Nonlinear chirped Doppler interferometry for χ (3) spectroscopy

Four-wave mixing processes are ubiquitous in ultrafast optics and the determination of the coeﬃcients of the χ (3) tensor is thus essential. We introduce a novel time-resolved ultrafast spectroscopic method to characterize the third-order nonlinearity on the femtosecond time-scale. This approach, coined as ”nonlinear chirped Doppler interferometry”, makes use of the variation of the optical group delay of a transmitted probe under the eﬀect of an intense pump pulse in the nonlinear medium of interest. The observable is the spectral interference between the probe and a reference pulse sampled upstream. We show that the detected signal is enhanced when the pulses are chirped, and that, although interferometric, the method is immune to environmental phase ﬂuctuations and drifts. By chirping adequately the reference pulse, the transient frequency shift of the probe pulses is also detected in the time domain and the detected nonlinear signal is enhanced. Nonlinear phase shifts as low as 10 mrad, corresponding to a frequency shift of 30 GHz, i.e. 0.01% of the carrier frequency, are detected without heterodyne detection or active phase-stabilization. The diagonal and/or non-diagonal terms of reference glasses (SiO 2 ) and crystals (Al 2 O 3 , BaF 2 , CaF 2 ) are characterized. The method is ﬁnally applied to measure the soft vibration mode of KTiOAsO 4 (KTA).


INTRODUCTION
The third-order susceptibility tensor χ (3) governs both resonant and non-resonant nonlinear processes involving four-wave mixing [1]. The real part of the diagonal terms, responsible for the well-known optical Kerr-effect (OKE), plays a fundamental role in ultrafast optics where frequency-degenerated nonlinear processes are routinely used to modify, control or characterize the spatial, spectral and temporal properties of intense pulses. The nondiagonal terms of the tensor χ (3) are also exploited in nonlinear processes such as cross-polarized wave generation [2] and are ubiquitous when intense waves of different polarizations are mixed in nonlinear media. This is, in particular, the case of most optical parametric amplifiers, which have become the backbone of third-generation femtosecond sources [3]. It is thus essential to determine the coefficients of the χ (3) tensor of nonlinear media ranging from common optical glasses to exotic nonlinear birefringent crystals.
Experimental methods suitable to measure the real part of the χ (3) tensor are numerous [1] and suffice it to say that OKE spectroscopy gathers time-resolved pumpprobe techniques tracking the changes (polarization, frequency, spatial phase or temporal phase) of a weak probe pulses under the effect of an intense pump pulse in the medium of interest, either without [4][5][6][7][8] or with [9,10] interferometric detection. Another related approach, twobeam coupling (2BC), relies on the mutual interaction between two noncollinear beams crossing in the medium [11][12][13][14][15]. All these methods have their own advantages and drawbacks, but most of them require noise reduction * Aurelie.Jullien@inphyni.cnrs.fr to isolate the contribution of weak nonlinearities (averaging and/or modulation with heterodyne detection).
In this paper, we report a novel time-resolved ultrafast transient spectroscopy method to characterize thirdorder nonlinearity on the femtosecond time-scale. We coin this method as "nonlinear chirped Doppler interferometry". As explained in the next section, "Doppler" refers to the observable, "interferometry" describes the detection method and "chirped" points out the fact that adequately chirped pulses are required.
The approach consists in monitoring the variation of the optical group delay of a transmitted probe under the effect of an intense pump pulse, instead of the phase changes. We demonstrate that, under certain chirp conditions, two distinct physical effects (spectral and temporal shifts) add up and that monitoring the optical group delay, via spectral interferometry, gives access to the nonlinear phase value and therefore to the nonlinear tensor terms of the medium of interest. We show, both theoretically and experimentally, that the method is: 1. interferometric but immune to environmental phase fluctuations and drifts: no active stabilization or shielding of the interferometer is required, 2. highly sensitive: nonlinear phase-shifts as low as 10 mrad can be detected without heterodyne detection, 3. selective: self-focusing although visible on the experimental data does not affect the measurement, 4. polarization sensitive: non-diagonal terms of the χ (3) tensor can be independently measured, 5. temporally-resolved: both instantaneous and delayed nonlinear processes can be investigated.
