## 1 Introduction  
Both spontaneous and coherent Raman scattering effects have been employed in spectroscopic systems applied to trace detection and identification of chemicals [1; 2] and as a method to measure concentrations of various components in gases [3; 4; 5], liquids and solids [6; 7]. In spontaneous Raman scattering, the signal is directly proportional to the number of molecules interacting with a single beam of input light. By contrast, coherent anti-Stokes Raman scattering (CARS) is a nonlinear process where the molecules are first put into a state of coherent oscillations, resulting in an increased probability for a probe pulse to scatter and produce Stokes or anti-Stokes shifted photons [8]. Therefore, CARS-based techniques possess an inherent ability to offer higher signal strength and faster collection speed compared to approaches based on spontaneous Raman scattering [9]. With the development of ultrashort pulsed laser sources, CARS imaging opened the possibility to study the dynamics of such rapidly changing systems as gas flows [10]. The great flexibility of the CARS technique makes it a popular instrument for performing thermometry measurements for various gases: nitrogen and oxygen [11; 12; 13], carbon dioxide [14], as well as methane and hydrogen [15]. CARS techniques allow concentration and temperature measurements in harsh environments and at high temperature during the combustion process [16; 17]. Moreover, one-dimensional [18; 19], as well as two-dimensional [20] single-shot thermometry and mapping of reactants and products in gas flows can be performed by different experimental approaches. The CARS signal is expected to scale proportionally to the second power of the concentration of target molecules in a sample; however, Raman linewidth changes [21], signal re-absorption [7], as well as nonresonant contribution from background molecules [22; 23] may alter the dependence of the CARS signal on the concentration, making it difficult to correlate the signal strength with the distribution of those molecules. Gas flows, in particular, are multi-component systems, usually containing a large number of background molecules contributing to the CARS signal background. Some of the aforementioned problems, i.e., Raman line broadening, can be resolved by taking into account line broadening coefficients [24] as well as dephasing rates for the gas species [25]. Here we show that the femtosecond adaptive spectroscopic technique (FAST) for CARS [26] maintains the proper dependence of the CARS signal on concentration and thereby assures a direct correspondence between CARS intensity image and molecular distribution. FAST CARS is a combination of methods aimed at optimizing the CARS signal and suppress the noise background. In the 'hybrid' implementation of FAST CARS, we use two ultrashort pump-Stokes excitation pulses to prepare a coherent oscillation of molecules (Figure 1a), in combination with a narrowband delayed probe pulse to provide near-perfect non-resonant four-wave mixing background suppression [27; 28]. Recently, in addition to gas-phase studies, this scheme was successfully used for microspectroscopy [29; 30], and for spectroscopic molecular sensing aided by plasmonic nanostructures [31; 32; 33]. In this work, we experimentally demonstrate that the hybrid CARS signal obtained for molecular oxygen scales as the square of the number of molecules in two scenarios. We should note that the determination of molecular concentration based on the quadratic dependence of the CARS signal intensity on the number of molecules has been performed before using ns-CARS [34] and hybrid CARS [35] techniques. However, here we focus on demonstrating that this dependence can be maintained at different experimental conditions without prior calibration of specific gases composition or concentration.