*3.6. Back Scattering Approximation*

Back scattering means that the incident ray is reflected in the exactly opposite direction, i.e., both rays go the same path. In case of the first-order BRDF, the incident and reflected rays have a common point. In this case, obviously, *c* = 0 or *ρ* → ∞. As the reflected ray approaches the incident path, *c* approaches 0 and *ρ* diverges. For very large *ρ*, integration effectively ends at *<sup>ζ</sup>* <sup>&</sup>lt; (<sup>∼</sup> 2.5) <sup>1</sup> <sup>2</sup>(1−*t*)*ρrF* 1, where <sup>∼</sup> *α*(*ζ*) ≈ 1.6361*ζ*, and so

$$\begin{array}{rcl} A(t,\rho,F) &=& \int\_0^{\frac{1}{2\rho}} \mathfrak{c}^{-4(1-t)\rho F(\zeta - \frac{(1-t)}{T}\widehat{a}(\zeta))} d\widetilde{\zeta} \\ &\approx& \int\_0^{\frac{1}{2\rho}} \mathfrak{c}^{-4(1-t)(1-0.52(1-t))\rho F\zeta} d\widetilde{\zeta} \\ &=& \frac{1-\mathfrak{c}^{-2(1-t)(1-0.52(1-t))F}}{4(1-t)(1-0.52(1-t))\rho F} \end{array}$$

While LTE (which corresponds to *α* = 0) gives

$$A(t, \rho\_\prime F) = \int\_0^{\frac{1}{2\rho}} e^{-4(1-t)\rho F \zeta} d\zeta = \frac{1 - e^{-2(1-t)F}}{4(1-t)\rho F}$$

And thus, the ratio of the first-order BRDFs (14) is

$$s = \frac{A\_{\text{acurate}}(t, \rho\_\prime \, F)}{A\_{\text{LTE}}(t, \rho\_\prime \, F)} \approx \frac{1}{1 - 0.52(1 - t)} \frac{1 - e^{-2(1 - t)(1 - 0.52(1 - t)) \, F}}{1 - e^{-2(1 - t) \, F}}$$

which for opaque flakes is

$$s \approx 2.08 \frac{1 - e^{-0.96F}}{1 - e^{-2F}} \tag{18}$$

This asymptotic highlights a remarkable fact: the difference between continuous medium (the LTE approach) and individual particles (the accurate approach) persists even for the tiniest flakes for illumination and observation close to normal.

#### **4. Results**

The following paint structure was used in our simulation experiments: the binder refraction index is 1.5, and the paint layer thickness is *H* = 100 μm. Flakes are thin platelets with thickness of 0.5 μm and specular reflectance of 50%. Their specular transmittance is 0% (calculations for opaque flakes) or 50% (calculations for semi-transparent flakes). The angle between the paint surface normal and normal of flakes has a Gaussian distribution with a variance of 2 degrees. The concentration and size of the flakes are described by two dimensionless parameters: relative radius *r* = *R*/*H* and pigment area concentration *F* = *DSH*, which equals to the total area of flakes in paint layer with unit area. In our calculations, *r* varies from 1 for large flakes to 0.01 for small flakes and *F* varies from 2 (high concentration) to 0.2 (low concentration). Flake area has a Gaussian distribution with center at 0 and two different widths: 4000 μm<sup>2</sup> and 40,000 μm2. Effective flake radius *R* (such a value that *π R* <sup>2</sup> equals the mean flake area) was 1 μm, 10 μm and 100 μm.

BRDF is a function of incident and outgoing directions. The azimuth of incidence is irrelevant because of the isotropy of our paint layer. The outgoing direction is taken in the coordinate system with the polar axis along the mirror reflection of the incident ray. This allows to choose high angular resolution for the sharp near-specular peak which is close to the pole regardless of *σ*. Hence, in our coordinate system, BRDF is a function of three angles: the angle between illumination and paint normal *σ* and two angles of observation, *ϕ* and *ϑ*, where *ϑ* is the angle between observation and direction of the mirror reflection and *ϕ* is the rotation of the observation direction about the mirror direction.

BRDF of the paint layer was calculated using three methods: Monte Carlo ray tracing in continuous medium (the LTE approach), ray tracing within paint layer with explicitly specified individual particles (the accurate approach) and the LTE approach with correction (15). For the accurate approach, we created a sample of paint geometry with randomly distributed and not overlapping flakes. Additionally, for comparison, we calculated the BRDF for fixed flake area (with the same mean area *S*). However, the plots for fixed area and for the Gaussian distribution of area are indistinguishable within the line thickness; the LTE results are completely independent from flake area while the accurate ray tracing results change but very slightly, see Section 3.5.

