3.1 Influence of ${\psi}_2$ on B-SRS

Figure 2 shows spectra of the back-scattered light under different ${\psi}_2$. It shows that the larger $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$ is, the stronger the back-scattering is. For both negative and positive ${\psi}_2$, the back-scattering spectra are almost identical for the same $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. When $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid \geqslant 1000$ ${\mathrm{fs}}^2$, clear Stokes sidebands with a regular interval were observed, indicating a substantial stronger excitation of B-SRS^[Reference Coverdale, Darrow, Decker, Mori, Tzeng, Marsh, Clayton and Joshi^32]. From the match condition, ${\omega}_0 = {\omega}_1\pm {\omega}_{\mathrm{pe}}$ (${\omega}_{\mathrm{pe}}$ is the plasma frequency), we deduce that B-SRS occurs in the density region ${n}_{\mathrm{e}}\approx 5\times {10}^{18}$ ${\mathrm{cm}}^{-3}$.

Figure 2 The back-scattered light spectra with various (a) positive and (b) negative second-order dispersions. The absence of light within 730–870 nm is due to the total reflection of the M1 mirror in front of the collection fiber (Fiber 1 in Figure 1(a)).

To quantitatively compare the results, the red-shifted back-scattering (840–1100 nm) was integrated and plotted as a function of pulse duration ${\tau}_{\mathrm{L}}$ (shown in Figure 3). The integrated B-SRS signal has an exponential-like distribution for both negative and positive chirps. Although from Equation (2), the laser pulses with the same $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$ have the same pulse duration, the integrated signal is dependent on the sign of laser chirp. A larger signal was acquired with a positive chirp.

Figure 3 Integrated B-SRS spectral signal (840–1100 nm) versus pulse duration. The red solid squares are experimental results of positive ${\psi}_2$ and the blue solid squares correspond to negative ${\psi}_2$. The error bars are due to shot-to-shot fluctuations. The red dashed line presents the theoretical calculation of ${e}^{\gamma_0{t}_\mathrm{g}}$ with ${\lambda}_0 = 835$ nm, whereas the blue dashed line is that with ${\lambda}_0 = 765$ nm. The inset shows the calculated ${\gamma}_0$ with duration for ${\lambda}_0 = 835$ nm (red solid line) and ${\lambda}_0 = 765$ nm (blue solid line), respectively.

In order to understand the influence of ${\psi}_2$ on B-SRS, a simple model is used. For an ultrashort laser propagating in a homogeneous plasma, the growth rate of SRS is given by the following^[Reference Wilks, Kruer, Williams, Amendt and Eder^3]:

(4)$$\begin{align}{\gamma}_0\approx \frac{k_{\mathrm{epw}}}{8}{v}_{\mathrm{osc}}{\left[\frac{\omega_{\mathrm{pe}}^2}{\omega_{\mathrm{epw}}\left({\omega}_0-{\omega}_{\mathrm{epw}}\right)}\right]}^{1/2},\end{align}$$

where ${v}_{\mathrm{osc}}$ is the electron oscillation velocity.

The analysis of the Stokes sidebands has shown that B-SRS arises in the region of non-relativistic laser intensity (${I}_0\leqslant 1.0\times {10}^{18}$ $\mathrm{W}/{\mathrm{cm}}^2$ for ${\psi}_2 = 0$) far before the focus. Thus, the electron oscillation velocity is estimated using ${v}_{\mathrm{osc}} = {a}_0c$, where $c$ is the speed of light. To evaluate ${\gamma}_0$, plasma temperature ${T}_{\mathrm{e}}$ is required to infer the thermal velocity ${v}_{\mathrm{th}} = \sqrt{T_{\mathrm{e}}/{m}_{\mathrm{e}}}$, which is further used to estimate the electron-plasma-wave frequency from the dispersion relation, ${\omega}_{\mathrm{epw}}^2 = {\omega}_{\mathrm{pe}}^2+3{k}_{\mathrm{epw}}^2{v}_{\mathrm{th}}^2$. The quasi-static model of above-threshold ionization (ATI) is applicable to get the plasma temperature^[Reference Corkum, Burnett and Brunel^33, Reference Burnett and Corkum^34]. In the model, the phase mismatch, $\Delta \phi$, between the electron releasing position and the crest of pump light, produces electrons with residual kinetic energy $\varepsilon = 2{E}_{\mathrm{q}}{\mathit{\sin}}^2\Delta \phi$, where ${E}_{\mathrm{q}}$ is the electron ponderomotive potential. Over a laser cycle, the averaged ATI energy is given by the following^[Reference Burnett and Corkum^34]:

(5)$$\begin{align}\left\langle \varepsilon \right\rangle = \frac{\int_0^{\frac{\pi }{2}}2{E}_{\mathrm{q}}W(t)\mathit{\cos}{\phi}^2 \mathrm{d}\phi}{\int_0^{\frac{\pi }{2}}W(t) \mathrm{d}\phi},\end{align}$$

where $W(t)$ is the tunneling ionization rate, which is dependent on the strength of the electric field ${E}_{\mathrm{l}}$ and the ionization potential ${E}_{\mathrm{i}}$ of the target ion^[Reference Ammosov, Delone and Krainov^35]. Then, ${T}_{\mathrm{e}}$ could be calculated by ${T}_{\mathrm{e}}\sim \frac{2}{3Z}\sum \left\langle {\varepsilon}_{\mathrm{i}}\right\rangle$. For a helium target, ${E}_{\mathrm{i}}$ is 13.6 and 54.4 eV to generate ${\mathrm{He}}^{+}$ and ${\mathrm{He}}^{2+}$, respectively. However, we ignore the contribution of ${\mathrm{He}}^{+}$ to ${T}_{\mathrm{e}}$ since ${\mathrm{He}}^{+}$ could be generated by the amplified spontaneous emission instead of the main pulse.

As SRS develops in the rising edge of the laser pulse, where different wavelength components are contained, the chirp effect is straightforward. To simplify the estimation and investigate the effect of the laser wavelength on SRS growth, two cases with different single wavelengths, simulating the positive and negative chirps, were used to calculate the growth rates. We take ${\lambda}_0 = $ 835 and 765 nm, corresponding to our laser bandwidth, for cases of positive and negative ${\psi}_2$, respectively.

The inset in Figure 3 shows the growth rate as a function of pulse duration. Here, ${\gamma}_0$ decreases with ${\tau}_{\mathrm{L}}$ and follows the scale of $\sim 1/\sqrt{\tau_{\mathrm{L}}}$. It is 8.2$\%$ higher for long-wavelength laser excitation, as ${a}_0$ is correlated with the laser wavelength. Based on the calculation, the total gain of B-SRS could be estimated by $G\approx \exp \left({\gamma}_0{\tau}_{\mathrm{g}}\right)$ with growth time ${\tau}_{\mathrm{g}}$. The best fit of the measurements is obtained with ${\tau}_{\mathrm{g}} = 0.75{\tau}_{\mathrm{L}}$ and plotted in Figure 3. This confirmed the earlier studies that B-SRS is excited at the pulse front^[Reference Tzeng, Mori and Decker^6, Reference Faure, Marquès, Malka, Amiranoff, Najmudin, Walton, Rousseau, Ranc, Solodov and Mora^21]. Interestingly, the crude estimation gives a similar growth time, which is shorter than the pulse duration. The growth of B-SRS with $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$ scales as $G\approx \exp \left({\gamma}_0{\tau}_{\mathrm{g}}\right)\propto \exp \left(\sqrt{\tau_{\mathrm{L}}}\right)$.