The paper is organized as follows. The theoretical lineout of nonlinear chirped Doppler interferometry is described in section 2. The experimental setup and nonlinear materials are then described in section 3. Section 4 presents measurement results : validation of the technique with known isotropic and χ (3) -anisotropic materials, followed by soft vibration mode measurement in an anisotropic crystal (KTA).

PRINCIPLE
The setup is essentially a frequency-degenerated pump-probe experiment with an interferometric detection. As justified below, the probe optical group delay as a function of pump-probe delay is monitored, rather than the phase.
An illustration of the principle is available in Fig. 1a. A strong pump beam (P ) and a weak probe beam (P r), of identical carrier frequency ω 0 , are weakly focused and cross in a thin optical sample. A small angle ( 1 • ) is introduced between the two beams to allow a spatial separation before/after the sample and a delay line controls the relative group delay between the two pulses (τ PPr ). Through cross-phase modulation (XPM), the probe pulse undergoes a transient nonlinear phase shift ϕ NL (t). The temporal dependence of ϕ NL causes a shift of the probe's instantaneous carrier frequency, ±Ω (Fig. 1b): an downshift (shift toward the "red") on the rising edge of the pump pulse and a up-shift (shift toward the "blue") on the trailing edge of the pump pulse (for a medium characterized by a positive nonlinear index n 2 > 0) [16]. Ω is proportional to the nonlinear phase, and thus to the involved χ (3) term [16]. An interpretation of this effect relies on the fact that, because of the nonlinear index of refraction, the probe pulse propagates in a medium of increasing/decreasing optical thickness. This is as if the radiation source were moving away from or toward the observer, causing an (unrelativistic) Doppler effect on the observed probe frequency. For this reason, the frequency shift Ω will be referred to as a the nonlinear Doppler shift hereafter. In nonlinear spectroscopy, ϕ N L can be extracted from the transient variation of Ω, although with little precision [17]. We assume in the following that this frequency shift, noted Ω = Ω(τ P P r ), is small compared to the optical bandwidth ∆ω such that |Ω| << ∆ω.
The coupling between the pump and probe beams is treated elsewhere, and can be described by the following propagation equation for the probe field [14] in the limit of a purely electronic nonlinearity: (2) where I P is the time-dependent pump intensity, n g,0 the group index at ω 0 , and γ is the nonlinear coupling coefficient. This expression holds as long as the input polarization states of the pump and probe beams are either parallel or perpendicular with respect to each other. The general expression of γ is rather complex and depends on the polarization states of the pump and probe pulses as well as on the crystallographic orientation and symmetry of the sample. For an isotropic medium, far away from any resonance, the expressions of γ in SI units, for respectively parallel and perpendicular polarizations, are: The propagation equation Eq 2 assumes slowly varying envelopes and neglects temporal dispersion which is compatible with the thin medium assumption and/or narrowband pulses. The effect of the pump field is threefold: the group index is increased by 4γI P (left member of Eq.2), and two nonlinear source terms contribute to the propagation (right member of Eq.2). The first source term corresponds to XPM and is responsible for the Doppler frequency shift Ω, while the second term induces gain and loss via an energy transfer between the two beams (2BC). The interplay between these three effects is rather complex but, to simplify, the change in wave velocity can be neglected while the XPM and 2BC may significantly reshape the transmitted probe pulse in both the spectral and time domains. As a general result, the optical group delay of transmitted probe pulse τ Pr is altered when the pump and probe pulses overlap. For the sake of clarity, the probe group delays with and without the pump beam are respectively noted τ Pr and τ 0 Pr . As τ Pr − τ 0 Pr cannot be measured directly, the transmitted probe pulse is recombined with a reference pulse (in this case a replica selected upstream) and the relative group delay τ RPr = τ R − τ Pr between reference and probe is measured instead. We note τ 0 RPr . The relative delay τ 0 RPr , which is kept constant during the experiment, is chosen adequately so as to be able to resolve the phase difference between the probe and reference pulses by spectral interferometry: ∆ω 1/|τ 0 RPr | < δω sp where δω sp is the spectral resolution of the spectrometer. For plane waves, the spectral interference pattern between the transmitted probe and the reference writes: S(ω) contains one non-oscillating term (DC term) and two conjugate oscillating terms (AC terms), the Fouriertransform of which is: If E Pr (ω) = E R (ω) (ie case, without the pump wave), thenŜ AC (t − τ 0 RPr ) is equal to the Fourier-transform of |E R (ω)| 2 , and the AC terms are centered at t = ±τ 0 RPr and well separated from the DC term at t = 0. To anticipate on the following paragraphs, four-wave mixing introduces additional contributions to the optical group delay of the probe pulse and tends to shift the location of the AC terms from ±τ 0 RPr to ±τ RPr (our observable). As shown below, this definition actually aggregates two distinct physical effects.
