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Abstract
In this paper, we propose a novel effective optical smoothing scheme to suppress laser plasma instabilities (LPIs) by time-dependent polarization rotation (TPR) on a picosecond timescale. The polarization rotation with time-dependent frequency is generated by the superposition of chirped light pulses with dynamic frequency shift and counter-rotating circular polarization. Compared to light without polarization rotation or pulse chirping, such superposed light with TPR has a broader spectrum and lower temporal coherence. Using the one-dimensional fluid laser-plasma-instability code (FLAME) and PIC simulation, TPR is demonstrated working well in suppressing parametric backscattering, which provides an effective approach to suppress LPIs. In the meantime, a significant improvement of irradiation uniformity of the chirped pulses is achieved by the introduction of proper spatial phase modulation and grating dispersion.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser plasma instabilities (LPIs) is one of the most important fatal obstacles in inertial confinement fusion (ICF) facilities [1]. How to effectively mitigate LPIs is one of the major challenges in the study of laser plasma interaction, hohlraum physics and optics. As potential methods, many optical smoothing technologies have been developed to improve laser irradiation uniformity to mitigate LPIs of the lasers, represented by spatial shaping of focal spots by continuous phase plate (CPP) [2], temporal smoothing by spectral dispersion (SSD) [3] and polarization smoothing [4]. To review these technologies, the spatial shaping that homogenizes the large-scale envelope of focal spot is dominant, while the temporal and polarization smoothing contribute to further smoothing the fine-scale speckles of focal spot. However, the combined implementation of the aforementioned technologies is still not enough to eliminate the risk of LPIs in current ignition designs, especially for the parametric backscattering (PB). In the experiments on NIF [5] and SG-III [6] facilities, there is about 10% of the total incident laser energy reflected by PB [7,8]. Commonly, backward stimulated Raman scattering (SRS) and backward stimulated Brillouin scattering (SBS) are two main types of PB. Backward SRS is a coupling process among incident light wave, backward-propagating scattered light wave and a forward-propagating electron plasma wave, while backward SBS is a coupling process among the light waves and a forward-propagating acoustic wave. In order to suppress SRS and SBS further, some ultrafast optical smoothing have been developed [9,10]. Recent studies on the spike trains of uneven duration and delay (STUD) pulses [11,12] indicate that, through the transient and intermittent turning on and off the light pulses on picosecond timescale, the STUD pulses can effectively break the accumulation paths of SRS and SBS. Later, methods of adding a constant rotation frequency to the polarization of incident light (CPR) [13], and the alternating-polarization pulses [14] are shown to be more effective than the STUD pulses. However, these works did not involve detail optical smoothing approach and laser irradiation uniformity analysis.
In this paper, a novel effective optical smoothing to suppress LPIs by time-dependent polarization rotation (TPR) is proposed. Light field with TPR can combine the advantages of CPR and broadband light [15] together, and can be generated by the coherent superposition of the chirped pulses with dynamic frequency shift and counter-rotation circular polarization. Simulations based on FLAME code [16] and PIC simulation demonstrate that the performance of light field with TPR has broader bandwidth and lower temporal coherence and can suppress parametric backscattering effectively. Furthermore, the novel optical smoothing scheme also involves temporal and spatial smoothing to achieve better irradiation uniformity than the traditional SSD. The paper is organized as follows: in Section 2, the novel optical smoothing is presented in detail. Sections 3 and 4 analyze the its performance in improving laser irradiation uniformity and suppressing parametric backscattering, followed by the conclusion in Section 5.
2. Physical model of optical smoothing scheme
The conceptual scheme to generate such light field with TPR is given in Fig. 1. The optical smoothing scheme can be divided into two aspects, i.e., pulse chirping and grating dispersion in the pre-amplifier, spatial shaping and polarization control of the beamlets before focused by the wedged focus lens (WFL). Firstly, the pulse for each beamlet is chirped independently by using the direction phase modulation [17,18], which provides time-dependent frequency shift among the beamlets that is enormously attractive in optical smoothing. Secondly, the non-adjacent beamlets in the laser quad are counter-rotating, circularly polarized by the polarization control plates (PCPs). Thus, the coherent superposition of such beamlets on focal spot can generate light field with time-dependent polarization rotation, as will be discussed below. On the other hand, inspired by traditional SSD, a grating is adopted in this novel scheme to introduce the temporal-spatial coupling of the beamlets to smooth the focal spot. Spiral phase plates or plasma-based q-plates [19] are also inserted in the beamline to smooth the focal spot in radial direction. It is worth noting that, the pulse chirping, for instance, is on picosecond timescale so that the intensity is smoothed on picosecond timescale. Eventually, the intensity smoothing and the polarization rotation can simultaneously contribute to the suppression of SRS and SBS.