3.2 Influence of ${\psi}_2$ on S-SRS

Figure 4 shows shadowgraphs of laser-produced plasma with different ${\psi}_2$. Filamentary structures were observed originating from the edge of the plasma channel due to the ionization of neutral atoms by side-scattered light^[Reference Matsuoka, Mcguffey, Cummings, Horovitz, Dollar, Chvykov, Kalintchenko, Rousseau, Yanovsky, Bulanov, Thomas, Maksimchuk and Krushelnick^8, Reference Sylla, Flacco, Kahaly, Veltcheva, Lifschitz, Sanchez-Arriaga, Lefebvre and Malka^36, Reference Arunachalam, Schwab, Sävert and Kaluza^37]. The side-filamentation is more noticeable, with a higher $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$ occurring both in upward and downward directions. It is absent with ${\psi}_2 = 0$ ${\mathrm{fs}}^2$(shown in Figure 1(c)). The filaments occupy a larger spatial region and have a longer length with a higher $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$, regardless of the sign of the chirp. Figure 5 plots the typical transverse profiles for ${\psi}_2 = 0$ and ${\psi}_2 = +500$ ${\mathrm{fs}}^2$ at the different spatial positions. The density profiles are quasi-Gaussian distribution in the transverse direction, which gives rise to the observed beam divergence (marked by the white solid lines in Figures 1(c) and 4). One should notice that the transverse density profiles are almost the same for ${\psi}_2 = 0$ and ${\psi}_2 = +500$ ${\mathrm{fs}}^2$, but the side-scattering was absent for ${\psi}_2 = 0$. This suggests the side-scattering could not be due to the beam divergence but rather the S-SRS.

Figure 4 Shadowgraphs showing side filaments at different second-order dispersions ${\psi}_2$. (a) ${\psi}_2$ = +500 ${\mathrm{fs}}^2$, (b) ${\psi}_2$ = +1000 ${\mathrm{fs}}^2$, (c) ${\psi}_2$ = +2000 ${\mathrm{fs}}^2$, (d) ${\psi}_2$ = –500 ${\mathrm{fs}}^2$, (e) ${\psi}_2$ = –1000 ${\mathrm{fs}}^2$ and (f) ${\psi}_2$ = –2000 ${\mathrm{fs}}^2$. The red arrows denote the filament direction at different spatial position. The white lines show the edges of the plasma channel.

Figure 5 The transverse plasma density profile in the cases of ${\psi}_2 = 0$ and ${\psi}_2 = +500$ ${\mathrm{fs}}^2$. The solid lines and the dash-dot lines represent the profiles obtained at $x = -800$ μm and $x = -1000$ μm, respectively.

The effect of ${\psi}_2$ on side-scattering can also be distinguished by comparing the length and number of side filaments. We found the difference in chirp is significant at a shorter pulse duration. For example, filamentation in the case of ${\psi}_2 = -500$ ${\mathrm{fs}}^2$ (Figure 4(d)) is extremely weak compared to that with ${\psi}_2 = +500$ ${\mathrm{fs}}^2$ (Figure 4(a)). However, for ${\psi}_2 = \pm 2000$ ${\mathrm{fs}}^2$ (Figures 4(c) and 4(f)), the difference between the two cases becomes indistinguishable. The group velocity dispersion induced by laser chirp could contribute to the observation. The pulse compression (for positively chirped pulse) or stretch (for negatively chirped pulse) induced by the group velocity dispersion would reinforce or suppress the Raman process. Moreover, this phenomenon is strongly dependent on the chirp rate^[Reference Dodd and Umstadter^26, Reference Pathak, Vieira, Fonseca and Silva^27]. A higher chirp rate has a stronger impact. Considering the chirp rate is 3.8 times larger for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 500$ ${\mathrm{fs}}^2$ than that for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 2000$ ${\mathrm{fs}}^2$ (shown in Table 1), it is reasonable to observe a significant difference for the former cases.

