1. Introduction

Ultrafast spectroscopy has proven to be an extremely powerful tool for the investigations of femtosecond timescale phenomena. It provides a femtosecond resolution and allow the probing of samples on large optical bandwidths that contain a wealth of information about atomic and molecular structures. The spectrally-resolved pump-probe technique combines these two complementary aspects and allows to resolve ultrafast dynamics while probing the sample on a large bandwidth. It has been used to study an extremely large variety of phenomena, ranging from ultrafast processes in molecules or semiconductors to chemical and biological reactions.

However, this technique gives rise to artifacts. The coherent artifact denomination mostly refers to a pump polarization coupling term appearing when pump and probe overlap[1]. The perturbed polarization artifact, on which we focus in this article, might more dramatically affect the time-dependent spectrum on a range of delays much longer than the pump and probe durations. It arises if the dynamics of the investigated optical transition is faster than their dephasing time[2, 3, 4]. Its main feature is the counter-intuitive appearance, at negative pump-probe delays, of spectral and temporal oscillations in the time-dependent spectrum. Such effects, reported in both the visible[5, 6] and the infrared range[7, 8], have a well-understood physical basis. They typically occur due to a fast decrease — triggered by the pump arrival — of the probe-induced free-induction decay, hence their name and their appearance at negative delays. In the frequency domain, this sharp decrease translates into a broad spectrum that smears the measured signal. Other fast perturbations of the polarization decay such as frequency shifts[5, 9], changes of oscillator strength or dephasing time lead to effects of the same kind[10]. In the spectrally-resolved pump-probe technique, the artifacts appear as a spectral interference between the probe and the field emitted later on during the perturbed free-induction decay. Consequently, they are inherently coherent and ill-localized in time. Yet, they are not a defect of this particular technique since other time-dependent spectra show similar artifacts[9]. Generally, artifacts are a consequence of the violation of a Fourier-transform limit[4].

In practice, the presence of artifacts prevents a straightforward analysis of the data. This is especially true around zero-delay where the artifact is spectrally extremely large. Interestingly, one can circumvent the artifacts by using spectrally-integrated — but still time-resolved — experiments carried out with Fourier-transform limited probe pulses of different central frequencies[11]. The joint time-frequency resolution of such experiments then respect the Fourier-transform limit. The response R(ω ₁,ω ₂,ω ₃) measured in multidimensional spectroscopy[12] is another way round the artifacts. It possesses all the information that can be derived from a time-dependent spectrum and does not possess artifacts. Nevertheless, keeping a time-dependent spectral picture remains attractive because of its very intuitive character. This motivates the development of schemes that remove the artifacts from time-dependent spectra.

Although, using a suitable model, experimental data with artifacts can be fitted to analyze the time-dependent spectra, a model-independent method that would directly extract a clear information from a time-dependent spectrum is lacking. The only attempt in that direction concerns the case of a time-dependent frequency[5]. In this article, we provide a general analysis.

In order to eliminate the perturbed polarization effects, a general clear-cut definition of the artifact is required. Exploiting the Fourier--transform $\widehat{\alpha }$ (Δω,Δt) of the time-dependent spectrum α(t,ω), we will show that coherent artifacts are characterized in the Fourier-conjugate domain as components of α̂(Δω,Δt) verifying |ΔωΔt|≥4. This definition directly follows from the fact that spectrally-integrated experiments do not give rise to artifacts. Eliminating these components leads to the suppression of artifacts. We will demonstrate this filtering technique in the case of an instantaneous frequency shift. We also show that the same technique, in a general manner, applies to noise removal in time-dependent spectra.

2. A resolution-dependent approach to perturbed polarization artifacts removal

2.1. Spectrally resolved experiments

The time-dependent spectrum directly derives from the medium non-stationary linear response function 𝓡(t,t ^′) which determines the polarization 𝒫(t) induced by a weak probe field 𝓔(t) :

This description of the medium response applies to pump-probe experiments. It highlights the probe perspective : the pump-field dependence that gives rise to a non-stationary response is implicit in 𝓡(t; t ^′). Spectrally-resolved experiments measure Im[𝓔⋆(ω)𝒫(ω)] as a function of the pump-probe delay[13]. For an infinitely brief probe (𝓔(t)∝δ(t-τ)), the time-dependent absorption spectrum is Im[α(τ-ω)] with

A typical appearance of perturbed polarization artifacts in the Im[α(τ,ω)] spectrum is shown in Fig. 1(a) in the case of an instantaneous frequency shift. Such shifts for instance appear in the ultrafast vibrational spectroscopy of ligand transfers in proteins[11].