Fig. 1. Conceptual optical smoothing for generating light field with TPR on picosecond timescale. Non-adjacent beamlets are counter-rotating, circularly polarized. The arrows represent the smoothing directions of focal spot.
For simplicity without loss of generality, we ignore the spatial shaping of CPPs at first. Each beamlet with periodic chirped pulses propagates through the grating, spatial phase plate and polarization control plate, thus the light field before focused by the WFL is expressed by
Realization of the time-dependent frequency of beamlets employs direct phase modulation on the pulse output from the fiber front end, like periodic linear chirp and sinusoidal phase modulation. Hereafter, the periodic linear chirp, for instance, is employed to the beamlets to generate time-dependent frequency shift. According to Ref. [17], after being dispersed by the grating along the x direction, the spatiotemporal field distribution Aj(r, t) of each beamlet can be expressed as
In Eq. (2), the instantaneous frequency of each beamlet is periodically linearly chirped so that the frequency shift among each beamlet varies periodically and can be precisely controlled by selecting proper time delay factor γ or different chirp factor b. Moreover, we can change the chirp factor and chirp period of each pulse to enlarge the bandwidth to achieve a better performance in both improving uniformity and suppressing parametric backscattering In the meantime, the light field is dispersed by the grating so that the focal spot is smoothed along the grating dispersion direction, i.e. transverse smoothing. Using the diffraction integral formula, the superposition of two counter-rotating, circularly polarized beamlets with different optical vortices and time delay on focal plane can be written as
Eq. (2) indicates that for a Gaussian pulse, the 1/e bandwidth is Δω=(1+b2)1/2/τ. The bandwidth of each pulse or the maximum rotation frequency, increases with the increasing chirp ratio b and the decreasing pulse duration τ. Hence, we can control the maximum rotation frequency by either changing the chirp ratio or the pulse duration.
Reference [17] used the combination of periodic chirped pulse, sinusoidal phase modulation and lens array to smooth the focal spot with a broader bandwidth of 1.2 nm, while the scheme presented here is focused on improving the irradiation uniformity of laser quad and suppressing parametric backscattering through time-dependent polarization rotation.
By assuming lj=−lj’=1, the analytical expression of the superposed light field on focal plane can be derived as
The intensity distributions of the polarized light in x and y directions can be written as
From Eq. (5), we can further conclude that both the intensity distributions of the polarized light in x- and y- directions rotate on picosecond timescale that is related to the time-dependent frequency shift. The rotation period of the intensity profile is Tr=lλ2/(cΔλ) so that the intensity rotation is strongly time dependent. Now we have applied transverse smoothing to the focal spot by the use of the grating, and radial smoothing by spiral phase plates (SPPs) to the linearly chirped light. Hence the irradiation uniformity can be significantly improved, as will be discussed in Section 3.
On the other hand, the polarization of the coherently superposed light field rotates rapidly, providing an amplitude modulation to each polarization direction of the light on picosecond timescale, which is beneficial to suppress PB. According to Eq. (5), the Jones Matrix of the superposed light field on focal plane is expressed by
From Eq. (6) we can conclude that the polarization of the focal spot rotates periodically. Further, when the CPPs are taken into consideration, the light field on focal plane is expressed by
Eq. (7) reveals that the total intensity distribution of the focal spot depends on the convolutions of two parts: the dominant intensity envelope generated by CPPs and the rapid rotating and transverse-sweeping intensity profiles by SPPs and grating, respectively.