A numerical resolution of Eq.2, described in the S.M., is proposed for a pump pulse duration of 180 fs FWHM, I P = 300 GW/cm 2 and a crystal of 1 mm length characterized by a nonlinear index γ = n 2 = 2.8 10 −16 cm 2 /W. The corresponding nonlinear XPM phase is 300 mrad.
Pump and probe pulses are equally chirped, with a chirp coefficient of either ϕ Pr + ∆ϕ (2) RPr , with either ∆ϕ (2) RPr = 0 fs 2 or ∆ϕ (2) RPr = +2000 fs 2 . The initial group delay between the probe and the reference pulses is τ 0 Pr =3 ps. As our toy model is unidirectional, spatial effects such as Kerr lens and self-diffraction are not simulated.
We first consider the case ϕ Pr = 0 fs 2 , i.e. all involved pulses are limited by Fourier transform. The spectral interferogram as a function of pump-probe delay, τ PPr = τ P − τ 0 Pr , is plotted in Fig. 1c. The Doppler transient spectral shift Ω appears for ∆Ωτ P P r 1, when the pump and probe pulses temporally overlap. For each pump-probe delay, the discrete Fourier transform of the spectral interferogram is computed and the relative group delay between the reference and probe pulses (τ RP r ) is retrieved by fitting the AC peak |Ŝ AC (t − τ 0 RPr )| with a Gaussian function. As shown on Fig. 1c, τ RPr − τ 0 RPr varies with the pump-probe delay τ PPr . When none of the three pulses are chirped, weak variations are observed, indicating that, to the first order, the optical group delay of the transmitted probe pulse is constant despite the spectral/temporal reshaping effects.
We then consider ϕ Pr = +5000 fs 2 , i.e. the three pulses are equally chirped. Because of this chirp, the instantaneous frequencies of the pump and Wigner-Ville distribution of the probe-reference electric fields for five different pump-probe delays τPPr. For each sub-plot, the reference pulse is on the left, and the (delayed) probe pulse is on the right. Both pulses are chirped and exhibit a time-dependent instantaneous frequency. The two pulses being chirped differently, the slopes of the individual representations differ. In between the two pulses appears the interferometric component. Upper plot: integration of the distribution along the frequency coordinate (temporal intensity, solid line) compared to the cross-section of the pump pulse (dotted line). Right plot: integration of the distribution along the time coordinate (interference spectrum S(ω)). The large arrow indicates the reference-probe group delay τRPr. For τPPr = ±450 fs (i.e. no temporal overlap between pump and probe) τRPr = τ 0 RPr . For τPPr 0, the nonlinear phase induces a red shift of the transmitted probe (red arrow), which increases τRPr. Symmetrically, for τPPr 0, a blue shift decreases τRPr. For τPPr = 0, there is no spectral shift and τRPr = τ 0 RPr . The nonlinear phase and chirp values were increased compared to Fig.1 for the sake of the illustration.
probe pulses are detuned with respect to each other when τ PPr = 0 and energy flows from one wave to the other during the nonlinear interaction (2BC becomes dominant). If the chirp coefficient is positive, the probe will gain energy for negative pump-probe delays and vice versa. As this energy transfer also scales with the pump intensity, the general effect is a reshaping of the temporal profile which is indistinguishable from an additional optical group delay (Fig. 1d). For negative pump-probe delays (resp. positive), the rear edge (resp. the leading edge) of the probe is strengthened, resulting in a overall increase (resp. decrease) of the group delay. As a result τ RPr exhibits a Z-shape, similar the transient probe transmission reported for two-beam coupling [11][12][13]15].
