1. Introduction

Coherent anti-Stokes Raman Scattering (CARS) is one of the most commonly used nonlinear-optical spectroscopic methods [1, 2]. Recent advances in lasers, optical materials and instrumentation have enabled the development of many powerful new CARS-based techniques including three-dimensional microscopy [3, 4], coherence-controlled CARS [5] and femtosecond time resolved and frequency resolved spectroscopy [6, 7]. In CARS, the two laser pulses, E_p (pump) and E_s (Stokes), induce the vibrational coherence A at frequency Ω:

where Ω_n, Γ_n are the frequency and relaxation rate of vibrational level n; C_n, C_NR are the amplitudes of the resonant (n-th level) and non-resonant non-linear susceptibility, respectively. The third probe pulse E_pr scatters off the induced coherence producing the anti-Stokes polarization p_as ⁽³⁾ with the frequency response given by

Experimental detection of |E_as(ω)|² ∝ |P_as ⁽³⁾(ω)|² or |E_as(t)|² ∝ |P_as ⁽³⁾(t)|² reveals the amplitude square of the spectral or temporal vibrational response of the medium, respectively. However, the phase of a complex function A is lost due to the lack of the phase of E_as. Knowledge of the phase is critical for full temporal and spectral reconstruction of the vibrational response. Recently, homodyne and heterodyne interferometric detection schemes have been employed for the detection of both the real and imaginary parts of the Raman response P_as ⁽³⁾(ω) [8,9]. Experimental noise, however, complicates the procedure of deconvolving A(Ω) from Eq. (2). In a recent paper [10] we have proposed and demonstrated an alternative method of re-constructing the complex response A, based on the technique of cross-correlation frequency resolved optical gating (XFROG). After introducing a time delay τ between the pump-Stokes pulse pair and the probe pulse, we applied an iterative XFROG algorithm to the experimentally measured two-dimensional CARS spectrogram, I_as(ω,τ) = |E_as(ω,τ)|². Since the XFROG method utilizes the full bandwidth of a femtosecond probe pulse, it offers higher signal levels and better stability against noise due to the redundancy of the measured data [10].

2. Theoretical analysis

Our numerical analysis shows that the XFROG algorithm becomes less susceptible to noise with the increased number of interference fringes in the input spectrogram. In this letter we present an experimental evidence of this result, and suggest an instructive and intuitive approach to optimizing the XFROG method in CARS. In XFROG algorithm the error of the retrieved result after k-th iteration is calculated as the absolute difference between experimental and calculated two-dimensional spectrograms [11]:

where N is the total number of grid points along the frequency axis (index i) and time axis (index j), and μĨ_as ^(k)(ω_i,τ_j) is the calculated two-dimensional spectrogram after k-th iteration multiplied by the normalization constant.

 

Fig. 1. (a). - Deviation of the retrieved spectra from the model Raman spectra as a function of the noise level in the two-dimensional spectrogram for various spectral shapes of the probe pulse: transform limited pulse (), π-step shaping (), sequence of two transform-limited pulses with decreasing amplitude (), sequence of two transform-limited pulses with increasing amplitude (), π-step shaping of the two-pulse sequence with increasing amplitude (). Insets to (a): illustrative example of the CARS spectra shifted in respect to each other (black and red lines) after being obtained with a broadband transform-limited probe pulse (I) and a broadband shaped probe pulse (II). (b) - model Raman spectrum with three Lorentzian lines showing the amplitude (black line) and phase (blue line) of the vibrational response A(Ω). (c) - retrieved result for the π-step shaping of the two-pulse probe sequence with increasing amplitude at 8% noise level, (d) - same result for a transform-limited pulse.

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In the impulsive CARS spectroscopy, i.e. CARS with ultra-fast pump-Stokes excitation, the spectrum of the experimentally detected anti-Stokes light reflects the spectrum of the corresponding probe pulse. This can be seen from the following analysis. The anti-Stokes field E(t), generated by the Raman scattering of the probe field E₀(t) off the vibrational mode of frequency Ω, decay rate γ, and strength C, can be expressed as a real part of

According to the Wiener-Khintchine theorem, the power spectrum of this field is equal to the Fourier transform of the field correlation function

where G₀(τ) is the correlation function of the probe field multiplied by the decay of vibrational coherence, E ₀(t)e ^-γt. Hence, the measured anti-Stokes spectrum is given by

where I ₀ is the spectrum of the modified probe, E ₀(t)e^-γt. Expression (6) shows that an unshaped probe pulse with a smooth Gaussian spectrum results in a similarly smooth anti-Stokes spectrum. The discrepancy between the detected and retrieved spectra in the XFROG algorithm is therefore small, as schematically shown in inset (I) of Fig. 1(a). If, on the other hand, the probe spectrum I ₀ exhibits sharp features, such as interference fringes, the anti-Stokes spectrum I(ω) becomes correspondingly structured. Now even a small mismatch between the measured CARS signal and the XFROG retrieval produces high error G ^(k) and more accurate output of the XFROG algorithm (inset (II) of Fig. 1(a).

