1. Introduction

According to the CIE International Lighting Vocabulary (ILV) [1], reflectance is defined as “the ratio of the reflected radiant flux or luminous flux to the incident flux in the given conditions”, while transmittance is defined as “the ratio of the transmitted radiant flux or luminous flux to the incident flux in the given conditions”. Both quantities describe the interaction of the radiant flux (${\varPhi }_\textrm {i}$) with the objects, but for given conditions. These conditions include the irradiated area of the surface ($A_\textrm {i}$), the collection area of the surface for which the optical radiation is evaluated ($A_\textrm {r}$), the irradiation and collection solid angles ($\omega _\textrm {i}$ and $\omega _\textrm {r}$) (Fig. 1, Table 1), and other conditions not addressed in this work such as polarization. Therefore, when reporting the value of reflectance or transmittance, these conditions must be specified explicitly. Quantities such as diffuse reflectance, specular (regular [1]) reflectance, diffuse transmittance, regular transmittance or others [2] are defined to implicitly include well-established geometrical conditions. Many other quantities might be defined, that would provide very different results, making the measurement of reflectance and transmittance in given conditions far from being a general characterization of the interaction of the radiant flux with objects.

 

Fig. 1. Variables used in the definition of the scattering quantities. Quantities and symbols are given in detail in Table 1 for a better understanding of the equations.

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Table 1. List of quantities and symbols.

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In order to obtain a more general description, quantities must be defined that will allow the reflectance and the transmittance to be derived for any condition. Nicodemus $et$ $al.$ [3–5] defined the Bidirectional Reflectance Distribution Function (BRDF) with the purpose of describing such a fundamental quantity for reflectance. This function describes the bidirectional reflectance for infinitesimal solid angles and for any irradiation ($\bf {r}_\textrm {i}$) and collection ($\bf {r}_\textrm {r}$) directions (Fig. 1, Table 1). The BRDF should allow reflectance to be calculated for any condition via integration, as thoroughly described in Ref. [5], where it is defined as:

Differentials are used to indicate that the irradiation is incident from a differential element of solid angle $\omega _\textrm {i}$ in direction of $\mathbf {r}_\textrm {i}$, with $\textrm {d}E(\mathbf {r}_\textrm {i})=L_\textrm {i}(\mathbf {r}_\textrm {i})\cos {\theta _\textrm {i}}\textrm {d}\omega _\textrm {i}$, where $L_\textrm {i}$ and $\theta _\textrm {i}$ are the incident radiance and the incidence angle, respectively. In this and in the following equations, Leibniz notation is used, for which quotients of differentials can be interpreted as derivatives (non-standard analysis). To our knowledge, this interpretation does not incur any contradictions in analogous expressions with radiometric quantities.

Nicodemus et al. [5] related the BRDF to a more fundamental distribution function, the Bidirectional Scattering - Surface Reflectance Distribution Function (BSSRDF, for which the symbol $f_\textrm {ssr}$ is used in this article; the subscript “ssr” stands for “Scattering - Surface Reflectance”):

The reason why a scattering quantity that depends on $\bf {x}_\textrm {i}$ and $\bf {x}_\textrm {r}$ must be defined is the existence of subsurface scattering. The scattering caused by an object is often due to two optical phenomena: reflection at the air-matter interface (here referred to as insurface reflection), and scattering within the bulk (referred to as subsurface scattering). These two phenomena generate different lateral light propagation distances, which can be defined as the distance between the position where the incident radiant flux irradiates the object and the furthest position where a proportion of it may exit. Insurface reflection produces a very short lateral propagation due to inter-reflections within the concavities in the micro-rough structure of the surface, and approaches zero for flat surfaces. However, subsurface scattering can produce considerable lateral propagation, and the irradiation at $\bf {x}_\textrm {i}$ gives rise to non-negligible radiance at non-irradiated positions $\bf {x}_\textrm {r}$. Very low scattering and absorption coefficients yield a transparent aspect, whereas, at the other extreme, very high scattering coefficients or absorption coefficients provide an opaque aspect. Translucent objects are those with appropriate intermediate scattering and absorption coefficients and thicknesses for permitting light to pass through but not allowing the objects on the opposite side to be clearly visible.

