Temporal analysis and control of Raman scattering dynamics

Abstract

Raman scattering underlies a broad range of spectroscopic and light-generation techniques, yet its conventional description, based on the Raman gain spectrum, accurately describes only long-pulse, steady-state dynamics. We present and develop a time-domain theoretical approach that provides a unified and physically-transparent description of Raman interactions across all temporal regimes. It enables direct visualization of Raman temporal dynamics and accounts for spectrotemporal aspects of Raman phenomena. We apply this theory specifically to Raman-shifting with ultrashort light pulses in gases, where the excitation is in the impulsive Raman regime and dephasing of Raman transitions is weak. The analysis, for the first time, exposes temporal and spectral distortions that arise from Raman scattering and which impact frequency-shifting performance detrimentally. Crucially, it also identifies how these distortions can be suppressed through temporal control of the nonlinear response. Numerical simulations of the soliton self-frequency shift (SSFS) in gas-filled hollow-core fibers show that molecules with strong Raman responses do not yield efficient frequency conversion, and predict that reducing the relative Raman contribution (compared to the electronic response) enhances the process. Experiments using gas mixtures with tunable Raman fraction of the nonlinear response confirm these predictions. An analytic expression for impulsive SSFS in gases, which departs significantly from the well-known formula for glasses, predicts the observed behavior when Raman-induced temporal distortion is suppressed. The new time-domain framework uncovers phenomena and provides physical insight that are inaccessible through the decades-old frequency-domain treatment of Raman scattering.

Figures (7)

• Figure 1: Physical pictures of Raman scattering in time and frequency domains. (a) and (b) are Raman dynamics in the time domain while (c) is in the frequency domain. (a) Raman temporal response $R(t)$ with high (top) and low (bottom) dephasing rates, respectively. $\tau_R=1/\nu_R$: Raman period, $\nu_R$: Raman transition frequency, and $T_2$: Raman dephasing time. (b) Raman regimes. $\triangle\epsilon_R$: Raman-induced index change, $\tau_0$: duration parameter of a Gaussian or soliton pulse [which is approximately half the full-width at half-maximum (FWHM) duration], $A(t)$: electric field. Inset shows the weak Raman-induced oscillatory waves, i.e., phonon waves Bauerschmidt2015aChen2024, superposed on the pulse-following index in the transient and steady-state regimes, due to beating between the pump and vacuum fluctuations. This leads to spontaneous Raman scattering. Without vacuum fluctuations and the subsequent beating, there will be no discrete-frequency Raman generation of Stokes signals far from the pump frequency. (c) Top: the relationship between the Raman gain spectrum (black) and the pulse (colored lines) in strong-dephasing (left) and weak-dephasing (right) media. Bottom: spectral profiles of the imaginary part of the Fourier transform ($\mathfrak{F}$) of the Raman convolution integral, which follows ${\Im\left[\mathfrak{F}\left[R\right]\mathfrak{F}\left[\abs{A}^2\right]\right]}$. Zoom-in views in (c) of the long-pulse convolution-integral profile are displayed to visualize the beating-induced index change in frequency due to vacuum fluctuations [as shown in the inset in (b)].
• Figure 2: Intrapulse continuous redshifting. (a) Simulated magnitude of impulsive Raman redshift (blue) with varying Raman period $\tau_R$ normalized by $\tau_0$. Propagation distance is minimized to exclude dispersion and electronic effects, for comparison to Eq. (\ref{['eq:SSFS_im']}). $\tau_0^{\text{opt}}={0.131\tau_R}$ (equivalently, $\tau_{\text{FWHM}}^{\text{opt}}={0.231\tau_R}$). Red dashed line is the result of Eq. (\ref{['eq:SSFS_im']}). Top figures show the temporal profiles of the pulse $\abs{A}^2$ (black) and the Raman-induced index changes $\triangle\epsilon_R$ (orange) for the values indicated on the plot below. (b) Raman time $T_R$ evaluated as a function of pulse duration $\tau_0$ normalized by $\nu_R$, under different dephasing times $T_2$.
• Figure 3: Long-pulse spectrotemporal Raman dynamics. (a) Temporal evolution of the beating-induced index modulation $\triangle\epsilon_R^{\text{osc}}(t)$. Index modulation extends beyond the pulse in the transient regime due to $\tau_0\ll T_2$, leading to integrated-energy-dependent and thus stronger Stokes generation in the trailing edge Chen2024. (b) Illustration of Raman spectral shaping. If ${\triangle\epsilon_R^{\text{osc}}}$ is narrowband (approximately a Dirac delta function in the frequency domain), the spectral convolution, from temporal multiplication, yields a nonlinear Stokes increment ${\triangle\epsilon_R^{\text{osc}}(t)A(t)}$ that has the same bandwidth as the pump pulse $A(t)$; otherwise, the Stokes spectrum is broadened. In fact due to the temporal convolution in $\triangle\epsilon_R^{\text{osc}}(t)$, the bandwidth of $\triangle\epsilon_R^{\text{osc}}$ is approximately the minimum of the Raman-gain bandwidth $\mathfrak{F}\left[R\right]$ and the bandwidth of the beating amplitude $\mathfrak{F}\left[C^{PS}\right]$, and therefore of the pump and Stokes fields themselves. In other words, the bandwidth of $\triangle\epsilon_R^{\text{osc}}$ (black lines) can only be equal to or narrower than the pump bandwidth (blue line), implying that the maximum bandwidth of $\triangle\epsilon_R^{\text{osc}}(t)A(t)$ is $\sqrt{2}$ times the pump bandwidth. This shows that the initial Stokes growth draws energy from anywhere between ${1/\sqrt{2}}$ and the full temporal extent of the pump pulse. The spectral phase information within ${\mathfrak{F}\left[\triangle\epsilon_R^{\text{osc}}(t)A(t)\right]}$ (red lines) is governed by the Raman temporal dynamics: it represents the Stokes field at the trailing edge of the pump in the transient regime or at the pump's temporal center with the highest instantaneous intensity in the steady-state regime [also see (a)] Chen2024. $\nu^{P/S}$ is pump (P) or Stokes (S) frequency.
• Figure 4: Reduced SSFS due to Raman-induced temporal distortion at high Raman fractions. (a) Induced index change with different Raman fractions from electronic (blue) and Raman (orange) nonlinearities, along with the total variation (yellow) and the pulse profile ($\abs{A}^2$; black). (b) Magnitude of Raman impulsive redshift, in a medium with weak dephasing, and output pulse duration versus input duration (with the corresponding energy of the fundamental soliton), for small (red) or large (blue) Raman fraction. The pulse propagates over an extended distance to include all effects. For a pulse that propagates as a fundamental soliton (under $N=1$), the normalized impulsive redshift $\tau_0\abs{\triangle\nu_s}\propto{\tau_0E_s}$ should be constant [Eq. (\ref{['eq:SSFS_im']})], as shown in the red line at short durations.
• Figure 5: Impulsive SSFS with different rotational Raman fractions. The rotational Raman fraction is tuned by introducing a gas mixture, with varying amounts of Ar and a fixed 1.2o̅f N2, into a hollow-core fiber. Measured and simulated output spectra produced by launching [number-unit-product=-]74 pulses are shown, with energy incremented by 0.33µJ across the color sequence: pink, light purple, purple, blue, green, yellow, orange, red, and brown. Frequency-resolved optical gating (FROG) measurements of the intensity profiles of filtered pulses at an injected energy of 0.67µJ (blue lines and labeled with stars) are shown in the rightmost column, for each value of the Raman fraction. In cases where multiple solitons are generated, their spectra are labeled with "1" for the first soliton and so on.