Here is my understand regarding the azimuthal component of Marschner’s model. All the analysis are carried on a cross section of the hair.
I posted two figures for later reference. Figure 1 shows the scattering geometry, and Figure 2 the cross section.
The goal of Marschner’s model is to solve for the outgoing intensity. Please refer to Figure 2 first, the incoming ray has been scattered into three ways, some part of the energy has been reflected directly; some enters the hair, and leaves at the other side of the surface; some enters the hair, reflected at the inner surface and final refracted out. Likewise, when given the outgoing angle phi(the relative azimuth phi_r – phi_i), we can reverse this process to find out the directions of light that contribute to the outgoing intensity. For instance, in Figure 3, the intensity of the outgoing direction (in blue) is the contribution from three (not necessary to be exactly three)incoming beams (in red).
Back to the second figure, setting aside attenuation for the moment, power from a small interval dh in the incident beam is scattered into an angular interval d/phi in the exitant intensity distribution
To complete this equation, we have to take into account attenuation caused by absorption and reflection by introduce an attenuation factor A(p, h) in front of the intensity contributed by a path.
where p = 0 stands for surface reflection R, 1 for refractive transmission TT, and 2 for internal reflection TRT. Refer to Figure 3 for a visualization of the three paths.
Consider Path 1 in Figure 3, the light reflects directly, so the Fresnel factor can be applied to account for the reflected energy, thus A(0, h) = F(eita, gamma_i);
For Path 2, the light first undergoes refraction, thus the energy enters the hair should be (1-F(eita, gamma_i)) multiple the original energy. Then it travels a distance of 2*cos(gamma_t) (with the radius of the hair being unit length), assume sigma_a to be the volumn absorption per unit length, the absorption factor T should be exp(…), finally, it refracts out of hair, so (1-F(eita, gamma_t)) is used to account for the refracted energy.
Path 3 is similar to Path 2, except that the light reflected once at the inner surface of hair, and travels two times of the distance as it in Path 2. Personally, I would write A(2, h) as
One more thing to mention. Before we carry on analysis, we have to project the scattering geometry from 3D to 2D, as well as various parameters, including the index of fraction, and each internal segment should be lengthened by a factor of 1/cos(theta_t).