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Abstract
As a diagonal projection of the traditional generalized two-dimensional correlation function, a one-dimensional second-order correlation function in the frequency domain is introduced. This function was applied in coherent Stokes and anti-Stokes Raman scattering (CSRS/CARS) processes. The experiment was performed by a high-average-power fiber amplified femtosecond laser system with a 1 MHz repetition rate. To measure spectral noise fluctuations, sets of 30 sample spectra of pyridine solution were recorded in the fast sequence mode. Experimental design combined with the use of appropriate notch filters allowed for simultaneous recording of the CARS/CSRS spectra. Asymmetry in the noise fluctuations between CARS and CSRS processes is revealed by both one- and two-dimensional (synchronous as well as asynchronous) noise correlation calculations, demonstrating a potential for a high-resolution spectroscopic tool.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Due to the limitations of spectral resolution and sensitivity of spectroscopic techniques, more in-depth spectral analysis is required to obtain insightful information. Recently, the higher order correlation analyses [1–4] have been developed for a better elaboration of the complex spectra with enhanced spectral resolution [5]. Namely, the two dimensional (2D) correlation analysis as an optical analogue to nuclear magnetic resonance imaging is a particularly powerful spectroscopic technique based on external perturbations such as temperature, concentration, noise, etc. [1,4]. This perturbation-based 2D incoherent Raman correlation spectroscopy has been extensively studied [6–9]. Its straightforward extension to the coherent Raman noise spectroscopy in frequency domain is demonstrated in this work. Since its first observation [10], coherent Raman scattering has become well-established spectroscopic tool with a number of variations. One of them the so-called hybrid femtosecond/picosecond (fs/ps) coherent Raman spectroscopy first developed in [11–13] is best suited for this study. In this case, a time delayed ps probe pulse is scattered off the molecules that are excited simultaneously by the fs broadband pump and Stokes pulses giving rise to enhanced CSRS/CARS signals (for a review, see [14]). The recorded signals are spectrally resolved as a result of a narrowband probe and are free of background due to a temporal mismatching of the broadband excitations and probing. This technique has superior sensitivity demonstrating e.g., bacterial spores detection [11,15] and temperature measurement of reacting and non-reacting flows in gas-phase [16]. Additionally, the present high-order correlation analyses can be adopted in a label-free coherent Raman microscopy [17–19], particularly, in the three-pulse multiplex CARS/CSRS microspectroscopy [20] allowing a possibility of high-resolution spectral analyses for species.
2. Time-resolved hybrid fs/ps coherent Stokes and anti-Stokes Raman spectroscopy
An experimental setup is shown in Fig. 1. A fs amplified fiber laser system (MXR-Clark Inc.) produces million pulses per second with a 10 μJ energy per pulse at a center wavelength of 1040 nm. These pulses (a laser beam) enter the slightly modified conventional (MXR-Clark Inc.) non-collinear optical parametric amplifier (NOPA) to produce two new beams in addition to the residual of the input laser beam. The NOPA produces Stokes (1040 nm), pump (900 nm) and probe (520 nm) pulses. The two portable spectrometers (Ocean Optics Inc.) are used to record their spectra. The pump and Stokes beams are broadband and their full width at half maximum (FWHM) are ∼ 60 and ∼ 5 nm, respectively. However the probe beam can be shaped a narrowband with less than 1 nm by using a homemade pulse shaper. This pulse shaper consists of a grating (500 grooves, Thorlabs Inc.), a lens with a 20 cm focal length and a mechanical slit spaced closely to the end-mirror in a folded 4-f configuration. The Stokes beam (the residual 1040 nm) is compressed using two prisms and a pump (900 nm) is generated by the BBO crystal and compressed inside the NOPA. The telescopes with a pair of two converging lenses are placed on the paths for pump and probe beams. The pulse width of Stokes (∼ 