According to Nicodemus et al. [5] [Eqs. (5)–(9) therein], the BRDF at a position $\bf {x}_\textrm {r}$ can be calculated by integrating the BSSRDF via the following reasoning.

Let $\textrm {d}L_\textrm {r}(\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r};{A}_\textrm {i})$ be an element of radiance at position $\mathbf {x}_\textrm {r}$ for given irradiation and collection directions ($\mathbf {r}_\textrm {i}$, $\mathbf {r}_\textrm {r}$), produced by irradiation of an area $A_\textrm {i}$. The area $A_\textrm {i}$ can be considered a collection of differential elements of area $\textrm {d}A_\textrm {i}$, each with an element of incident radiant flux, $\textrm {d}{\varPhi }_\textrm {i}(\mathbf {x}_\textrm {i},\mathbf {r}_\textrm {i}$) (see Fig. 1, Table 1). According to the definition in Eq. (3), $\textrm {d}{\varPhi }_\textrm {i}(\mathbf {x}_\textrm {i},\mathbf {r}_\textrm {i})$ contributes to $\textrm {d}L_\textrm {r}(\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r};{A}_\textrm {i})$ with a proportion given by:

Thus, $\textrm {d}L_\textrm {r}(\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r};{A}_\textrm {i})$ is expressed as the integral of these elementary radiances, i.e.:

According to Nicodemus et al. [5], this equation allows the relation between the BRDF and the BSSRDF to be established under the condition with a (as stated by the authors) “uniform irradiance over a large enough area ($A_\textrm {i}$) of a uniform and isotropic surface”. A “large enough area” $A_\textrm {i}$ is required to make the BRDF invariant around $\mathbf {x}_\textrm {r}$. Because of the dependence of the BSSRDF on $\mathbf {x}_\textrm {i}$ and $\mathbf {x}_\textrm {r}$, this condition is required even in the absence of local variations of the scattering properties of the surface, or under the assumption that any spatial inhomogeneity of the reflectance is spatially averaged by the detector. Therefore, due to this “large enough area” condition, the BRDF definition excludes situations faced with smaller irradiation areas; this leaves the following question unanswered: How can the radiance generated at an object’s surface be described when the irradiated area is not “large enough”?

The radiance of the surface is produced under two different conditions. On the one hand, the radiance at $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$ is caused by both insurface reflection and subsurface scattering, while on the other hand, the radiance at $\bf {x}_\textrm {r}\neq \bf {x}_\textrm {i}$ is caused only by subsurface scattering. Therefore, a discontinuity in the BSSRDF is expected at $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$. This is a very special and important condition that needs to be closely examined.

This article gives insight into the condition in which the radiance is evaluated within the irradiated area and, consequently, is produced by both the insurface reflection and the subsurface scattering. By explicitly considering both contributions, two other scattering quantities are defined: one accounting exclusively for the insurface reflection and the other only for the subsurface scattering. In this regard, these quantities might be considered more fundamental than the BRDF and the BSSRDF, although they are coincident with these two functions apart from the above-mentioned special condition and for materials with negligible subsurface scattering.

2. Discontinuity and dependence on the irradiated area

As introduced above, the radiant flux $\textrm {d}{\varPhi }_\textrm {i}(\bf {x}_\textrm {i},\bf {r}_\textrm {i})$ that irradiates each elementary area $\textrm {d}A_\textrm {i}$ contributes to $\textrm {d}L_\textrm {r}(\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r};{A}_\textrm {i})$ with a proportion given by Eq. (4). In the following derivation, the discontinuity of $\textrm {d}L_\textrm {ssr}(\mathbf {x}_\textrm {i},\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r})$ at $\mathbf {x}_\textrm {i}=\mathbf {x}_\textrm {r}$ is explicitly taken into account.

