# Raman spectroscopy based on spectral covariance in stochastic light pulses

Nonlinear optics aims at revealing signals originating from the interplay between several spectral components in a broadband pulse driving a response from a sample. As such interactions involve small scattering cross sections, the output signal is typically very weak. It is common wisdom that to reveal such features one must use a source delivering very stable pulses and acquire a large number of repeated measurements to increase the signal to noise ratio.

In this work we flip this paradigm taking advantage of the unique random features of each pulse, whose spectral components are uncorrelated. For each realization the signal originating from the sample introduces a coupling that translates in the form of a statistical correlation and can be quantified by the covariance. In this perspective, noise is no longer a liability to be mitigated but can be exploited as a powerful asset to retrieve the sample nonlinear response. We enhance the spectrally narrow pulse-to-pulse fluctuations and show that properly chosen analytical tools can access information that is missed by standard average value based experiments.

We demonstrate the feasibility of our approach by applying it to stimulated Raman scattering in α-quartz. As a proof of principle we employ the Pearson coefficient ρ = [<I(ω** _{i}**) I(ω

**)>-<I(ω**

_{j}**)>< I(ω**

_{i}**)>]/(σ**

_{j}**σ**

_{i}**), calculated on the measured single shot spectrum I(ω).**

_{j}Figure 1 shows ρ for a measurement employing pulses with spectrally uncorrelated fluctuations applied along the whole spectral bandwidth. ρ in Figure 1a is calculated on a set of reference pulses, revealing that every component in the bandwidth is solely correlated with itself. ρ in Figure 1b is calculated on the spectra transmitted by quartz. We recognize features offset from the diagonal. The frequency distance between correlated components matches the Raman shift of a spontaneous Raman scattering spectrum (panel c).

**Figure 1**. Pearson correlation coefficient for the frequencies within the pulse spectrum before (a) and after (b) the quartz sample. (c) Spontaneous Raman spectrum of quartz (adapted from Rundquist A. et al., J. Mod. Opt. 52:2501–2510 (2005)).

We explored several variations in how the noise is introduced. We applied the noise to the whole bandwidth, as in the measurement of Figures 1 and 2a, then to half of the spectrum, Figure 2b, then to half of the spectrum reducing the intensity of the other half to zero. Typical spectra are shown in the insets of Figure 2, with the corresponding rmaps (rotated to express the bottom axis as frequency distance from the diagonal).

The Raman signatures lie between regions with shifting background and are more evident in the block with a noisy and a non-noisy spectral component (regions in dashed line in Figure 2b-c). These off diagonal features have a dispersive lineshape when the full spectrum is present while are purely positive when half of the spectrum is zero, confirming the sensitivity of the technique to the total field phase.

**Figure 2**. Rotated Pearson correlation coefficient for three sets of measurements: modulation (a) along the whole spectrum, (b) along half of the spectrum, (c) same as (b) but the coherent part has zero intensity. The inset show a few spectra with the corresponding modulation type.

Our results indicate that the combined use of stochastic light pulses and a covariance-based technique provides a very powerful platform to measure a nonlinear response. This methodology will be applied in wavelength ranges where stable, transform-limited pulses are not available, such as X-ray free-electron lasers which suffer from intrinsically strongly noisy spectra. While here we apply a covariance based method to study phonons we stress that a similar approach can be used to reveal low-energy modes of electronic and magnetic origin and adapted to different techniques.

In summary in this work we establish femtosecond covariance spectroscopy as a technique that uses ultrashort stochastic light pulses to measure nonlinear response of a material. By using pulses with spectrally uncorrelated fluctuations we can leverage on the noise (rather than fighting against it!) and consider each repetition of the experiment as a measurement under different conditions. In this limit we demonstrate that nonlinear processes in the sample can be retrieved by measuring the spectral correlations in different pulses. We validate the approach by studying stimulated Raman scattering in α-quartz.

**This research was conducted by the following research team:**

J. Tollerud ^{1,2}, G. Sparapassia ^{2}, A. Montanaro ^{1,2}, S. Asban^{3}, F. Glerean^{1,2}, F. Giustia ^{1,2}, A. Marciniak^{ 1,2}, G. Kourousias^{2}, F. Billè^{2}, F. Cilento^{2}, S. Mukamel^{3}, D. Fausti^{1,2,4}

^{1} Physics Department, University of Trieste, Trieste, I-34127

^{2} Elettra-Sincrotrone Trieste S.C.p.A., Trieste, I-34149

^{3} Chemistry Department, University of California, Irvine, CA 92617

^{4} Department of Chemistry, University of Princeton, Princeton, NJ 08544

**Contact persons:**

### Reference

Jonathan Owen Tollerud, Giorgia Sparapassi, Angela Montanaro, Shahaf Asban, Filippo Glerean, Francesca Giusti, Alexandre Marciniak, George Kourousias, Fulvio Billè, Federico Cilento, Shaul Mukamel, Daniele Fausti |