Radiative properties of materials with surface scattering or volume scattering: A review

Abstract

Radiative properties of rough surfaces, particulate media and porous materials are important in thermal engineerit transfer between surfaces and volume elements in participating media, as well as for accurate radiometric temperature measurements. In this paper, recent research on scattering of thermal radiation by rough surfaces, fibrous insulation, soot, aerogel, biological materials, and polytetrafluoroethylene (PTFE) was reviewed. Both theoretical modeling and experimental investigation are discussed. Rigorous solutions and approximation methods for surface scattering and volume scattering are described. The approach of using measured surface roughness statistics in Monte Carlo simulations to predict radiative properties of rough surfaces is emphasized. The effects of various parameters on the radiative properties of particulate media and porous materials are summarized.

Keywords

aerogel / fiber / particle scattering / radiative properties / soot / surface roughness

Introduction

Radiative properties of rough surfaces are very important for many applications in various industries [1-4]. In some cases the contact measurement of surface temperature is not feasible. Hence, non-contact temperature measurement techniques, such as radiation thermometry, are desired. The measurement error of a radiation thermometer is related to the uncertainty in emissivity [2]. Characteristics of surface scattering are also essential in modeling radiative heat transfer in an enclosure [1]. Both exact solutions and approximate methods have been applied to model surface scattering problems [5-7]. Exact solutions are obtained by solving Maxwell’s equations, but they usually consume overwhelming computational resources. Approximation methods can often reveal physical insights and are computationally less intensive. Among the approximation methods, the geometric-optics approximation (GOA) is commonly used. Many researchers have applied both analytical models and Monte Carlo numerical simulations based on GOA [8-13]. In most studies, surface roughness statistics are assumed to be Gaussian. However, this hypothetical distribution function may not be a reasonable approximation [14]. It is of critical importance to incorporate actual surface roughness statistics in the modeling of surface scattering. In Section 2, rigorous approaches and several approximations are summarized and comparisons of numerical simulation predictions using real surface statistics and measurement results are emphasized.

Particulate media and porous materials are commonly used in energy conversion systems and cryogenic systems for various purposes [15-17]. For example, fibrous materials are commonly used in thermal insulation to prevent heat loss from energy conversion equipments. Soot in flames can enhance radiative heat transfer in boilers and internal combustion engines [18]. Radiative properties of fibrous materials and soot are important in calculating radiative heat transfer. Aerogel is a quasi-homogenous and light-weight porous material, which can have extremely low thermal conductivity for thermal insulation [19]. Radiative transfer in aerogel is largely dominated by scattering of nano- to micro-porous structures. Another example is light scattering in biological media, where optical methods can be used for disease diagnostics and laser therapy. Section 3 deals with volume scattering and discusses various factors affecting scattering by dispersed media. Finally, an outlook of future research directions for surface and volume scattering is provided in Section 4.

Radiative properties of rough surfaces

Temperature is a critical parameter for many industrial processes. For example, the microstructural development of aluminum alloys is highly temperature dependent. Accurate measurement of temperature is required at virtually every stage of fabricating an aluminum part to achieve the desired attributes [4,20]. Similarly, an accurate temperature measurement is important in achieving temperature uniformity of a silicon wafer and batch repeatability in rapid thermal processing, which is a key technique in semiconductor manufacturing [3,21]. It is difficult to implement thermocouples to fast-moving aluminum parts and rotating wafers. Due to their quick response and non-intrusiveness, radiation thermometers are frequently chosen to monitor the temperature of surfaces for which contact methods are not feasible. However, in radiation thermometry, variations of surface emissivity can remarkably affect the measurement uncertainty [2]. The emissivity of rough surfaces is dependent on wavelength, temperature, and surface conditions including roughness and coating [4,22]. Therefore, significant improvement of emissivity models is required to achieve an acceptable uncertainty in temperature measurement.

Bidirectional reflectance distribution function (BRDF) describes the distribution of scattered radiation over a hemisphere above a rough surface. BRDF is defined as the ratio of the reflected intensity I_r to the incident radiant power on the spectral basis [1,22,23]:

(1)f_{\mathrm{r}}(\lambda ,{\theta }_{\mathrm{i,}}{\phi }_{\mathrm{i}};{\theta }_{\mathrm{r}},{\phi }_{\mathrm{r}})={\displaystyle \frac{\mathrm{d}{I}_{\mathrm{r}}(\lambda ,{\theta }_{\mathrm{i}},{\phi }_{\mathrm{i}};{\theta }_{\mathrm{r}},{\phi }_{\mathrm{r}})}{I_{\mathrm{i}}(\lambda ,{\theta }_{\mathrm{i}},{\phi }_{\mathrm{i}})\cos {\theta }_{\mathrm{i}}\mathrm{d}{\omega }_{\mathrm{i}}}}(\mathrm{s}{\mathrm{r}}^{-\mathrm{1}}).

In the spherical coordinates shown in Fig. 1, (θ_i, ϕ_i) and (θ_r, ϕ_r) denote incidence and reflection directions, respectively, and ωstands for the solid angle. For some cases, the polarization of light scattered by rough surfaces contains useful information for material characterization and then, polarized BRDF is used to describe the properties of the light scattered from the surface by means of the Mueller matrix [24,25]. If a surface is not opaque, bidirectional transmittance distribution function (BTDF) can be similarly defined by replacing the reflected intensity in Eq. (1) with the transmitted intensity. Knowledge of BRDF and BTDF enables the calculation of radiative heat transfer between rough surfaces or the determination of an effective emittance at a given radiation environment. Both BRDF and BTDF are fundamental radiative properties, and other hemispherical properties can be obtained from their integration. For example, spectral directional-hemispherical reflectance R is given by

(2)R={\displaystyle {\int }_{\mathrm{2}\pi }{f}_{\mathrm{r}}(\lambda ,{\theta }_{\mathrm{i}},{\phi }_{\mathrm{i}};{\theta }_{\mathrm{r}},{\phi }_{\mathrm{r}})\cos {\theta }_{\mathrm{r}}\mathrm{d}{\omega }_{\mathrm{r}}.}

Similarly, spectral directional-hemispherical transmittance T can be calculated from BTDF. Based on energy conservation and Kirchhoff’s law, spectral directional emittance is ϵ=1-R-T.