In addition, with the increase of $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$, the lateral dimension of the plasma channel reduces as a result of reduced on-axis laser intensity due to pulse stretching. It is approximately 280 $\pm 15$ μm for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 500$ ${\mathrm{fs}}^2$, approximately $200\pm 15$ μm for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 1000$ ${\mathrm{fs}}^2$ and approximately 140 $\pm 10$ μm for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 2000$ ${\mathrm{fs}}^2$, respectively.

3.3 Influence of ${\psi}_2$ on F-SRS

Figures 6(a) and 6(b) show the transmitted spectra under positive ${\psi}_2$ and negative ${\psi}_2$, respectively. The spectra of transmitted light are mainly composed of the laser spectrum and a blue-shifted component due to the ionization-induced frequency shift^[Reference Wood, Siders and Downer^38, Reference Koga, Naumova, Kando, Tsintsadze, Nakajima, Bulanov, Dewa, Kotaki and Tajima^39]. The red-shifted spectrum is intriguingly absent. The influence of $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$ on F-SRS significantly differs from that on the backward and side-scattering. There is no pronounced enhancement on SRS with increasing $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. For positive ${\psi}_2$, anti-Stokes sidebands emerge at ${\psi}_2 = +500$ ${\mathrm{fs}}^2$ and become more obvious at ${\psi}_2 = +1000$ ${\mathrm{fs}}^2$. A further increase in ${\psi}_2$ will not affect the intensity of anti-Stokes sidebands, while slightly modulating their intervals. Sidebands in the cases of ${\psi}_2 = +1000\ {\mathrm{fs}}^2$, ${\psi}_2 = +2000\ {\mathrm{fs}}^2$ and ${\psi}_2 = +3000$ ${\mathrm{fs}}^2$ have comparable signal intensity, which may suggest saturation of F-SRS at longer pulse durations.

Figure 6 The spectra of transmitted light with (a) positive second-order dispersion and (b) negative second-order dispersion.

In addition, the laser chirp was found to have a significant impact on F-SRS. Sections 3.1 and 3.2 show similar features of B-SRS and S-SRS for both negative and positive chirps. For F-SRS, however, the transmitted optical spectrum depends significantly on the sign of the laser chirp. As shown in Figure 6(b), in all cases with a negative laser chirp, the spectra are similar to that with ${\psi}_2 = +500$ ${\mathrm{fs}}^2$ in Figure 6(a). There is no distinguishable sign of anti-Stokes sidebands in F-SRS with a negative chirp, even though the pulse duration is stretched to 550 fs (${\psi}_2 = -3000$ ${\mathrm{fs}}^2$). This indicates that F-SRS is strongly suppressed with a negatively chirped pulse. This observation is consistent with theoretical estimation^[Reference Dodd and Umstadter^26], where a laser with 12$\%$ bandwidth can eliminate Raman forward scattering.

3.4 Side-scattering angle dependence on ${\psi}_2$

As shown in Figure 4 , there is a trend that the inclined angle of filaments relative to the laser axis gets smaller with the laser propagation (denoted by red arrows). In Figure 7, we plot the angles of both upward and downward filaments as a function of spatial position for different ${\psi}_2$. It shows that a relatively larger side-scattering angle is observed with a higher $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. At $x = -1150$ μm, for instance, the angle increases from $31.0^{\circ }\pm 4^{\circ }$ for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 500$ ${\mathrm{fs}}^2$ to $41.0^{\circ }\pm 4^{\circ }$ for $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid = 2000$ ${\mathrm{fs}}^2$. In Figures 7(b) and 7(c), the upward and downward side-scattering angles under positive and negative chirps are compared. The chirp effect is negligible, since the filament angles follow the same trends for both negative and positive ${\psi}_2$. Meanwhile, we note that the side-scatterings are asymmetric where the downward filaments have angles several degrees larger than that in the upward direction at the same spatial position. Similar observation has been reported by Matsuoka et al.^[Reference Matsuoka, Mcguffey, Cummings, Horovitz, Dollar, Chvykov, Kalintchenko, Rousseau, Yanovsky, Bulanov, Thomas, Maksimchuk and Krushelnick^8]; however, the angular variation and underlying physics remain elusive.

