1. Introduction

A variety of nonlinear effects arise during intense laser propagation in dielectrics. Among these is stimulated Raman scattering (SRS), which transfers laser energy to both the medium and optical sidebands. Significant efforts have been expended to understand and measure SRS, including in water [1–3], where it may serve as a tool for frequency conversion [4] or an important loss mechanism [5]. Better understanding of SRS could also benefit applications from communications [6,7] to industrial processing [8,9] and microscopy [10].

Stimulated Raman scattering is long known to be affected by parametric four-wave mixing (FWM) between the laser and the two Raman sidebands [11]. The coupled system can experience either suppressed or enhanced Raman gain, both of which have been investigated theoretically [12–15] and experimentally [13,14,16,17] in fibers with laser power of order 100 W. The coupled gain depends sensitively on the phase mismatch, which was optimized in experiments by tuning the laser wavelength [14,17], power [13,17], pump-probe propagation angle [16], and/or using purpose-built fibers [17]. These results may be important for fiber applications, where both the laser and material parameters can be carefully controlled. However, in non-phase-matched systems such as in laser nanosurgery [18], water-based frequency conversion [4] and underwater acoustic generation [5,19,20], the impact of FWM is less clear.

At laser powers exceeding the Kerr critical power (of order 1 MW in condensed media), the influence of FWM is obscured by several highly nonlinear effects. The laser will self-focus until it ionizes the medium and plasma defocusing counteracts self-focusing. The presence of the plasma, as well as the range of propagation angles inherent to a focusing beam, greatly complicate the SRS gain. We can progress theoretically, however, by limiting our analysis to the weakly-ionized ($<1$%) regime, where SRS is still determined largely by the medium response. Analyzing the laser intensity clamping condition further allows us to link the plasma and Kerr response and predict the coupled SRS gain. Finally, we turn to the SNOPROP simulation code to solve the full three-envelope nonlinear propagation equations with SRS, FWM, the optical Kerr effect, and multiphoton and collisional ionization in 2D cylindrical geometry [21,22].

We find that the self-generated plasma can significantly increase the SRS gain. Analysis demonstrates that at high intensities the self-generated plasma modifies the phase matching for enhanced Raman gain. Multidimensional simulations of a few-MW picosecond laser focusing in water confirm the SRS gain enhancement and agree well with theory. Laser energy losses to the Stokes Raman band can increase due to the enhanced gain.

2. Theoretical analysis

We consider an envelope propagation model including beams at the Raman Stokes, laser, and Raman anti-Stokes frequencies $\omega _S=\omega _L-\omega _v$, $\omega _L$, and $\omega _A=\omega _L+\omega _v$ respectively, where $\omega _v$ is the vibrational Raman frequency. The total electric field has the form

We begin in the strong, static pump approximation by assuming $|A_L| \gg |A_S|$, $|A_L| \gg |A_A|$, and $|A_L|$ is constant, and we neglect transverse effects, ionization losses, and self-phase modulation. In this case the propagation equations for the Stokes and anti-Stokes waves reduce to the coupled equations [11,23]

The general solution to a system of the form in Eqs. (2a) and (2b) was first given by Bloembergen [11]. The dominant mode $A_+$, a linear combination of $A_S$ and $A_A$, will evolve according to $A_+ \propto e^{gz}$ where

2.1 Stimulated Raman scattering

Equation (3) has been studied both analytically and experimentally in various limits [11,13,17,23]. To understand the combined effects of SRS, FWM, the optical Kerr effect, and ionization, it is useful to start by reviewing each of the effects in turn. Beginning with only SRS (and neglecting FWM, Kerr, and plasma) requires $\alpha _1 = -\alpha _R$, $\alpha _2 = {C}\alpha _R$, and $\kappa _1=\kappa _2 = 0$, where $\alpha _R = 3i\omega _S \chi _{RS}|A_L|^2/(cn_S)$ is the Raman gain coefficient in the absence of four-wave mixing, ${C}=\omega _A n_S/(\omega _S n_A)$ a constant, $c$ the speed of light, and $n_n = ck_n/\omega _n$ the linear index of refraction at frequency $\omega _n$. Note that because $\chi _{RS}$ is negative imaginary, the gain coefficient $\alpha _R$ is positive real. In this simplest case, $\alpha _1$ and $\alpha _2$ have opposite signs, and it follows from Eqs. (2a) and (2b) that the Stokes sideband gains energy from the pump pulse while the anti-Stokes sideband loses energy to the pump.

