## Results: 7

1.

(a) Schematic of the experimental setup. The expanded incident laser beam passes through a 20/80 beamsplitter and is delivered to the scattering sample through an objective. Light scattered from the sample is collected by the same objective and is reflected by a beamsplitter toward the imaging CCD. (b) Experimentally collected spatially resolved reflectance.

2.

The medium is illuminated with a narrow collimated beam. In addition to unscattered photons, low-angle scattered photons are also shown. Any point along the beam can be a source of a single large angle turn, which is followed by low-angle scattered photons propagating toward the surface. The large angle turn is described by the phase function, providing phase function correction. Here .

3.

Red circles - experiment, the bars on the experimental points show the range of measurements. Black dashed line - standard diffusion approximation (DA), blue dashed line - PFC diffusion theory. Inset: Expanded view of the critical region near the point-of-entry to show the range of measurements more clearly. The PFC diffusion theory demonstrates excellent agreement with the experiment while the DA deviates significantly near the POE.

4.

(a) Angles between vectors

**s**,**s**′, and**j**_{d}. The vector**s**defines the*z*′. The vectors**s**and**j**_{d}(**r**) define the (*y*′,*z*′) plane. (b) Contributions of the neglected integral term, PFC term, and diffuse reflectance term to the reflectance. Black line - radially dependent diffuse reflectance*R*_{d}(ρ), blue line - contribution of the PFC term*R*_{p}(ρ), red line - contribution of the neglected integral term Δ*R*_{p}. The contribution of the neglected integral term, Δ*R*_{p}(ρ), is less than 4% of*R*_{p}(ρ) for . The contribution of Δ*R*_{p}(ρ) is less than 7% of*R*_{d}(ρ) for . The calculations are performed for Henyey-Greenstein phase function,*g*=0.9 and .5.

The phase function correction to the fluence rate within the medium (solid lines) is calculated for the Henyey-Greenstein phase function with

*g*=0.9. The Monte Carlo data (circles) are obtained taking the difference in the fluence rate from a simulation with isotropic scattering (*g*=0) and a simulation with*g*=0.9. In all cases, The depth values corresponding to the colors black, green, red, blue and magenta are 0.01, 0.06, 0.1, 0.24 and 0.5 respectively.6.

(a) Dimensionless reflectance for Henyey-Greenstein (H-G) and Mie phase functions with

*g*=0.95 and . Blue lines and circles - Henyey-Greenstein phase function, red lines and circles - Mie phase function, where lines are for PFC diffusion theory and circles are for Monte Carlo simulations. Black dashed line - the standard diffusion approximation (DA). (b) Comparison of PFC diffusion theory with the standard diffusion approximation, the*P*_{3}and δ-*P*_{1}approximations and Monte Carlo simulations for the Henyey-Greenstein phase function with*g*=0.95 and . The error plot shows the percentage error ((*R*−*R*_{MC})/*R*_{MC}·100%), between each of the approximations,*R*, and Monte Carlo simulation,*R*_{MC}.7.

Dimensionless reflectance for the PFC diffusion theory compared with Monte Carlo simulations (MC) and the standard diffusion theory (DA) for the case of the Henyey-Greenstein phase function with different absorptions; (a) blue lines and circles -

*g*=0.9 and relatively weak absorption (), red lines and circles -*g*=0.9 and relatively high absorption (). Dashed lines - standard diffusion approximation, solid lines - PFC diffusion theory, circles - Monte Carlo simulations. Though the standard diffusion theory gives significant errors for the PFC diffusion approximation does not suffer from this problem and agrees with the Monte Carlo simulations for all distances; (b) effect of different anisotropy factors*g*. Red represents*g*=0 and blue represents*g*=0.9. The PFC diffusion theory (solid lines) agrees with Monte Carlo simulations (circles) for all distances, while the standard diffusion theory (black dashed line) does not have any dependence on the anisotropy factor*g*. (Note that the standard diffusion theory and the PFC diffusion theory curves exactly overlap for*g*=0.) Inset: the separate contributions of the diffuse reflectance (black line) and PFC reflectance (blue line) and their asymptotic behavior (dashed lines).