Several books and excellent reviews exist on the theories and numerical simulation methods of light scattering on rough surfaces [5-7,26]. In general, surface roughness is assumed to satisfy Gaussian statistics. Moreover, surface roughness statistics of a two-dimensional (2D) rough surface is mostly assumed to be isotropic so that the autocorrelation function is independent of direction. However, the Gaussian distribution may miss important features of real surfaces because this function does not allow any abrupt event in the rapidly decreasing tails [14]. In some cases, the measurement results cannot be explained if the surface is treated as a Gaussian rough surface [27]. Therefore, it is important to determine the actual roughness statistics of surfaces using state-of-art techniques and evaluate the significance of incorporating the measured roughness statistics into analytical models and numerical simulations.

Surface topographic characterization

Since light scattering is strongly dependent on surface roughness statistics, it is crucial to understand the true statistics of a rough surface. The following parameters are described for one-dimensional (1D) rough surfaces. Similar definitions can be extended to 2D rough surfaces [28]. Surface topography is considered as the variation of height along a sampling direction, z(x). The mean surface height is determined by

(3)\langle z\rangle =\underset{l\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{l}}{\displaystyle {\int }_{\mathrm{0}}^{l}z}(x)\mathrm{d}x,

where l is the sampling length. The root-mean-square (rms) roughness σ is defined by

(4)\sigma =\sqrt{\underset{l\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{l}}{\displaystyle {\int }_{\mathrm{0}}^l{[z(x)-\langle z\rangle ]}^{\mathrm{2}}\mathrm{d}x}}.

The autocorrelation function (ACF) correlates the deviation from the mean value with a translated version by a distance τ [29],

(5)\mathrm{ACF(}\tau \mathrm{)=}{\displaystyle \frac{\mathrm{1}}{{\sigma }^{\mathrm{2}}}}\underset{L\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{l}}{\displaystyle {\int }_{\mathrm{0}}^l[z(x)-\langle z\rangle ][z(x+\tau )-\langle z\rangle ]dx.}

The autocorrelation length is defined as the value ofτwhen ACF(τ) is equal to 1/e.

Surface topography of a rough surface can be measured with an atomic force microscope (AFM) and is mapped into a data array of size M×N, in two orthogonal directions, x and y, respectively. Figure 2 shows an image reproduced from the AFM data.[30] The rough surface can be divided into numerous miniature surfaces as shown in Fig. 3. These miniature surfaces, microfacets, are smooth and their orientations are well represented by the 2D surface slope, which is estimated from the heights of four neighboring points, i.e.,

(6a){\zeta }_x(m,n)={\displaystyle \frac{z_{m+\mathrm{1},n}-z_{m,n}}{\mathrm{2}d}}+{\displaystyle \frac{z_{m+\mathrm{1},n+\mathrm{1}}-z_{m,n+\mathrm{1}}}{\mathrm{2}d}},

(6b){\zeta }_y(m,n)={\displaystyle \frac{z_{m+\mathrm{1},n}-z_{m,n}}{\mathrm{2}d}}+{\displaystyle \frac{z_{m+\mathrm{1},n+\mathrm{1}}-z_{m+\mathrm{1},n}}{\mathrm{2}d}}.

Figure 4 plots the 2D slope distributions for two samples, denoted by Si-1 and Si-2.[31] . The rms roughness is 0.51 μm and 0.63 μm, respectively, and the autocorrelation length is 4.4 μm and 3.1 μm [12]. It is apparent that the slope distribution function (SDF) is significantly different between two samples. The slope distribution of Si-1 is nearly isotropic, as shown in Fig. 4(a), while that of Si-2 is strongly anisotropic. As shown in Fig. 4(b), there are four side peaks located at |ζ_x|≈|ζ_y|≈0.36 besides the dominant peak at the origin (zero inclination).

The unique side peaks in the SDF of Si-2 imply that there are a number of microfacets tilted around 27° with respect to the mean surface. By examining the crystalline structure of silicon, the side peaks can be associated with the {311} planes, since angles between any of the four {311} planes and the (100) plane are 25.2° [32]. On the contrary, the SDF of Si-1 does not show side peaks. The reason may be attributed to the different conditions in silicon wafer production as discussed in detail by Zhu and Zhang [30].

Modeling of scattering from rough surfaces

Rigorous methods and approximations

Experimental demonstration of backscattering enhancement from rough surfaces has inspired many researchers to develop rigorous approaches for scattering simulations [33]. Because backscattering enhancement is caused by multiple scattering from very rough surfaces, small perturbation methods have difficulty in accounting for backscattering enhancement. With advances of computational resources, Maxwell’s equations are numerically solved and reasons of backscattering enhancement are well understood [34,35]. Backscattering enhancement occurs because of the integration of even-order multiple scattering between steep valleys in surface topography. Boundary integral methods, based on the extinction theorem and Green’s second integral theorem, are widely developed and applied to surface scattering problems. Early rigorous methods only deal with 1D perfectly conducting surfaces, and subsequently they are extended to handle scattering from dielectric surfaces [36] and thin-film coated surfaces [37,38]. Other than methods based on the integral equation, the finite-difference-time-domain (FDTD) method relies on the differential numerical scheme to solve Maxwell’s equations [39]. This method calculates fields at discrete regions and a near-to-far field transformation links the computed electromagnetic fields near the surface to the radiative properties in the far field. The rigorous solutions not only have explained backscattering enhancement but also have served as references for approximate methods. Various rigorous approaches by virtue of numerical schemes can be found in Saillard and Sentenac [6] and Warnick and Chew [7].

Despite the success of rigorous solutions, high complexities in formulation and overwhelming requirements in computational resources hinder their applications in surface scattering problems, especially for 2D rough surfaces. On the other hand, analytical approximations are of great importance because they can provide physical insights on light scattering. Two physical-optics-based approximations are the small-perturbation method (SPM) and the Kirchhoff approximation (KA) [26]. SPM is the oldest approximation, often called the Rayleigh-Rice approximation. It takes advantage of a perturbative expansion of the electromagnetic field and can be applied to surfaces of a small rms roughness with respect to the wavelength of interest. KA, also known as the tangent-plane approximation, is valid for large radii of curvature in the surface irregularities, i.e., a locally smooth surface [5]. In KA the electromagnetic field at any point of the rough surface is the sum of the incident field and the reflected field multiplied by Fresnel’s reflection coefficient of the tangent plane at the considered point. In the short wavelength limit λ→0, KA reduces to the geometric-optics approximation (GOA). Both SPM and KA have been continuously improved, resulting in a variety of modified methods, and are extended to scattering from surfaces with thin-film coatings as well [40-42]. Other approximate methods can be found in Elfouhaily and Guérin [43].

