1. Introduction

Coherent anti-Stokes Raman scattering (CARS) is a four-wave mixing process that offers a unique tool for detecting molecular vibrations. One of its most appealing implementation is multiplex CARS microspectroscopy [1–3], which offers label-free and chemically specific imaging for the study of a wide variety of materials and biological systems [4–7]. It enables distinguishing of species in complex and heterogeneous samples by their vibrational signatures with high spatial and spectral resolutions. By employing a suitable phase retrieval technique [8–13] on the normalized CARS spectrum, and computing the corresponding Raman line-shape using the retrieved phase spectrum, multiplex CARS microspectroscopy has the potential to accomplish a quantitative extraction of resonant vibrational responses of those molecules that occupy the diffraction-limited focus volume element [14–16].

The quantitative analysis of a CARS spectrum is complicated by experimental errors encountered in obtaining a normalized CARS line-shape spectrum that exclusively depends on the vibrational properties of the molecular sample of interest. The latter assumption is often not satisfied in actual experiments, resulting in an additive, incoherent background and an erroneous, non-additive, and low-frequency modulation error contribution to the CARS line-shape. While the additive errors can readily be determined by independent measurements and eliminated beforehand, the latter results in an error-phase component, which manifests itself as a slowly varying function of frequency when compared with the retrieved phase spectrum. Conventionally, the error-phase term is often estimated and removed by manually fitting a polynomial baseline to the retrieved phase spectrum [4–6, 15, 16] or, more conveniently, by semi-automatic baseline correction methods [17–19]. However, in order to work to satisfactorily, these baseline correction methods require that the spectral lines are sufficiently well separated, which is not often the case in congested spectral regions commonly encountered in liquid-phase samples. Moreover, the presence of a nonconstant error-phase spectrum causes not only phase distortions but also amplitude distortions in the reconstructed Raman line-shape spectrum. Therefore, an additional amplitude correction including a Hilbert transform is required for an accurate and quantitative Raman line-shape reconstruction [20].

In this work, we present a new approach where experimental artefacts in the normalized CARS line-shape spectrum are converted from multiplicative to additive components, which enables their separation by a wavelet prism (WP) decomposition procedure and yields an error-free CARS line-shape spectrum that only depends on material properties of the sample. In order to validate our WP-based approach, we demonstrate accurate quantitative extraction of resonant Raman responses in equimolar aqueous solutions of D-fructose, D-glucose, and their disaccharide combination sucrose from normalized CARS spectra in the presence of experimental artefacts measured in the fingerprint spectral region.

2. Theoretical background

2.1 Quantitative reconstruction of the resonant Raman response from a CARS spectrum

In the frequency domain, the detected CARS intensity spectrum is most generally written as [21]:

For the analysis of the experimental CARS spectrum of the sample,$I_{CARS}^{sample}(\nu ),$it is beneficial to eliminate the proportionality constants and laser pulse spectral amplitudes involved in its absolute CARS intensity of Eq. (1) by normalization with a reference CARS intensity spectrum,$I_{CARS}^{ref}(\nu ),$measured under near identical experimental conditions:

In order to extract quantitatively the resonant Raman response that is proportional to the number of Raman-active scatterers in the sample, we need to reconstruct the imaginary part of ${\chi }_R^{(3)}(\nu )$from$S(\nu )$(cf. Equation (4)). For this purpose, numerical phase retrieval algorithms based on either the maximum entropy (ME) method [8, 10] or the time-domain Kramers-Kronig (TDKK) method [11, 12] are commonly applied. These methods do not require additional instrumentation or prior information about the vibrational resonances of the substances under study. The imaginary part of the complex${\chi }_R^{(3)}(\nu )$is then obtained by:

2.2 Correcting errors in the normalized CARS spectrum using the WP decomposition method

In actual experiments, the above assumptions commonly made for a genuine reference spectrum measurement are often not fulfilled, and$S_{exp}(\nu )\ne S(\nu ).$ Besides simply using a reference spectrum for normalization in Eq. (3) not arising from a purely nonresonant third-order nonlinear susceptibility, such that${\chi }_{ref}^{(3)}(\nu )\ne {\chi }_{ref}^{(3)nr}$ [20], slowly varying modulation errors that affect$S_{exp}(\nu )$may also be introduced by distinct chirps of the broadband laser pulses inside the sample and reference materials. The latter results in distinguishable laser pulse spectral amplitudes influencing the detected$I_{CARS}^{sample}(\nu )$and$I_{CARS}^{ref}(\nu )$spectra [2]. Since for an unknown heterogeneous sample it is virtually impractical to determine and correct for one or the other error source, we introduce a total slowly varying modulation error function,$\epsilon (\nu ),$such that the experimental normalized CARS spectrum can be expressed as:

In this work, we present a new approach that is based on the WP decomposition in order to extract the error-free CARS line shape,$S(\nu ),$from the measured normalized CARS spectrum,$S_{exp}(\nu ),$in which the slowly varying experimental artefacts,$\epsilon (\nu )$are estimated and removed at the initial stage of the measured CARS power spectrum.