We now consider the case of unchirped pump and probe pulses ϕ Pr = 0 fs 2 with a positively chirped reference pulse ϕ (2) R = +2000 fs 2 (Fig. 1e). The reference pulse being chirped, the Doppler spectral shift Ω is temporally encoded in the interferogram and (also) appears as a delay τ RPr . As plotted in Fig. 1e, this effect produces a similar Z-shape behaviour, although less pronounced than in the former case -but scaling linearly with ∆ϕ Pr . To illustrate the principle of spectral encoding, we represent in Fig. 2 the Wigner-Ville distributions of the transmitted probe and of the delayed and chirped reference pulse. As developed in the supplementary material, the linear relationship between the nonlinear Doppler shift (Ω) and the relative chirp between probe and reference (∆ϕ (2) RPr ) can be retrieved analytically from Eq.6.
We have evidenced here two phenomena: temporal reshaping and temporal encoding of the nonlinear spectral shift. They originate from different mechanisms but are the two sides of the same coin. Although the differences between the spectrograms in Fig. 1 are not visible to the naked eye, the Fourier analysis shows that both mechanisms result in similar transient shifts of τ RPr , over the same temporal scale (the correlation width of the pump pulse), and with similar amplitudes (a few fs). With the right chirp parameters, these two contributions may add up and increase significantly the global signal-to-noise ratio of the measurement, as shown in Fig. 1f. The delay swing δτ = max(τ RPr ) − min(τ RPr ) is then evolving linearly with the nonlinear phase amount, as plotted in Fig. 1g. As will be demonstrated experimentally in the next section, measuring τ RPr instead of phase changes not only makes the detection insensitive to phase fluctuations but also gives additional means to enhance the sensitivity and specificity (Fig. 1g) of the detection, without resorting to heterodyne detection. In the present experiments, a Pharos laser system (PH1-SP-1mJ, Light Conversion) delivers 180 fs FTL pulses, with a central wavelength of 1034 nm, a repetition rate of 10 kHz, and pulse energy up to 500 µJ. The laser chirp can be tuned by adjusting the compressor. Each pulse is split into three separated pulses: an excitation pump pulse, a probe pulse ( 20% of the pump energy), and a reference pulse selected before the nonlinear stage (Fig. 3). The pump and probe pulses are focused (f = 1.5 m) and overlap in the focal plane under a small angle with a pump beam size of about 550 µm. The pump-probe delay (τ PPr ) is controlled with a delay stage equipped by a motorized actuator with K-Cube controller (models Z825B and KDC101, Thorlabs). After the interaction, the probe is selected and recombined with the reference pulse. The nominal group delay between the reference and the probe pulses (τ 0 RPr ) is set to 3.5±0.15 ps for all measurements. The two pulses, probe and reference, are respectively chirped through the addition of various bulk plates: SF11 (20 mm, 2500 fs 2 ), CaCO 3 (10 mm, 430 fs 2 ) [18] and Al 2 O 3 (5 mm, 156 fs 2 ) [19]. The resulting interference pattern is collected by a spectrometer (Avantes, spectral resolution 0.07 nm) for each step of the optical delay stage in the pump arm. The polarization and energy of each pulse are controlled by half-wave plates and thin-films polarizers (TFP). The different components of the χ (3) tensor can then be measured by changing the polarization state of the three pulses.

EXPERIMENTAL METHODS
The detailed procedure for data acquisition and analysis can be found in the S.M. To summarize, for each acquisition scan, τ PPr is scanned (single scan) from -2.6 ps to 2.6 ps with temporal steps of 13 fs (400 spectra per scan). The integration time of the spectrometer is 1 ms (total acquisition time ¡1 mn, limited by the delay stage). The probe-reference delay and relative chirp are measured before each acquisition, out of pump-probe temporal overlap. τ RPr is computed by fitting the modulus of the AC peak with a Gaussian function after discrete Fourier transform. The relative phase between the probe and reference pulses was also retrieved by Fourier filtering, so as to compare our analysis with usual nonlinear phase measurements.

RESULTS
The method is validated in three steps. We first compare the toy model with experimental data and measure the swing amplitude of τ RPr −τ 0 RPr with respect to the relative chirps and nonlinear phase. We then measure the diagonal third-order tensor coefficient of a set of wellknown isotropic samples. Last, we vary the polarization states of the pump and probe beams, characterize the nonlinear anisotropy of barium fluoride and compare the measured coefficients with the reported values. Finally, we investigate the instantaneous and delayed nonlinear properties of potassium titanyl arsenate crystal (KTA).
