1. Introduction

Raman spectroscopy and microscopy are indispensable tools for chemical analysis in a wide variety of fields such as biology, chemistry, and material science [1,2]. By using these techniques, the spectrum of spontaneous Raman scattering from a sample can be analyzed to specify molar ingredients and to quantify their concentrations. This means that, unlike conventional microscopy, Raman microscopy can obtain not only morphological information but also quantitative chemical information from a sample, which can offer deep insight into the components of complex structures such as biological specimen. However, because the spontaneous Raman scattering has extremely low scattering cross section, Raman spectroscopy/microscopy are not suitable for analysis of dynamics or large amounts of samples. Thus, coherent anti-Stokes Raman scattering (CARS), which has much higher efficiency than the Raman scattering, attracted great attention as a more efficient alternative to spontaneous Raman scattering in spectroscopy and microscopy. Especially, CARS microscopy has been extensively developed in the past fifteen years [3–7], along with a similar technique of stimulated Raman scattering (SRS) microscopy that has been developed recently [3,8–10].

CARS is a third order nonlinear optical process that produces light with frequency of 2ω_p - ω_s, where ω_p and ω_s are the frequencies of the pump and the Stokes beams, respectively. Unfortunately, a CARS spectrum is not the same as the corresponding Raman spectrum but a coherent superposition of resonant Raman signal and a nonresonant background that originates from electronic transitions of a molecule, which hampers direct access to quantitative information of chemical species. Therefore various techniques have been developed to effectively remove the influence of the nonresonant background. Methods for removing the nonresonant contribution in the detected signal have been developed, such as epi-detection [11], polarization CARS [12], and time-gating [13,14]. However, these techniques are associated with reduction of resonant signals, which results in a decrease of measurement sensitivity. Although some other techniques such as heterodyne detection, single-pulse CARS, Fourier-transform CARS, and dual-comb CARS can remove the nonresonant background without critical deterioration of the resonant signal levels [15–20], the experimental setups associated with these methods are somewhat complicated and thus accompanies experimental complexity. Data processing approaches such as the Kramers-Kronig transformation and the maximum entropy method reconstruct Raman spectrum from a raw CARS spectrum [21–25]. They are considered to be promising approaches for CARS microscopy because they never complicate the experimental setup and assure no deterioration of signal levels. However, the data processing is time consuming and thus calculation time may be a potential problem especially when CARS images with large number of spatial pixels are involved. As an alternative approach, several methods were proposed to infer the Raman signal level from CARS intensity levels at different wavenumbers [26–29]. They are applicable to an isolated resonant Raman signal. Li et al. quantified concentration of deuterated or hydrogenated lipid in a mixed isotope lipid bilayer by calculating I_max - I_min, where I_max and I_min are respectively the maximum and the minimum intensity values of a CARS spectrum around the resonance wavenumber [26]. FM-CARS has also been developed, which enables I_max - I_min in a fast manner [28]. Similarly, Zimmerly et al. quantified deuterated glycine concentration in human hair by calculating I_max/I_min [27]. These approaches are easy to implement both experimentally and computationally, and thus may be suitable for dynamic or high-throughput data analysis with CARS microscopy. However, the above methods properly quantify molar concentration only when the concentration is sufficiently low so that resonant signal level is much weaker than the nonresonant signal level, which greatly limits their applications. Zimmerly et al. made a calibration curve from standard samples prior to the measurement for the sample of interest to avoid this limitation [23]. However, the process of calibration itself complicates the experiment and care must be taken to choose the standard samples because resonant Raman spectrum may change with the environmental condition of the sample [7]. As a similar approach, Fang et al. calculated quantitative molar concentration by measuring CARS signal levels at three different frequencies around the resonant frequency [29]. This method does not require prior assumption of the concentration level. However, it requires prior knowledge of the resonant frequency, which may cause several problems. First, when the spectrometer is not properly calibrated, the quantitative values may include errors. More seriously, the resonant frequency often drifts due to environmental conditions, which may cause unpredictable errors in calculated quantitative values [7]. Although SRS microscopy does not suffer from the nonresonant contribution and properly quantify molar concentration, CARS microscopy is still often preferred mainly because of its broadness of the spectrum.

