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Abstract
In this paper, a theoretical model is developed to demonstrate that fluctuations in the Stokes signal and occurrence position contribute to the final compression ratio in stimulated Brillouin scattering (SBS). This theoretical analysis can be applied to the investigation of the temporal characteristics of SBS pulse compression. This model agrees well with the experimental results in a two-stage SBS compressor.
© 2017 Optical Society of America
1. Introduction
Pulse compression through stimulated Brillouin scattering (SBS) is considered a simple yet efficient way in improving the peak power of laser pulses [1,2]. While possessing advantages such as phase-conjugation, high quantum efficiency, and high gain, SBS is now widely used in the compression of nanosecond laser pulses down to the sub-nanosecond scale in many research labs [3–5].
The pursuit for an SBS compressor is to achieve a high conversion efficiency and compression ratio. In previous studies, the compact two-cell structure is widely used because of its easy operation [1,6]. Many achievements have been made on it such as energy capacity of SBS compression beyond 1 J [7] and Stokes pulse width shorter than 200ps [8]. It is always by increasing the pump intensity to raise the gain of the Stokes pulse, aiming to obtain narrower Stokes pulse width. However, fluctuations in the Stokes signal is always ignored.
Unless seed injection is applied, the backward Stokes pulse in SBS will be generated from the noise. The stochastic initiation of SBS leads to fluctuations in the Stokes beam, which will reduce the coherence length of the scattered beam [9], when the transit time through the interaction region is much longer than the phonon lifetime, fluctuations in the Stokes intensity occur [10,11]. The pump wave is scattered from thermally excited density fluctuations and the Stokes signal has an inherent randomness in its phase and intensity. Although the random component is small compared to the stimulated coherent contribution, its effect on the conjugated signal is a measurable quantity [9,12]. In SBS pulse compressors, these fluctuations affect the pulse duration and intensity.
In this paper, we attempt to explain the fluctuations’ contribution to pulse compression. To this end, we use theoretical analysis and experiments to elucidate the generation of Stokes pulse and its time duration characteristics in FC-770 at 1064nm pump laser.
2. Experimental setup and results
In the experiments, FC-770, which is composed of C4F9NO, is selected as the SBS medium. The experimental parameters of FC-770 are summarized in Table 1 in which the phonon lifetime, gain coefficient were calculated theoretically. The pump is generated by a Q-switched Nd:YAG single longitudinal mode laser system, which outputs a 6 ns and 100 mJ Gaussian pulse with a beam diameter of 3 mm.
Table 1. Parameters of FC-770
The experimental arrangement is presented in Fig. 1. The output laser beam is linearly polarized. After reflecting twice from the mirrors, the linearly polarized output beam passes through an isolation system, which is composed of two polarizers (P1, P2), a Faraday rotator (FR), and half-wave plate (H1), and is then injected into a two-stage SBS generator-amplifier pulse compressor, cell 2 is the generation cell, cell 1 is the amplification cell. The injected pump energy can be controlled by a half-wave plate (H2) and polarizer (P3). A quarter-wave plate (Q1) is added before the SBS pulse compressor to change the polarization of the beam from linear to circular. As the reflected Stokes beam is phase-conjugated, its polarization is perpendicular to the pump after passing Q1 and gets extracted at P3.
Fig. 1 Experimental optical path; green line: the Stokes signal in the generator cell that recorded by photon detector 3; indigo line: the final compressed pulse recorded by photon detector 2.
In the SBS pulse compressor, the two cells are both 60 cm in length. A convex lens (L1) with a focal length of 30 cm is added in front of the generation cell. To find the conditions to achieving a high compression ratio, a beam splitter is added before the lens to monitor the generation of Stokes pulses. We use a wedge plate as a beam splitter (BS1) to measure the waveform of the Stokes signal in cell 2.
In the experiments, the Stokes and pump pulse are detected and measured by an optical probe (Ultrafast UPD-50-UP). The results are displayed on an oscilloscope (Tektronix DPO71254C) and the energy meter used is the PE50 by OPHIR. The waveform of the Stokes signal in generator and the waveform of the final compressed laser pulse are recorded separately. In the compact two-cell SBS pulse compression structure, a Stokes signal is generated in cell 2 and then gets amplified in cell 1 to achieve high peak power and high compression ratio.
Figure 2 presents the behavior of the energy extraction efficiency as the pump intensity increases. As the pump intensity increases from 5 to 50 MW/cm2, the extraction efficiency increases rapidly; when the pump intensity is higher than 50 MW/cm2, the increase in efficiency slows in the gain saturation region; as the pump intensity reaches 80 MW/cm2, the Stokes extraction efficiency remains around 82%.