Comparison with the toy model All polarization directions are first set to horizontal. The pump energy is 40 µJ, which corresponds to a pump peak intensity estimated to 300 GW/cm 2 on the sample. The measured chirp coefficients are ϕ (2) P = ϕ (2) Pr = (+5000±500) fs 2 and ∆ϕ (2) RPr = (+2000±100) fs 2 . Fig. 4 gathers typical experimental results acquired for a 1 mm c-cut sapphire crystal. As expected (Fig. 4a), in the vicinity of τ PPr = 0, a red-shift is observed for τ PPr < 0 while a blue-shift is observed for τ PPr > 0. Self-focusing is visible on the experimental data as a transient signal decrease but does not affect the measurement. A close-up of the Fourier transform near +3.5 ps is shown in Fig. 4b. The transient shift of the peak position (τ RPr ) is clearly distinguishable. The latter, plotted in Fig. 4d, shows the characteristic zshape, in excellent agreement with Fig. 1. The delay swing δτ = max(τ RPr ) − min(τ RPr ) = 21 fs is obtained for τ PPr = ±100 fs with a linear dependence of the signal between these two extrema (slope of -0.1). For the sake of comparison, Fig. 4c represents the relative phase between the reference and the transmitted probe pulses extracted from the spectrogram by Fourier filtering. This metric is the quantity usually measured to determine the value of the non-linear phase. However, as can be seen on Fig. 4c, ϕ NL can hardly be distinguished from the phase noise (fluctuations and drifts) added by the interferometer. The comparison between figures 4c,d helps to appreciate the improvement in signal-to-noise ratio of our method. Monitoring the group delay rather than the spectral phase is indeed insensitive to phase fluctuations and makes it possible to measure a nonlinear signal at least an order of magnitude times weaker.
Starting from this working configuration, we then characterize δτ as a function of (i) input chirp (controlled by the grating compressor), (ii) ∆ϕ (2) RPr (controlled by adding/removing bulk in the reference beam path) and, (iii) pump energy -all other experimental parameters being kept constant. The acquired raw data are shown in Fig. 5. As explained in section 2, the input chirp mainly triggers 2BC, meanwhile ∆ϕ (2) RPr tunes temporal encoding of the Doppler shift. The toy model reproduces well the variations of τ RPr and Ω (normalized with respect to ω 0 ) with the input chirp (Fig. 5a). The Doppler shift follows the parabolic evolution of pump intensity, with a maximum close to pulse compression. Conversely, δτ varies mostly linearly (V-shape) around a minimum, staggered from the compression. We can distinguish several trends, as labelled in Fig. 5a. For moderate positive input chirp (1), the Doppler shift remains mostly constant (i.e. so is the nonlinear phase) while δτ exhibits a linear increase with the chirp value, emphasizing how the signal to noise ratio is enhanced by 2BC. This area then defines the efficient working conditions of our method. For negative input chirp (2), similar trend is observed, but with a reversed Z-shape, as shown in the raw data in Fig. 5b. Reversing the sign of the chirp is in fact equivalent to modify the sign of the energy couplings between the pump and the probe and thus to reverse the signal. The minimum of δτ V-shape (3) corresponds to a weak signal with ill-defined shape (Fig. 5b). The large offset from pulse compression is both predicted by the model and observed. It matches the reference-probe relative chirp, -2000 fs 2 , and results from the compensation of the temporal and spectral contributions detailed before. The numerical fit agrees quite well with the experimental data, except for the large chirps (4). This discrepancy is attributed to higher orders of spectral phase (neglected in the model).
The increase of δτ with ∆ϕ (2) RPr is shown in Fig. 5c: the detected signal varies linearly with the relative chirp value, even if the pump energy remains the same. This additional degree of freedom, easy to implement experimentally, is be particularly useful to increase δτ in order to detect a weak amount of nonlinear phase. Finally, we characterize how δτ and Ω scale with the pump intensity by scanning the pump energy from 5 µJ and 80 µJ. Fig. 5d evidences the linear dependence of both δτ and Ω with the pump energy up 50 µJ. This measurement also exemplifies that Doppler shifts spanning over more than an order of magnitude (from 30 GHz to 400 GHz, i.e. from 0.01% to 0.2% of the carrier frequency) can be detected in a single scan.