In the novel optical smoothing scheme, besides the rapid temporal modulation of laser intensity in each polarization with the total constant pulse intensity, the time-dependent rotational frequency can result in the broader bandwidth and lower temporal coherence. It does not mean that the bandwidth of each beamlet is broadened, but the coherent superposition of such beamlet with time-dependent frequency shift on focal plane can generated light field with broader bandwidth, as will be discussed in Section 4. Theoretically, this scheme not only disturbs accumulative temporal growth of PB but also limits the effective coupling length of the three-wave interaction in PB process. Moreover, the intensity variation and polarization rotation frequency can be tuned according to the requirement for suppressing LPIs. So, the present optical smoothing scheme performs well in improving the irradiation uniformity in Section 3, and also suppressing SRS and SBS in Section 4.
3. Analysis of irradiation uniformity improvement
With capable wavelength shift of non-adjacent beamlets, the beamlets interfere on focal plane. The initial modulation depth of ∼1.2 of the near-field intensity distributions and wavefront distortions with peak-to-valley values of ∼2λ are introduced to the beamlets. After spatial phase modulated by CPP and SPP, each beamlet with an aperture of 360 × 360 mm is finally focused by a wedged focus lens with a focal length of 7.7 m [21]. For sake of comparison, the performance of traditional two-dimensional SSD is also simulated, in which the modulation depth and angular frequency of the EO modulator are δ=2.33 and ωm=17 GHz for a total bandwidth of ∼0.3 nm. The fractional power above intensity (FOPAI) curve [22] and the contrast of focal spot are adopted to evaluate the illumination uniformity. In this section, the maximum bandwidth of the chirped pulse is also selected to be 0.3 nm to make comparison with the typical SSD. For each chirped pulse, once the pulse duration is fixed, the chirp ratio is also decided. We are also aware that the chirp ratio can be increased to broaden the bandwidth of the chirped pulse. According to Ref. [17], the irradiation uniformity of the focal spot can be further improved by increasing the bandwidth of the chirped pulse.
3.1 Laser irradiation characteristics
In Fig. 2(a), the coherent superposition of the counter-rotating, circularly polarized beamlets with time-dependent frequency shift and opposite topological charges leads to the rapid variation of the intensity distribution. The rapid intensity variation contains two aspects: the time-dependent intensity rotation generated by topological charges and frequency shift, of which the linearly polarized light in x- and y- direction are naturally conjugate with each other; the transverse motion of speckles caused by time-varying frequency and grating dispersion. From Fig. 2(b) we can see that the novel optical smoothing scheme performs better than the combination of traditional SSD with polarization rotation. Moreover, when the maximum frequency shift is the same, the decay time of focal-spot contrast decreases with the shorter chirp period. This is because the speckles are swept faster for shorter chirp period and the irradiation uniformity is thus improved faster. We also note that the steady uniformity is almost the same for different chirp periods, which is consistence with the aforementioned analysis. The FOPAI curves in Fig. 2(c) gives the same results that is consistence with the above analysis
Fig. 2. (a) Concept of the optical smoothing scheme. (b) Contrast with the integration time and (c) FOPAI curves of the focal spot at different chirp periods.
Figure 3 depicts the simulated intensity variation of a single speckle within the focal spot smoothed light field with different chirp period. We here note that the intensities of the orthogonal polarized light in x- and y-directions are conjugate with each other and vary in same periods [see Figs. 3(a) and 3(b)], which agrees quite well with Eq. (6). When the chirp period is shorter, we can clearly see that the variation periods of intensities in both x-and y- directions are shorter, as depicted in Fig. 3(a). Overall, the intensities of the speckles within the focal spot turn on and off on picosecond timescale, which can effectively break the accumulation of the parametric instabilities in the sense of the STUD pulses.
Fig. 3. Intensity variation of one point in the focal spot: (a) Chirp period τ=20 ps, (b) τ=40 ps. The black solid line represents the total intensity.
According to Eqs. (5) and (6), the intensity variation of the focal spot includes two aspects: The first is the rapid rotation of both linearly polarized light in x- and y-directions. As shown in Fig. 3, the intensities of either polarization changes rapidly synchronously, but the total intensity almost remains unchanged. Hence, we have demonstrated the polarization rotation of the focal spot, and even can conclude that such intensity rotation cannot improve the irradiation uniformity of focal spot. In Figs. 3(a) and 3(b), we can clearly see that within one chirp period, the intensity variation period becomes shorter with the time. This is because the frequency shift increases linearly with the increasing time. Moreover, the chirp period can be shorter to decrease the decay time of the focal-spot contrast, while the chirp ratio can be larger to increase the maximum frequency shift so as to speed up the polarization rotation.