The variation of the output Stokes pulse width is shown in Fig. 3. Before the SBS process reaches the gain saturation region, the output Stokes pulse width narrows quickly as the pump intensity increases. The shortest output pulse width reached is 430 ps (shorter than the phonon lifetime of FC-770). However, the pulse width does not remain stable as the pump intensity increases; it broadens.
As shown in Fig. 4, the rising time (from 10% to 90% of the peak power) of the Stokes pulse is measured with the increase in pump intensity. As the pump pulse is Gaussian-shaped, its power reaches a maximum in the middle of the pulse. The generation position of the Stokes signal is close to the middle of the pump pulse with a low pump intensity. As the pump intensity increases, the generation position moves forward, the front part of the pump pulse in the time domain will reach the SBS threshold. On one hand, this will increases the intensity of the spontaneous scattering of the pump, On the other hand, the delay effect in SBS interaction causes pulse distortion [13]. amplification of Stokes pulse decreases the rising time. With a low pump intensity, the amplification is the main contributing factor to the rising time. However, when the pump intensity is deep in the gain saturation region, the generation position becomes the man factor.
As the series of output Stokes pulses is measured with the increase in pump intensity, it is clear that a pulse width shorter than the medium phonon lifetime can be achieved through the compact two-cell structure. However, Stokes pulse width broadening exists at a high pump intensity. In the following section, we try to provide a theoretical explanation of this phenomenon.
3. Theoretical analysis and discussion
In the SBS process, two optical fields (laser) and (Stokes) are coupled through an acoustic field . In this theory, these fields represent the plane waves, processing time, and propagation coordinates only.
The equations describing the pump and Stokes optical fields are derived from Maxwell’s equations. Moreover, the acoustic field in the medium is due to the Navier–Stokes equation. The slowly varying envelope approximation is applied in solving the optical fields’ equations. As the bandwidth of the acoustic field is similar to its frequency, the second-order term in the acoustic field equation cannot be ignored. Thus, the wave equations can be described as [14]:
where n is the refraction index of the medium, c is the speed of light, and γe is the electrostrictive coefficient.Equation (2c) can be solved by Fourier transformation; thus, the wave equations can be expressed as [15]:
Here, is the steady-state gain coefficient andFor t < 0, f(t) = 0.In the equations above, τB = 1/ΓB is the phonon lifetime, ΩB is the Brillouin shift, t represents the time axis, and z is the pump propagation coordinates. The Stokes signal is generated from spontaneous noise in the medium. Under the influence of the acoustic field, energy transfers from the pump to the opposite-propagating Stokes signal.
We use a split-step method to solve the wave equations. In the simulation, the SBS structural parameters are consistent with the experimental conditions as well as the medium parameters. The injected pump light possesses a Gaussian shape with a pulse width of 6 ns and a flat-top intensity distribution. As our theoretical model involves one-dimensional coupled wave equations, we use the “q” transformation [16] in the calculation to describe the intensity variation of the focused pump light after passing through the lens.
There are two descriptions of the initial Stokes signal. One is that the Stokes signal is generated by the spontaneous scattering of the pump light [17]. The other is that the initiation of SBS is caused by Langevin noise in the medium [18].
In this paper, we combine the two models together; both spontaneous scattering and Langevin noise are considered. The Langevin noise added to the spontaneous scattering signal is considered as the modulation of the initial signal. This is used as the boundary condition of the Stokes signal in the theoretical analysis, and it subsequently gets amplified in the transmission opposite to the pump.
The phonon lifetime is regarded as the physical limit to the compressed pulse duration according to the traditional theory [1]. This is because the phonon lifetime indicates the decay time of the acoustic field, which interacts with the pump and Stokes signal. However, a compressed pulse with a duration shorter than the phonon lifetime has widely been achieved [15]. Consequently, we believe for the experiment results and simulation setup that the phonon lifetime is not a major physical limit to SBS pulse compression.
The duration of the output backward Stokes pulse, which mainly originates from the initial signal with modulation, is determined by two factors. One is the duration and modulation depth of the initial signal, and the other is the gain of the initial pulse. The asymmetry of amplification at the front and tail part of the Stokes pulse results in further compression. In general, the cycle time of the initial signal is shorter than the medium phonon lifetime. Thus, the limit of pulse compression based on SBS is shorter than the phonon lifetime.