Many studies have been devoted to identifying the domain of validity for various approximate methods [36,44-47]. Despite efforts of many researchers, a decisive criterion to select a proper approximation for various scattering problems does not exist. Most validity domains are established for 1D perfectly conducting surfaces with Gaussian statistics, because of the limit of rigorous solutions for 2D rough surfaces. Even for 1D rough surfaces, overwhelming computational requirements do not allow a thorough investigation for all roughness parameters, and besides, different researchers have their own criteria to develop the validity domain. Furthermore, the validity domain is not only dependent on roughness parameters, such as rms roughness and autocorrelation length, but also on the incidence angle and the optical constants of materials. Therefore, care must be taken to select an approximate method for surface scattering problems.

Analytical BRDF model based on GOA

Because GOA is valid within a relatively wide domain and has been employed in many applications, attention will be focused on modelings based on GOA with real surface roughness statistics. In GOA, interference of electromagnetic waves is neglected and optical phenomena such as reflection and transmission are described by the concept of light rays. Analytical models applying the ray-tracing scheme make convenient calculations of BRDFs possible. In-plane BRDF, i.e., the BRDF in the plane of incidence (either ϕ_r=ϕ_i or ϕ_r=ϕ_i+180°), can be expressed as follows [12]:

(7)f_{\mathrm{r}}({\theta }_{\mathrm{i}},{\phi }_{\mathrm{i}};{\theta }_{\mathrm{r}},{\phi }_{\mathrm{r}})={\displaystyle \frac{p_{\mathrm{s}}({\zeta }_x,{\zeta }_y)S({\theta }_{\mathrm{i}},\sigma /\tau )S({\theta }_{\mathrm{r}},\sigma /\tau )}{\mathrm{4}\cos {\theta }_{\mathrm{i}}\cos {\theta }_{\mathrm{r}}{\cos }^{\mathrm{4}}\alpha }}\rho (n_{\mathrm{s}},),

where p_s and S are the SDF and the shadowing function, respectively, n_s is the (complex) refractive index of materials, ϕ is the local incidence angle, and α is the microfacet inclination angle. The 2D slopes in the x and y directions can be written in terms of the incidence and reflection angles,

(8a){\zeta }_x={\displaystyle \frac{\partial z}{\partial x}}={\displaystyle \frac{\sin {\theta }_{\mathrm{i}}\cos {\phi }_{\mathrm{i}}+\sin {\theta }_{\mathrm{r}}\cos {\phi }_{\mathrm{r}}}{\cos {\theta }_{\mathrm{i}}+\cos {\theta }_{\mathrm{r}}}},

(8b){\zeta }_y={\displaystyle \frac{\partial z}{\partial y}}={\displaystyle \frac{\sin {\theta }_{\mathrm{i}}\sin {\phi }_{\mathrm{i}}+\sin {\theta }_{\mathrm{r}}\sin {\phi }_{\mathrm{r}}}{\cos {\theta }_{\mathrm{i}}+\cos {\theta }_{\mathrm{r}}}}.

The microfacet reflectivity　 ρ(n_s,ϕ) is the modulus of the complex Fresnel’s reflection coefficient for given polarization.

In the ray-tracing scheme, shadowing and re-striking (masking) can exist for reflections on microfacets. Re-striking can occur successively, and thus the intercepted ray can be reflected back and forth between microfacets, causing multiple scattering. Multiple scattering becomes significant for surfaces with large rms microfacet slopes or for large angles of incidence or reflection. However, most analytical models including Eq. (7) address first-order scattering only. Although some analytical expressions can account for multiple scattering, additional assumptions are required and they cannot fully capture the characteristics of multiple scattering [9]. Even for first-order scattering, moreover, some analytical BRDF models do not satisfy the energy conservation.

Monte Carlo methods

Ray tracing can be implemented by means of a numerical method, called the Monte Carlo method, in which a large number of rays are illuminated on a rough surface. The propagating direction of reflected rays is determined from Snell’s law, and each ray is traced until it leaves the rough surface. Radiative properties of rough surfaces are obtained from an algebraic sum of total energy of rays. An advantage of the Monte Carlo ray-tracing method is the thorough consideration of multiple scattering, and thus backscattering enhancement can be explained in the aspect of GOA [8]. Although multiple scattering is inherently considered in the rigorous solution of Maxwell’s equations, it does not distinguish the first-order scattering from multiple scattering.

Two ray-tracing algorithms associated with the Monte Carlo method have been developed, namely, the surface generation method (SGM) [8,46] and the microfacet slope method (MSM) [11,13,48]. The major difference between SGM and MSM lies in how to simulate rough surfaces. SGM needs an ensemble of surface realizations in advance, and then light rays are traced according to the physical dimension of each surface realization. In MSM each time a ray hits a microfacet, the orientation of the microfacet is statistically determined as the ray is traced, instead of generating a random rough surface a prior. Shadowing and multiple scattering are inherently taken into account of in SGM while MSM resorts to shadowing functions, which are not applicable for very rough surfaces at oblique incidence [49]. However, MSM takes less computational time and has the advantage for multiscale problems such as light scattering from semitransparent materials [48].

The microfacet reflectivity is calculated based on the local coordinate and is unity for a perfectly conducting surface. Dependence of the refractive index on wavelength is implicitly considered in the microfacet reflectivity. Furthermore, depolarization, i.e., polarization state of the scattered wave is different from that of the incident, can be considered in the calculation of the microfacet reflectivity as well [32]. For a 2D rough surface, even though the incident light is purely s or p polarized, both polarization components generally coexist in the local coordinates. Accordingly, all terms of microfacet reflectivities for co-polarization and cross-polarization should be calculated. Note that depolarization is not considered in Eq. (7) because it does not occur in the in-plane BRDF with first-order scattering only.

Extension of the Monte Carlo methods from opaque rough surfaces to coated surfaces is straightforward [41,49], because the presence of a thin coating changes the microfacet reflectivity only. If a coating is sufficiently thin and has a uniform thickness, the microfacet reflectivity can be calculated from thin-film optics, i.e., Airy’s formula [23]. In addition to the assumption of uniform film thickness, the application of Airy’s formula also requires that the microfacet is sufficiently larger compared with the film thickness and the autocorrelation length is much larger than the rms roughness. Otherwise, the reflected waves by different microfacets may interfere with each other, which is significant for the very precipitous surface roughness, relatively thick coatings, and at large incidence or reflection angles [49].