240 fs) and pump (∼ 50 fs) are measured by a homemade collinear interferometric autocorrelator. These three beams overlap both in time and space in a collinear configuration. Temporal coincidence of all beams is controlled with motorized delay stages (Newport Inc.). The beams are focused with an achromatic lens (Thorlabs Inc.) with a focal length of 10 cm on pyridine sample solution (99.8%, Sigma Aldrich Inc.). The 2-mm (free-path) cuvette (Starna Cell, Inc.) with about 1-mm quartz windows is used for pyridine solution. Since pyridine is a transparent solution, the forward coherent Raman signal is detected by a spectrograph with an EMCCD (Andor Inc.). Powers of the input beams before sample are 100 mW (Stokes), 40 mW (pump) and 0.5 μW (probe). Both notch and short pass filters block the input beams to detect CSRS/CARS signals simultaneously. The signal is focused in to the entrance slit of the spectrometer. The Andor spectrometer is designed to take spectral data in a fast sequence mode. In this work to study the effect of overall laser noise converted into signal noise the integration time and number of data samples in sequence mode are chosen to be 0.2 s and 30, respectively. For the weak probe pulse, the signal is detected with noise fluctuations mainly from the laser source as well as from the background (spectrometer dark current levels, lab condition variations, etc.); however, the simultaneous recording of both CARS and CSRS signals enables us to compare them as free as possible from systematic fluctuations (i.e., background variations). A set of 30 spectra is recorded at a selected delay position between τ = −2 and 10 ps. The recorded raw spectra as functions of the probe pulse delay with respect to the broadband excitations are shown in four spectrograms as in Fig. 2. The constant background about 300 counts was detected and no gain parameter was used in the signal recording. After subtraction of the background the unit of the spectral intensity reads as photoelectron counts. Later, the correlation functions are defined as fluctuations, thus any constant background contribution is automatically subtracted. Although both the CSRS/CARS spectra are recorded simultaneously, they are depicted in separate figures with the intention to study cross-correlation between CSRS and CARS signals. The averaged (mean) spectra and the standard deviations as functions of probe delay are shown in (A,B) and (C,D), where CARS and CSRS signals are separated in (A,C) and (B,D), respectively. The experimental setup is optimized to observe incoherent Raman (also referred to as ordinary Raman) spectra, in addition to the coherent Raman signals. A quantitative ratio between incoherent and CSRS signals is reported in [21]. In this experiment, for a low probe power, the incoherent Raman signal is also observed noticeably when only probe is present while the other two are blocked. Thus, the CSRS signal is accompanied with an incoherent Raman signal as seen from Fig. 2(B), 2(D). Namely, when the probe precedes (for negative delay) only incoherent Raman signal is present. This is also true when probe is delayed farther (e.g., for more than 10 ps delay) and the macroscopic vibrational coherence is already dephased. The coherent Raman signal for pyridine is optimized by tuning the NOPA output signal to display the two main peaks at 992 and 1031 cm−1, which correspond to ring-breathing (992 cm−1) and in-plane ring-bending (1031 cm−1). The unwanted broadband, non-resonant four-wave mixing (FWM) occurs when all three pulses are present simultaneously. As probe is delayed the CARS/CSRS signals are first enhanced [14,22] and then exponentially decrease [23] without FWM background contamination.
Fig. 1 Experimental setup: A fs laser with the NOPA produces broadband pump (900 nm), Stokes (1040 nm) and narrowband shaped probe (520 nm) pulses that are temporally and spatially overlapped in pyridine solution. The generated CSRS/CARS signals are recorded simultaneously with a spectrograph with EMCCD.
Fig. 2 Spectrograms for the hybrid fs/ps coherent Raman spectroscopy: (A) CARS mean spectrum and (C) its standard deviation as functions of probe delay τ and anti-Stokes Raman shift ν1; (B) CSRS mean spectrum and (D) its standard deviation as functions of τ and Stokes Raman shift ν2.