The radiance $\textrm {d}L_\textrm {ssr}$ is expressed as the sum of the radiance produced by the insurface reflection ($\textrm {d}L_\textrm {is}$) and the radiance produced by the subsurface scattering ($\textrm {d}L_\textrm {ss}$):

Since $\textrm {d}L_\textrm {is}$ is zero when $\bf {x}_\textrm {r}\neq \bf {x}_\textrm {i}$, Eq. (4) can be written as:

The BSSRDF can be obtained directly from Eq. (8) as:

The differential $\textrm {d}A_\textrm {i}$ at the denominator in the second term of Eq. (10) describes the fact that the variation of the radiance with respect to the radiant flux increases for smaller irradiation areas, and that this variation is not completely defined unless the second term of the equation is negligible with respect to the first.

On the other hand, $f^{\ast }_\textrm {r}$ is obtained from Eq. (8) via integration over $A_\textrm {i}$ to obtain $\textrm {d}L_\textrm {r}(\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r};{A}_\textrm {i})$. In analogy with Eq. (5),

Finally, $f^{\ast }_\textrm {r}$ can be expressed as:

Whereas the first condition of the equation is related to the BRDF, the interpretation of the second one ($\mathbf {x}_\textrm {r}\notin A_\textrm {i}$) is not obvious. At the back of a planar sample with parallel interfaces, the second condition is related to the scattering quantity usually referred to as the Bidirectional Transmittance Distribution Function (BTDF) [9–11]. In such a case, it can be expressed as:

3. Fundamental scattering quantities for reflectance and transmittance

The analysis in the previous section demonstrates the convenience of defining new fundamental quantities to explicitly emphasize the spatial distribution dependence of the irradiation and to completely determine the reflectance and the transmittance of objects without the restriction imposed by such dependence.

The function $f_\textrm {is}(\mathbf {r}_\textrm {i};\mathbf {r}_\textrm {r})$, which is referred to here as the Bidirectional Insurface Scattering function (BIS), is defined, in analogy to $f^{\ast }_\textrm {r}$, by Eq. (11) and has units $sr^{-1}$. In this equation, as commented on above, $L_\textrm {is}$ refers to the radiance produced exclusively by insurface reflection. In contrast with the function $f^{\ast }_\textrm {r}$ defined in Eq. (2), this function is independent of the irradiated area $A_\textrm {i}$ because the radiance involved excludes by definition any radiance produced by subsurface scattering, which is proportional to the total radiant flux on the surface ($E\times A_\textrm {i}$).

The function $f_\textrm {ss}(\mathbf {x}_\textrm {i},\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r})$, which is referred to here as the Bidirectional Bipositional Subsurface Scattering function (BBSS), is defined, in analogy with the BSSRDF, by Eq. (9) and has units $m^{-2}\cdot sr^{-1}$. In this equation, $L_\textrm {ss}$ refers to the radiance produced exclusively by the subsurface scattering. Unlike the BSSRDF, it excludes by definition any radiance produced by the insurface reflection.

The explicit dependence of the scattering quantities $f^{\ast }_\textrm {r}$, $f_\textrm {ssr}$ and $f^{\ast }_\textrm {t}$ on the irradiation area and on these proposed fundamental scattering quantities is given in Eq. (13), Eq. (10) and Eq. (14), respectively, where the “large enough area” condition is not applied.

