Log-Gaussian gamma processes for training Bayesian neural networks in Raman and CARS spectroscopies.

We propose an approach utilizing gamma-distributed random variables, coupled with log-Gaussian modeling, to generate synthetic datasets suitable for training neural networks. This addresses the challenge of limited real observations in various applications. We apply this methodology to both Raman and coherent anti-Stokes Raman scattering (CARS) spectra, using experimental spectra to estimate gamma process parameters. Parameter estimation is performed using Markov chain Monte Carlo methods, yielding a full Bayesian posterior distribution for the model which can be sampled for synthetic data generation. Additionally, we model the additive and multiplicative background functions for Raman and CARS with Gaussian processes. We train two Bayesian neural networks to estimate parameters of the gamma process which can then be used to estimate the underlying Raman spectrum and simultaneously provide uncertainty through the estimation of parameters of a probability distribution. We apply the trained Bayesian neural networks to experimental Raman spectra of phthalocyanine blue, aniline black, naphthol red, and red 264 pigments and also to experimental CARS spectra of adenosine phosphate, fructose, glucose, and sucrose. The results agree with deterministic point estimates for the underlying Raman and CARS spectral signatures.

Evaluating different deep learning models for efficient extraction of Raman signals from CARS spectra.

The nonresonant background (NRB) contribution to the coherent anti-Stokes Raman scattering (CARS) signal distorts the spectral line shapes and thus degrades the chemical information. Hence, finding an effective approach for removing NRB and extracting resonant vibrational signals is a challenging task. In this work, a bidirectional LSTM (Bi-LSTM) neural network is explored for the first time to remove the NRB in the CARS spectra automatically, and the results are compared with those of three DL models reported in the literature, namely, convolutional neural network (CNN), long short-term memory (LSTM) neural network, and very deep convolutional autoencoders (VECTOR). The results of the synthetic test data have shown that the Bi-LSTM model accurately extracts the spectral lines throughout the range. In contrast, the other three models' efficiency deteriorated while predicting the peaks on either end of the spectra, which resulted in a 60 times higher mean square error than that of the Bi-LSTM model. The Pearson correlation analysis demonstrated that Bi-LSTM model performance stands out from the rest, where 94% of the test spectra have correlation coefficients of more than 0.99. Finally, these four models were evaluated on four complex experimental CARS spectra, namely, protein, yeast, DMPC, and ADP, where the Bi-LSTM model has shown superior performance, followed by CNN, VECTOR, and LSTM. This comprehensive study provides a giant leap toward simplifying the analysis of complex CARS spectroscopy and microscopy.

Effect of non-resonant background on the extraction of Raman signals from CARS spectra using deep neural networks.

We report the retrieval of the Raman signal from coherent anti-Stokes Raman scattering (CARS) spectra using a convolutional neural network (CNN) model. Three different types of non-resonant backgrounds (NRBs) were explored to simulate the CARS spectra viz (1) product of two sigmoids following the original SpecNet model, (2) Single Sigmoid, and (3) fourth-order polynomial function. Later, 50 000 CARS spectra were separately synthesized using each NRB type to train the CNN model and, after training, we tested its performance on 300 simulated test spectra. The results have shown that imaginary part extraction capability is superior for the model trained with Polynomial NRB, and the extracted line shapes are in good agreement with the ground truth. Moreover, correlation analysis was carried out to compare the retrieved Raman signals to real ones, and a higher correlation coefficient was obtained for the model trained with the Polynomial NRB (on average, ∼0.95 for 300 test spectra), whereas it was ∼0.89 for the other NRBs. Finally, the predictive capability is evaluated on three complex experimental CARS spectra (DMPC, ADP, and yeast), where the Polynomial NRB model performance is found to stand out from the rest. This approach has a strong potential to simplify the analysis of complex CARS spectroscopy and can be helpful in real-time microscopy imaging applications.

Bayesian Quantification for Coherent Anti-Stokes Raman Scattering Spectroscopy.

We propose a Bayesian statistical model for analyzing coherent anti-Stokes Raman scattering (CARS) spectra. Our quantitative analysis includes statistical estimation of constituent line-shape parameters, the underlying Raman signal, the error-corrected CARS spectrum, and the measured CARS spectrum. As such, this work enables extensive uncertainty quantification in the context of CARS spectroscopy. Furthermore, we present an unsupervised method for improving spectral resolution of Raman-like spectra requiring little to no a priori information. Finally, the recently proposed wavelet prism method for correcting the experimental artifacts in CARS is enhanced by using interpolation techniques for wavelets. The method is validated using CARS spectra of adenosine mono-, di-, and triphosphate in water, as well as equimolar aqueous solutions of d-fructose, d-glucose, and their disaccharide combination sucrose.

