences of curve waveform of Gaussian pump, unilateral semi-cut pump, unilateral full-cut pump on the Stokes pulse width, energy reflectivity, effective energy reflectivity, and the pump peak power. As shown in Figure 3b, with the increasing of the pump peak power, there is a tail modulation in the Stokes profile obviously when higher Gaussian pump is injected. Unilateral semi-cut pump and the corresponding Stokes pulse depending on pump peak power are shown in Figure 3c, 3d separately. Compared with Figure 3b, the Stokes tail modulation is suppressed significantly and the main peak is amplified evidently, which leads to narrower pulse width and higher peak power. The peak power and the Stokes waveform with the injection of unilateral full-cut pump demonstrate (Fig. 3e, 3f) that, compared with Gaussian pump, unilateral full-cut pump method can suppress the tail modulation but without amplifying the output Stokes pulse sufficiently caused by the excessive pump reduction.
The wave form of (a, b) Gaussian pump and corresponding Stokes pulse. (c, d) Unilateral semi-cut pump Gaussian pump and corresponding Stokes pulse. (e, f) Unilateral full-cut pump Gaussian pump and corresponding Stokes pulse.
3. EXPERIMENTAL SETUP AND RESULTS
The schematic layout is shown in Figure 4. A Q-switched, injection-seeded Nd: YAG laser (Continuum PRII9010) at 1064 nm is operated in a single longitudinal mode; its pulse duration is 8 ns with repetition of 1 Hz. The cell is filled with fluorinated liquid (FC-72) medium with a short phonon lifetime (1.2 ns). The combination of a Faraday rotator FR and a polarizer P1 as an isolator avoids the backscattered radiation back to the laser resonator. A polarizer P3 and a half wave plate are used together to adjust the pump energy. The circularly polarized laser beam is focused into the cell1 using a lens L1 (focal length = 300 and 600 mm). Reflected by the polarizer P3 and a mirror M, the output Stokes light is injected into the second cell (1 m) by the L2, 60 cm convex lens. The final Stokes is extracted by a polarizer P4.
Experimental setup of double compressor; P, polarizing beam splitter; L, convex lens; H, half wave-plate; Q, quarter-wave plate; E, energy meter; Cell, Brillouin compressor; D, photo detector.
In experiment, we use Israel Ophir energy detector ED500 (E1, E2, and E3) to monitor the pump energy, the first-stage compression and the Stokes in the second-stage compression, respectively. Stokes pulse shapes both in the first and the second stage are monitored using two PIN photodiodes and a digital phosphor oscilloscope TDS3032B. The effective energy reflectivity can be obtained through the area ratio of the main peak to entire waveform.
4. DISCUSSION
In the experiment, waveforms of unilateral semi-cut Gaussian pump and unilateral full-cut Gaussian pump can be achieved when the focal length of L1 is 300 and 600 mm, respectively. When pump energy is 300 mJ, the reshaped pump waveform and corresponding Stokes pulses are shown in Figure 5. One can see from Figure 5a, 5b that there is serious tail modulation in a temporal trace of the Stokes pumped by Gaussian pump, which is in reasonable agreement with the result of simulations shown in Figure 3b. But in Figure 5c, 5d, the experimental Stokes waveform pumped by unilateral semi-cut pump pulse demonstrates high peak value, narrow pulse width, and no tailing modulation. In Figure 5e, 5f, it is coincides with the expectation that the Stokes temporal trace of unilateral full-cut pump injection creates no tailing modulation. The output pulse width pumped by Gaussian pump, unilateral semi-cut Gaussian pump, and unilateral full-cut Gaussian pump are 257, 102, and 198 ps, respectively. This manifests that the reduction in the effective pump width results in a significant suppression of the Stokes tail modulation, which makes it possible to achieve shorter Stokes pulse. It should be noted that when effective pump pules width is controlled in a rational range, excessive cut of pump is unfavorable to the further compression instead.
Oscillograms of the pump pulses (on the left) and corresponding Stokes pulses (on the right) (a) Gaussian pump and corresponding Stokes (257 ps). (b) Unilateral semi-cut Gaussian pump and compressed Stokes (102 ps) and (c) unilateral full-cut Gaussian pump and corresponding Stokes (198 ps).
Figure 6 shows that the Stokes width of unilateral cut pump gets a more significant compression and is only a half of it compressed from Gaussian pump in the same condition. This is related to their sloping leading-edge, which causes more rapid establishment of acoustic field and faster generation of Stokes pulse.
Dependences of the Stokes pulse width on incident pump energy.
Effective energy reflectivity is an appropriate parameter to analyze modulation degree quantitatively. Figure 6 presents the experimental measurement of effective energy reflectivity and Stokes pulse width pumped by different pump waveforms. While the Gaussian pump energy reflectivity could increase as the increasing of pump energy until reaching 90% in 300 mJ in Figure 7a, the corresponding effective energy reflectance, as shown in Figure 7b, merely emerges a slow escalating trend until the pump energy reaches 50 mJ, and then sustains to decline with the pump energy increasing. Consecutive drop of effective energy reflectivity in high-energy indicates an increasingly serious tailing modulation. In the case of Gaussian pump, the effective pulse width of pump continues to grow with the pump energy, as shown in Figure 2b, and the unwanted amplification of Stokes tailing edge becomes inevitable. By contrast, the effective energy reflectivity of unilateral cut pump manifests an exponential rising trend and can reach a high value (nearly 50% in pump energy of 300 mJ).
Dependences of (a) energy reflectivity and (b) effective energy reflectivity on incident pump energy.
In the process of unilateral cut pump, the Gaussian pump is converted during the first-stage compression into a short pulse. The effective pulse width of unilateral cut pump can remain at a proper value and is not sensitive to the increase of the pump energy any more. So the effective pulse width of pump is not long enough to amplify the Stokes tailing edge even at high pump energy, and the tailing modulation can be suppressed. It can be said that if the effective pump duration is in a rational range, unilateral cut pump is more favorable to achieve highly effective energy conversion efficiency due to better amplification and longer interaction length of Stokes and pump.
5. CONCLUSION
In conclusion, we have investigated a method using the effective pump width control to suppress the tail modulation in the process of high-energy SBS pulse compression. It has been demonstrated that the effective pump width control can be achieved by one compressor. It has been shown that when effective pump width is controlled in a rational range, the tail modulation can be suppressed well. Moreover, appropriate control of effective pump width is conducive to achieve high compression radio and effective energy reflectivity.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grants No. 61378007, 61378016, and 61138005), the Fundamental Research Funds for the Central Universities (Grant No. HIT. IBRSEM. A. 201409).