Note that, for objects for which the optical radiation transmitted at the air-matter interface is negligible with respect to the reflected optical radiation, the following is obtained:

4. Relations between the proposed fundamental scattering quantities and $f^{\ast }_\textrm {r}$

The radiance produced by insurface reflection ($L_\textrm {is}$) and the radiance produced by subsurface scattering ($L_\textrm {ss}$) are not directly evaluable by measurement when $\mathbf {x}_\textrm {r}\in A_\textrm {i}$; as a consequence, neither $f_\textrm {is}$ nor $f_\textrm {ss}$ is directly measurable except in the case described by Eq. (15). However, they can be related to $f^{\ast }_\textrm {r}$ as explained in the following.

Over a range of $\left |\bf {x}_\textrm {i}-\bf {x}_\textrm {r}\right |$ length values much smaller than the absorption mean free path, the optical radiation flux crossing the air-matter interface at position $\bf {x}_\textrm {i}$ and then undergoing multiple scattering within the bulk can go back across the air-matter interface at any position within this range with a very similar probability. Therefore, assuming $\sqrt {A_\textrm {i}}$ to be much smaller than the absorption mean free path, $f_\textrm {ss}$ can be regarded as independent of the spatial dimensions $\mathbf {x}_\textrm {i}$ and $\mathbf {x}_\textrm {r}$ [$f_\textrm {ss}(\mathbf {r}_\textrm {i};\mathbf {r}_\textrm {r})=f_\textrm {ss}(\mathbf {x}_\textrm {i},\mathbf {r}_\textrm {i};\mathbf {x}_\textrm {r},\mathbf {r}_\textrm {r})$]; for $\mathbf {x}_\textrm {r}\in A_\textrm {i}$, Eq. (13) can be written as:

By taking the derivative with respect to $A_\textrm {i}$, the following is obtained:

The dependence of $f_\textrm {is}$ on $f^{\ast }_\textrm {r}$ can be expressed from Eqs. (17) and (18) by:

Equations (18) and (19) relate $f_\textrm {ss}$ and $f_\textrm {is}$, respectively, to $f^{\ast }_\textrm {r}$. Equation (18) expresses the fact that the function $f_\textrm {ss}$ is the derivative of $f^{\ast }_\textrm {r}$ under the specific condition $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$. Since, at this condition, the radiance $L_\textrm {ss}$ is not always directly measurable because both the insurface reflection and the subsurface scattering contribute to the reflected radiant flux, it can be more practical to define $f_\textrm {ss}$ as:

For measurements, Eq. (18) and Eq. (19) can be approximated as:

According to Eq. (10), $f_\textrm {ssr}$ (usually obtained as $L_\textrm {r}/{\varPhi }_\textrm {i}$) equals $f_\textrm {ss}$ when it is much larger than $f_\textrm {is}/\textrm {d}A_\textrm {i}$, that is, when the condition $\textrm {d}A_\textrm {i}\gg f_\textrm {is}/f_\textrm {ss}$ is fulfilled. This condition, which involves an infinitesimal element, may be confusing, but makes sense in a BSSRDF measurement, where $\Delta A_\textrm {i}$ instead of $\textrm {d}A_\textrm {i}$ would denote the finite-size irradiated area. The condition implies that $L_\textrm {ss}\gg L_\textrm {is}$ and that most of the total radiance is produced by the subsurface scattering term. On the other hand, according to Eq. (17), $f^{\ast }_\textrm {r}$ (usually obtained as $L_\textrm {r}/{E}$) equals $f_\textrm {is}$ for small enough values of $A_\textrm {i}$, i.e., when $A_\textrm {i}\ll f_\textrm {is}/f_\textrm {ss}$ ($L_\textrm {is}\gg L_\textrm {ss}$).

From these considerations, a very simple formula can be given for the measurement of $f_\textrm {is}$ and $f_\textrm {ss}$ at the condition $\mathbf {x}_\textrm {r}\in A_\textrm {i}$:

Here, the increment $\Delta A_\textrm {i}$ is used instead of the infinitesimal $\textrm {d}A_\textrm {i}$ because finite intervals are involved in the measurements.