Influence of Fano resonance on SERS enhancement in Fano-plasmonic oligomers.

Plasmonic oligomers can provide profound Fano resonance in their scattering responses. The sub-radiant mode of Fano resonance can result in significant near-field enhancement due to its light trapping capability into the so-called hotspots. Appearance of these highly localized hotspots at the excitation and/or Stokes wavelengths of the analytes makes such oligomers promising SERS active substrates. In this work, we numerically and experimentally investigate optical properties of two disk-type gold oligomers, which have different strength and origin of Fano resonance. Raman analysis of rhodamine 6G and adenine with the presence of the fabricated oligomers clearly indicates that an increment in the strength of Fano resonance can improve the Raman enhancement of an oligomer significantly. Therefore, by suitable engineering of Fano lineshape, one can achieve efficient SERS active substrates with spatially localized hotspots.

Wavelet prism decomposition analysis applied to CARS spectroscopy: a tool for accurate and quantitative extraction of resonant vibrational responses.

We propose an approach, based on wavelet prism decomposition analysis, for correcting experimental artefacts in a coherent anti-Stokes Raman scattering (CARS) spectrum. This method allows estimating and eliminating a slowly varying modulation error function in the measured normalized CARS spectrum and yields a corrected CARS line-shape. The main advantage of the approach is that the spectral phase and amplitude corrections are avoided in the retrieved Raman line-shape spectrum, thus significantly simplifying the quantitative reconstruction of the sample's Raman response from a normalized CARS spectrum in the presence of experimental artefacts. Moreover, the approach obviates the need for assumptions about the modulation error distribution and the chemical composition of the specimens under study. The method is quantitatively validated on normalized CARS spectra recorded for equimolar aqueous solutions of D-fructose, D-glucose, and their disaccharide combination sucrose.

Optical and terahertz spectra analysis by the maximum entropy method.

Phase retrieval is one of the classical problems in various fields of physics including x-ray crystallography, astronomy and spectroscopy. It arises when only an amplitude measurement on electric field can be made while both amplitude and phase of the field are needed for obtaining the desired material properties. In optical and terahertz spectroscopies, in particular, phase retrieval is a one-dimensional problem, which is considered as unsolvable in general. Nevertheless, an approach utilizing the maximum entropy principle has proven to be a feasible tool in various applications of optical, both linear and nonlinear, as well as in terahertz spectroscopies, where the one-dimensional phase retrieval problem arises. In this review, we focus on phase retrieval using the maximum entropy method in various spectroscopic applications. We review the theory behind the method and illustrate through examples why and how the method works, as well as discuss its limitations.

Phase retrieval of reflectance for nanoparticle optical identification.

We present a method for optical identification of dielectric and metal nanoparticles in a liquid matrix using phase retrieval of reflectance with TE- and TM-polarized light. A formula is derived for extracting the effective complex dielectric function of a nanoparticle colloid based on different complex reflectance components. The phase retrieval is performed using the maximum entropy method. We observe excellent accuracy both for dielectric and metallic nanoparticles with volume fractions up to 10%.

Causality-based method for determining the time origin in terahertz emission spectroscopy.

We propose a method for determining the time origin on the basis of causality in terahertz (THz) emission spectroscopy. The method is formulated in terms of the singly subtractive Kramers-Kronig relation, which is useful for the situation where not only the amplitude spectrum but also partial phase information is available within the measurement frequency range. Numerical analysis of several simulated and observed THz emission data shows that the misplacement of the time origin in THz waveforms can be detected by the method with an accuracy that is an order of magnitude higher than the given temporal resolutions.

Quantitative coherent anti-Stokes Raman scattering (CARS) microscopy.

The ability to observe samples qualitatively at the microscopic scale has greatly enhanced our understanding of the physical and biological world throughout the 400 year history of microscopic imaging, but there are relatively few techniques that can truly claim the ability to quantify the local concentration and composition of a sample. We review coherent anti-Stokes Raman scattering (CARS) as a quantitative, chemically specific, and label-free microscopy. We discuss the complicating influence of the nonresonant response on the CARS signal and the various experimental and mathematical approaches that can be adopted to extract quantitative information from CARS. We also review the uses to which CARS has been employed as a quantitative microscopy to solve challenges in material and biological science.

Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy.

