CARS spectral fitting with multiple resonant species using sparse libraries

The dual pump CARS technique is often used in the study of turbulent flames. Fast and accurate algorithms are needed for fitting dual-pump CARS spectra for temperature and multiple chemical species. This paper describes the development of such an algorithm. The algorithm employs sparse libraries, whose size grows more slowly with number of species than a conventional library. The method was demonstrated by fitting synthetic "experimental" spectra containing 4 resonant species (N2, O2, H2 and CO2), both with noise and without it, and by fitting experimental spectra from a H2-air flame produced by a Hencken burner. In both studies, weighted least squares fitting of signal, as opposed to least squares fitting signal or square-root signal, was shown to produce the least random error and minimize bias error in the fitted parameters. I. Introduction Coherent anti-Stokes Raman spectroscopy (CARS) is a non-linear spectroscopic technique in which three laser beams, two pump beams and a Stokes beam, are focused and crossed at their focal point, and a fourth signal laser beam is generated at the intersection through a four-wave mixing process. Resonances associated with Raman active molecular rotational-vibrational transitions strengthen the CARS signal in a manner dependent on gas composition and temperature. The wavelength of excitation or "Raman shift" corresponds to the difference in energy between a pump beam and a Stokes beam (longer wavelength) photon. In the broadband CARS technique one of the laser beams, the Stokes beam, is spectrally broad whereas the other pump beams are spectrally narrow, and this enables multiple resonances to be excited. The signal beam is spectrally broad and carries with it the spectral signature of the gases present (over some range of wavelengths). This signal beam is dispersed by a spectrometer and spectra are compared to analytical models to obtain information on composition and temperature. In the dual-pump CARS technique, originally developed by Robert Lucht and coworkers, 1,2 the two pump beams are at different wavelengths and two different ranges of Raman shift may be accessed. In recent years, efforts have been made to utilize the CARS technique to obtain experimental data for development and validation of computational models of supersonic combustion. Data sets have been acquired in a hydrogen-fueled supersonic combustor using the CARS technique 3 to measure temperature, and the dual-pump CARS technique has been used to measure temperature and composition; 4 in Ref. 3 the resonant species was N2 and in Ref. 4 they were N2, O2, and H2. Both mean flow and turbulence statistics (variances and covariances) were derived from the data, although the uncertainty in the latter was high due both to instrument error and to the small number of measurements on which to base the statistics. More recently, data have been acquired in a hydrogen- combustion heated supersonic coaxial jet with the intent to obtain turbulence statistics of lower uncertainty. 5 These efforts required analysis of large numbers of experimental spectra containing up to three different species. Analysis of dual-pump CARS experimental spectra is typically performed with the Sandia CARSFT code as modified by Lucht 1,2,6 and O'Byrne. 4 The CARSFT code is able to accurately compute spectra for a range of molecular species and, in the fitting mode, to read an experimental spectrum and fit it to a theoretical spectrum by repeatedly computing spectra and minimizing the residual between the theoretical and experimental spectra. This fitting process becomes very slow when the number of molecular species increases, due to a large spectral range, many energy level transitions, and a large number of iterations. Additionally, when fitting multiple variables, CARSFT has problems in finding a global minimum, sometimes returning a result dependent upon the initial guess. At NASA Langley, with the exception of Tedder et al., 5 CARS data has usually been fitted using libraries of spectra. 3,4 Libraries are computed ahead of time using CARSFT. Theoretical spectra are then interpolated from the library rather than being calculated from scratch. However, conventional library size scales to the power of the number of parameters varied (temperature plus resonant chemical species) and tends to become uneconomically