Besides the selection of an appropriate approach for surfaces scattering problems, surface roughness statistics plays a crucial role. Topography measurement by the AFM allows the precise determination of roughness statistics, as discussed in Section 2.1. Most studies merely import roughness parameters from topography measurement and instead, assume Gaussian statistics for surface roughness. However, Zhu and Zhang [12,30] demonstrated that the roughness statistics deviates from those assumptions for some surfaces. They applied the anisotropic SDF obtained from AFM measurements to Eq. (7). Rather than relying on the AFM measurement, Lee and Zhang [50] obtained SDFs from BRDF measurements of rough surfaces by virtue of an inverse method. Their method allows the error reduction due to surface characterization and takes less time compared with the AFM topography measurement.

Lee et al. [32] developed a Monte Carlo ray-tracing algorithm that incorporates measured surface topography into the simulation of rough surfaces. The surface topographic data are in the form of a 2D array storing heights and locations, similar to surface realizations. Since the number of topographic measurements is limited in the SGM simulation, proper average methods to reduce fluctuations in BRDF calculations are applied. On the other hand, the SDF obtained from topographic measurements is used in MSM and the rejection method is applied to determine the orientation of microfacets. Both the analytical model and the Monte Carlo methods demonstrate that anisotropic roughness effect is so prominent that the BRDF of anisotropic silicon wafers changes drastically, which will be discussed in details in Section 2.4.

Bruce [51] introduced the concept of phase ray-tracing method as an extension of the energy ray-tracing method to calculate the Mueller matrix elements for rough surfaces with large characteristic lengths. The idea is to trace both the field amplitude and phase for rays reflected from the microfacets of a rough surface. Lee and Zhang [52] performed a critical assessment of the applicability of this method for scattering from rough surfaces by comparing the four Mueller matrix elements of 1D dielectric and highly reflecting surfaces. They compared both ray-tracing methods with KA for 1D rough surfaces. When the characteristic length of roughness is much longer than the incident wavelength, these three methods yield essentially the same results. At small characteristic lengths, the phase method predicts coherent peaks resulted from wave interferences. However, the phase method always underpredicts the coherent peaks compared with KA and, furthermore, fails to predict peaks for some elements. Therefore, the phase ray-tracing method cannot accurately model wave features associated with light scattered from rough surfaces of small characteristic length scales due to its inherent assumption from geometric optics, i.e., specular reflection at surface microfacets [52].

Measurement instruments

The instrument for BRDF measurement is usually called optical scatterometer or goniometric reflectometer [53,54]. Although the system configuration may change depending on the purpose of instruments, optical scatterometers have several common components such as a goniometric table, a light source, a detector, and a data acquisition system. The Spectral Tri-function Automated Reference Reflectometer (STARR) at the National Institute of Standards and Technology (NIST) is a high-accuracy reference instrument for the in-plane BRDF measurement in the visible and near-infrared spectrum [53].

The Three-Axis Automated Scatterometer (TAAS) can measure both the in-plane and out-of-plane BRDF [54]. Figure 5 illustrates the system setup of the TAAS using three rotary stages in its design. One rotates a vertically mounted sample around the y-axis to change the incidence angle θ_i, the second rotates detector A in the x-z (horizontal) plane to change the reflection angle θ_r, and the third rotates the arm where detector A is attached to change the azimuthal angle ϕ_r. Manual rotation of the sample around the z-axis adjusts the azimuthal angle ϕ_i. The incident laser beam is located in the x-z plane. A diode laser system serves as the optical source, and a laser controller connected to a lock-in amplifier modulates the incident radiant power. The diode laser is mounted on a thermoelectrically cooled plate to maintain wavelength stability.

A multi-spectrum BRDF measurement system was developed by Dai et al. on the basis of the single reference method [55,56]. Wavelengths of four lasers are 0.6328 μm, 1.34 μm, 3.39 μm and 10.6 μm, respectively, covering from visible to mid-far infrared spectrums. The distance between the detector and the sample can be adjusted for different incident wavelengths. The rotating range of the detector arm is±180° and that of the sample holder is 360°. The angular resolution is 0.036° and the measurement uncertainty of this instrument is about 6.42%. A scatterometer capable of polarized BRDF measurement was developed by Feng and Wei [57]. The optical source is mounted on the rotating stage and the detection system is fixed in the stationary platform. Rotations of two waveplates are automatically controlled. The discrete Fourier transform method is used to retrieve the Mueller matrix of the sample.

BRDF can be measured over the whole hemisphere and the emittance can be calculated from the hemispherical reflectance on the basis of Eq. (2). However, such a large amount of BRDF measurements are not practically feasible. Therefore, an integrating sphere is typically used for measuring the directional-hemispherical reflectance, and subsequently, the emittance is deduced from the measured reflectance based on Kirchhoff’s law. Figure 6 shows the experimental setup of a customized integrating sphere with a center-mount scheme [58]. The inner wall of the 200-mm-diameter sphere is coated with polytetrafluoroethylene (PTFE), and the entrance aperture has a diameter of 25 mm. A silicon detector is mounted at the bottom of the sphere. The sample measurement can be taken after it is confirmed that all light hits the sample surface. The reference measurement is obtained when the sample is rotated out of the beam path. The reflectance of the measured surface is the ratio of these two measurements.

Comparisons of BRDFs

In this section, simulation results of radiative properties by means of GOA are compared with experimental measurements for the two silicon samples mentioned above. Figure 7 shows the in-plane BRDFs of sample Si-1 for random polarization while the observation angle θ_obs is defined as θ_rwhen ϕ_r=ϕ_i=180° and -θ_r when ϕ_r=ϕ_i. These two Monte Carlo methods yield essentially the same results and agree well with the measurements. The BRDF of Si-1 differs from a bell-shaped curve for a Gaussian surface even though its SDF appears isotropic at a glance. Figure 8 shows comparisons for sample Si-2, which is strongly anisotropic and has larger values of σ and rms slope. The Monte Carlo methods capture major features and general trends of the measured BRDF. The modeling results shown in Fig. 8(b) demonstrate the ability to predict two prominent side peaks, associated with the side peaks in the SDF of Si-2 at |ζ_x|≈|ζ_y|≈0.36. Moreover, a small side peak associated with |ζ_x|≈|ζ_y|≈1.15 appears at θ_i=45° [32]. The noticeable difference between modeling and measurement in Figs. 7 and 8 is probably due to limitations of GOA and artifacts in the AFM measurements. Figure 9 shows comparisons of out-of-plane BRDFs of Si-2. Both the predicted and measured BRDFs clearly show variations around the side peaks, reinforcing the importance of topography measurement. Additionally, it should be noted that BRDFs for both polarizations are essentially the same at ϕ_r=0°. Although the incidence is purely s or p polarized, it is evenly decomposed into s and p components in the local coordinate of the microfacet.

