Recent advances in optical spectroscopy of human epithelium, high resolution imaging and early cancer detection demand an accurate description of light propagation near the point where light enters the tissue

^{1–8}. However, an analytical method capable of describing the propagation of photons near their point of entry (POE) in turbid media has eluded researchers in diverse fields for generations

^{9–12}. Propagation of electrons in semiconductors

^{13}, anisotropic light diffusion in liquid crystals

^{14}, and coherent backscattering

^{15} are only a few examples that can benefit from such a method. The most accurate treatment of such problems, when interference can be neglected, uses the radiative transport equation,

where

*L*(

**r**,

**s**) is the radiance of light at position

**r** traveling in the direction of unit vector

**s**,

*S*(

**r**,

**s**) describes the distribution of internal sources, and μ

_{a} and μ

_{s} are the absorption and scattering coefficients of the medium, respectively. The unity normalized function

*p*(

**s, s**′) is the phase function of the medium, which describes the angular distribution for a single scattering event (here,

**s**′ is the incoming and

**s** the outgoing direction of scatter)

^{9}.

Unfortunately, no analytical solution to this equation has been discovered for practical scenarios. An often used numerical method is Monte Carlo simulation. Monte Carlo simulation gives accurate results but does not provide any physical insight and often requires excessive computational time. Recent GPU based Monte Carlo methods significantly reduce computational time

^{16–19} thus extending the utility of Monte Carlo simulations to practical applications, but inverse problems with a large number of parameters will greatly benefit from the analytical approach provided here. In the absence of suitable numerical methods, the most widely used approach is to obtain an approximate solution using the diffusion approximation (DA) to the radiative transport equation. The DA works best when scattering is isotropic or near isotropic

^{20}. It is not sensitive to the particular form of the phase function, reducing phase function information to a single parameter,

, called the average cosine of the phase function or the anisotropy factor, which in turn is incorporated into the reduced scattering coefficient,

. Hence,

becomes the only parameter in the DA to describe scattering. While the diffusion approximation can provide analytical solutions in some cases, it does not provide accurate results for the critically important region near the POE.

This limitation of the DA has been a major impediment, particularly for light scattering in biological media. Many investigators have attempted to amend the DA by solving the radiative transport equation using more complicated phase functions

^{21–23}. One such approach, called the δ-

*P*_{1} approximation, is to solve the equation for the delta-Eddington phase function, which results in some improvements for characterizing forward scattering media

^{24}. Another approach is to decompose the phase function into a series of Legendre polynomials and to construct a system of radiative transport equations corresponding to each Legendre term. Because it becomes very cumbersome to solve the system of partial differential equations as the number of Legendre terms increases, the

*P*_{3} approximation uses only the 0

^{th} through 3

^{rd} terms in the Legendre series

^{25}. Both approximations are substantial improvements over the DA because they take the phase function into account and broaden the parameter space with which the approximation can be used

^{26,27}. However, until now, no proposed analytical method has addressed the important regime of light scattering near the POE (e.g., at source-detector separations less than

) well enough to be widely used as common practice.

Here, we introduce a novel approach that results in the first successful attempt to analytically describe spatially dependent diffuse reflectance near the POE in turbid media with arbitrary phase function and absorption. The phase function corrected diffusion approximation (PFC) takes advantage of the delta-isotropic phase function, described below, to derive a two-part expression for the diffuse reflectance. The first part is identical to the DA for a pencil beam illumination, while the second part approximates the changes in reflectance due to the specific form of the phase function. The resulting predictions are shown to be in excellent agreement with experiment and also with Monte Carlo simulations at all distances, including the critical area near POE.