To apply the WP decomposition method, it is useful to make the slowly varying modulation error term in Eq. (7) additive by taking the logarithm:

In order to avoid the dyadic loss of frequency resolution of discrete wavelet coefficients, the so-called wavelet prism algorithm has been introduced by Tan and Brown [27]. It is an implementation of the Mallat pyramid algorithm [28] coupled with a particular wavelet reconstruction. This algorithm is arranged in a hierarchical way, such that a coarse approximation,$A_j(\nu ),$and detail components,$D_j(\nu )\left(j\le J\right),$are reconstructed separately at each scale level$j,$which exhibit a frequency resolution identical to that of the original function. As a result, this algorithm enables the representation of the original spectrum,$\ln S_{exp}(\nu ),$as a sum of mutually orthogonal components, which are the so-called wavelet prism spectra:

Different wavelet prism spectra represent different frequency ranges, such that the lowest scale-level detail component,$D_1(\nu ),$and the highest scale-level approximation component,$A_J(\nu ),$correspond to the highest and lowest frequency ranges, respectively. If the value of$J$is high enough, the approximation component separates an almost constant DC-offset from all AC detail components. In the decomposition of the$\ln S_{exp}(\nu )$ spectrum, the few first detail components represent the noise spectrum,${\left[\ln S_{exp}(\nu )\right]}_{noise},$ whereas the next components constitute the informative signal part,${\left[\ln S_{exp}(\nu )\right]}_{signal}.$The following detail components of higher scales are assigned to the logarithm of the slowly varying modulation error spectrum,$\ln \epsilon (\nu ),$and the approximation component is the DC-offset,${\left[\ln S_{exp}(\nu )\right]}_{DC}.$Assuming that these different spectral contributions are located within non-overlapping frequency ranges [27], Eq. (13) can be re-written as

The principle of the WP decomposition method is illustrated in Fig. 1, where the algorithm has been applied to the logarithm function of the normalized CARS spectrum, $\ln S_{exp}(\nu )$, recorded for an aqueous solution of sucrose. Figure 1(a) depicts the reconstructed detail, $D_{j\le J}(\nu ),$ and approximation, $A_J(\nu ),$components for $J=14,$ according to Eq. (13). As shown in Fig. 1(b), noise, signal, modulation error, and DC-offset spectra have been constructed according to${\left[\ln S_{exp}(\nu )\right]}_{noise}={\displaystyle {\sum }_{j=1}^3{D}_j(\nu ),}$${\left[\ln S_{exp}(\nu )\right]}_{signal}={\displaystyle {\sum }_{j=4}^p{D}_j(\nu ),}$$\ln \epsilon (\nu )={\displaystyle {\sum }_{j=p+1}^J{D}_j(\nu ),}$ and${\left[\ln S_{exp}(\nu )\right]}_{DC}=A_J(\nu ),$ respectively, where the scale level $p=6$ was chosen to differentiate between the signal and slowly varying modulation error components. Hence, in this example, an estimation of the modulation error spectrum is provided by $\epsilon (\nu )=\exp \left\{{\displaystyle {\sum }_{j=7}^{14}{D}_j(\nu )}\right\}.$

 

Fig. 1 Illustration of the principle of the WP method applied to the$\ln S_{exp}(\nu )$spectrum of an aqueous solution of sucrose. (a) WP decomposition into discrete detail components,$D_{j\le J},$and approximation,$A_J,$for$J=14.$ (b) Spectral contributions of noise, signal, modulation error function, and DC-offset constructed from the corresponding groups of detail and approximation components from the input$\ln S_{exp}(\nu )$spectrum.