Figure 7 The side-scattering angle at different spatial positions with (a) ${\psi}_2 = +500$ ${\mathrm{fs}}^2$, (b) ${\psi}_2 = \pm 1000$ ${\mathrm{fs}}^2$ and (c) ${\psi}_2 = \pm 2000$ ${\mathrm{fs}}^2$. Orange circles and green squares correspond to the measurements of the upward scattering angle with positive and negative ${\psi}_2$, respectively. Blue triangles are the measurements of the downward scattering angle with negative ${\psi}_2$. The orange (blue) dashed line is the calculation based on the maximum spatial growth rate with ${n}_{\mathrm{e}} = 1.8\times {10}^{19}$ ${\mathrm{cm}}^{-3}$ (${n}_{\mathrm{e}} = 2.4\times {10}^{19}$ ${\mathrm{cm}}^{-3}$).

To explain the spatially-dependent scattering angles, a model incorporating the growth of Raman scattering and the energy damping processes was used^[Reference Wilks, Kruer, Estabrook and Langdon^2]. The energy damping rate of plasma waves is expressed as follows^[Reference Wilks, Kruer, Estabrook and Langdon^2]:

(6)$$\begin{align} \begin{array}{ll}{\Gamma}_{\mathrm{p}} = \!\!\!\!\!\!\!\!& {\left(\dfrac{\pi }{8}\right)}^{1/2}\dfrac{\omega_{\mathrm{pe}}}{{\left({k}_{\mathrm{p}}{\lambda}_{\mathrm{D}}\right)}^3}\mathit{\exp}\left[-\dfrac{1}{2}{\left({k}_{\mathrm{p}}{\lambda}_{\mathrm{D}}\right)}^{-2}-\dfrac{3}{2}\right]\\[6pt] {}& +{\left({\omega}_{\mathrm{pe}}/{\omega}_{\mathrm{epw}}\right)}^2{v}_{\mathrm{ei}},\end{array}\end{align}$$

where ${\lambda}_{\mathrm{D}} = {v}_{\mathrm{th}}/{\omega}_{\mathrm{pe}}$ is the Debye length and ${v}_{\mathrm{ei}}$ is the electron–ion collision frequency. The first term on the right-hand side in Equation (6) represents the Landau damping and the second term is the inverse bremsstrahlung absorption. The inverse bremsstrahlung absorption also causes an energy dissipation of the EM wave, giving ${\Gamma}_{\mathrm{s}} = {\left({\omega}_{\mathrm{pe}}/{\omega}_1\right)}^2{v}_{\mathrm{ei}}$. Combining the two damping processes, the spatial growth rate of SRS is given as follows^[Reference Wilks, Kruer, Estabrook and Langdon^2]:

(7)$$\begin{align}\kern-5pt\kappa = {\left[\frac{\gamma_0^2}{v_{1\mathrm{x}}{v}_{2\mathrm{x}}}+\frac{1}{4}{\left(\frac{\Gamma_{\mathrm{s}}}{v_{1\mathrm{x}}}-\frac{\Gamma_{\mathrm{p}}}{v_{2\mathrm{x}}}\right)}^2\right]}^{1/2}-\frac{1}{2}\left(\frac{\Gamma_{\mathrm{s}}}{v_{1\mathrm{x}}}+\frac{\Gamma_{\mathrm{p}}}{v_{2\mathrm{x}}}\right),\end{align}$$

where ${v}_{1\mathrm{x}}$ and ${v}_{2\mathrm{x}}$ represent the group velocity of the scattered light and the plasma wave, respectively. They are expressed as follows:

(8)$$\begin{align}{v}_{1\mathrm{x}} = {k}_1{c}^2 \cos\theta /{\omega}_1,\end{align}$$

(9)$$\begin{align}{v}_{2\mathrm{x}} = 3\left({k}_0-{k}_1 \cos\theta \right)\left({v}_{\mathrm{th}}^2/{\omega}_{\mathrm{pe}}\right),\end{align}$$

where $\theta$ is the angle between the scattered light and the pump light. According to Equations (7)–(9), the spatial growth rate intrinsically depends on $\theta$.