Notice that Eqs. (23) and (24) require at least rough previous knowledge of the order of magnitude of the values of $f_\textrm {is}$ and $f_\textrm {ss}$ to select the adequate irradiated area. Otherwise, an iterative procedure can be used.

One may question why a value of $\Delta A_\textrm {i}$ that is larger than $A_\textrm {i}$ needs to be used for measuring $f_\textrm {ss}$, since $\textrm {d}A_\textrm {i}$ is included in $A_\textrm {i}$ (Fig. 1, Table 1). However, it should be noted that $f_\textrm {is}$ is not related to $f_\textrm {ss}$, and that both measurements described in Eqs. (23) and (24) are independent. The difference in the nomenclature of both irradiated areas simply allows these equations to be related to Eqs. (10) and (17).

5. Illustration by means of a measurement

Experimental results are shown in Fig. 2 to illustrate the conclusions from this article. The measurements were carried out by means of the goniospectrophotometer described in [12,13], with a camera for spatial resolution in the detection path. The radiance of the reflected optical radiation, the size of the irradiated area and the irradiance on the sample were measured, allowing both $f^{\ast }_\textrm {r}$ and $f_\textrm {ssr}$ to be obtained from the same measuring procedure. The irradiated area was uniform and always much larger than the collection area, which is given by the field-of-view area of the pixels of the camera ($\Delta A_\textrm {r}=2\times 10^{-3}$ mm$^2$). The data presented in Fig. 2(a) consist of BSSRDF measurements of a translucent sample made of a polycarbonate with organic scattering particles. The data was obtained at a fixed polar angle of incidence of 15$^\textrm {o}$, a fixed polar angle of collection of 10$^\textrm {o}$, fixed azimuth angles of incidence and collection of 0$^\textrm {o}$ with respect to the incidence plane, and a fixed wavelength of 600 nm. The figure represents the BSSRDF as a function of $a$, which is the distance from the center of the irradiation spot ($\mathbf {x}_\textrm {i}$) to the collection position ($\mathbf {x}_\textrm {r}$) (Fig. 1, Table 1). Two sets of measurements are shown for two different irradiated areas ($\Delta A_\textrm {i,1}$ and $\Delta A_\textrm {i,2}$), which are circular areas of 2.1 mm$^2$ and 8.4 mm$^2$, respectively. Note that the $a$-axis is on a logarithmic scale to better show the results at the irradiated area. The vertical lines in the plot show the positions of the edges of the irradiation spots. It must be noted that, although the very small collection area allows better insight into the spatial distribution, only one integrated $f_\textrm {ssr}$ value should be reported within the irradiation spot, which corresponds to the elementary irradiated area $\Delta A_\textrm {i}$ for assessing the BSSRDF. This $f_\textrm {ssr}$ value at $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$ can be given as the ratio between the average radiance within the irradiated area and the total radiant flux.

 

Fig. 2. $f_\textrm {ssr}$ (BSSRDF) and $f^{\ast }_\textrm {r}$ measurements supporting the conclusions of this article.

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The data presented in Fig. 2(b) consists of the BSSDRF data in Fig. 2(a) multiplied by the size of the irradiated area. Therefore, the importance of this additional figure lies in its absolute value. The data represents measurements of $f^{\ast }_\textrm {r}$ and can only be interpreted as BRDF ($f_\textrm {r}$) measurements for “large enough” irradiated areas, and at irradiated positions ($\mathbf {x}_\textrm {r}\in A_\textrm {i}$), on the left of the vertical lines, but not when $\mathbf {x}_\textrm {r}\notin A_\textrm {i}$. Note that, to better relate the experimental results to the previous theoretical analysis, the irradiated areas are denoted as $\Delta A_\textrm {i}$ for BSSRDF measurements and as $A_\textrm {i}$ for $f^{\ast }_\textrm {r}$ measurements, although they represent the same area.