Lee and Zhang [50] investigated SiO₂ coating effects on BRDF using the SDF obtained from the BRDF measurement of bare surfaces instead of that from AFM measurement. The BRDF of Si-2 at normal incidence is shown in Fig. 10 for different coating thicknesses. Figures 10(b) and (c) show that the GOA model predictions agree with the experiment results even at large reflection angles. When the coating thickness h is 324.6 nm, the model remarkably underpredicts at |θ_obs|≤45°. The disagreement is related to the evaluation of reflectivity of thin-film coated microfacets. As mentioned in Section 2.2, coating effects are considered in the calculation of microfacet reflectivity with the application of Airy’s formula. When h=324.6 nm, the autocorrelation length of Si-2 may not be large enough to guarantee that multiple reflections within a thick coating layer on a microfacet are independent of those emerging from adjacent microfacets. Another reason is that the coating thickness may not be uniform over the surface and the coated surface becomes even rougher.

Comparisons of hemispherical properties

The detailed information on surface roughness statistics, rather than Gaussian statistics, is also essential for an accurate prediction of emittance [58]. The emittance spectra of Si-1 and Si-2 for random polarization at near normal direction are presented in Fig. 11. The anisotropic modeling results are obtained using the AFM measurements while the Gaussian modeling results are obtained using hypothetical Gaussian surface realizations with the same values of σ and τ. Comparisons in Fig. 11(a) reveal negligible difference between the two modeling results and they both agree well with measurements. The large error at λ=1000 nm is attributed to transparency that is not considered in the model. The measured emittance of Si-2 is enhanced by 7.8% on average, compared with that of a smooth surface. The agreement of modeling results and measurements is much better for the anisotropic modeling than for the Gaussian modeling (See Fig. 11(b)). The reason is that Si-2 which has much more steep microfacets than the Gaussian surface, enhances the cavity effect of trapping the incident radiation by means of multiple scattering. Meanwhile, the difference between anisotropic modeling results and measurement values increases with wavelength. Nevertheless, a recent study indicates that GOA has a larger domain of validity for the emittance modeling than for the prediction of BRDF [59].

Scattering of particulate media and porous materials

The study of light scattering by small particles is important in astronomy, meteorology, biophysics and biomedicine, remote sensing, fire and flame, combustion, thermal insulation, color and appearance, etc. As early as 1881, Lord Rayleigh formulated electromagnetic wave scattering by dielectric spheres whose diameter was much smaller than the wavelength. In 1908, Gustav Mie derived the general solution for scattering by spheres of any size with or without absorption. Mie scattering theory reduces to the Rayleigh limit for small particles and the geometric-optics limit for particles larger than the wavelength [22]. The Rayleigh scattering theory has been extended by other researchers for particles of irregular shapes under conditions that the refractive index of the particle is close to that of the free space and that the particle is relatively small so that the phase shift can be neglected. This type of formulation is often called the Rayleigh-Debye-Gans (RDG) theory, along with other names. Detailed discussions of light scattering by individual particles can be found from classical texts [60-62].

Schuster [63] studied the light radiation through a foggy atmosphere and developed the original two-flux method. In color industry where color matching is of great interest, the Kubelka-Munk model is widely used for the calculation of light reflectance of turbid media, such as paints [64]. Researchers have extended the two-flux approximation and developed the three-flux method and the four-flux method to account for collimated light. However, the angle-resolved scattering of a turbid medium could not be predicted until the more general radiative transfer equation (RTE) was developed [65]. Afterwards, volume scattering has been included in analysis of radiative heat transfer in thermal engineering systems. Radiative transfer and properties of particulate and porous media have been widely studied in the last fifty years or so [1,22].

In order to solve the RTE of any particular system, knowledge of the temperature- and wavelength-dependent absorption coefficient, scattering coefficient, and scattering phase function is required. If light travels in a particulate medium without being scattered more than once, i.e., in the optically thin limit, absorption and scattering of a cloud of particles can be superimposed. This is called single-scattering approximation. On the other hand, a light beam can be scattered more than once and the scattered light can be scattered again by other particles. When multiple scattering becomes important, the exact electromagnetic-wave equation is difficult to solve; nevertheless, RTE can be used to describe energy propagation and exchange [66,67]. If the interference effect of the scattered waves is negligible, the process is categorized as independent scattering. In such cases, the scattered waves are said to be incoherent. When the scattered fields are strongly correlated (i.e., interfere with each other coherently), it is categorized as dependent scattering. The interactions between scattered waves are either through interference of the propagating wave in the far field or through the coupling of evanescent waves in the near field, because diffraction results in both propagating and evanescent waves (whose amplitude decay away from the surface exponentially) [23]. It remains a daunting task to accurately determine the scattering phase function for dependent scattering. In most cases, approximate or semi-empirical phase functions are commonly employed. A comprehensive review of earlier research on scattering in particulate media was provided by Tien and Drolen [68].

Some approximate analytical expressions were derived from RTE to describe the bidirectional reflectance of a turbid medium in closed forms for special cases [65,69-72]. Mueller and Crosbie [73] used an integral transform method to solve 3D radiative transfer in an anisotropically scattering parallel medium. On the other hand, a number of numerical methods have been developed and used to solve RTE in different geometries, such as the discrete-ordinates method [74-76], the spherical-harmonics method [77], the finite-element method [78-80], the finite-volume method [81], the Monte Carlo method [82-87], the spectral element method [88], and the ray-tracing analytical method [89]. Detailed discussion of formulations of these methods can be founded in books by Tan et al. [90] and Liu et al. [91].

In recent years, along with the increasing computational speed and capability, there have been significant advancements in rigorous solutions of wave scattering by nonspherical particles as well as wave propagation in periodic and random inhomogeneous media [92,93]. Kahnert [94] gave a review of the numerical methods used in solving electromagnetic scattering problems. In addition to the separation of variables method and finite element method, FDTD has been widely applied as a powerful method for simulating electromagnetic wave interaction with structural objects considering both near- and far-field effects [95]. Volume-integral method, such as the method of moments (MoM) and discrete-dipole approximation (DDA) are commonly used to study scattering from particle aggregation and irregular shaped objects. Draine and Flatau [96] provided a detailed formulation of DDA. Other methods include surface-integral method, null-field method, the T-matrix formulation, and surface Green’s function formulation [92-94]. Some online codes as well as commercial software packages are available for rigorous solution of Maxwell’s equations in complex structures.