Previous work shows that the Stokes signal with a steep front edge contributes to the high pulse compression ratio [19]. As the front edge of the Stokes pulse extracts more energy than the trailing edge, the Stokes pulse shortens in the amplification while transports backward. At the Rayleigh area in the generation cell, the generated Stokes signal exhibits a modulation at the front edge and the pump pulse has a Gaussian distribution. Firstly, SBS cannot be generated because the power density at the front of the pump pulse is lower than the generation threshold of SBS. However, when the top of the pump pulse with a power density near the threshold enters the cell, the Stokes pulse will be excited under the interaction between the pump and acoustic field. It is noticed that a high-power pulse is generated with a shorter duration at the top of the pump pulse, which is caused by the modulation of the acoustic field. As the front side of the Stokes signal has the peak power, we can obtain a highly compressed short pulse.
Theoretical simulation of the compressed pulse width varies with the input pump intensity is shown in Fig. 5. The compressed pulse width has a has a minimum value when the SBS process has just entered the gain saturation, and it broadens as the pump intensity continues increasing. The simulation and the experimental agrees well with each other.
Fig. 5 Comparison between simulated pulse width obtained varies with input pump intensity and the experimental results; solid red squares: experimental results; cyan: simulation results.
If the Stokes signal has a short rise time and high power at the front edge, it can be amplified more effectively than the tail part in the same condition while propagating opposite to the pump light. Consequently, energy is transferred mostly to the Stokes front edge, and a compressed Stokes pulse can be output. Moreover, with the increase in injected pump intensity, more energy would be transferred to the Stokes pulse and a shorter pulse width would be obtained, shown in Fig. 6.
Fig. 6 Description of Stokes pulse generation and amplification in a two-stage SBS compressor. The injected pump intensity is 15 MW/cm2. (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell.
As the pump intensity continues increasing, the generating position of the Stokes signal moves forward. Meanwhile, the energy transfer efficiency enters the gain saturation region and the Stokes pulse cannot fully extract the energy of the pump pulse. Furthermore, a strong acoustic field exists in the medium, thus, the tailing part of the Stokes signal will get amplified. However, the output Stokes pulse can have the shortest pulse for a high gain in the medium in this condition, shown in Fig. 7.
Fig. 7 (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell. The injected pump intensity is 30 MW/cm2.
When the pump intensity increases further, the generating position of the Stokes signal moves further forward. The power at the front edge of the Stokes signal is lower than that at the peak. This will result in the front edge of the Stokes pulse becoming less steep than that with a low pump intensity. However, the modulated tailing part will attain a higher gain. These will result in the broadening of the Stokes pulse width, shown in Fig. 8.
Fig. 8 (a) Final output Stokes pulse calculated by theoretical analysis. (b) Stokes signal in generation cell. (c) Output Stokes pulse obtained in experiment. (d) Experimental output Stokes pulse from generation cell. The injected pump intensity is 140 MW/cm2.
The theoretical modulation and experimental results demonstrate that the output Stokes pulse width will broaden when the pump intensity is much higher than the threshold. Next, a theoretical model is used to explain this phenomenon. As fluctuations exist in the initial Stokes signal, the front peak of the fluctuations becomes distinct from the others; it attains a high gain and gets amplified quickly. Consequently, a compressed narrow pulse is obtained. However, with a low pump intensity, the position of the initial Stokes signal is close to the middle peak of the Gaussian pulse. The Stokes pulse has a steep leading front edge, which is desired for pulse compression [19]. As the pump intensity increases, the occurrence position moves forward. The leading edge of the Stokes pulse become soft, the tailing end receives more energy, and the pulse width broadens. According to the above research, while the compact two-cell structure is used for SBS pulse compression, the pump intensity in the generation cell should be correctly controlled to reach the gain saturation region of the highest compression ratio.
4. Conclusions
The generation of hundreds of picoseconds laser pulses achieved by stimulated Brillouin scattering was investigated. A numerical model was used to describe the occurrence of the Stokes signal and its performance in the backward-propagating amplification. Distinguished from traditional SBS pulse compression studies, we proposed that the fluctuations in the Stokes signal and its generation position at the pulse peak contribute to the high compression ratio. The results show that, it is easy to obtain a narrowest Stokes pulse when the pump intensity has just entered the gain saturation region due to the fluctuations effect. And as the pump intensity grows higher in gain saturation, a wider Stokes pulse width is obtained. This is because the increasing pump intensity changes the fluctuations’ position and power modulation depth in time domain, which changes the pulse of shape of the Stokes signal, and then results in pulse broadening of the final Stokes light. The theoretical simulation agrees well with the experimental results.
Funding
National Natural Science Foundation of China (NSFC) (Grant No. 61622501, No. 61378007 and No.6137806).
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