Figure 5 shows the results of calculations using these three models for the case *F* = 2 (high concentration) and *r* = 1 (large flakes) when the difference is maximal. To fit them all in one image, we show BRDF only in the plane of incidence (*ϕ* = 0◦) and for three angles of incidence *σ* = 0 ◦ , 30◦ , *and* 60◦, because the other BRDF plots demonstrate nearly the same relation.

**Figure 5.** BRDF calculated using the accurate method (solid green), LTE (solid black) and the corrected LTE model (dashed dark green). Left column is for opaque flakes (t = 0), and right column is for semi-transparent flakes (t = 0.5). Effective mean flake radius *R* is 100 μm, F = 2.

In Figure 5, we can see that the correction (15) significantly improves the accuracy. Results of the accurate approach and the LTE with correction are close. Additionally, we can see that the role of size is weaker for semi-transparent flakes than for the opaque ones. This is because BRDF depends on flake size through the term *α*, see (17), and this term is scaled by (1 − *t*), so for *t* → 1 it vanishes.

The difference between the LTE and the accurate approaches increases with the increase in *F* or *r* parameters. Figure 6 shows the results of the increasing (from left to right and from top to bottom) parameters for opaque flakes. Figure 7 demonstrates the same tendency for semi-transparent flakes. It is seen that the difference from LTE increases with F and r and that the correction significantly improves the accuracy.

**Figure 6.** Dependence of BRDF on the size and concentration for opaque flakes. BRDF calculated using the accurate method (solid green), LTE (solid black) and the corrected LTE model (dashed dark green). We only show the section in the plane of incidence when the angle of incidence σ = 30◦.

**Figure 7.** Dependence of BRDF on the size and concentration for semi-transparent flakes (t = 0.5). BRDF calculated using the accurate method (solid green), LTE (solid black) and the corrected LTE model (dashed dark green). We only show the section in the plane of incidence when the angle of incidence σ = 30◦.

One can see that our method that calculates BRDF from scaling of the LTE results provides good accuracy in the wide range of flake size and concentration. Naturally, as the flake size decreases, the accurate model for individual flakes and the LTE for continuous medium converge (Figure 6, left column).

As expected from the derived formulae, the case of normal incidence is special. Indeed, for normal incidence and observation close to normal, we have nearly the back scattering when correlation effects do not vanish even for tiny flakes, see Section 3.6. Therefore, now the difference between the accurate approach (individual flakes) and the LTE (continuous medium) must persist even for tiny particles. Figures 8 and 9 demonstrate that this is really so. The effect is more pronounced for opaque flakes (Figure 8, top row) than for semi-transparent flakes (Figure 9, top row). The effect is, however, gone for *σ* = 10◦ (bottom rows of Figures 8 and 9).

**Figure 8.** Singularity of the case of normal incidence for opaque flakes (t = 0). BRDF for the smallest flakes (*R* = 1 μm) calculated using the accurate method (solid green), LTE (solid black) and the corrected LTE model (dashed dark green).

**Figure 9.** Singularity of the case of normal incidence for semi-transparent flakes (t = 0.5). BRDF for the smallest flakes (*R* = 1 μm) calculated using the accurate method (solid green), LTE (solid black) and the corrected LTE model (dashed dark green).

Again, the effect of size is much weaker for semi-transparent flakes than for the opaque ones.

It should be noted that the back scattering occurs for any *σ*. It corresponds to *ϕ* = 0, and *ϑ* = 2*σ*. Therefore, the first order BRDF for *ϑ* → 2*σ* must deviate seriously from the LTE results according to (18), for example, the difference can be up to twofold for *F* = 2. However, this applies to the first-order BRDF, which for this angle is less than 10% of the full one. Meanwhile, for higher scattering orders, the reflected ray does not have a common point with the incident ray. Hence, it is spatially separated even if its direction is exactly opposite to the direction of incidence. Therefore, this effect does not apply to higher scattering orders while they dominate for, for example, *σ* = 10◦, and *ϑ* = 20◦. This is why we do not see significant divergence of the two curves at this point (Figure 9, bottom row).