3.2 Redistribution characteristic of speckles
At first glance, it can be seen that the novel optical smoothing exhibits attractive behavior in laser beam propagation without considering the laser plasma interactions. By drawing the laser intensity evolution over time on the focal plane in Figs. 4(a)–4(b), we can clearly see the transversal movement or redistribution of the speckles. The speckles are swept in x direction periodically that is consistence with the chirp period. Moreover, even during one period, the speckle motion is interrupted rather than continuous, which is rather complicated owing to the combined movement of transverse sweep and rotation in two dimensions. This is contributed by the coherent superposition of the beamlets with time-dependent frequency shift and opposite helical. However, the light paths in the traditional 1D-SSD are continuous, as shown in Fig. 4(c). The comparison indicates that the novel optical smoothing scheme shows remarkable potential in interrupting the accumulation paths of SRS and SBS.
Fig. 4. Spatiotemporal evolution of the laser intensities of (a) the optical smoothing with chirp period τ=10 ps (b) the optical smoothing with chirp period τ=20 ps. (c) Spatiotemporal evolution of the laser intensity of traditional SSD.
In Fig. 4, we can clearly see the transversal movement or redistribution of the speckles with the same maximum frequency shift but different chirp periods. The key factors affecting the transversal movement of the speckles are the chirp period and the maximum frequency shift (or the bandwidth). By comparing Figs. 4(a)–4(c), we can see that the sweeping period of the speckle is exactly the same with the chirp period, which is consistent with the analysis in Fig. 2. The sweeping or redistribution ranges of the speckles are the same in Figs. 4(a)–4(c) due to the same bandwidth. In addition, the sweeping range can be enlarged by increasing the bandwidth, which can further improve the irradiation uniformity but may cause the “hole closure effect”.
4. Suppression of parametric backscattering
4.1 Characteristics of TPR and CPR
Theoretically, polarization rotation of laser beam can suppress PB when the rotation frequency is comparable to the temporal growth rate of PB without PR, because the change of amplitude of electromagnetic waves in either polarization can modulate the temporal growth rate to be smaller than the case without PR, and the effective interaction length can be reduced. In Ref. [13], Ido Barthe et al. have demonstrated that polarization rotation with a suitable constant frequency can effectively reduce the reflectivity of SRS and SBS. However, due to the large difference in mass between electron and ion, the growth rate of SRS is commonly larger than SBS. Correspondingly, the required frequency of PR to suppress SRS and SBS is different, such as 1600 × 2π GHz for suppressing SRS and 500 × 2π GHz for SBS, respectively. However, such rapid rotation frequency of 1600 × 2π GHz for SRS reflectivity reduction is difficult to realize in current experiment.
In our TPR scheme, the rotation frequency is time-dependent by controlling the chirp ratio or pulse duration between the coherently superposed linearly chirped pulse. In this case, the time-dependent rotation frequency in single chirp period can be expressed by
where B is chirp ratio difference between the chirped pulses, τ is chirp period.Hereafter, in order to better illustrate how the TPR scheme reduces the parametric backscattering, the chirp ratio of the chirped pulse is not constant but increasing with the pulse stack number. As shown in Fig. 5(a), the time-dependent frequency in the TPR scheme varies with different pulse duration. When the pulse duration is 40 ps, the maximum frequency shift equals to 3THz, while it gradually increases and finally is four times of the first pulse when the pulse duration is 10 ps. So, the period chirp on the light can significantly broaden its spectrum and thus decrease the temporal coherence, as shown in Fig. 5(b). When the rotation frequency is constant, the light only contains two individual frequency components. When pulse is chirped, its spectrum is broadened and the power of these frequency components decrease. Moreover, the bandwidth increases with the increasing chirp ratio and the decreasing chirp period. Such light with broadened spectrum can not only break the wave vector matching of LPIs, but also decease the accumulative temporal growth and spatial amplification of PB. Overall, we have introduced an effective physical mechanism to suppress PB by using time-dependent polarization rotation, as will be demonstrated below.
Fig. 5. Performance of TPR and CPR, (a) Instantaneous rotation frequency versus time, (b) Spectrum of the incident light. (c) Intensity variation of one point in the focal spot, Chirp period τ=20 ps.