In the following subsections, scattering properties of fibrous materials, soot, aerogel, biological materials and PTFE are summarized and recent studies are emphasized.

Fiber

Fibrous material is commonly used to provide thermal insulation in many high temperature applications, such as thermal protection of the space shuttle during reentry and combustion chamber liners [16]. Even at moderate temperatures of 300-400 K, radiative heat transfer through fibrous media is as important as that by conduction. The pertinent factors affecting scattering of thermal radiation in fibrous materials are the fiber diameter, fiber orientation, spacing between fibers, and the wavelength [17]. Fibers are typically several millimeters in length and a few micrometers in diameter. Because the characteristic wavelength of thermal radiation from 300 K to 1000 K is around 3-10 μm, respectively, fibers are generally considered as infinite long cylinders [97]. The fiber diameter is comparable to the thermal radiation wavelength. Fibers can be either randomly oriented in space or aligned in a plane. The insulation effect of fibrous materials is evaluated by reflectance and transmittance of thermal radiation. Various numerical methods have been applied to solve the radiative transport in fibrous material, such as the two-flux model [97,98], the spherical-harmonics method [99], the discrete ordinates method [100,101], the Monte Carlo method [102], and the ray-tracing method [103].

Fibrous materials used at moderate temperature are of high porosity, i.e., low volume fraction. For example, a typical volume fraction for glass fiber is under 2%. The space between fibers is much larger than fiber diameters and radiation wavelengths. Hence, scattering by fibrous materials can be treated as independent scattering [97]. The radiation diffusion coefficient determined without considering both dependent scattering and interference effects agrees well with the experimental data for arbitrarily oriented quartz fibers with a density of 144 kg/m³ [98]. However, woven fabrics used at high temperature have very low porosity. As the space between fibers decreases, near-field multiple scattering and far-field interference of scattered radiations gradually become dominant. It has been shown that the extinction efficiency and phase velocity of electromagnetic waves in the dependent scattering regime deviate from those in the independent scattering regime [104]. Lee [105] employed a rigorous solution of Maxwell’s equations to obtain extinction efficiency of a dense medium consisting of parallel fibers. The results indicate that independent scattering prevails if the volume fraction is much less than 1% and as the volume faction increases, coherent scattering occurs for closely packed scatters. Moreover, the boundary of scattering regimes is roughly independent of the refractive index for nonabsorbing fibers and is strongly affected by the refractive index for absorbing fibers. Lee [106,107] applied the effective field and quasi-crystalline approximation to treat the multiple scattering interactions in the finite dense layer of cylinders at normal incidence and oblique incidence.

Inspired by the low radiant conductivity of opacified powders, Wang and Tien [97] theoretically investigated radiative heat transfer through opacified cylindrical fibers and found that fine metal fibers could provide high thermal radiative resistance due to their large extinction coefficients and that there was no optimal fiber diameter and the radiative heat flux dropped with decreasing fiber diameter. Effects of coatings on the radiative properties of silica fibers have been examined [16,108]. The reflectance of silica fibers can be enhanced by coatings provided suitable coating thicknesses are used. It is found that silicon is a more effective coating material than alumina at most cases. Tian et al. [103] applied an absorbing film of 50-100 nm to a large optical fiber with a diameter of around 125 μm. It is found that the thin absorbing film has profound effects on the thermal radiative properties. Yang et al. deposited an indium tin oxide (ITO) thin film onto fibers using the sol-gel method and found that the ITO film could act a reflective coating to reduce radiative heat transfer [109]. Yan et al. fabricated composite-material coatings on fibers and found a three-lay stack could improve the heat insulation of fibrous materials [110].

Radiative properties of fibrous media are strongly dependent on fiber orientation. They are strongly affected by the polar orientation of the fibers but are independent of the azimuthal direction of the fibers. There is no backscattering of radiation if all fibers are oriented perpendicular to the boundaries. On the contrary, less radiative heat transfer can be obtained with fibers oriented parallel to the boundaries [98,100]. The orientation-dependent radiative properties of fibrous material are verified by experiment measurements. Measurement results of spectral reflectances of polypropylene fibers in the infrared spectrum of 0.25-15.5 μm show that reflectance is higher when fibers are aligned parallel to the boundaries and lower when oriented perpendicular to the boundaries [111].

In some applications the fiber diameter is much larger than the radiation wavelength. Therefore, the geometric-optics approximation is applied to analyze radiative properties of large fibers. The interaction of diffraction and reflection is neglected. Scattered radiation by a large fiber can be approximated by superimposition of diffraction by a rectangular aperture and specular reflection off the fiber surface [101]. The analysis of dense fibrous materials with size parameters in the geometric optics limit reveals that the shadowing effect has almost no influence on the scattering properties [102]. Radiative properties of fibers with non-circular cross-sectional shapes have been investigated. The cross-sectional shape of a fiber does not strongly affect the scattering and extinction efficiencies provided that two fibers have the same cross-sectional area [112].

Soot

Soot is another kind of particulate media. It is formed in combustion of almost all hydrocarbons and can enhance radiative heat transfer in combustion system due to the continuum radiation in the infrared spectrum [22]. It is well known that radiative heat transfer in combustion processes can be dominated by absorption and emission from soot particles [18]. Generally speaking, soot consists of primary nearly monodispersed spherical particles whose diameters are between 0.01 μm and 0.1 μm [17]. The structure of soot aggregates can be described by fractal scaling laws. A critical parameter of predicting radiative properties of soot is the volume fraction, f_v, which lies between 10^-6 and 10^-4 [1]. In order to obtain detailed information on the soot aggregates and volume fraction, non-intrusive techniques such as laser diagnostics are essential [17,113,114].

Rigorous methods, such as the T-matrix method and DDA method, have been applied to simulate radiation scattered by soot aggregates [115-117]. On the other hand, much research has been devoted to numerical simulations using the RDG theory for soot aggregates. In the RDG theory the electromagnetic coupling within aggregates is neglected [118]. Exact methods have also been used to study radiative properties of soot aggregates and the results are employed to examine the validity of the RDG theory and the equivalent volume Mie sphere [117,119-122]. It is found that the RDG theory is a good compromise between accuracy and simplicity of application. Error of the RDG theory is related to size parameter and it is less than 10% if size parameter is smaller than 0.3 [123]. Normally the RDG theory underpredicts absorption cross section of aggregates. The error is small in the visible spectrum and becomes significantly larger in the near- to mid-infrared spectra [124]. As fractals evolve from chain-like to more densely packed morphologies, scattering cross section and single-scattering albedo increase monotonically. It indicates that scattering interaction among spheres is of increasing importance and the assumption of a volume-equivalent soot sphere without electromagnetic interactions between the monomers is inaccurate [117,119,121]. Therefore, applications of the volume-equivalent soot sphere for aggregates can lead to questionable results. Corrections for the RDG formulation have been provided to consider internal scattering of primary particles in aggregates [122]. If fractal parameters of a soot cluster are fixed, random variations in the geometrical configuration of monomers have a rather weak effect on scattering and absorption in the visible spectrum [119].