In this report, following the same approach as in the Refs [26–29], we propose a new simple index that quantifies molar concentration for an isolated resonant signal. The index is applicable to an arbitrary concentration of the measured molecule, does not require prior calibration, and is robust against environmental change. The method is very simple, that is, to calculate I_max^1/2 - I_min^1/2. Although the index is applicable to an isolated single resonant signal theoretically, we demonstrate that it can also be applied to a mixed Raman spectrum. We will present an application of this index in quantifying lipid concentration distribution in a live murine adipocyte. The index can be applied to a broad range of experimental implementations such as multiplex CARS, single CARS, and FM-CARS.

This manuscript is organized as follows: Section 2 includes theoretical background and comparison with other similar indices. In Section 3, we present a proof-of-concept experiment with aqueous hydrogen peroxide solution. Section 4 describes an application to a live-cell CARS imaging, mainly a quantification of lipid distribution in a murine adipocyte.

2. Theoretical background

The proposed index is simply explained by depicting the trajectory of complex electric field of a CARS spectrum. In general, electric field of CARS is expressed as a superposition of resonant signal that corresponds to vibrational resonance and nonresonant signal component that corresponds to electronic transition. Thus a power spectrum of CARS is expressed as:

 

Fig. 1 (a) A typical CARS spectrum for an isolated resonant vibrational level. (b) Trajectory of CARS electric field in the complex plane.

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For a better understanding of the proposed index, we will present a comparison with previously introduced indices. In previous studies [26,27], the authors used the assumption: ${\chi }_{\text{R}}\ll {\chi }_{\text{NR}}$ to assure the validity of the indices introduced in the studies. This assumption corresponds to the situation where the trajectory circle is far from the origin. In this case, the maximum and the minimum points in the complex plane nearly align in a horizontal line (in parallel with the real axis). Then the maximum and the minimum intensity values in the CARS spectrum are expressed as

The approximations and the rigorous expressions are drawn in Fig. 2. As expected, the approximation values had a large deviation from the rigorous values for large values of A that correspond to higher concentration. Thus these indices cannot be applied to highly concentrated molecules. On the contrary, the proposed index is always proportional to molar concentration according to Eq. (3) and thus can be applied to samples with an arbitrary concentration.

 

Fig. 2 Theoretical values of quantitative indices of (a) I_max - I_min and (b) I_max/I_min versus molar concentration. Solid lines and dashed lines are rigorous representations and linear approximations at $A\ll 1$, respectively. Parameters were set as Γ = χ_NR = 1.

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On the other hand, even if the resonant frequency Ω varies with the environmental condition or concentration, the trajectory never changes and thus the proposed index gives precise values. This is not the case for the index in [29] that requires precise information of resonant frequency. Thus the proposed index is reliable and robust against environmental change compared to the index in [29].

3. Proof-of-concept experiment

We performed a proof-of-concept experiment to validate the proposed index. A schematic of the experimental setup is shown in Fig. 3. The setup is similar to the one in Ref [25]. An output beam from a microchip laser (HorusLaser HLX-I-F040) was divided by a polarization beam splitter (PBS) into two arms to form the Stokes and the pump beams. Around 70% of the power was allocated to the pump beam path. One of the beams is coupled into a photonic crystal fiber (NKT Photonics SC-5.0-1040, length: ~5.5 m) to generate supercontinuum light. Two longpass filters (cutoff wavelength: 1100 nm, OD4) was used on the supercontinuum beam path to select out the long wavelength components to form the Stokes beam. The second beam, which was used as a pump beam, was recombined with the Stokes beam with nearly zero delay. The combined light beam is routed to the back aperture of the objective (Nikon Plan Apo IR 60x NA1.27WI). The CARS signal generated at the sample is collected by another objective (Nikon S Plan Fluor ELWD 40x NA0.60) and it is directed thorough a shortpass filter (cutoff wavelength: 1025 nm, OD4) and a notch filter (center wavelength: 1064, OD6) to a spectrometer (Princeton Instruments Acton SP-2358). The CARS power spectrum is then recorded by a CCD camera (Princeton Instruments PIXIS 400BR) attached to the spectrometer. The sample was aqueous hydrogen peroxide (H₂O₂), and measurement was performed for various concentrations between 0 and 30%. The acquired spectra were normalized by the water spectrum, which was assumed to be a purely nonresonant signal. The purpose of the normalization process was to correct the non-uniformity of the Stokes power spectrum and sensitivity spectrum of the detection system including the CCD camera. After the normalization, the maximum and the minimum values of the spectrum were read out and the indices were calculated.