4.2 Performance in suppressing SRS and SBS
4.2.1 Case one
In this subsection, the one-dimensional fluid laser-plasma-instability code (FLAME) [16] is applied to simulate the reflectivity of SRS and SBS. Based on a multi-fluid plasma model combined with full electromagnetic wave equations, FLAME code inherently covers the coupling of LPIs for the inhomogeneous plasma condition and has advantages of controllable noises and the lower computing-resource consuming compared with PIC. In this paper, these fluid simulations adopt the typical laser and plasma parameters in direct- or hybrid-driven ICF experiments [23]. The laser wavelength and peak intensity are λ=351 µm and I=1015W/cm2. The plasma length is 600 µm. Profiles of the plasma density, flow velocity and the electron (ion) temperature are not uniform on the path, as shown in Fig. 6. In all simulations, the resolution dx=0.0112 µm and dt=0.00335fs are maintained the same, as well as the total simulation time 40.2ps. In FLAME simulations, the magnitude of the volume noise is 10−9nc in order to describe the thermal noise [24]. Hereafter we show the reflectivity of SBS and SRS by incident light with invariant polarization, CPR with constant frequency of 3THz, and TPR, as shown in Figs. 7(a)–7(f). The averaged reflectivity 〈R〉 over time of SBS and SRS have also been compiled in Fig. 7.
As depicted in Figs. 7(a) and 7(b), the averaged reflectivity of SBS and SRS over time are 〈R〉 SBS=11.95%, 〈R〉 SRS=10.1%, respectively when incident by light with invariant polarization. In CPR scheme with constant frequency of Ω=3THz, the SBS reflectivity of electric fields of two linear polarizations (x- and y- directions) are reduced by a factor of ∼5, whereas the SRS reflectivity gains a little. This is because the rotation frequency of 3THz is too slow compared to growth rate of SRS. The suppression of SBS would decrease the depletion of the incident light, which on the contrary increases the SRS reflectively due to the competition between SBS and SRS [25]. The result of CPR indicates that our FLAME code is effective in calculating parametric backscattering in such laser plasma conditions. In TPR scheme, the SBS reflectivity is also reduced almost the same with that of CPR. On the other hand, we delightedly note that SRS reflectivity is reduced from 10.6% to 8.7%. Though the maximum rotation frequency is still not large enough, the time-dependent rotation frequency broadens the spectrum of incident light so that SRS is suppressed to an extent.
Fig. 6. Plasma parameters versus the propagation length. (a) Electron density ne/nc and plasma flow velocity vx/c versus propagation length. (b) Ion and electron temperature.
Fig. 7. Reflectivity of (a) SBS and (b) SRS versus time by incident light with invariant polarization, CPR and TPR. For invariant polarization, 〈R〉 SBS=11.95%, 〈R〉 SRS=10.1%. In CPR with constant frequency of 3THz, 〈R〉 SBS=2.6%, 〈R〉 SRS=10.6%. In TPR, the chirp period is τ=5 ps, 〈R〉 SBS=2.65%, 〈R〉 SRS=8.7%. The curve of reflectivity in CPR and TPR cases show the summary of reflectivity of two polarized directions.
Next, Fig. 8 is plotted to explore the influences of rotation frequency and chirp period τ on the PB reflectivity. For simplicity, the Ω in Fig. 8 denotes to the maximum frequency in the first period in Fig. 5(a).
Fig. 8. Reflectivity of parametric backscattering of invariant polarization, CPR and TPR in reducing SRS and SBS reflectivity. (a) SBS reflectivity versus maximum rotation frequency, (b) SRS reflectivity versus maximum rotation frequency.
In Fig. 8(a), the SBS reflectivity reduces significantly with the increasing rotation frequency in CPR and TPR when the maximum frequency is below 1 THz. It is worth noting that, the SBS reflectivity is reduced significantly quicker with shorter chirp period in the TPR scheme. This is mainly benefit from the broader spectrum and lower temporal coherence of the light with TPR. When the maximum rotation frequency is fixed, the spectrum of the incident light is broader with the shorter chirp period. However, when the rotation frequency keeps growing, the SBS reflectivity no longer decreases and there seems exists a minimum SBS reflectivity. This is because of the fact that the decrease of temporal growth rate and interaction length are saturated when the modulation of incident light is too fast and the average modulated effect in both time and space reaches a constant to the accumulative growth of SBS within one e-folding.