Magnitudes of scattering cross section and absorption cross section are dependent on the size of aggregates [121]. The absorption increases with the number of spheres in the aggregate and reaches an asymptote for aggregates containing 100-200 spheres [120]. For small aggregates, scattering is negligible compared with absorption, and the Rayleigh approximation can be used. For large aggregates, the scattering cross section is in the same order of the absorption cross section. Furthermore, the phase function becomes highly peaked in the forward direction and the Rayleigh approximation is invalid [121]. The limiting shapes of conglomeration are the sphere and the long cylinder [125] and the extinction coefficient of soot agglomerates lies between those of spherical and cylindrical particles [126].

Aerogel

Aerogel is a high porous, quasi-homogenous material [127-129]. It has low thermal conductivity and high transmittance in the solar spectrum. Therefore, aerogel can be incorporated into glazings for solar thermal systems or building windows as transparent insulation materials [130]. Aerogel can also be considered as a dispersed medium because it is comprised of nanometer solid skeleton particles and nanoscale open pores. Much research has been performed to study the transmittance of solar energy by aerogel and evaluate radiative heat transfer via aerogel-based glazing.

When electromagnetic waves pass through aerogels, nanoscale pores and skeleton particles can act as scatters. The size of individual particles is smaller compared with the wavelength of the visible spectrum, satisfying one criterion of Rayleigh scattering. However, the distance between particles in aerogel is not large enough for scattering to be considered as independent scattering. Nevertheless, bulk scattering from aerogel shows some characteristic features of Rayleigh scattering. Therefore, a monolithic aerogel slab looks bluish when viewed against a dark surrounding and yellowish when viewed in a bright surrounding. Since aerogel absorbs little in the near-ultraviolet and visible spectra, bulk scattering dominates optical loss in these spectra. In the wavelength range of 200 nm<λ<600 nm, spectral scattering coefficient is dependent on the wavelength with a relation ofλ^-4[131]. Whenλ>600 nm, spectral scattering coefficient deviates from theλ^-4dependence [132]. Besides bulk scattering, scratch and crack in the surface can also scatter the incident radiation. Therefore, total scattering is comprised of bulk scattering and surface scattering. Since bulk scattering is approximately isotropic while surface scattering is favorably forwarded, the pattern of total scattering is forwarded.

The spectral transmittance and reflectance of aerogel are commonly measured with a UV/VIS/NIR spectrometer. Measurement of regular transmittance is straightforward. By means of an integration sphere, hemispherical transmittance and reflectance can also be quantified. Figure 12 displays the measurement results by Zhu et al. [133]. The hemispherical transmittance is around 90% in the range of 800 nm<λ<1800 nm. The regular transmittance and diffuse transmittance are also shown. The dips around 1400 and 1900 nm are attributed to absorption bands of residual water. The refractive index of aerogel is dependent on the bulk density as n≈1+0.21ρ, where ρ is the bulk density, g/cm³ [134]. Since aerogel is usually very light, its refractive index should be close to 1. Thus, reflection by monolithic aerogel slabs is insignificant.

Solar transmittance can be evaluated by averaging the spectral transmittance with the standard reference data. Generally, solar transmittance of monolithic aerogels is better than that of granular aerogels. On the other hand, granular aerogel is sufficient for the day lighting purpose if the image quality is not important, such as in skylights and bathroom windows. Granular aerogel can also introduce diffuse irradiation into rooms, which is an advantage [135].

Biological materials and PTFE

The diffusion theory was often used to approximate RTE when the medium is mostly scattering, especially in the field of optical scattering in tissues [136-138]. The adding-doubling method can effectively solve RTE for directional-hemispherical reflectance and transmittance, although it cannot predict bidirectional distributions [139]. The Henyey-Greenstein function is commonly used as the scattering phase function [139,140] and is given by [22]

(9){\mathrm{\Phi }}_{\mathrm{HG}}(\cos \theta )={\displaystyle \frac{\mathrm{1}-g^{\mathrm{2}}}{{(\mathrm{1}+g^{\mathrm{2}}-\mathrm{2}g\cos \theta )}^{\mathrm{3}/\mathrm{2}}}},

where the scattering angle θ is the angle between the propagation directions of the incident light and the scattered light, and g is called the asymmetric parameter (from-1 to 1). The Henyey-Greenstein function yields isotropic scattering when g=0. If g>0, there are more forward scattered photons than backward scattered photons. If g approaches 1, all photons are scattered in the direction parallel to the incident light.

Measurements of R and T have been made using a single integrating sphere or a a_λdouble-sphere arrangement [141,142], which allows the determination of absorption coefficient and a_λ reduced scattering coefficient σ′_λ=σ_λ(1-g) because the scattering coefficientσ_λ and the asymmetric parameter in the phase function are coupled [139]. An additional measurement of the transmittance of unscattered collimated light, called direct transmittance, is needed to determine the scattering coefficient [139,140,143]. Care must be taken to subtract the scattered light in the collimated direction from the overall transmittance. Cheong et al. [144] offered a comprehensive review of the optical properties of biological materials, which are generally characterized by highly scattering withσ_λ from 50 to 400 cm^-1, low absorption withσ_λ from 0.5 to 5 cm^-1, and forward scattering with gfrom 0.88 to 0.99. The absorption coefficient and reduced scattering coefficient were also obtained in the near infrared for human skin and mucous tissues using the adding-doubling method [145,146].

Another method to determine absorption and scattering properties is to use the Monte Carlo method, which can predict both directional-hemispherical properties and bidirectional properties. A suitable inverse method allows the determination of absorption and scattering properties. Wang et al. [147] developed a Monte Carlo modeling tool for simulation of radiative transport in multilayered tissues, for which the interfaces are assumed to be perfectly smooth. Liu and Ramanujam [148] developed a sequential estimation technique based on a Monte Carlo simulation for a two-layer epithelial tissue in the visible range. It was found that the bottom layer has a slightly higher a_λ andσ′_λ, by a factor of 2-3, than those of the top layer. The use of Henyey-Greenstein function is merely for convenience and there is a lack of validation and microscopic justification for this model.