 

Fig. 3 A schematic of experimental setup. HWP, half-wave plate; PBS, polarization beam splitter; PCF, photonic crystal fiber, QWP, quarter-wave plate; LPF, longpass filter; DM, dichroic mirror; Obj., objective; SPF, shortpass filter; NF, notch filter.

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A normalized CARS spectrum obtained by 30% aqueous hydrogen peroxide is shown in Fig. 4. As shown in the Refs [30] and [31], the isolated O-O stretching vibrational resonance is observed around 880 cm⁻¹. Calculated indices corresponding to the maximum and the minimum intensity values of the spectra for different concentrations are shown in Fig. 5. Figure 5(a) and 5(b) are the cases where I_max - I_min and I_max/I_min were chosen as the index according to [26] and [27], respectively. It was obvious that for high concentrations, the values greatly deviated from the linear approximations, which indicates that these values are not appropriate as quantitative indices. On the other hand, the proposed index shown in Fig. 5(c), were nearly proportional to the concentration both for the cases where actual maximum/minimum values were used and where values with fixed wavenumbers were used, which demonstrates the validity of the proposed index for quantifying arbitrary molar concentration.

 

Fig. 4 CARS spectrum for 30% H₂O₂ normalized by nonresonant CARS spectrum of water.

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Fig. 5 H₂O₂ concentration dependence of (a) I_max - I_min, (b) I_max/I_min, and (c) I_max^1/2 - I_min^1/2. Red diamond: index calculated by true maximum and minimum of a CARS spectrum, blue circle: index calculated by CARS intensities at fixed wavenumber (860 cm⁻¹ for maximum and 881 cm⁻¹ for minimum). The solid lines represents theoretical curve by which the red diamond data are fitted and dashed line represents linear approximation of the theoretical curves at low concentration limit.

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The calculated and theoretical index values showed good agreement as illustrated in Fig. 5(a) and 5(b) and thus it seems that concentration can be deduced from them. However, it should be noted that the theoretical curves have fitting parameters of χ_NR and Γ, which means that data calibration by preparing several standard samples with known concentrations is necessary to deduce concentration from the calculated values as shown in [27].

In the above results, the index values calculated by actual maximum/minimum and those calculated by fixed wavenumbers were very close and almost indistinguishable. For the latter case, the wavenumbers of the maximum and the minimum for the highest concentration (30%) was chosen as the fixed ones. On the contrary, when the wavenumbers of the maximum and the minimum at a lower concentration were chosen, the index values showed large deviations from the linear line at higher concentrations as shown in Fig. 6. These results implies that, the fixed wavenumbers should be carefully chosen by considering the measurement range of concentration, and the wavenumbers of the maximum and the minimum for higher concentration should be chosen as the fixed wavenumbers. When the Raman spectrum of a target sample is known, the fixed wavenumbers for calculation of the proposed index can be determined in the above method. Otherwise, spectrum measurement by multiplex CARS may be necessary to find the maximum and the minimum, although only several spectral data points should be acquired.

 

Fig. 6 H₂O₂ concentration dependence of I_max^1/2 - I_min^1/2. Indices were calculated by CARS intensities at fixed wavenumbers (851 cm⁻¹ for maximum and 874 cm⁻¹ for minimum). The solid line is the same as that in Fig. 5(c).

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4. Application to live-cell CARS imaging

Next we report an application of the proposed index to quantify the lipid distribution in a live murine adipocyte. The experimental setup is the same as the one described in Section 3. The sample was placed on a piezo stage (Mad City Labs Nano-LPS100) and the focused beam position on the sample was scanned by moving the sample with the stage. The cell was an adipocyte differentiation of the C3H10T1/2 cell line. Lipid in a murine adipocyte has a strong Raman resonance around 2900 cm⁻¹ that is a superposition of CH₂/CH₃ symmetric/asymmetric stretching and symmetric = C-H stretching [32]. We investigated if the proposed index properly quantifies the lipid concentration through complex resonance signals. First, a CARS image of a murine adipocyte was acquired. Then the quantitative indices were calculated from the maximum and the minimum of a CARS spectrum around 2900 cm⁻¹ for each pixel. The maximum entropy method was also applied to the observed CARS spectra and the values of the resonance signals around 2900 cm⁻¹ in reconstructed Raman spectra were read out as references of the lipid concentration [23–25]. The pixel numbers, pixel size, and the exposure time for each pixel were 100 × 100, 500 × 500 nm², 30 ms, respectively.