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The advantages of the WP method include its applicability to any spectral form of a slowly varying error-term and its ability to handle rather small signal to background ratios. Moreover, there is no need for assumptions about the slowly varying modulation error distribution or the chemical composition of the specimen. Thus, the modulation error function estimation is unsupervised in a sense that no a priori knowledge about control points within nonresonant spectral regions is required. Thus, for an appropriate choice of scale levels$J$and$p,$ this approach is suitable for automated modulation error function estimation for $S_{exp}(\nu )$ spectra and a derivation of the corresponding error-free CARS line shape spectra, $S(\nu )$.

3. Experimental details

3.1 Samples

D-fructose, D-glucose, and their disaccharide combination, sucrose (α-D-glucopyranosyl-(1→2)-β-D-fructofuranoside) were dissolved in buffer solutions (50 mM HEPES, pH = 7) at equal molar concentrations of 500 mM [15].

3.2 Multiplex CARS spectroscopy

All CARS spectra were recorded using a multiplex CARS spectrometer, the detailed description of which can be found elsewhere [29]. In brief, a 10-ps and an 80-fs mode-locked Ti:sapphire lasers were electronically synchronized and used to provide the narrowband pump/probe and broadband Stokes laser pulses in the multiplex CARS process. The center wavelengths of the pump/probe and Stokes pulses were 710 nm and 767 nm, respectively, which corresponds to probing CARS spectra within a wavenumber range from 700 to 1200 cm⁻¹. The linear and parallel polarized pump/probe and Stokes beams were made collinear and focused with an achromatic lens into a tandem cuvette. The latter could be translated perpendicular to the optical axis to perform measurements in either of its two compartments, providing a multiplex CARS spectrum of the sample and of a non-resonant reference under near identical experimental conditions. Typical average powers used at the sample were 95 mW and 25 mW for the pump/probe and Stokes laser, respectively. The anti-Stokes signal was collected and collimated by a second achromatic lens in the forward-scattering geometry, spectrally filtered by short-pass and notch filters, and focused into a spectrometer equipped with a CCD camera. The acquisition time per CARS spectrum was 200 ms.

3.3 Spontaneous Raman scattering spectroscopy

Spontaneous Raman scattering spectra were recorded using a Raman microscope (Renishaw RM1000, spectral resolution: ~5 cm⁻¹) with cw-excitation at 632 nm and a power of 6 mW [15]. A polarizer in the detection channel ensures that only scattered light was detected with a plane of polarization parallel to that of the exciting radiation. The acquisition time per spectrum was 50 s.

3.4 Computational methods

The spectral data analysis was software-implemented and performed in MATLAB (The MathWorks, Inc.). In the WP analysis, the wavelet type Daubechies ($db16$) was used, and the decomposition up to the scale level $14$ was performed on the logarithm of the normalized CARS spectrum (cf. Equation (13)). For the phase retrieval, the ME method with a squeezing parameter $K=1$ was applied, in which the number of autocorrelation coefficients was $M=M_{\max }$ [9].