2.2. Spectrally integrated experiments

We now consider spectrally-integrated experiments that do not lead to artifacts and take 𝓔(t) to be a finite-bandwidth Fourier-transform limited pulse. E leads to an integrated absorption signal S=Im[∫dω𝓔⋆(ω)𝒫(ω)]. Varying the delay τ and central frequency Ω of the probe leads to a time-dependent spectrum S(τ,Ω) linked to a(τ,ω) :

S̃(τ,ω) appears as a convolution of a(t,ω) with 𝓔(t,ω)=𝓔(t)𝓔⋆(ω)exp(iωt) — known as the Kirkwood distribution[14] — which represents the probe field. Consequently, the oscillating artifacts appearing in α(α,ω) are smoothed in S(τ,Ω), as shown in Fig. 1. An essential difference between these two spectra is their time-frequency resolution. The idea that resolution issues play an important role to define well-behaved time-dependent spectra has been previously highlighted[15]. In S(τ,Ω), the time and frequency resolution are set by the duration and the spectral bandwidth of the probe pulse so that its joint time-frequency resolution respects the Fourier-transform limit. In contrast, the definition of α(τ,Ω) is does not rely on a particular time and frequency resolution. Although in practice, the time and frequency resolution of the measured α(τ,ω) are limited, no condition binds them, so that the Fourier-transform limit can be violated. As a basis for the following analysis, we consider S(τ,Ω) as reference spectra because they are artifact-free and possess a transparent physical interpretation.

Although spectrally integrated measurements do not bear artifacts, their experimental implementation might be impractical : one needs as many acquisitions as resolutions; it requires not only to scan the pump-probe delay but also the probe frequency, which is inconvenient, especially if one could otherwise benefit from a multiplex detection. Since α(t,ω) theoretically contains all the information needed to derive any of the S (τ,Ω) spectra, a numerical retrieval of the S(τ,Ω) spectra from a single Im[α(t,ω)] measure is worth considering. This requires retrieving α(t,Ω) from its imaginary part, as detailed in 3.1. Figure 1 and the attached movie show retrieved S(τ,Ω) spectra in the case of an instantaneous frequency shift.

 

Fig. 1. (a) Simulation of a Im[α(t,ω)] time-dependent spectrum in the case of a transition (dephasing time 1/Γ=1ps) going through an instantaneous frequency change of Δν=6THz at time t=0. Artefacts appear at negative pump-probe delays. Around the initial frequency (set to 0), the spectrum shows spectral oscillations and a slow decrease of the absorption peak (time-constant 1/Γ) typical of the sharp extinction of the polarization decay. Around the final frequency (6THz), the spectrum shows oscillations along the delay axis at frequency Δν. They follow from coherence transfer[9]. (b) The S(τ,Ω) spectrum, derived from the Im[α(τ,ω)] spectrum of Fig. (a), is here shown for a probe duration of 400fs (FWHM). The attached movie (file size 2.2 KB) shows the similarly derived S(τ,Ω) spectra for probe durations ranging from 25 fs to 2 ps: independently of the duration, the artifacts vanish. For short probe durations, S(τ,Ω) gives access to fast evolving features at the expense of frequency resolution whereas for probe durations on the order of the dephasing time 1/Γ=1ps, the transition line is fully resolved. [Media 1]

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3. Fourier-filtering technique : a resolution-independent approach

3.1. Shift-space analysis

Significant insight into perturbed polarization effects is gained when considering the time-dependent spectrum in its bidimensional Fourier-domain. We refer to this domain as the shift-space and note (Δω,Δt) the conjugate variables of (t,ω). Shift-space quantities are obtained from their time-frequency counterparts through the following Fourier-transform :

α̂(Δω,Δt) quantifies how much the field emitted by the medium in response to an excitation is shifted, both in frequency and time, from the exciting field : Δω and Δt are spectral and time shifts. For instance, in a stationary case where the emission of new frequencies is forbidden, α̂(Δω,Δt)=2πδ (Δω)𝓡 _stat(Δt), 𝓡 _stat being the time-domain stationary response.

We now expose how S(τ,Ω) follows from Im[α(t,ω)]. Causality requires, for Δt<0, α̂(Δω,Δt)=0. This allows to obtain α̂(Δω,Δt) from the Fourier-transform of the absorption spectrum since $\left[\widehat{\mathrm{Im}\left(\alpha \right)}\right]$(Δω,Δt) coincides with α̂(Δω,Δt)/2i on Δt≥0. In shift-space, the convolution equation 3 reads $\hat{\tilde{S}}(\Delta \omega ,\Delta t)=\widehat{𝓔}(\Delta \omega ,\Delta t)\hat{\alpha }(\Delta \omega ,\Delta t)$ so that, knowing 𝓔̂(Δω,Δt) for Fourier-transform limited pulses, S(τ,Ω) can be retrieved by a inverse Fourier-transform.