An observed CARS spectrum and a reconstructed Raman spectrum around 2900 cm⁻¹ are shown in Fig. 7(a) and 7(b), respectively. The CARS spectrum was normalized by that of medium around the cell (alpha-MEM). Although it is slightly different from ideal one for an isolated Raman resonance such as those shown in Fig. 1(a) and Fig. 4, it exhibited the maximum and the minimum around 2900 cm⁻¹. Figure 7(b) is similar to a well-known Raman spectrum of lipid [32], which means that the reconstruction process of the maximum entropy method was successfully performed. Correlation between calculated indices and Raman intensity at 2950 cm⁻¹ were plotted in Fig. 8. The red points are the calculated indices from the actual maximum and the minimum, and the blue points are the indices calculated from values at 2865 cm⁻¹ for maximum and values at 3002 cm⁻¹ for minimum. According to the discussion in Section 3, these wavenumbers were chosen as those of the maximum and the minimum at the pixel of the strongest signal. As well as the results in Section 3, calculated indices of I_max - I_min and I_max/I_min shown in Fig. 8(a) and 8(b) showed large deviations from the linear approximations for high concentrations, and the proposed index values shown in Fig. 8(c) were nearly proportional to the concentration both for the cases where actual maximum/minimum values were used and where values with fixed wavenumbers were used. This result demonstrates that the proposed index can also be applied to a mixed Raman resonance to quantify arbitrary molar concentration.

 

Fig. 7 (a) An observed CARS spectrum normalized by nonresonant CARS spectrum of medium. (b) Corresponding Raman spectrum reconstructed by maximum entropy method.

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Fig. 8 Lipid concentration dependence of (a) I_max - I_min, (b) I_max/I_min, and (c) I_max^1/2 - I_min^1/2. Red dots: index calculated by the true maximum and minimum of a CARS spectrum, Blue points: index calculated by CARS intensities at fixed wavenumber (2865 cm⁻¹ for maximum and 3002 cm⁻¹ for minimum). Solid lines are theoretical curves by which the red dot data are fitted and dashed lines are linear approximation of the theoretical curves at low concentration limit.

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A lipid concentration map generated by the proposed index is shown in Fig. 9. The index values were normalized by the maximum value. Strong signals from lipid droplets and relatively weak signals that may be attributed to lipid bilayer were observed. Similarly to other quantitative CARS microscopy techniques [7], this result demonstrates that application of the proposed index to imaging can provide deeper insight into the structural components of a sample compared to a single point quantitative analysis and qualitative imaging such as wide-field microscopy.

 

Fig. 9 Lipid concentration map of a murine adipocyte generated by I_max^1/2 - I_min^1/2. Scale bar: 10 μm.

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5. Discussion

Here we discuss the applicability of the proposed index to a real congested spectrum. In general, a trajectory of χ_R for a congested Raman spectrum appears in a complex plane as a closed loop in the upper half plane that begins and ends at zero. In the case of ${\chi }_{\text{R}}\ll {\chi }_{\text{NR}}$, I_max^1/2 and I_min^1/2 corresponds to the maximum and the minimum of Re[χ_R + χ_NR], respectively, as shown in Fig. 10(a). Thus the index represents the “width” of the trajectory of χ_R. In this condition, the proposed index is proportional to the molar concentration because the “size” of the trajectory of χ_R is proportional to the concentration. In the case of ${\chi }_{\text{R}}\gg {\chi }_{\text{NR}}$, the index represents the maximum of |χ_R| (namely, I_max^1/2) as shown in Fig. 10(b), which is again proportional to the concentration. As a result, for very high and low concentrations that satisfy ${\chi }_{\text{R}}\gg {\chi }_{\text{NR}}$ and ${\chi }_{\text{R}}\ll {\chi }_{\text{NR}}$, respectively, the proposed index properly quantifies them. On the other hand, index values in the middle range between ${\chi }_{\text{R}}\ll {\chi }_{\text{NR}}$ and ${\chi }_{\text{R}}\gg {\chi }_{\text{NR}}$ depend on the shape of the trajectory. Thus in general case the proposed index does not quantify molar concentration. However, when the shape of the trajectory is close to a circle, the index may approximately quantify the concentration. If a CARS or Raman spectrum of a target molecule (with a certain concentration) is known, one can calculate the index versus molar concentration as shown in Fig. 10(c) to evaluate the error from the proportional relationship and check if the index is available for a specific purpose.