2.2 Four-wave mixing

The dynamics become more complicated when the anti-Stokes and Stokes sidebands are allowed to exchange energy with each other due to four-wave mixing. We consider both SRS and FWM by setting $\kappa _1 = \alpha _R$ and $\kappa _2 = -{C}\alpha _R$ with $\alpha _1 = -\alpha _R$ and $\alpha _2 = {C}\alpha _R$ as before. Here both the Stokes and anti-Stokes waves will experience gain with a coefficient depending on the phase mismatch $\Delta k$, illustrated in Fig. 1. Figure 1(a) shows the gain spectrum in water, for a range of phase mismatches around 355 nm (|Δk| = -550 cm^-1), and for several laser intensities.

 

Fig. 1. Coupled gain spectrum including four-wave mixing. (a) Coupled gain (solid lines) and decoupled gain (dashed lines) in water for a 355 nm laser with three different intensities. (b) Phase mismatch $\Delta k$ depicted for copropagating beams in a material with normal dispersion. (c) The phase mismatch $\Delta k$ may be eliminated if the Stokes and anti-Stokes beams propagate off-axis from the laser. (d) The normalized coupled gain spectrum clearly shows a change in behavior at $|\Delta k| \sim \alpha _R$.

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Figure 1(a) shows the gain spectrum in water, for a range of wavevector mismatches, including at 355 nm ($\Delta k = -550~\rm {cm}^{-1}$), and for several laser intensities. In the case of a large phase mismatch, $|\Delta k| \gg \alpha _R$ (laser intensities $I_L \ll 10^{12}$ W/cm$^2$), the coupled gain, Re[$g$], asymptotes to the uncoupled Raman gain coefficient $\alpha _R$; although both waves experience gain, the dominant coupled mode $A_+$ will be composed almost entirely of Stokes light with very little anti-Stokes light [23]. In the phase-matched case where $|\Delta k| = 0$, the waves can transfer energy extremely efficiently, as demonstrated experimentally in silica fibers [24]. However, in this case the coupled gain drops to 0; since the Stokes and anti-Stokes waves are perfectly coupled, any laser energy given to the Stokes wave is easily transferred to the anti-Stokes wave and then back to the laser. This gain suppression when $\Delta k \ll \alpha _R$ near perfect phase matching has also been demonstrated experimentally [13].

The magnitude of the phase mismatch depends on the material dispersion as well as the relative orientation of the laser, Stokes, and anti-Stokes beams. In the copropagating case shown in Fig. 1(b), the phase mismatch is always negative (positive) in media with normal (anomalous) dispersion near the laser wavelength, and vanishes only near the zero-dispersion wavelength. Experiments with two input beams can phase match by orienting the laser and Stokes beams at a small angle $\theta _S$ with respect to each other, which will produce an anti-Stokes beam oriented slightly off-axis at another angle $\theta _A$ as depicted in Fig. 1(c). As indicated previously, the coupled gain vanishes when phase-matched, rendering this useful mostly for frequency-conversion experiments from Stokes to anti-Stokes. A focusing beam contains a range of wavevectors, and choosing the correct F-number (including rays propagating at the angle $\theta _S$ from each other) can nearly satisfy the phase matching criterion while still allowing some coupled gain. With strong self-focusing, however, the propagation angle near focus can differ significantly from the initial beam, making it difficult to carefully control the phase mismatch.