Several observations support the proposed formalism:

6. Discussion

The BRDF is well accepted as a fundamental scattering quantity for the determination of reflectance. However, for objects allowing the optical radiation to be transmitted under the surface, the BRDF needs to be defined with “large enough” irradiated areas because it is invariant for larger irradiated areas only at this condition. Consequently, the BRDF does not completely describe the reflectance of these objects for any irradiation condition. For the “large enough” condition, the radiance caused by the subsurface scattering prevails over the radiance from insurface reflection. For the “small enough” condition, the insurface reflection would prevail over the subsurface scattering contribution. For arbitrary intermediate sizes, both contributions are important.

The BSSRDF can be regarded exclusively as a fundamental scattering quantity with the exception of the condition $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$. For this condition, the radiance is produced both by the insurface reflection (proportional to the irradiance) and by the subsurface scattering (proportional to the radiant flux).

Given these considerations, insurface and subsurface scattering quantities need to be given separately, as they are quantities that are more fundamental for reflectance than the BRDF and the BSSRDF.

For transmittance, the condition $\bf {x}_\textrm {r}=\bf {x}_\textrm {i}$ is always excluded. However, the BTDF, as well as the BRDF, needs to be defined with a large enough irradiation area to be invariant for larger irradiation areas and makes the Bidirectional Bipositional Subsurface Scattering function [BBSS, Eq. (9)] a more fundamental quantity for transmittance.

The proposed fundamental scattering quantities can be related to the radiance at the surface of an object in a more practical way as:

In this work, we conclude that the BRDF and the BSSRDF, as defined in Eqs. (1) and (3), can be regarded as fundamental scattering quantities under most of the conditions, but the following points must be considered:

To obtain representative measurements of samples, the size of the evaluated area must be selected such that the result is spatially invariant. In this article, it is assumed that the diameters of both irradiated and collection areas contain many times the wavelength of the optical radiation and the correlation length of any internal structure. This is fulfilled for the shown experimental results, where the material does not have a surface structure and the surface roughness is very small (the amplitude parameter $R_\textrm {a}$ is around 5 nm).

In addition, uniform irradiation has been assumed, meaning that radiance and irradiance are well specified. In some practical cases, uniform irradiation on the scale required for irradiance and radiance measurements is not possible, and these quantities must be determined as the average values of radiant fluxes. In such cases, the measurement equation for the BRDF and the BTDF is obtained from Eq. (2) as:

However, when the object is translucent, this simplification could be wrong, under conditions in which a significant part of the scattered radiant flux emerges from an area larger than $A_\textrm {i}$. In these cases, such a measurement approach is not recommended; here the measurement of the BBSS is a better option, since it does not require a uniform irradiation but a known value of the radiant flux.

7. Conclusions

In this work, relating radiance to irradiation conditions at the surface of objects has been studied theoretically. The Bidirectional Reflectance Distribution Function (BRDF) is only invariant for “large enough” irradiated areas, and cannot be used to predict radiance for smaller irradiated areas. The Bidirectional Scattering - Surface Reflectance Distribution Function (BSSRDF) can include both subsurface and insurface scattering contributions when the radiance is evaluated at an irradiated position, which makes it dependent on the irradiation area. Therefore, the BRDF and the BSSRDF cannot be regarded as fundamental scattering quantities for determining reflectance and transmittance under every irradiation condition. We have proposed the definition of two different scattering quantities, one which exclusively describes insurface reflection and the other which exclusively describes subsurface scattering; these quantities are completely independent of the irradiation conditions and can therefore be regarded as more fundamental than the BRDF and the BSSRDF. Therefore, the proposed quantities should allow the reflectance and transmittance of an object to be calculated under any given irradiation condition. The conclusions drawn from the formalism presented have been supported by measurements. In addition, relating these quantities to the Bidirectional Transmittance Distribution Function (BTDF) has been examined and some practical measurement issues discussed. New procedures to provide traceability to measurements of BIS and BBSS need to be developed, and that will be part of future work.