 

Fig. 10 (a) The trajectory of χ_R + χ_NR in the case of ${\chi }_R\ll {\chi }_{NR}$ for a fictitious congested Raman spectrum. The assumed spectrum was that of triple resonance. The dashed line is the trajectory for smaller concentration. (b) Similar plot as (a) in the case of ${\chi }_R\gg {\chi }_{NR}$. (c) An example of the values I_max^1/2 - I_min^1/2 versus molar concentration (solid line) for the fictitious Raman resonance used in Fig. 10(a) and 10(b). The dashed and dotted lines are linear approximations for ${\chi }_R\ll {\chi }_{NR}$ and ${\chi }_R\gg {\chi }_{NR}$, respectively.

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Although an ultrabroadband multiplex CARS setup was employed in the above experiments to obtain the proposed index [25], the index can be obtained by using other types of setups such as multiplex CARS systems based on optical parametric oscillator (OPO) or other swept sources [33,34]. Additionally, if the fixed-wavenumber approach is taken, the index can be applied to broader range of experimental setups such as two-color, three-color, and FM-CARS setups [35–37]. In FM-CARS, in addition to the modulation amplitude, DC levels should be measured to obtain I_max^1/2 and I_min^1/2. However, in this approach, care must be taken to choose wavenumbers and to calibrate the wavenumber [27].

The proposed index can be obtained by much simpler calculation than the time domain Kramers-Kronig (TD-KK) transformation and the maximum entropy method (MEM). This feature may be beneficial in terms of throughput when spatial mapping is taken or large amount of samples must be processed. Another potential advantage of the proposed method over the TD-KK and MEM is that the number of spectral data points can be greatly reduced. In fact, at most several spectral points may be sufficient to find the maximum and the minimum of a CARS spectrum, while larger number of points may be needed for TD-KK and MEM so that the spectral shape is revealed. This feature can substantially improve measurement time in several implementations of multiplex CARS such as the cases where a OPO or a swept light source are used [33,34].

Throughout the experiments, normalization by a nonresonant CARS spectrum was performed to obtain CARS spectra. However, this process can be greatly simplified by prior calibration of sensitivity at the wavenumber region of interest. Moreover, if the nonuniformity of sensitivity is sufficiently small, the normalization/calibration process may be omitted.

Here we will mention the correlation between the present study and the report by Jurna et al. in which the trajectory of nonlinear susceptibility in a complex plane has been used for quantitative measurement of molar concentration [38]. In [38], positions of spectral data points in a complex plane are obtained experimentally by using vibrational phase contrast CARS technique, which corresponds to the situation where TD-KK or MEM are applied to a CARS spectrum. On the other hand, in the present study, only the absolute values of the spectral data points were used to quantify molar concentration, and thus, phase information were not explicitly used. This fact greatly simplified the method including experimental setup, while making it difficult to analyze complicated samples such as mixture of multiple chemicals.

6. Conclusion

We proposed a new simple quantitative index of molar concentration for CARS spectroscopy/microscopy. Unlike previously reported similar indices, it can be applied to an arbitrary concentration of the target molecule and it is robust against environmental change. In a proof-of-concept experiment, it was validated that the index properly quantified concentration of aqueous hydrogen peroxide through an isolated Raman resonance of C-O stretching at ~880 cm⁻¹. The index was applied to analyze a live murine adipocyte and successfully quantified the distribution of lipid concentration through a mixed Raman resonance around 2900 cm⁻¹. The index can be obtained in a broad range of CARS setups and it is readily applicable to quantitative CARS microscopy for deep inspection of samples such as biological specimens.