## 1 Introduction  
The measurement of gas concentration using absorption spectroscopy is a technique with great environmental adaptability and commercial potential. To improve the detection sensitivity, a long optical path length is necessary for trace gas measurement because the signal to noise ratio (SNR) improves with increasing optical path length in absorber [1]. Many methods have been developed to extend the optical path length, such as multi-pass cells [2; 3], cavity ring-down spectroscopy (CRDS) [4], gas in media absorption spectroscopy (GASMAS) and applications of diffuse integrating cavities [5; 6; 7]. Among these methods, integrating cavities have proven advantageous in good stability, effortless laser beam alignment, few interference fringes, and low cost [8; 9]. Besides, a diffuse integrating cavity is an attractive choice for spectral techniques with incoherent light sources, such as Fourier transform infrared (FTIR) spectroscopy [10].  
Many achievements have been obtained on the light transmission law of diffuse integrating cavities [11; 12]. The Beer-Lambert law is usually used to describe the output light radiation from a diffuse cavity, where the effective optical path length (EOPL) is usually introduced to describe the equivalent path length of a photon in the cavity. From previous publications, the EOPL formula has been presented as being related to the launch geometry condition, \(L_{0}\), reflectivity, \(\rho\), of the inner surface, the single pass average path length, \(L_{\rm{ave}}\), and the port fraction, \(f\), of the cavity [13; 14]. Thus, the EOPL should be independent of the detected gas concentration. The experimental methods used to measure the EOPL and its related parameters, such as \(\rho\), \(L_{\rm{ave}}\), and \(f\) have been established and implemented [15; 16]. However, detected absorption signals usually show a nonlinear relation with gas concentration at high gas concentration values [17; 18] in applications. To explain such non-linear phenomenon, the EOPL was considered as a variation with gas concentration in some previous research [19; 20], in which an absorption term (including gas concentration information) was introduced to describe the varied EOPL in a modified formula. Nevertheless, a varying EOPL makes practical gas concentration measurements complicated. Hence, the EOPL should be independent of the detected gas. That is, a restriction might exist while using the Beer-Lambert law to describe the output radiance from a diffuse cavity, which has not been considered in previous research. If the condition is not satisfied, a non-linear phenomenon will be observed. Thus, two concerns should be addressed preferentially. Firstly, what is the restrictive condition? Secondly, how is the condition value determined from an experiment?  
In this study, the relation between absorbance and concentration in gas detection was investigated when a diffuse integrating cavity was used as a gas cell. A beam reflection analysis was implemented to analyze the light transmission process for an arbitrary-shaped diffuse cavity. An approximate condition was proposed and used in the derivation to ensure that the output radiation as a form of the Beer-Lambert law. When this condition was not satisfied, at high gas concentrations, a non-linear phenomenon was observed. Different oxygen concentrations were detected by scanning the P9 absorption line at 763.8 nm based on tunable diode laser absorption spectroscopy (TDLAS). A \(35\times 35\times 35\) cm diffuse integrating cavity was designed and used as the gas cell. A relationship between absorbance and gas concentration was obtained. The absorbance value at the non-linear transition was determined by experiments.