4. Results and discussions

In order to validate our WP-based approach for the correction of experimental errors, we analyzed normalized CARS spectra recorded for equimolar aqueous solutions of fructose, glucose, and sucrose in the fingerprint spectral region. Figures 2(a), 2(b), and 2(c) show the normalized CARS spectrum obtained from the experiment, $S_{exp}(\nu )$ (solid blue curve), the slowly varying modulation error spectrum extracted from our WP decomposition according to $\epsilon (\nu )=exp\left\{{\displaystyle {\sum }_{j=7}^{14}{D}_j}(\nu )\right\}$ (dashed blue curve), and the error-corrected CARS line shape according to Eq. (15), $S(\nu )$ (solid red curve), for sucrose, fructose, and glucose, respectively. The middle row of Fig. 2 depicts for each compound the maximum entropy (ME) phase spectra, ${\varphi }_{exp}(\nu )$ and $\varphi (\nu )$, as retrieved from the corresponding $S_{exp}(\nu )$ and $S(\nu )$ spectra, respectively. For the ${\varphi }_{exp}(\nu )$ spectra (blue dashed lines), slowly varying error-phase modulations are observed, which originate from the propagation of the modulation error contributing to $S_{exp}(\nu ),$and possess phase-amplitudes exceeding that of the actual vibrational resonances of interest. Therefore, for a quantitative Raman line shape interpretation based on ${\varphi }_{exp}(\nu )$ and corresponding $S_{exp}(\nu )$ spectra, an estimation of an error-phase function for ${\varphi }_{exp}(\nu )$accompanied by its inverse Hilbert transform resulting in an estimation of an amplitude rescaling function is required [20]. In contrast, the $\varphi (\nu )$ spectra (red lines) are free of such phase modulation artefacts and exhibit only small constant phase-offset values acquired during the numerical ME phase retrieval that are measurable within the vibrationally nonresonant spectral regions, i.e.${\varphi }_{err}\approx \text{constant}.$ Because the Hilbert transform of a constant phase yields zero, which corresponds to an amplitude correcting factor of unity, the quantitative reconstruction of the Raman line shape based on $\varphi (\nu )$ and the corresponding error-corrected $S(\nu )$ spectra avoids the additional assumptions and accumulation of propagation errors made in the subsequent phase retrieval and inverse Hilbert transform. Therefore, by estimating and correcting the modulation error contribution to the experimental normalized CARS spectrum in the first place, it not only significantly simplifies the quantitative reconstruction of the sample’s Raman response but also circumvents the need for additional amplitude corrections. This is demonstrated in Figs. 2(g), 2(h) and 2(i), where the reconstructed $\mathrm{Im}[{\chi }_R^{(3)}(\nu )]$ spectra (red curves) are directly compared with the spontaneous Raman spectra (green curves) of the same samples of sucrose, fructose, and glucose, respectively. Additionally, the $\mathrm{Im}{\left[{\chi }_R^{(3)}(\nu )\right]}_{exp}$spectra (dashed blue curves) representing the imaginary parts reconstructed from uncorrected experimental CARS data are shown for comparison in the same graphs. Here, the $\mathrm{Im}[{\chi }_R^{(3)}(\nu )]$spectra were reconstructed according to Eq. (5), where the constant error-phase offset values of −0.014, −0.0065, and −0.008 were taken into account for sucrose, fructose, and glucose, respectively. To compare qualitatively the $\mathrm{Im}[{\chi }_R^{(3)}(\nu )]$ spectra with the respective, independently measured, spontaneous Raman spectra, a single scaling factor was found to provide a very good match in terms of spectral line shapes and relative peak amplitudes for all three sugar samples. The latter observation provides direct evidence for a successful WP-based correction and quantitative reconstruction of $\mathrm{Im}[{\chi }_R^{(3)}(\nu )]$ spectra from normalized CARS spectra that are affected by experimental slowly varying modulation errors.

 

Fig. 2 Experimental normalized CARS spectrum,$S_{exp}(\nu ),$the WP-extracted slowly varying modulation error spectrum,$\epsilon (\nu ),$and the error-corrected normalized CARS line shape,$S(\nu ),$for aqueous solutions of (a) sucrose, (b) fructose, and (c) glucose. D, E, F: The corresponding ME phase spectra,${\varphi }_{exp}(\nu )$and$\varphi (\nu ),$as retrieved from the$S_{exp}(\nu )$and $S(\nu )$ spectra, respectively. g, h, i: The corresponding reconstructed $\mathrm{Im}{\left[{\chi }_R^{(3)}(\nu )\right]}_{exp}$and$\mathrm{Im}\left[{\chi }_R^{(3)}(\nu )\right]$spectra in comparison with the spontaneous Raman spectra,$I_{Raman}(\nu ),$of the same samples obtained by independent measurements.

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A second, and more stringent, validation of a quantitative reconstruction of $\mathrm{Im}[{\chi }_R^{(3)}(\nu )]$ spectra is performed by directly exploiting the fact that the disaccharide sucrose is inherently made up of an $\alpha$-D-glucopyranosol group and a $\beta$-D-fructofuranosyl group, as occurring in the monosaccharides $\alpha$-D-glucose and $\beta$-D-fructose, respectively, and linked by a C-O-C disaccharide bridge (see insert in Fig. 3). Figure 3 directly compares the $\mathrm{Im}{[{\chi }_R^{(3)}(\nu )]}_{fruc}$ and $\mathrm{Im}{[{\chi }_R^{(3)}(\nu )]}_{gluc}$spectra obtained for the fructose and glucose samples, respectively (see Figs. 2 (h) and 2(i)), with the reconstructed spectrum obtained for the sucrose sample, $\mathrm{Im}{[{\chi }_R^{(3)}(\nu )]}_{suc}$(see Fig. 2(g)). Differences in spectral details of the three sugars are most pronounced in the spectral range from 800 to 1000 cm⁻¹, to which C-C stretching in the ring skeletons and C-H deformation vibrations predominantly contribute [30,31]. In this region, the differences are mostly attributed to: (i) the existence of equilibria of isomers in the aqueous solutions of D-glucose and D-fructose with characteristic vibrational bands that are partly absent in sucrose [32]; and (ii) the influence of the disaccharide link in sucrose on the ring skeletons [33]. The binding of the monosaccharides inflicts constrains on the pyranoid and furanoid rings that result in frequency shifts of the C-C stretching vibrations [34]. In contrast, the spectral region from 1000 to 1200 cm⁻¹ is dominated by out-of-ring vibrations from C-O stretching (1064 cm⁻¹) and C-OH deformation (1120 cm⁻¹) modes that are mostly insensitive to isomers and to the disaccharide link [30,31]. Therefore, only within this spectral region, we can assume that the spectral Raman features observed for the fructose and glucose are well-preserved in the spectrum of the sucrose. Provided that the normalized CARS spectra of the three different sugars were measured at identical molar concentrations and experimental conditions, the spectra obtained for the fructose and glucose should then add up by equal weights to yield the reconstructed spectrum obtained for the sucrose:

