LIGHT SCATTERING AT THE BOUNDARYBETWEEN TWO MEDIA

A. P. Ivanov and V. V. Barun UDC 535.36

Coefficients of diffuse reflection and transmission of light by smooth Fresnel boundaries of an optically thicklayer illuminated by scattered radiation are calculated. Consideration is given to the absorption effect on an-gular diagrams of light at the boundaries. An analysis is made of asymptotic equations of radiation transferin a medium with reflecting boundaries, and the range of their applicability is studied. Using methods of geo-metric optics, coefficients of reflection of directional and diffuse radiation from a rough surface are studied.The results can find use in various areas of science and technology, specifically, in optical diagnostics of bio-logical tissues.

Keywords: diffuse and specular reflection, coefficients of reflection, transmission, and brightness, surface mi-crosites, diagnostics.

Introduction. Scattering of electromagnetic waves at the boundary between different media (reflection fromthese boundaries and transmission by them) in a wide spectral interval has for many decades drawn the attention ofscientists. This has to do with solving problems of spectroscopy, thermal physics, lighting engineering, photometry,viewing, etc. The theory of radiation propagation in disperse systems also requires knowledge of these characteristics.According to the structure of the boundary between media, wave or geometric-optical approaches are used in calcula-tions. As a result, a good deal of data have been accumulated and various calculating procedures have been proposed.Still, practice encounters problems necessitating more detailed knowledge of these characteristics, especially when thesurfaces are not smooth and the surface is irradiated with light of complex angular structure. In the current paper, co-efficients of reflection of light from various boundaries between two media with reference to several optical problemsare calculated using known methods.

Reflection (transmission) of light by the boundary between two media depends on the relative refractive indexof these media, structure of their surface and its roughness, and the angular structure of the energy and polarizationcharacteristics of incident radiation. Consideration is only given to the coefficients of reflection from the boundary,since the light transmitted by it is a portion of incident light. Let us examine two types of boundaries, namely, smooth(specular) and rough ones.

1. Specular Boundary. We first present the well-known concepts for various cases of illumination.I. Directional illumination. In this case, the coefficients of specular reflection of monochromatic light is de-

scribed by the Fresnel equations

rp (ϕ) = tan

2 (ϕ − ψ)

tan2 (ϕ + ψ)

, rs (ϕ) = sin

2 (ϕ − ψ)

sin2 (ϕ + ψ)

, (1)

where rp and rs are the reflection coefficients in oscillations of the light vector in the incidence plane and normal toit; sin (ϕ) = n sin (ψ). With normal light incidence,

rp∗ (0) = rs

∗ (0) =

⎛⎜⎝

n − 1n + 1

⎞⎟⎠

2

. (2)

If light is partly polarized, the coefficient of its reflection is defined as

B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, 68 Nezavisimost Ave., Minsk,220027, Belarus; email: barun@dragon.bas-net.by. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 84, No. 1, pp.22–31, January–February, 2011. Original article submitted March 16, 2010.

1062-0125/11/8401-0023�2011 Springer Science+Business Media, Inc. 23

Journal of Engineering Physics and Thermophysics, Vol. 84, No. 1, January, 2011

r (ϕ) = arp (ϕ) + (1 − a) rs (ϕ) . (3)

To completely depolarized light corresponds a = 0.5.In laboratory experiments, liquid media in a cell covered with transparent plane-parallel plates is frequently

studied. Such complex boundaries are characterized by two refractive indices — of the plate (substrate) nc and the in-vestigated medium nm — relative to air. The coefficient of reflection from the boundary surfaces is determined fromexpression (3); however, considering multiple light rereflections in the plate, we obtain

rp (ϕ) = rp1 (ϕ) + (1 − rp1 (ϕ))2

rp2 (ψ)1 − rp1 (ϕ) rp2 (ψ)

, rs (ϕ) = rs1 (ϕ) + (1 − rs1 (ϕ))2

rs2 (ψ)1 − rs1 (ϕ) rs2 (ψ)

, (4)

where rp1(ϕ) and rs1(ϕ), and rp2(ψ) and rs2(ψ) are the reflection coefficients at the air–plate and plate–mediumboundaries. These coefficients are calculated from Eqs. (1) and (3) involving the refractive indices nc and nm

⁄ nc, re-spectively.