3. Spectral noise correlations
3.1. Second-order correlation function in the frequency domain
Noise in nonlinear processes have been studied by the higher order correlation tools [24–30]. However these tools are not directly related to the 2D spectroscopy which is defined in frequency domain. In this work, I study spectral noise and introduce a 1D second-order correlation function in frequency domain as
where I(ν) and I′(ν − Δν) are coherent Raman spectral intensities and ν and Δν are frequency and its difference. For example, Δν = |ν1|−ν2 is a difference between anti-Stokes (|ν1|) and Stokes (ν2) Raman shifts. For functions X and Y, 〈. . . 〉 stands for integration as and {. . .} stands for averaging by external parameter, e.g., noise is an external parameter here as where M is a total number of traces in a set and δY = Y − {Y} is a variation of Y where {δY} = 0. In fact, Eq. (1) can be further normalized by a factor defined as [〈{δI(ν)2}〉 〈{δI′(ν)2}〉]−0.5 but in the present study I prefer an unnormalized form which provides one-to-one comparison with the conventional two-dimensional correlation spectroscopy results. (or ) defines CARS-CARS (or CSRS-CSRS) noise auto-correlations. defines CARS-CSRS noise cross-correlation. The 1D auto- and cross-correlations as functions of probe time delay calculated from the observed spectral data collection are shown in Fig. 3. At some delays, the calculations for noise correlation functions show some artifact (see, few stripes in Fig. 3), although their overall delay dependences are clearly demonstrated. Auto-correlations either for CARS (A) or CSRS (B) noise spectra and cross-correlations either between CARS and CSRS (D) or CSRS and CARS (C) noise spectra with probe delay consist of three peaks separated by about 40 cm−1 as a difference of two Raman peaks for pyridine. Moreover, the auto-correlations (A,B) are insensitive to the relative phase between FWM and coherent Raman signals, whereas cross-correlations (C,D) are sensitive to that and the two are symmetric as they should be. For negative delay of probe in Fig. 3, no indication of cross-correlations because no CARS signal present (C,D), while auto-correlations (A,B) remain with a sole central narrow peak. The central peak for auto-correlations remains when the probe is delayed but the side peaks (at about 40 cm−1 apart from the centre) disappear, whereas the central peak for cross-correlations disappears eventually after their side peaks disappear. This is evident in Fig. 4. Coherent Raman spectra with their standard deviations (A,D,G,J), auto- (B,E,H,K) and cross- (C,F,I,L) correlations for four selected delay positions of τ = −2, 0.4, 0.8 and 10 ps are shown in Fig. 4 in four rows, respectively. For negative (τ = −2 ps) delay, only Stokes Raman peaks are visible, though all 1D correlations exhibit no informative spectral feature. As the probe is delayed, these correlations exhibit side peaks in addition to the central ones. The central peaks have a certain spectral width and the heights of the side peaks can be the same (see, (E,H)) or different (see, (F,I)). For large delay (τ = 10 ps) when the macroscopic coherence almost vanishes and the auto-correlations become again narrow and featureless and the cross-correlations disappear. This indicates that the 1D correlations have useful information about the macroscopic coherence associated with corresponding Raman bands. Extracting macroscopic coherence is in general, cumbersome, if there are significant four-wave mixing contaminations and/or many overlapped Raman bands particularly in samples composed of many species, for example, biological tissue or hydrogen-bonded mixture. One of the main findings of this work is that the side peaks for the cross-correlations exhibit a clear asymmetry at different delays. At this stage, an explanation of the observed asymmetry in connection to the pyridine vibrating molecules is open for a speculation. However, this asymmetry in noise for CARS versus CSRS is visualized further in terms of the generalized 2D correlation spectroscopy.Fig. 3 Spectral noise correlations: (A) CARS and (B) CSRS noise auto-correlations as functions of τ and differences Δν1 and Δν2, respectively; (C,D) noise cross-correlations as functions of τ and relative differences |ν1| − ν2 and ν2 − |ν1|, respectively.
Fig. 4 Selected samples for four different delay positions of probe: (A,B,C) at τ = −2 ps; (D,E,F) at τ = 0.4 ps; (G,H,I) at τ = 0.8 ps and (J,K,L) at τ = 10 ps. First column (A,D,G,J) for CARS (black curves and error bars) and CSRS (red curves and error bars) spectra with the standard deviations; second column (B,E,H,K) for auto-correlations for CARS (black curves) and CSRS (red curves) Raman noise spectra; third column (C,F,I,L) for CARS-CSRS noise spectral cross-correlations.
