Diffuse Reflectance · Giles Blaney

Background

Semi-infinite diffuse reflectance with extrapolated boundary condition

Diagram of semi-infinite medium geometry showing pencil beam source, detector, real isotropic source at depth z₀, image source at height z₀′, and extrapolated boundary at z_b — Geometry of the semi-infinite medium model. The pencil beam (red) is replaced by a real isotropic source (purple) at depth $z_0$. An image source (purple) is placed symmetrically above the extrapolated boundary (dashed) at height $z_0'$. The detector (blue) sits on the surface at source–detector separation $\rho$.

Consider a pencil beam incident on the surface of a semi-infinite, homogeneous, turbid medium. Within the diffusion approximation, the pencil beam is replaced by an isotropic point source placed one reduced scattering mean free path below the surface at depth $z_0 = 1/\mu_s'$. Generally, the diffusion approximation is valid when $\rho \gg 1/\mu_s'$ and $\mu_a \ll \mu_s'$. Within this approximation, the diffusion coefficient is $D = 1/(3\mu_s')$. To satisfy the extrapolated boundary condition, an image (virtual) source mirrors the real source across the extrapolated boundary at $z_b = -2AD$, giving image height $z_0' = -z_0 + 2z_b$. The parameter $A$ accounts for the refractive-index mismatch between the medium ($n_{in} = 1.4$) and air ($n_{out} = 1$), giving $A \approx 2.95$. With the detector on the surface at source–detector separation $\rho$, the distances to the real and image sources are $r_1 = \sqrt{\rho^2 + z_0^2}$ and $r_2 = \sqrt{\rho^2 + z_0'^{\,2}}$. Given this setup, the following are the Green's functions for the diffuse reflectance.

Continuous-Wave (CW)

The effective attenuation is $\mu_{eff} = \sqrt{\mu_a/D} = \sqrt{3\mu_a\mu_s'}$ and the CW diffuse reflectance is:

$$R_{CW}(\rho) = \frac{1}{4\pi} \left[ z_0 \frac{\mu_{eff} + 1/r_1}{r_1^2}\,e^{-\mu_{eff} r_1} - z_0' \frac{\mu_{eff} + 1/r_2}{r_2^2}\,e^{-\mu_{eff} r_2} \right]$$

which has units mm⁻² (n.b., $R_{CW}$ is normalized by source power).

Frequency-Domain (FD)

For an intensity-modulated source at angular frequency $\omega = 2\pi f_{mod}$, the effective attenuation becomes complex,

$$\tilde\mu_{eff} = \sqrt{\frac{\mu_a}{D} - \frac{i\,\omega}{v\,D}}, \qquad v = c/n_{in}$$

and the complex reflectance $\tilde R_{FD}(\rho)$ takes the same form as $R_{CW}$ with $\mu_{eff}$ replaced by $\tilde\mu_{eff}$. Amplitude $|\tilde R_{FD}|$ and phase $\angle \tilde R_{FD}$ are the typical measured data types. Similar to CW, $\tilde R_{FD}$ has units mm⁻² (n.b., $\tilde R_{FD}$ is normalized by source power).

Time-Domain (TD)

For an impulse source, the diffuse reflectance versus time $t > 0$ is:

$$R_{TD}(\rho,t) = \frac{1}{2}\, \frac{e^{-\mu_a v t}}{(4\pi D v)^{3/2}\, t^{5/2}} \left[ z_0\,e^{-r_1^2/(4Dvt)} - z_0'\,e^{-r_2^2/(4Dvt)} \right]$$

which has units (ps·mm²)⁻¹ (n.b., $R_{TD}$ is normalized by the energy of the source impulse).

Citation

Translated from MATLAB code originally published with: Giles Blaney, Angelo Sassaroli, and Sergio Fantini, Spatial sensitivity to absorption changes for various near-infrared spectroscopy methods: A compendium review, Journal of Innovative Optical Health Sciences, Vol. 17, No. 04, 2430001 (2024). https://doi.org/10.1142/S1793545824300015