An efficient way of introducing interference fringes into the probe spectrum (and therefore, into the CARS spectrogram) involves splitting the probe pulse into a train of pulses by means of the spectral pulse shaping [12]. The corresponding anti-Stokes pulses generated by the probe train interfere with each other as long as the coherence is preserved in the medium due to the presence of Raman resonances. Here we consider two different pulse shaping methods which result in the interference of the vibrational modes both on the short and long time scales, determined by the excitation pulse duration and the decay time of molecular coherence, respectively. Utilization of multiple time scales maximizes the amount of interference fringes in the CARS spectrogram and improves the performance of the XFROG method.

It has already been shown that the “π-step shaping” (i.e., half of the frequency band of the pulse has a π phase shift relative to the other half of the band) results in better spectral resolution in comparison with the transform limited pulses [5, 13]. Similarly, as was pointed out previously [10], applying π-step shaping allows to increase the performance of the XFROG retrieval. π-step shaping splits an originally transform limited probe pulse into a pair of phase coherent pulses, closely following each other in time with a time delay of order of the original pulse length. Spectral interference of these two sub-pulses produces sharp “spectral markers” in the two-dimensional CARS spectrogram and therefore increases the accuracy of the XFROG algorithm. The latter is not particularly sensitive to the exact size of the phase step as long as it results in the generation of a pair of temporally close pulses of comparable amplitude.

Utilizing the vibrational coherence on a time scale longer than the excitation pulse duration is possible by splitting an original probe pulse into a series of transform limited pulses. Two consecutive probe pulses have been used recently to increase the spectral resolution of CARS [14]. Here we generate longer probe sequences by means of the periodic phase modulation of the initial probe pulse. By controlling the period of the spectral phase modulation, we make the time interval between the pulses in the train comparable to the decay time of the vibrational coherence, introducing fine structure in the CARS spectrogram.

Numerical simulations of the robustness of the XFROG retrieval for a model Raman medium and various shapes of the probe pulse are shown in Fig. 1(a) as a function of the noise level, defined as the ratio of the noise amplitude to the maximum CARS signal. We considered tree Lorentzian lines on top of a nonresonant background as a model system (Fig. 1(b). To investigate the stability of the method, we added white noise to the calculated two-dimensional spectrogram and compared the retrieved spectrum with the model spectrum by calculating the spectral error:

where K is the total number of frequency points, E_as(ω_i) is the model Raman spectrum, Ẽ_as ^(k)(ω_i) is the retrieved Raman spectrum after k-th iteration which results in the smallest G ^(k) error of the XFROG algorithm.

Figure 1 demonstrates increasingly higher performance of the retrieval algorithm from the transform limited probe, to the π-step shaped probe, to the train of probe pulses, and finally a combination of both. The best and the worst spectrum retrievals are shown for comparison in Fig. 1(c) and Fig. 1(d), respectively.

2. Experiment

Our experimental demonstration of the XFROG performance in CARS with different probe pulse shapes was based on a series of measurements on toluene. The experimental setup (Fig. 2) was described in [10]. Briefly, it is based on a commercial Ti:Sapphire laser system (Spitfire, Spectra-Physics) and an optical parametric amplifier (OPA) (Topas, Light Conversion), generating probe pulses of 130 fs duration at 800 nm and 100 fs pulses at 1111 nm and 1250 nm central wavelengths for the probe, pump and Stokes beams respectively. This corresponds to an excitation bandwidth of ∼200 cm^-1 with a central frequency around 1000 cm^-1. The probe pulse was spectrally shaped in a home made pulse shaper based on a liquid crystal spatial light modulator. All three beams, with ∼10 μJ pulse energy each, were focused by a lens with a focal length of 50 cm, and overlapped in BOXCARS geometry in a quartz cuvette containing the toluene sample.

The CARS spectrum was measured with a spectral resolution of 0.05 nm as a function of the time delay between the pump-Stokes pair and the probe pulse, scanned in steps of 20 fs. We used the commercial XFROG software from Femtosoft Technologies to process the measured CARS spectrograms.

 

Fig. 2. Experimental setup for XFROG CARS with the spectrally shaped probe pulses.

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3. Results and discussion

Experimental results are presented in Fig. 3. The decaying CARS signal at positive time delay between the pump-Stokes pulse pair and the probe pulse reflects the relaxation of the excited vibrational coherence in the ground electronic state of toluene. Oscillations in the time domain, observed with the unshaped probe and clearly visible in Fig. 3(a) (marker 1), correspond to the beating between the three excited modes of toluene at 783, 1000, and 1027 cm^-1. The two-dimensional spectrogram obtained with the π-step shaping of the probe pulse is shown in Fig. 3(b). The length of the shaped pulse becomes longer, resulting in the non-zero signal at negative time delays which are measured in respect to the center of the pulse (marker 2, Fig. 3(b).

