Motivation
Diffuse reflectance spectroscopy consists of determining the radiative properties of an absorbing and scattering sample from diffuse reflectance measurements. It has been applied to noninvasive healthmonitoring of in vivo biological tissues, quality control in agricultural applications, and remote terrestrial sensing, for example. In many biological applications, the irradiated medium can be modeled as a strongly scattering multilayer medium whose radiative properties are constant within each layer but differ from layer to layer. For example, the human cervix, colon, and skin have been modeled as twolayer optical systems. Skin consists of an outer layer called the epidermis and of an underlying layer called the dermis. The epidermis is characterized by strong absorption in the ultraviolet and visible part of the spectrum due to melanin. On the other hand, the blood and connective tissues are responsible for absorption and scattering in the dermis. In addition, the absorption characteristics of blood depend on the concentrations of oxyhemoglobin and deoxyhemoglobin. The objective of this study is to develop simple and accurate expressions for the diffuse reflectance of semiinfinite and twolayer absorbing and scattering media. Such expressions could be combined with an inverse method to retrieve the radiation properties and thickness of these media based on spectral diffuse reflectance measurements. Approach
Simple analytical expressions for the diffuse reflectance of the semiinfinite and twolayer media considered were first derived using the two flux approximation. Then, parameters appearing in the analytical expression previously derived were instead fitted to match results from more accurate Monte Carlo simulations. A single semiempirical parameter was sufficient to relate the diffuse reflectance to the radiative properties and thickness of the semiinfinite and twolayer media. All media were assumed to be nonemitting, strongly forward scattering. The incident light source was a collimated, monochromatic, and normally incident beam of infinite radius. 

SemiInfinite Medium 
TwoLayer Medium 

First, a homogeneous semiinfinite slab characterized by μ_{a}, μ_{s}, g, and n_{1} was considered. A wide range of property values was explored namely 0.3 < μ_{s}(1g)/[μ_{a}_{ }+ μ_{s}(1g)] < 0.99, 0.70 < g < 0.90, and n_{1} =1.00, 1.33, 1.44, 1.77, and 2.00. 
A twolayer medium was considered. It consisted of a planeparallel slab (Layer 1) characterized by μ_{a,1}, μ_{s,1}, g_{1}, n_{1} and thickness L_{1} supported by a semiinfinite sublayer (Layer 2) characterized by μ_{a,2} and μ_{s,2}, and μ_{g,2}. The thickness of Layer 1 was considered between 0 and infinity. The transport single scattering albedo of the slab μ_{s,1}(1g_{1})/[μ_{a,1 }+ μ_{s,1} (1g_{1})] and of the sublayer μ_{s,2}(1g_{2})/[μ_{a,2 }+ μ_{s,2} (1g_{2})] varied between 0.50 and 0.99 which covers a wide range of biological tissues such as skin. 

The semiempirical model describing reflectance from a single layer system was given by
where the reflectivity at the medium/air interface accounting for the fact that intensity inside the medium
The approximate diffuse reflectance was approximated where the values of A_{i} and B_{i} for N=3 can be found in Ref. [1]. 
The semiempirical model describing reflectance from a two layer system was given by
where R_{} is given above and R^{*} is
Here, Y_{1} is the optical thickness parameter of Layer 1. The parameter α was found empirically and given for various values of n_{1} in the figure below. 

Conclusions We presented computationally efficient and accurate semiempirical models of light transfer suitable for realtime diffuse reflectance spectroscopy. The models predict the diffuse reflectance of both (i) semiinfinite homogeneous and (ii) twolayer media exposed to normal and collimated light. The twolayer medium consisted of a planeparallel slab of finite thickness over a semiinfinite layer with identical index of refraction but different absorption and scattering properties. The model accounted for absorption and anisotropic scattering, as well as for internal reflection at the medium/air interface. The present model can be used for a wide range of applications including noninvasive diagnosis of biological tissue. Publications D. Yudovsky, A. Nouvong, K. Schomacker, and L. Pilon, 2011. Monitoring Temporal Development and Healing of Diabetic Foot Ulcer Using Hyperspectral Imaging, Journal of Biophotonics, Vol. 4, No. 7–8, pp. 565–576. doi: 10.1002/jbio.201000117. D. Yudovsky and L. Pilon, 2011. Retrieving Skin Properties From In Vivo Diffuse Reflectance Measurements on Human Skin, Journal of Biophotonics, Vol. 4, No.5, pp.305314, doi: 10.1002/jbio.201000069. D. Yudovsky, A. Nouvong, K. Schomacker, and L. Pilon, 2011. Assessing Diabetic Foot Ulcer Development Risk with Hyperspectral Tissue Oximetry, Journal of Biomedical Optics, Vol.16, No.2, 026009. doi: 10.1117/1.3535592. D. Yudovsky and L. Pilon, 2010. Modeling of Local Excitation Fluence Rate and Florescence Emission in Absorbing and Strongly Scattering Multilayered Media. Applied Optics, Vol. 49, No. 31, pp. 6072–6084. doi: 10.1364/AO.49.006072 D. Yudovsky , A. Nouvong, and L. Pilon, 2010. Hyperspectral Imaging for Diabetic Foot Wound Care, Journal of Diabetes Science and Technology, Vol.4, No.5, pp. 10991113. 