Since side filaments were found to originate from the plasma channel edge, the local plasma density ${n}_{\mathrm{e}}$ was used in calculating the growth rate $\kappa$. Deduced from the interferogram, ${n}_{\mathrm{e}}$ is $\left(1.8\pm 0.3\right)\times {10}^{19}$ ${\mathrm{cm}}^{-3}$ at the upper edge and $\left(2.4\pm 0.3\right)\times {10}^{19}\ {\mathrm{cm}}^{-3}$ at the lower edge. Considering the laser focusing geometry, the laser intensity increases with the propagation. Plasma temperature ${T}_{\mathrm{e}}$ is estimated by the ATI ionization model described above.

Figure 8(a) shows the calculated plasma temperature distribution in the spatial position for different $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. Here, ${T}_{\mathrm{e}}$ increases with the laser propagation and decreases with $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. The calculated angular-dependent spatial growth rate $\kappa$ is plotted in Figure 8(b). It shows that S-SRS has the largest growth rate along a cone angle near the forward direction^[Reference Wilks, Kruer, Estabrook and Langdon^2, Reference Higginson, Zhang, Bailly-Grandvaux, McGuffey, Bhutwala, Winjum, Strehlow, Edghill, Dozières, Tsung, Lee, Andrews, Spencer, Lemos, Albert, King, Wei, Mori, Manuel and Beg^40]. The cone angle reduces with an increase in ${T}_{\mathrm{e}}$ or a decrease in ${n}_{\mathrm{e}}$. We evaluated the influence of chirp on the spatial growth rate by changing the laser wavelength ${\lambda}_0$. The angle with maximum growth rate ${\theta}_{\mathrm{m}}$ is weakly dependent on ${\lambda}_0$. For example, at $x = -1300$ μm, ${\theta}_{\mathrm{m}}$ varies from ${29.6}^{\circ }$ to ${30.1}^{\circ }$ when ${\lambda}_0$ changes from 760 to 840 nm. The insensitivity of the scattering angle to the laser wavelength is consistent with our observations. Thus, the effect of chirp on the angle is trivial.

Figure 8 (a) The plasma temperature ${T}_{\mathrm{e}}$ for different $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$. (b) The typical angular distributions of the spatial growth rate that correspond to the black crosses in (a). LD is for ${n}_{\mathrm{e}} = 1.8\times {10}^{19}$ ${\mathrm{cm}}^{-3}$, and HD for ${n}_{\mathrm{e}} = 2.4\times {10}^{19}$ ${\mathrm{cm}}^{-3}$.

Since filaments should be most evident near ${\theta}_{\mathrm{m}}$, we plot ${\theta}_{\mathrm{m}}$ as a function of spatial position (shown in Figures 5(a)–5(c)). The calculations successfully reproduce the measurements in all cases, which suggest that the reduced filament angle is mainly due to the increased ${T}_{\mathrm{e}}$. During the laser propagation and focusing, the spot size decreases and ${T}_{\mathrm{e}}$ increases. Similarly, a larger ${\theta}_{\mathrm{m}}$ is expected for a higher $\mid \kern-1pt\!{\psi}_2\!\kern-1pt\mid$, as the longer pulse duration will lower the temperature ${T}_{\mathrm{e}}$. The measured asymmetric side-scattering in the upward and downward directions is evidently attributed to the ${n}_{\mathrm{e}}$ difference of plasma channel edges in the transverse direction.