We have developed a scheme for determining the time origin by the maximum entropy method (MEM) in time-domain terahertz (THz) emission spectroscopy. By applying the MEM to trial damped sinusoidal waveforms, we confirmed that the MEM gives true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is +/-15 fs for 100-fs-class excitation/sampling optical pulses. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with a finite sampling interval for the Bloch oscillation in a semiconductor superlattice, a misplacement of the time origin was indeed detected with an accuracy limited by the worse of the MEM inherent uncertainty and the sampling interval.

Quantitative CARS spectroscopy using the maximum entropy method: the main lipid phase transition.

The maximum entropy method for phase retrieval of multiplex coherent anti-Stokes Raman scattering (CARS) spectra is described in detail and applied to the time-resolved measurement of the main lipid phase transition of small, unilamellar 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) vesicles subject to a 3 min temperature sweep. Since the--thus derived--imaginary part of the third-order CARS susceptibility can be directly related to the linear vibrational spectrum, the multiplex CARS spectral data can be analyzed quantitatively and without prior knowledge of the sample. It is shown that the maximum entropy model provides an exact description of the original data, including the noise, if all available autocorrelation functions are used. Our findings confirm the acyl-chain order-disorder phase-transition behavior of small, unilamellar lipid vesicles.

Spectroscopic analysis of the oxygenation state of hemoglobin using coherent anti-Stokes Raman scattering.

A method for noninvasively determining blood oxygenation in individual vessels inside bulk tissue would provide a powerful tool for biomedical research. We explore the potential of coherent anti-Stokes Raman scattering (CARS) spectroscopy to provide this capability. Using the multiplex CARS approach, we measure the vibrational spectrum in hemoglobin solutions as a function of the oxygenation state and observe a clear dependence of the spectral shape on oxygenation. The direct extraction of the Raman line shape from the CARS data using a maximum entropy method phase retrieval algorithm enables quantitative analysis. The CARS spectra associated with intermediate oxygenation saturation levels can be accurately described by a weighted sum of the fully oxygenated and fully deoxygenated spectra. We find that the degree of oxygenation determined from the CARS data agrees well with that determined by optical absorption. As a nonlinear optical technique, CARS inherently provides the 3-D imaging capability and tolerance to scattering necessary for biomedical applications. We discuss the challenges in extending the proof of principle demonstrated to in vivo applications.

Comparison of subtractive Kramers-Kronig analysis and maximum entropy model in resolving phase from finite spectral range reflectance data.

The maximum entropy model (MEM) and Kramers-Kronig (K-K) analysis were compared with the aim of phase retrieval from reflectance. The object was to test two different phase-retrieval methods when reflectance is known at a finite frequency range and data fitting is not performed beyond the finite frequency band. In addition, it was assumed that the phase is known only at one or two anchor points. As an example, we study the terahertz reflection spectrum related to a semiconductor and an optical spectrum of potassium chloride. It is shown that the MEM resolves the complex refractive index of a medium, in the vicinity of initial and final points of the spectra, better than singly and doubly subtractive K-K relations. Both methods give only satisfactory results in the event of one anchor point, but in the case of two anchor points, the MEM is better than doubly subtractive K-K. It is proposed that the MEM should be used instead of K-K analysis, for a priori information of phase at two anchor points, for the purpose of resolving the complex refractive index of a medium from reflectance with high accuracy.

Direct extraction of Raman line-shapes from congested CARS spectra.

We show that Raman line-shapes can be extracted directly from congested coherent anti-Stokes Raman scattering (CARS) spectra, by using a numerical method to retrieve the phase-information hidden in measured CARS spectra. The proposed method utilizes the maximum entropy (ME) model to fit the CARS spectra and to further extract the imaginary part of the Raman susceptibility providing the Raman line-shape similar to the spontaneous Raman scattering spectrum. It circumvents the challenges arising with experimentally determining the real and imaginary parts of the susceptibility independently. Another important advantage of this method is that no a priori information regarding the vibrational resonances is required in the analysis. This permits, for the first time, the quantitative analysis of CARS spectra and microscopy images without any knowledge of e.g. sample composition or Raman response.

Simulation on wavelength-dependent complex refractive index of liquids obtained by phase retrieval from reflectance dip due to surface plasmon resonance.

A method for the calculation of the wavelength-dependent complex refractive index of absorbing liquid from reflectance in the vicinity of surface plasmon resonance (SPR) is presented. The calculation is based on the maximum entropy method (MEM). As an example, phase retrieval from a simulated SPR reflectance of a red colored liquid solution is carried out. It is proposed that MEM can be applied to wavelength-dependent complex refractive index assessment from reflectance of absorbing liquids in SPR measurement in wavelength scanning mode.