We can simplify the gain equation with four-wave mixing. By normalizing the coupled gain and the phase mismatch to the uncoupled gain $\alpha _R$ and approximating ${C} \approx 1$ (which is generally accurate for small vibration frequencies $\omega _v \ll \omega _L$), Eq. (3) becomes

This normalization allows us to plot a single gain curve valid for any laser intensity as demonstrated in Fig. 1(d). We can clearly see that the FWM-related gain suppression only manifests for phase mismatches $|\Delta k| \lesssim |\alpha _R|$. At high laser intensities, gain is suppressed over a wider range of laser wavelengths.

2.3 Kerr Dielectric Response

Next we consider a system subject to SRS, FWM, and the optical Kerr effect. The gain in Eq. (3) is calculated by setting $\alpha _1 = -(\alpha _R+\alpha _K)$, $\alpha _2 = -{C}(-\alpha _R+\alpha _K)$, $\kappa _1=(\alpha _R+\alpha _K)$, and $\kappa _2= {C}(-\alpha _R+\alpha _K)$, where $\alpha _K = 3i\omega _S \chi _{NR}|A_L|^2/(cn_S)$ is the positive imaginary Kerr phase contribution due to the strong laser field. The Kerr effect enhances the coupled gain when $\Delta k>0$ but suppresses gain when $\Delta k<0$ as shown in Fig. 2(a). Both gain suppression and gain enhancement have been demonstrated in silica fibers by tuning the laser about the zero-dispersion wavelength [13,17]. Notably, at wavelengths with normal dispersion, $\Delta k$ is always negative, so the Kerr effect will only suppress the coupled gain. We again consider the case of water at 355 nm ($\Delta k = -550$ cm$^{-1}$). As shown in Fig. 2(a), this wavelength always exhibits suppressed gain regardless of the laser intensity.

 

Fig. 2. Coupled gain spectrum including four-wave mixing and the Kerr nonlinearity. (a) Coupled gain (solid lines) and decoupled gain without FWM (dashed lines) in water for a 355 nm laser with three different intensities. (b) The approximate coupled gain spectrum normalized to the decoupled gain $\alpha _R$ is valid at any intensity.

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We can also simplify the gain equation when the Kerr effect is included. Again taking ${C} \approx 1$, Eq. (3) becomes

In Fig. 2(b) we plot Eq. (5) for the ratio $|\chi _{NR}/\chi _{RS}|=2.94$ characteristic of water at 355 nm. The enhanced gain occurs at a phase mismatch of $\Delta k \approx 2\alpha _K \approx 2\alpha _R|\chi _{NR}/\chi _{RS}|$ and reaches a gain of ${\textrm{Re}} [g]/\alpha _R \approx \sqrt {1+|\chi _{NR}/\chi _{RS}|^2}$. A similar enhancement factor has been demonstrated in silica fibers at laser wavelengths above the zero-dispersion wavelength [14,17]. Importantly, the maximum Raman gain depends strongly on the nonresonant suceptibility $\chi _{NR}$.

Previous works demonstrating gain suppression and enhancement were limited to relatively low, $\sim$100 W laser powers in silica fibers, reaching intensities of $\sim 10^9$ W/cm$^2$ [13,17]. At these intensities, gain enhancement occurs only at a low, positive phase mismatch of order $\Delta k \sim m^{-1}$, corresponding to a wavelength range on the order of 10 nm wide in silica. Based on these results, gain enhancement is unlikely in materials with normal dispersion ($\Delta k < 0$), e.g. in water at visible or UV wavelengths. Further, at high intensities, ($-\alpha _R \lesssim \Delta k \lesssim 0$) indicates gain suppression over much of the UV and/or visible range. Figure 2(a) demonstrates this, showing that the coupled gain dips far below the uncoupled gain as the intensity increases (where the visible/UV range corresponds to $\Delta k=$ few $\times 100~\rm {cm}^{-1}$). It is unclear how any system with normal dispersion could experience enhanced gain.