3.2. Generalized two-dimensional noise correlation spectroscopy
For the generalized 2D noise correlation spectroscopy, anti-Stokes and Stokes Raman shifts become two independent variables to form a (x, y) plane where a variance of recorded spectrum will be the third spectral variable along z axis. In this work, the conventional 2D correlation spectroscopic technique [4,9] for ordinary Raman spectral data is adopted for the CARS/CSRS spectroscopy. Two types of correlations are defined here. The synchronous correlations demonstrate the coincident increase or decrease, while the asynchronous correlations demonstrate the sequential variations in the noise of the two variables. By definition as in [4], the synchronous 2D spectral noise correlation is given by
and the asynchronous counterpart is given by Where 𝒩jk are the discrete Hilbert-Noda transformation matrix elements defined as zero if j = k, otherwise 𝒩jk = 1/[π(k − j)] [31]. For the auto-correlation functions, the diagonal elements of this matrix are zero, but off-diagonal elements are non-zero and asymmetric with respect to the main diagonal. In this work the spectral noise by means of conversion of the mainly input laser noise into the coherent Raman spectral noise is studied. Sequential recordings of 30 spectra with an integration time of 0.2 s reveals noise characteristics which are calculated by Eqs. (2) and (3) for different time delay positions of the probe. In Fig. 5, the results for τ = 0.4 ps (A to F) and 0.8 ps (G to L) are, in particular, chosen to demonstrate their distinct contrast. In the first and third row of Fig. 5, the synchronous CARS-CARS (), CSRS-CSRS (), and CARS-CSRS () noise correlations exhibit two peaks along the main diagonal and the other two along the upper and lower off-diagonals. The off-diagonal peaks demonstrate correlation between the pyridine Raman peaks. The peaks are all positive in this case revealing simultaneous increase or decrease of the coincident correlated noise. However, the rest of Fig. 5, ,, and exhibit rich variability. The asynchronous auto-correlations in Fig. 5 (D,E,J,K) are out of phase (with positive versus negative peaks) from each other. This means that at approximately τ = 0.4 ps and 0.8 ps both noise of the CSRS and CARS peaks at 992 cm−1 and 1031 cm−1, respectively, are sequentially out-of-phase. Overall, out-of-phase signature is visually demonstrated in the cross-correlations at these particular delays as seen from Fig. 5 (F,L).Fig. 5 Generalized two-dimensional spectral noise correlations for two different delay positions of probe: (A to F) at τ = 0.4 ps and (G to L) at τ = 0.8 ps where (A,B,C,G,H,I) for the synchronous and (D,E,F,J,K,L) asynchronous 2D spectral noise correlations.
3.3. Diagonal projection of the synchronous 2D spectral noise correlations
The synchronous 2D correlations in Eq. (2) construct 2D matrix in terms of discrete variables. The main diagonal elements contain the spectral noise obtained from the recorded set of spectra whereas off-diagonal elements contain correlation nature in the noise. The 1D correlation function is defined in Eq. (1) can be derived directly from Eq. (2). Namely, a sum of matrix elements along on- and off-diagonals for the synchronous correlations reduces 2D into 1D (i.e., the 2D correlations into the 1D correlations). The results in Fig. 4(E,F,H,I) are straightforwardly derived from the results in Fig. 5(A,B,C,G,H,I). The side peaks in the 1D correlations represent correlation peaks in the 2D correlations. Thus, for the auto-correlations the side peaks are symmetric (same heights), however, it is asymmetric (different heights) for the cross-correlations. The spacing of side peaks are about 40 cm−1 in the 1D correlations since correlation peaks should be on the off-diagonal corners of the composed squares in (x-y) plane for the 2D spectroscopy [4].
4. Conclusions
A new 1D second-order correlation function in frequency domain is introduced. It is identical to a diagonal projection of conventional synchronous 2D spectroscopic correlation functions. Utilizing these functions, spectral noise fluctuations for coherent Raman scattering processes were studied in detail. In the experiment, simultaneous coherent Stokes and anti-Stokes Raman spectra of pyridine solution, recorded at various time delay positions of the probe pulse relative to the broadband pulse excitations and their auto- and cross-correlations were analyzed. At the selected time delays the spectral noise between coherent Stokes and anti-Stokes Raman spectra is either coincident (in-phase) or sequential (out-of-phase). The results demonstrate that the 1D correlation function is a useful spectroscopic diagnostic tool, in particular, for distinguishing coherent processes from incoherent counterparts and revealing asymmetry in spectral noise fluctuations for coherent nonlinear processes. The findings of this study has the potential to contribute to high-resolution ultrafast spectroscopy.
Acknowledgments
I would like to thank to Dr. Dmitry Pestov for his valuable discussions and Z. Gong for his help in writing a code for controlling delay stages and S. Nagpal and P. Adhikari for measuring pulse durations and calibrating the spectrometer.
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