In Fig. 8(b), in CPR scheme, the SRS reflectivity firstly increases and then decreases with the increasing of the rotation frequency. In the region of Ω<1THz, significant reduction of SBS mitigates the depletion of incident light and inversely increases SRS reflectivity [25]. When Ω>1THz, the SRS reflectivity decrease very slowly with the increasing rotation frequency because such a rotation frequency is too low compared to the growth rate of SRS, even the maximum of 4.59THz. However, a considerable reduction of SRS reflectivity is achieved by TPR with short chirp periods like 5 ps and 2.5 ps. Therefore, TPR scheme provides a potential approach to effectively suppress strong SRS of the inner cone on NIF facility as well as the direct- and hybrid-drive ICF experiments.
In Ref. [26], V. M. Malkin et al. employed chirping to reduce parametric backscattering of light with constant (non-rotating) polarization in Raman amplifiers, in which the mechanism is detuning of the incident light such that the reflected light would be out of resonance. In the TPR scheme, the mechanism is not only the detuning of the incident light induced by pulse chirping, but also the amplitude modulation of the incident pulse in either polarization caused by polarization rotation. The TPR scheme should have better performance in reducing parametric backscattering with optimized parameters.
4.2.2 Case two
In case two, the laser wavelength and peak intensity are λ=351 µm and I=2 × 1015W/cm2. The plasma length is L=500λ, the electron and ion temperatures are Te=2 keV and Ti=320 eV, ne=(0.123∼0.127)nc. Hereafter we compare the reflectivity of SBS and SRS by incident light with invariant polarization, CPR with constant frequency of 4.59 THz, and TPR scheme, with both FLAME code and 1D-PIC [27], as shown in Figs. 9–11. In the 1D-FLAME code, the random fluctuating source with magnitudes approximating to thermal noise are realized by adding at each time step in space random numbers with controllable amplitudes and different random number seeds [16]. In PIC simulations, the duration of time step is 0.18/ω0 and the spatial length of each cell is 0.2c/ω0. There are 200 electrons and 200 ions in each cell.
The backscattered light is generally regarded as a counter-propagating wave (noise) with the incident light so that its travelling time should be about 2 ps. However, in our model, the noise is appended to the density perturbation of the plasma. Such a noise, physically acting as the thermal noise of the plasma, always exists during the parametric process. The excited plasma wave co-propagates along with the incident light and interacts with the subsequent backscattered light. Therefore, neither the growth time nor the period of the parametric backscattering is 2 ps, but the backscattered light continuous to resonance after achieving nonlinear saturation. The results have been given in Fig. 9.
Fig. 11. Comparison of invariant polarization, CPR and TPR in reducing SBS and SRS using the FLAME code. (a) averaged SBS reflectivity and (b) averaged SRS reflectivity with different chirp period.
From Figs. 9(a)–9(b), we can find that both the results by the FLAME code and 1D-PIC simulation indicate that CPR and TPR can effectively suppress SBS when compared with that with invariant polarization. In the TPR scheme, the SBS reflectivity reduces dramatically with the decreasing chirp period. The reason has already been given in case one: When the maximum frequency shift is fixed, the spectrum of the incident light is broader with the shorter chirp period.
In Figs. 10(a)–10(b), when comparing the SRS reflectivity of invariant polarization with CPR, the SRS reflectivity is not reduced but even increased. This is because such a rotation frequency of 4.59 THz is too slow compared to the growth rate of SRS. The suppression of SBS would decrease the depletion of the incident light, which on the contrary increases the SRS reflectivity due to the competition between SRS and SBS. Both the results of invariant polarization and CPR by the FLAME code and 1D-PIC agree well with that in case one. The simulation result here complies that our 1D-PIC simulation is consistent with the result by 1D-FLAME code, and can also reflect the parametric backscattering in the TPR scheme.
In the TPR scheme, the FLAME code shows that TPR can effectively suppress the growth of SRS when the chirp period is short as 2.5 ps, which agrees with the results in case one. The results of 1D-PIC simulation slightly show the suppression of SRS when the chirp period is 2.5 ps. However, the growths of the parametric backscattering are different in FLAME and 1D-PIC. The reason is that, in the FLAME code the magnitude of the volume noises is 10−9 nc (nc is the critical laser density) in order to describe the thermal noises, while in PIC the numerical noise are many orders of magnitude larger than thermal noises [28,29], which acts as a kind of artificial seeds of instabilities. So, SRS in PIC simulations is much stronger than in fluid simulations.