2.4 Plasma dielectric response

Plasma generation is a natural consequence of focusing megawatt-class lasers in condensed media, yet to our knowledge its effects on SRS in weakly-ionized condensed media have not previously been explored. We show that the presence of a low-density plasma can lead to strong gain enhancement, almost without regard to the phase mismatch $\Delta k$. To capture the effects of the plasma response in Eqs. (2a) and (2b), we set $\alpha _1 = -(\alpha _R+\alpha _K+\alpha _p)$ and $\alpha _2 = -{C}(-\alpha _R+\alpha _K+{C_p}\alpha _p)$ where $\alpha _p = -i \omega _p^2/(2 c \omega _S n_S)$ is the negative, imaginary phase contribution due to the plasma, $\omega _p = \sqrt {N_e e^2/{m_{\rm {eff}} \epsilon _0}}$ the plasma frequency, $N_e$ the electron density, $-e$ the electron charge, $m_{\rm {eff}}$ the effective electron mass, $\epsilon _0$ the vacuum permittivity, and ${C_p}=\omega _S^2/\omega _A^2$ a constant. We keep $\kappa _1=(\alpha _R+\alpha _K)$ and $\kappa _2= {C}(-\alpha _R+\alpha _K)$ as before.

Inserting these coefficients into Eq. (3) and approximating ${C}\approx {C_p} \approx 1$, we arrive at the normalized gain

We see in Eqs. (6) and (7) that the plasma effectively increases the phase mismatch $\Delta k$ by $|\alpha _p|$, thereby lowering the system’s zero-dispersion wavelength. Fig. 3(a) illustrates this effect at laser intensity $I_L=10^{12}$ W/cm$^2$, showing that higher plasma density allows Raman gain enhancement even for $\Delta k<0$ where the medium has normal dispersion. In Fig. 3(b) shows the gain for fixed laser intensity and fixed $\Delta k = -550$ cm$^{-1}$ with varied plasma density. We see that increasing the plasma density may permit gain enhancement up to a certain optimal plasma density, but further increasing the plasma density degrades the phase matching and the coupled gain asymptotes to the uncoupled limit. Note that for high laser intensities where $\alpha _K \gg \Delta k$, Eq. (7) suggests that the optimal plasma density will scale roughly proportionally to $I_L$, meaning that for $I_L \gtrsim \rm {few} \times 10^{13}~\rm {W/cm}^2$ the medium would need to be significantly ionized (>1%) for enhanced gain. To understand whether a self-focusing laser will generate the optimal plasma density for enhanced gain, we consider the process of intensity clamping.

 

Fig. 3. Coupled gain spectrum including four-wave mixing, the Kerr effect, and plasma response. (a) Coupled gain (solid lines) and decoupled gain (dashed lines) in water for a 355 nm laser with intensity $I_L=10^{12}$ W/cm$^2$ for three different plasma electron densities $N_e$. (b) Coupled gain at the copropagating phase mismatch $\Delta k=-550$ cm$^{-1}$ for different plasma densities with laser intensity $I_L=10^{12}$ W/cm$^2$.

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2.5 Intensity clamping

When the laser power $P_L$ exceeds the Kerr critical power ($P_{cr}\approx 0.19\lambda _L^2 n_L \epsilon _0 c/\chi _{NR}$), or about 300 kW in water at 355 nm, self-focusing overpowers diffraction and the laser intensity will rise until plasma defocuses the beam. Near focus, the spot size reaches a minimum and the laser will approximately obey the 1D propagation equation

We apply the clamping condition by requiring that the focusing and defocusing effects must approximately cancel, giving