To summarize this sub-section: (i) experimental data are found in very good agreement with the toy model, (ii) the expected trends of the signal dependence on involved chirp parameters have been recovered, (iii) suitable experimental chirps and pump energy ranges so as (0.5 ± 0.02) (0.6 ± 0.1) [20]  to monitor nonlinear phase changes have been identified.

Application to isotropic and anisotropic crystals
Although the method can in principle provide the absolute value of the nonlinear index, experimental sources of error are numerous (actual peak intensity, uncertainties on chirps...) and, in this work, we characterize δτ relatively to a reference sample, hereafter fused silica. The experimental parameters are those shown in Fig. 4. Table I gathers the measured χ (3) ratio between the samples to characterize (Al 2 O 3 , BaF 2 , CaF 2 ) and fused silica (χ Then, BaF 2 with holographic orientation was characterized. The ratio between δτ obtained for two distinct polarization configuration leads to a determination of the ratio between SPM and XPM χ (3) terms, and thus to the nonlinear anisotropy of χ (3) (σ, eq. 7) [23][24][25]. Our results are summarized in Table II, and once again, a good agreement between the values found in the literature is obtained.
Finally, we study the nonlinear properties of a Potassium Titanyl Arsenate (KTA) crystal along θ=47 • and φ=0 • (X-Z plane, thickness of 2 mm). The pump energy is reduced to 15 µJ, so as to keep the nonlinear phase within the linearity range indicated in Fig. 5d. The polarization configurations are successively set to (i) and (ii), to measure the nonlinear indices of the fast and slow axes. Our results are gathered in Tab. III and found slightly lower than previous measurements reported using the Z-scan technique [27].
Although outside the range of validity of the model developed above, a proof-of-principle measurement show that delayed linear phenomena can also be investigated with our method. The temporal scanning range is then increased to +5 ps, so as to evidence the delayed nonlinear answer of KTA (fast axis). The results are shown in Fig. 6. For this measurement, the pump energy is increased to 75 µJ to increase the signal-to-noise ratio at positive pump-probe delays. The group delay swing shows an asymmetry, followed by pseudo-periodic oscillations. This measurement evidences two main virbation modes, with respective wave numbers of 66 cm −1 and 75 cm −1 , matching A1 vibrational frequencies of KTA, as reported in [28]. In case of a delayed nonlinear response, the signal seems mainly originating from temporal transcription of the phonon-induced spectral shift, as shown in Fig. 6d, where temporal and spectral oscillations perfectly match. This final measurement illustrates the potential application our method to low-frequency Raman spectroscopy [29].

CONCLUSION
To conclude, we have introduced a jitter-free timeresolved spectroscopic method to characterize the thirdorder nonlinearity on the femtosecond time-scale. This approach, coined as "nonlinear chirped Doppler interferometry", consists in monitoring the variation of the optical group delay of a transmitted probe under the effect of a strong pump pulse, rather than the phase, via spectral interferometry between the probe and a reference pulse sampled upstream. We have shown that the optical group delay transient changes originate from two different mechanism, coherently added up : (i) 2BC triggers energy exchanges between chirped pump and probe pulses and induces temporal reshaping; (ii) the XPM frequency-shift undergone by the probe is encoded in time by adequately chirping the reference pulse. Thanks to a good agreement between experimental data and a 1D numerical model, we have been able to define experimental working area of our method, in terms of chirp of involved pulses and nonlinear phase amount.
Thus, we have demonstrated that monitoring this quantity instead of phase changes makes the detection insensitive to phase fluctuations and provides additional means to enhance the sensitivity and specificity of the detection, without resorting to heterodyne detection. Nonlinear phase shifts as low as 10 mrad, corresponding to a frequency shift of 30 GHz, i.e. 0.01% of the carrier frequency, can thus be detected. The method is suited to perform non-resonant χ (3) spectroscopy in isotropic or anisotropic nonlinear media, but also to survey resonant and delayed nonlinear processes. This original approach could be easily exported to the detection of spectral shifts driven by other linear or nonlinear processes, among them fluorescence, and Raman spectroscopy.
Finally, the sensitivity reported here could be pushed further by multiple scans acquisition, higher averaging and extending the involved spectral bandwidths.