A simple analysis allows to identify components of the response that do not contribute to any S(τ,ω) spectra. Since, for 𝓔 being any Fourier-transform limited pulse, almost the entire part of |𝓔̂(Δω,Δt)| is enclosed within the |ΔωΔt|≤4 domain (see Fig. 2), none of the components of α̂(Δω,Δt) verifying |ΔωΔt|≥4 significatively intervene in any of the Ŝ(Δω,Δt) spectra. We identify these components as being the origin of the perturbed polarization artifacts in α(t,ω). While α̂(Δω,Δt) retains all the information about the non-stationary response, the Ŝ(Δω,Δt) spectra are not sensitive to some physical processes : these are coherent emissions shifted too far away from the exciting Fourier-transform limited pulses to lead to their absorption.

3.2. Artefact removal using a cross-shaped filter

Filtering out the large Fourier-product components of α̂(Δω,Δt) leads to a spectrum free of artifact. We use the filter f(Δω,Δt)=exp(-|ΔωΔt|/2)exp(iΔωΔt/2) to reject the undesired |ΔωΔt|≥4 components beyond the 1/e ² limit of |f| (see Fig. 2) and back-Fourier-transform f(Δω,Δt)α̂(Δω,Δt) into the time-frequency domain. $\widehat{\alpha }$ derives from $\widehat{\mathrm{Im}\left(\alpha \right)}$ as shown in 3.1.

 

Fig. 2. Shift-space filter |f(Δω,Δt)|=exp(-|ΔωΔt|/2). f equals one along the axes (red) and zero for large ΔωΔt Fourier-product components (blue). Essentially, f rejects the components of α̂(Δω;Δt) verifying |ΔωΔt|≥4. The |ΔωΔt|=4 hyperbolas (white) mark its 1/e² limit. In shift-space, Fourier-transform limited pulses parametrized by their pulse duration Δt ₀ (field rms), have a Gaussian shape : 𝓔̂_Δt0(Δω;Δt)=exp(iΔωΔt/2)exp(-(Δω ²Δ$t_0^2$+Δt ²/Δ$t_0^2$)/4). The ellipses (black) delineating their 1/e ² limit all fall within the |ΔωΔt|≤4 domain; the legend indicates their durations (FHWM).

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A filtering procedure example is shown in Fig. 3. Like S(τ,Ω), the filtered spectrum is free of artifacts. Like α(t,ω), it is independent of a given time or frequency resolution. Importantly, because f(Δω,Δt=0)=1, the filtered spectrum preserves the oscillator strength value. To reproduce the behavior of the reference spectra $\hat{\tilde{S}}=\widehat{𝓔}\hat{\alpha }$, the filter includes the phase factor arising in 𝑑̂ (see caption of Fig. 2). This allows to avoid small but unjustified negative values in the filtered spectrum of our example. Like S(τ,Ω), the filtered spectrum forbids spectral lines to move faster than their inverse linewidth. This is why components at the final (initial) frequency remain at negative (resp. positive) time-delays in Fig. 3(b).

 

Fig. 3. Shift-space filtering procedure. The procedure is demonstrated on the frequency shift case introduced in Fig. 1. Using the filter shown in Fig. 2, the large Fourier-product components of α̂(Δω,Δt) are removed. The artifacts that appear in Im[α(t,ω)] at negative delays (a) are fully eliminated in the filtered spectrum (b). The same procedure applies to noise rejection. In (c), 30% noise is added to the data of (a). The filtered spectrum (d) obtained from the data of (c) is almost the same than the noise-free spectrum shown in (b).

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As shown in Fig. 3(c–d), the filtering procedure is also highly effective for noise removal. This efficiency comes from the fact that the retained artifact-free signal is highly localized along the axes of the shift-space. On the contrary, a white noise fills the entire Fourier-space and is considerably reduced by filtering. The constant background difference between Fig. 3(b) and 3(d) is due to the unfiltered low frequency components of the noise.

However, as it stands, the procedure might be invalidated by the use of non-Fourier transform limited probes. In particular, the resulting spectrum is distorted when the large Fourier-product components (|ΔωΔt|>4) of such pulses match the ones of α̂(Δω,Δt).

4. Conclusion

Using the Fourier-conjugate domain of the time-dependent spectrum, we identified components responsible for the perturbed polarization artifact. These components reflect correlations between the excitation and the emitted field that do not lead to the absorption of any Fourier-transform limited pulse. Removing these components by a simple filtering technique yields a time-dependent spectrum free of perturbed polarization artifacts. This analysis, demonstrated in the case of an instantaneous frequency shift, is general and model-independent. Furthermore, it does not require to set a given time or frequency resolution. The shift-space analysis we presented not only applies to the spectrally-resolved pump-probe technique but also extends to other techniques measuring time-dependent spectra.

I sincerely thank A. Alexandrou and M. Joffre for their help and support, stimulating exchanges and their critical reading of the manuscript.