By assuming a high laser power $P_L \gg P_{cr}$ and approximating ${C_L} \approx {C_{KL}} \approx 1$, Eq. (9) indicates that $\alpha _{p} \approx -\alpha _{K}/2$, which is smaller than the optimal value for enhanced gain suggested by Eq. (7). Yet substituting $\alpha _p = -\alpha _K/2$ into the gain curve in Eq. (6) and taking the limit of $|\Delta k| \ll |\alpha _K|$ predicts $\sim$84% of maximum gain, or ${\textrm{Re}} [g] \approx 2.7\alpha _R$ in water. Thus, at high enough intensity, strongly enhanced gain is always expected, and the clamping condition prevents the plasma from exceeding the optimal density seen in Fig. 3(b). Additionally, the phase mismatch could be increased by off-axis propagation as previously mentioned, or even due to diffraction within the Stokes wave which is not considered here. Both of these effects could push the gain even closer to the maximum enhancement of ${\textrm{Re}} [g] \approx 3.2 \alpha _R$ in water.

The condition $|\Delta k| \ll |\alpha _K|$ for enhanced gain can be easily converted to a laser intensity threshold by substituting the definition of $\alpha _K$; we find that when the laser intensity exceeds

3. Full spatiotemporal dynamics

The peak laser intensity, and thus the SRS gain, depends on multiple nonlinear, spatiotemporal processes including pump depletion and ionization. We turn to numerical methods. The three-wave nonlinear envelope propagation equations have been derived previously including both Raman and Kerr susceptibilities [25], four-wave mixing, and ionization [26] with 2D axisymmetry. The pulse-frame longitudinal, temporal and radial coordinates are $z$, $\tau = t - z/v_{gL}$, and $r$, with $v_{gn} = 1/ \beta '_n$ the group velocity at frequency $\omega _n$ and $\beta '_n = \partial \beta (\omega ) / \partial \omega \rvert _{\omega =\omega _n}$.

We model the plasma dynamics including both multiphoton and collisional ionization. The electron density $N_e$ evolves according to

The collisional ionization rate $\nu _i$ and electron collision frequency $\nu _e$ obey

Finally, the paraxial, slowly-varying envelope propagation equations are written [21]

4. Numerical simulation

We solve the system in Eqs. (11), (14a), (14b), and (14c) numerically with the SNOPROP simulation code [21,22]. The system is initialized with a temporally and radially gaussian laser pulse, while the Stokes and anti-Stokes fields are initialized to a uniform background intensity $I_{\rm {BG}}=10^{-2}$ W/cm$^2$ [1,32]. Our first example consists of a 4 MW ($\sim 15P_{\rm {crit}}$), 355 nm, 0.5 ps laser pulse with 192 $\mu$m FWHM focusing in water with F-number of 42 (geometric focus at $z=0.8$ cm). All additional simulation parameters are reported in Table 1. We perform two simulations: the first solves the equations as written, while the second disables four-wave mixing by setting $e^{i\Delta k z}\to 0$ and $e^{-i\Delta k z}\to 0$ in Eqs. (14a), (14b), and (14c).

Table 1. Simulation parameters for 0.5 ps pulse

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The energy exchange between the three waves is shown in Fig. 4 with and without four-wave mixing enabled. In panel (a) we see that the the Stokes power output increases from 12% to 39% of the input power when four-wave mixing is enabled, and the output laser power decreases from 60% to 35% of the input power. Anti-Stokes light is understandably only produced when four-wave mixing is enabled. The log-scale power evolution in panel (b) illustrates the Stokes gain enhancement, raising the peak Stokes growth rate from 160 cm$^{-1}$ to 540 cm$^{-1}$. This gain enhancement factor of 3.4 is slightly higher than the factor 2.7 expected for copropagating beams, but similar to the maximum enhancement factor $\sqrt {1+|\chi _{NR}/\chi _{RS}|^2}\sim 3.2$ with arbitrary phase matching. Panel (c) confirms that the peak laser intensity remains well above the threshold $I_T = 3.6\times 10^{12}$ W/cm$^2$ for enhanced SRS as the laser propagates through focus.

