*2.3. Light Penetration*

In diffuse optics, the light penetration depth *δ* in biological tissues can be attained from the response to an infinitely narrow photon beam normally incident on a semi-infinite medium. For the case where that photon's propagation depth *z* is larger than the light penetration depth, internal fluence distribution predicted from diffusion theory should be [28]:

$$\mathcal{Q}(z) = \mathcal{Q}\_0 \, k \, \exp(-z/\delta) \tag{6}$$

where *k* is a scalar that depends on the amount of backscattered reflectance, ∅<sup>0</sup> is the incident irradiance, and ∅(*z*) represents a function of photon fluence. The light penetration depth is defined as [29]:

$$\delta = \frac{1}{\sqrt{3\mu\_a(\mu\_a + \mu\_s(1-\mathfrak{g}))}} = \frac{1}{\mu\_{eff}}\tag{7}$$

where *μ<sup>a</sup>* is the absorption coefficient, *μ<sup>s</sup>* is the scattering coefficient, *g* is anisotropy factor, and *μeff* is the effective attenuation coefficient.

According to this, the light penetration depth is estimated to be 1.50–6.00 mm for apple tissues with typical *μ<sup>a</sup>* and *μ<sup>s</sup>* coefficients of 0.01–0.05 mm−<sup>1</sup> and 9.00–28.00 mm<sup>−</sup>1, respectively. However, the estimated depth does not always stand for the actually detectable depth for a general imaging system under spatially extended wide-field or broad-beam illumination. The fluence rate and reflectance properties of spatially modulated photon density plane waves in the SFDI are described in the study of Cuccia, Bevilacqua, Durkin, Ayers, and Tromberg [22], in which the effective penetration depth *δ eff* is concisely defined as:

$$
\mu'\_{eff} = \left(\mu\_{eff}^2 + k\_x^2 + k\_y^2\right)^{1/2} = \frac{1}{\delta'\_{eff}}\tag{8}
$$

where *μ eff* is a scalar attenuation coefficient, *kx* and *ky* are variable coefficients related to spatial frequencies *fx* and *fy* (*kx* = 2*π fx*, *ky* = 2*π fy*), and *δ eff* is the effective penetration depth, which is inversely proportional to spatial frequency. The above mathematical formula just provides a simple conceptual framework to understand the transmission of modulated scalar photons in a turbid medium. In practice, the detected signal is mostly due to the photons backscattered close to the illumination source, which corresponds to a far more superficial depth of tissue interrogation than that derived from diffuse light attenuation [30]. This is equivalent to calculating the relative probability that a photon will visit a certain location in tissue before its detection. In the reflectance measurement geometry with spatially modulated illumination, the reemitted light intensity decays by many orders of magnitude within millimeters. Therefore, in using the formula to calculate the light penetration depth, there directly exists some unreasonable aspects. In this study, well-designed experiments were conducted, coupled with two evaluation parameters (image contrast and PVR), to explore the light penetration depth in apple tissues using SFDI.