 

Fig. 3 Measured two-dimensional CARS spectrograms I_as(ω, τ) using (a) a transform-limited probe pulse, (b) π-step shaping of the probe pulse, (c) sinusoidal phase modulation of the probe pulse, (d) combination of the sinusoidal and π-step modulations of the probe pulse. See text for the detailed description of the interference fringes marked by white circles.

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In addition to the strong beating observed with the transform limited pulse, three distinct interference fringes appear at around 753, 741, and 739 nm as the “frequency markers” of the Raman resonances of toluene, corresponding to 783, 1000 and 1027 cm^-1 energy shifts (marker 3, Fig. 3(b).

The results for a probe pulse train are presented in Fig. 3(c). The train was obtained by a sinusoidal modulation of the spectral phase of the original probe pulse:Φ(λ) = 1.1∙sin[2π(801.5-λ)/2.16], where λ is measured in nanometers. This shaping splits the probe pulse into a sequence of 5 pulses equally separated in time by ∼1ps (marker 4, Fig. 3(c), and results in the appearance of fine structure in the interference pattern of the CARS spectrogram (marker 5, Fig. 3(b). The period of this interference pattern along the wavelength axis is dictated by the period of the sinusoidal modulation of the probe pulse and is equal to ∼33 cm^-1 (or ∼2nm at the central wavelength of 740nm). Combining the π-step shaping with the sinusoidal phase modulation results in the richest interference pattern shown in Fig. 3(d).

 

Fig. 4. XFROG retrieval of the Raman spectrum of toluene (amplitude - red line, phase - blue line), obtained with (a) a transform-limited probe pulse, and (b) a combination of the π-step and sinusoidal phase modulation of the probe pulse. Thin black line corresponds to the reference spectrum of spontaneous Raman scattering.

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The performance of the XFROG retrieval with the pulse shaping described above was assessed by comparing the retrieved Raman spectra, |A(Ω)|², with the reference spectrum of spontaneous Raman scattering (Fig. 4). Correspondence of the frequencies of the three resonances and the relative strength of the neighbor lines at 1000 and 1027 cm^-1 were used as the benchmarks. Note that a distant week line at 783 cm^-1 is excited less efficiently due to the limited bandwidth of the pump-Stokes pulse pair, and therefore cannot be used as a performance indicator. The best performance and the highest signal-to-noise ratio was achieved in the case of the combined π-step and sinusoidal phase modulation (Fig. 4(b). The retrieved phase of the Raman modes corresponds to the Lorentzian model of the resonant molecular response as can be seen by comparing the blue lines in Fig. 1(b) and Fig. 4(b). The availability of the vibrational phase provides additional information about the system and extends our capabilities of studying its dynamics. Knowledge about the phase of molecular vibrations is important in such applications as coherent control and the separation between the Raman active modes of complex molecular systems [10].

Femtosecond pulse shaping enables not only to generate longer pulse sequences, but also to tailor the pulse amplitudes in the train. In particular, increasing the energy from pulse to pulse in the series of probe pulses can be useful in compensating the decay of the vibrational coherence and therefore improve the contrast of the interference pattern. In order to obtain the maximum fringe contrast in the CARS spectrogram, the amplitude of the pulses in the probe pulse train should increase exponentially, with the same rate as the exponential decay of vibrational coherence. We have verified this statement experimentally, confirming that higher stability against noise can be achieved with such rising-amplitude pulse sequences. Pulse sequences with gradually increasing time delay between the pulses represent another possibility for improvement. Note, that in contrast to the pure phase modulation discussed here (i.e. π-step and sin), further tailoring of the probe pulses will in general require modulation of the spectral amplitudes, thus resulting in a certain energy loss.

4. Conclusion

In conclusion, we have demonstrated the improvement in the performance of the recently suggested method of cross-correlation frequency resolved optical gating of coherent anti-Stokes Raman scattering (XFROG CARS), which offers an efficient way of measuring both the amplitude and phase of molecular vibrations [10]. The accuracy of the XFROG method and its stability against noise is increased by means of the spectral shaping of the probe pulse which splits an original femtosecond probe into a series of pulses, separated in time on two characteristic time scales. The splitting on the short time scale is dictated by the original pulse length, and can be achieved with a simple sharp step-like spectral phase modulation. The splitting on the long time scale should match the decay time of the Raman modes, and can be easily adjusted to the medium of interest by choosing an appropriate period of the periodic phase modulation. When all relevant time scales are spanned with the probe pulse sequence, spectral interference of the correspondingly generated anti-Stokes pulses results in a rich interference pattern in the measured CARS spectrogram and significantly improves the accuracy and robustness of the XFROG retrieval.