 

Fig. 4. Evolution of a 0.5 ps laser pulse propagating through focus in water. (a) Power of each of the three waves in simulations with and without FWM enabled. (b) Stokes power growth with approximate exponential fits. (c) Peak laser intensity compared to the threshold for gain enhancement.

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Oscillations in the Stokes gain (c.f. Fig. 4(b)) correspond to spatiotemporal modulation of the laser profile (c.f. Fig. 5) by an ionization instability [33]. The instability corrugates both the laser envelope and the electron density with a spatial scale near the plasma wavelength; the peak electron density of $N_e\sim 10^{20}~\rm {cm}^{-3}$ corresponds to a plasma wavelength $\lambda _p = 2\pi c/\omega _p \sim 3~\mu$m, which is similar to the scale in Fig. 5(c,d,g,h). The behavior is similar with and without four-wave mixing, which only affects the peak intensity due to pump depletion. As the center of the pulse ($\tau =0$) has the highest power (well above $P_{\rm {crit}}$), it will self-focus first, followed later by slices near the head of the pulse. This causes the instability-generated corrugations to travel along the pulse in the $-\tau$ direction. We see in Fig. 5 that over a distance $\sim$64 $\mu$m, two intensity nodes cross the center of the pulse at $\tau =0$, simultaneous with two gain oscillations in Fig. 4(b)). This confirms that the gain modulations are related to the ionization instability and underscores the importance of a full spatiotemporal treatment of the separate waves to accurately predict SRS.

 

Fig. 5. Intensity profiles of a 0.5 ps laser pulse focusing in water with FWM disabled (a,b,c,d) and enabled (e,f,g,h). The laser envelope is shown at four different $z$ positions, with the head of the pulse at negative $\tau$.

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To understand the effect of the gain enhancement at longer pulse durations where collisional ionization becomes more significant, we simulate an additional system where a 5 ps laser pulse focuses with a larger F-number of 84 and longer propagation distance of 3.2 cm. The temporal box and cell size are increased by a factor of 10, but all other parameters are as in Table 1. The energy evolution in Fig. 6(a) differs dramatically from the previous case; with the 5 ps pulse, enabling FWM only increases the output Stokes power by 2% and decreases the laser output power by 5%. However, Fig. 6(b) shows a clear enhancement in the exponential Stokes gain rate by a factor of 3.3. Fig. 6(c) shows that the peak laser intensity is only comparable to the threshold $I_T$, thus a few intensity lobes (c.f. Figure 5) may experience enhanced gain, but the bulk of the laser energy cannot experience enhanced gain, and so the total energy converted to the Stokes wave is not significantly changed when enabling FWM. We confirm that the intensity must strongly exceed $I_T$ given in Eq. (10) for enhanced gain to significantly affect the final Stokes energy.

 

Fig. 6. Evolution of a 5 ps laser pulse propagating through focus in water. (a) Power of each of the three waves in simulations with and without FWM enabled. (b) Stokes power growth with approximate exponential fits. (c) Peak laser intensity and highest temporally-averaged laser intensity with FWM compared to the threshold for gain enhancement.

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5. Conclusion

We have shown that four-wave mixing can significantly enhance Raman gain when high-power lasers focus in condensed media. Our analysis agrees well with multidimensional nonlinear laser propagation simulations and confirms that above a certain threshold intensity $I_T$, the laser-generated plasma provides phase matching for strong gain enhancement over a broad wavelength range. This can lead to many times higher Stokes Raman production and significantly increased laser energy losses. Accurate prediction of the Raman gain at high intensities may require a fully-spatiotemporal treatment of each of the Stokes, laser, and anti-Stokes waves. Underwater laser experiments similar to Ref. [3] but with shorter laser pulses could experimentally probe this gain enhancement. Although we primarily consider laser propagation in water, the phenomenon may also occur in other dielectrics which are weakly ionized by a high-intensity, several-critical-power laser.