In Fig. 11, we compared the averaged SBS and SRS reflectivity in the three schemes. For SBS, reducing the chirp period can effectively reduce the SBS reflectivity in the TPR scheme. For SRS, the SRS reflectivity first increase and then decreases with the decreasing chirp period. This is because of the fact that the decrease of SBS reflectivity would reduce the depletion of the light and thus increase the SRS reflectivity due to their competition. What’s more, in Fig. 11(b), when the chirp period of the TPR scheme is 2.5 ps, SRS reflectivity is reduced below 1%.
In conclusion, the results of fluid simulations and PIC simulations agree well qualitatively with each other. Besides, the better performance of TPR scheme is consistently presented both in case one and case two.
In Fig. 12, we can see that in both TPR (τ=2.5 ps) and CPR schemes, the final scattering level is the lowest, the amplitudes of the corresponding electron wave and ion wave are the smallest. The results in Fig. 12 further proves that the TPR and CPR schemes can effectively reduce the parametric backscattering, and agree with the former analysis in Figs. 9,10.
Fig. 12. Spatiotemporal evolution of electron plasma wave (a) TPR, τ=2.5 ps, (b) CPR, (c) without PR. Spatiotemporal evolution of ion acoustic wave (d) TPR, τ=2.5 ps, (e) CPR, (f) without PR.
We analyzed the coherence factor of different smoothing schemes including the STUD pulses, SSD, CPR and TPR, as plotted in Fig. 13. The pulse duration τs in STUD is set to be 2.5 ps, the modulation frequency in SSD is 40 GHz but the total bandwidth remains 0.3 nm.
Fig. 13. Comparison of coherence time in TPR with STUD, CPR, SSD. τs is pulse duration in STUD pulses, ωm is the modulation frequency in SSD.
From Fig. 13, we can see that the laser in the CPR and the STUD is periodically coherent, while SSD shows low coherence to some extent. Compared to those three schemes, the TPR scheme shows much lower coherence. Hence, we can preliminarily estimate the remarkable progress of the TPR scheme compared to the other schemes.
Both polarization smoothing in Ref. [4] and polarization rotation in Ref. [13] have been demonstrated to be more effective in in suppressing laser plasma backscattering than intensity modulation only. This is because the change of amplitude of electromagnetic waves in either polarization can modulate the temporal growth rate to be smaller than the case without PR, and the effective interaction length can be reduced. In the traditional SSD, the polarization state of the focal spot is not time-independent so that SSD should be less effective than polarization rotation as far as we know.
In Ref. [13], the CPR method has been demonstrated to be more effective than the STUD pulses in suppressing parametric backscattering. In Ref. [14], the alternating-polarization pulse is also proved to be more effective than the STUD pulse. Moreover, in order to maintain the same incident energy, the instantaneous power of the STUD pulses is much higher than that in the other methods including TPR, CPR and alternating-polarization. Therefore, we can conclude that the TPR method should perform better than the STUD pulses in suppressing laser plasma backscattering.
5. Conclusion
We propose a novel effective optical smoothing for suppressing LPIs by TPR in ICF facilities. The presented approach to generate TPR is based on the coherent superposition of the beamlets with time-dependent frequency shift and counter-rotating circular polarization. Compared to light without polarization rotation or pulse chirping, the bandwidth of the incident light is broadened and the temporal coherence is reduced in the novel TPR scheme. With regard to SBS, the TPR scheme is helpful to reduce the requirement of the maximum rotational frequency. Moreover, the TPR scheme can provide considerable reduction of SRS reflectivity when the chirp period is short enough. On the other hand, the irradiation uniformity of the focal spot is significantly improved by the introduction of gratings, SPPs and CPPs to the chirped pulses of the beamlets. We believe the novel optical smoothing scheme has remarkable potential applications in laser plasma study and ICF experiments.
Funding
National Natural Science Foundation of China (11875093, 11875091, 61905167); National Major Science and Technology Projects of China (JG2017149, JG2019292, JG2019299).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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