 

Fig. 3 Comparison of the reconstructed spectrum obtained for the sucrose sample,$\mathrm{Im}{\left[{\chi }_R^{(3)}(\nu )\right]}_{suc}$(dark gray curve), with the best fit (red curve) according to Eq. (16) within the spectral region from 1000 to 1200 cm⁻¹. The linear spectral unmixing model consisting of the $\mathrm{Im}{\left[{\chi }_R^{(3)}(\nu )\right]}_{fruc}$(dashed blue curve) and$\mathrm{Im}{\left[{\chi }_R^{(3)}(\nu )\right]}_{gluc}$(dashed green curve) spectra obtained for the fructose and glucose samples, respectively, yields molar weights of $C_{fruc}=0.98$ and $C_{gluc}=0.98$. Insert: Chemical structure of sucrose (α-D-glucopyranosyl-(1→2)-β-D-fructofuranoside).

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5. Summary and conclusions

A new method for eliminating errors in quantitative phase retrieval and reconstruction of molecular Raman responses from experimental CARS spectra due to inaccurate normalization was introduced. The method is based on the WP decomposition analysis of the logarithm of the measured normalized CARS spectrum and allows separating its slowly varying modulation error function originating from experimental artefacts while preserving its DC-offset. The corrected CARS line shape spectrum can then be used to retrieve the phase spectrum and, thereby, the imaginary part of the Raman susceptibility providing a quantitative Raman line-shape spectrum. Unlike in conventional methods that are based on an error-phase estimation and correction of the retrieved phase spectrum, here we address the experimental artefacts at the original level of the measured CARS power spectrum. Because our method circumvents the nonlinear propagation of experimental errors through the phase retrieval algorithm, and the subsequent estimation of a slowly varying error-phase, the concomitant error-amplitude correction for a quantitative Raman line shape by performing another Hilbert transform is redundant. This significantly simplifies the quantitative reconstruction of the sample’s Raman response from measured CARS spectra. An additional benefit of the WP decomposition method is its ability to handle rather small signal to background ratios in CARS spectra within congested regions of Raman resonances. Here, we do not need to fulfill the requirement that at least some Raman lines are well enough separated. Thus, the modulation error function estimation is unsupervised in a sense that no a priori knowledge about control points within nonresonant spectral regions is required. For an appropriate choice of scale levels$J$and$p,$this approach is suitable for a fast and automated modulation error function estimation in typical hyperspectral CARS imaging applications. For this purpose, work is in progress to find and evaluate a general criterion for choosing the scale level p in a truly unsupervised manner, such as for example the so-called MIGC criterion [27].

The application of the proposed method to aqueous solutions of fructose, glucose, and sucrose in the fingerprint spectral region showed excellent quantitative agreement between the Raman line-shapes extracted from corrected normalized CARS spectra and corresponding spontaneous Raman scattering spectra. The demonstrated agreement between the reconstructed Raman response of sucrose with its sum of equimolar contributions from fructose and glucose confirms that the proposed method eliminates any potential amplitude distortions and scaling errors. Beyond CARS spectroscopy, the WP decomposition method also offers a tool for accurate and quantitative phase retrieval and reconstruction of complex material responses probed by other nonlinear optical spectroscopies, such as for the correction of experimental errors in measured third-harmonic generation, sum-frequency generation, and degenerate four-wave mixing power spectra [13].