PTFE is known as a strongly scattering and diffusely reflecting material. Due to its characteristics of being a nearly diffuse reflector, PTFE has been used as a diffuse-reflectance standard in the spectral range from 200 nm to 2500 nm [149], as a calibration standard for onboard sensors on satellites in remote sensing [150-153], as whiteness standards in colorimetry [154,155], and as a coating layer in integrating spheres [53,58]. In addition, since both PTFE and most biological tissues are strongly scattering materials, researchers have used PTFE as tissue phantoms to simulate layers of skin for the study of burn depth [156]. Early et al. [157] reported round-robin BRDF measurements with diffuse reflectors, including Spectralon, pressed PTFE, and sintered PTFE. These studies, however, did not obtain the scattering parameters for PTFE. In order to describe light propagation using RTE, the scattering coefficient, absorption coefficient, and the scattering phase function need to be estimated.

Huber et al. [158] reported the scattering coefficient, absorption coefficient, and the asymmetric parameter g in the Henyey-Greenstein phase function of PTFE films with thicknesses from 190 to 845 μm. The scattering parameters of PTFE reported in their study are very similar to those of biological tissues [144]. For example, the scattering and absorption coefficients at 633 nm wereσ_λ=240 cm^-1and a_λ=3.6 cm^-1, respectively. However, it appears that these authors did not distinguish the scattered light in the parallel direction from the collimated light transmission. Thus, the diffusely scattered light emerging from the other side of the film in the direction parallel to the incidence might have been taken as the unscattered light and the scattering coefficient was obtained from the measured transmittance

(10)T=e^{-(a_{\lambda }+{\sigma }_{\lambda })d}.

Li et al. [159] reported the measurements of BRDF and BTDF of five PTFE films, whose thicknesses varied from 109 μm to 10.1 mm, at a laser wavelength of 635 nm. Integrating the measured BRDF and BTDF over the corresponding hemisphere yields R and T for normal incidence. The reduced scattering coefficient was determined to be (167±20) cm^-1by fitting the ratio of R to T of the thin-film samples with those calculated from the adding-doubling method. Directional-hemispherical spectral measurements were performed with an integrating sphere, which resulted in reduced scattering coefficient at different wavelengths, as shown in Fig. 13 for the transmittance and reflectance of four samples and the reduced scattering coefficient as a function of wavelength. The square marks represent the data integrated from the scatterometer measurements and diamond marks represent the data obtained with the diode laser scatterometer at 635 nm wavelength and the integrating sphere. The thicknesses for samples 1 to 4 are 109, 259, 522, 1057 μm, respectively. It was also shown that the absorption coefficient of sintered PTFE is less than 0.01 cm^-1 in order to agree with the reported transmittance and reflectance data of a 10-mm thick sample. The value of a_λ is one to two orders smaller than that of human tissues.

Li et al. [159] found that even for the 109-μm sample, the scattering coefficient is too large for any directly transmitted light to be observable in the collimated direction (because they are overshadowed by the scattered light). The laser scatterometer [54] was used to measure the transmitted radiant power at observation angles from 0° to 6° with a separation of every 1°. The measured result suggests that at λ= 635 nm, σ_λ is larger than 1200 cm^-1, which is nearly five times that reported by Huber et al. [158]. Hence, careful distinction between the directly transmitted light and scattered light towards the direction parallel to the incidence is essential for future research of light scattering in biological media and phantoms. A Monte Carlo simulation considering only volume scattering was performed [159]. While it can predict the directional-hemispherical properties well, it cannot describe the bidirectional behavior well. Surface scattering may be important and depends strongly on polarization and angle of incidence. Furthermore, the Henyey-Greenstein phase function of PTFE may overpredict forward scattering but underpredict backward scattering. The microstructure of PTFE depends on the crystallization, particle cluster size, porosity, and density, etc. Several studies have attempted to deal with both surface scattering and volume scattering with RTE or rigorous electromagnetic-wave solutions for layered inhomogeneous materials [160-162]. Future study is also needed to investigate the effect of microstructure on the interaction of light with PTFE and other scattering media, and to develop more comprehensive theoretical models that can include scattering by rough surfaces as well as volume scattering.

Concluding remarks

Radiative properties of rough surfaces, particulate media and porous materials are essential for thermal engineering. Simulation results using GOA with the measured roughness statistics show better agreement with measurement results than those using hypothetical Gaussian functions. Moreover, the ray-tracing algorithm that incorporates measured surface topographies into simulation of rough surfaces results in a reasonable agreement with the experiment results. It has been obviously demonstrated that surface roughness is not Gaussian and anisotropic as well for some surfaces. Therefore, it is recommended that a thorough characterization of surface roughness should be necessary to obtain true roughness statistics and these detailed information should be used in surface scattering modeling instead of a simple application of Gaussian distribution. Future research can be done to further develop inverse methods to obtain accurate roughness statistics from the BRDF measurement since the AFM topography measurement is time-consuming. On the other hand, the measured roughness statistics and topographies may also be applied with other approximation methods besides GOA to determine whether this novel method is valuable in a wide range of roughness parameters.

Modeling of radiative properties of particulate media and porous materials are very complex. For dispersed media with a low volume fraction, scattering of the media may be approximated by the summation of intensities of overall individual scatters. However, for dense media the space between nearby scatters is smaller compared with the radiation wavelength. Therefore, besides scattering by individual scatters, near-field interaction and far-field interference of scattered waves are important too. Rigorous solutions are usually necessary to take into account the interaction and interference of scatted waves. The RDG theory has been used to model radiative properties of soot and aerogel. It has been shown that the RDG theory is a good compromise between accuracy and simplicity of application. However, because the refractive index of soot is considerably large, it is questionable whether the RDG theory is applicable for soot. Therefore, both well designed exact solutions and experiment investigation should be conducted to determine the validity domain of this theory for soot, aerogel, and other disperse media. For PTFE and biological materials, most reported studies are based on Henyey-Greenstein phase function using either the adding-doubling method or the Monte Carlo method to model radiative transport. The applicability of this phase function needs to be further studied. Furthermore, surface roughness and polarization effect should be investigated. In order to determine the true scattering coefficient, the specimen must be made sufficiently thin and scattered light must be excluded from the collimated transmission.

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