II. Partly diffuse illumination. With partly diffuse illumination of the medium, when the light brightness B(ϕ)is only dependent on the incidence angle ϕ, the reflection coefficient is defined as

R =

∫ 0

π ⁄ 2

B (ϕ) r (ϕ) sin 2ϕdϕ

∫ 0

π ⁄ 2

B (ϕ) sin 2ϕdϕ

. (5)

III. Totally diffuse illumination. In this case, where the brightness B is constant in all directions in Eq. (5) weshould put B(ϕ) = const.

If totally diffuse light is incident from a less dense medium with a relative refractive index n, then, accordingto [1], Eq. (5) is transformed to the form

R1 = 3n

8 − 10n

7 − 6n

6 + 4n

5 + 16n

4 − 10n

3 + 2n

2 + 1

3 (n4 − 1)2 −

n2 (n2

− 1)2

(n2 + 1)3 ln

⎪⎪⎪

n + 1

n − 1

⎪⎪⎪ +

8n4 (n4

+ 1) ln n

(n4 − 1)2

(n2 + 1)

. (6)

It should be noted that Eq. (6) is valid for the case of light scattering in the approximation of geometric optics [2–4].The coefficient of reflection R2 from a less dense medium can be calculated using Eq. (6), although n should be sub-stituted for 1 ⁄ n. In [1], proceeding from thermodynamic considerations, the relation

R2 = 1 − 1 − R1

n2 . (7)

was also obtained. It should be pointed out that Eq. (7) is valid for a boundary of any structure, not only for asmooth one.

1.1. Boundary Conditions in the Asymptotic Theory of Radiation Transfer. The asymptotic theory of ra-diation transfer [5, 6] became widely employed to describe the light propagation in scattering and absorbing media. Itis used, for example, for calculating measured characteristics of radiation and subsequent reconstruction of structural,biophysical, and optical parameters of blood erythrocytes [7, 8], for noninvasive diagnostics of characteristics of bio-logical tissues [9, 10], and for determining the dimensions of cloud droplets [11] and snow grains [12] from the coef-ficients of reflection and (or) transmission of the medium. The range of applicability of this theory (with an error ofabout 10%) is set by the following conditions [6]:

k

ε′ =

k

ε (1 − cos γ____

) ≤ 0.18 , τ′ = ε′l ≥

2

3 . (8)

Here, ε′ = ε(1 − cos γ____

), cos γ____

is the average cosine of the scattering indicatrix of an elementary volume.

24

When liquid suspensions or solid light-scattering matrices are investigated in laboratory conditions by opticalmethods, reflection arising at the boundary between media affects to an extent reflection and transmission of the entiresystem. Therefore, it is important to take account of this factor. Above it has been noted that it is dependent on theangular structure of incident radiation. Let us assess possible changes in this structure. According to the asymptotictheory [6], when the medium is illuminated by a wide light beam under the angle ϑ0 relative to the normal to its sur-face, the brightness coefficients of reflected and transmitted light rτ and tτ of a layer of the medium with an opticalthickness τ′ are defined, disregarding reflection at its boundaries, as

rτ = r∞ − tτ exp [− (x + y)] , tτ = g (μ0) g (μ) sinh (y)sinh (x + y)

, (9)

where

r∞ = r∞(0)

exp ⎡⎢⎣

⎢⎢−

g (μ0) g (μ) y

r∞(0)

⎤⎥⎦

⎥⎥ , r∞

(0) = 0.49

1 + 4μμ0

1 + μμ0

. (10)

Here, r∞ and r∞(0) are the brightness coefficients of light reflected respectively from the absorbing and nonab-

sorbing layer of infinite thickness, x = τ′√⎯⎯⎯⎯3k ⁄ ε′ , y = (4 ⁄ 3)√⎯⎯⎯⎯3k ⁄ ε′ , g(μ) = 3(1 + 2μ) ⁄ 7, and ϑ is the angle of light

incidence from within the scattering medium measured normal to its surface, μ = cos ϑ. With illumination of the me-

dium normal to its surface, μ0 = 1 and g(μ0) = 9 ⁄ 7. Figure 1 presents normalized indicatrices of the brightness coef-

ficients of reflected light at μ0 = 1 (Eqs. (9) and (10)), which embrace all possible situations where the above

equations are operative. The values of the brightness coefficient at ϑ = 0 are normalized. For the light transmitted by

the medium, angular dependences are not presented, since under condition (8) they are independent of τ′ and, accord-

ing to Eq. (9), are defined by a simple function g(μ). From Fig. 1 which illustrates the τ′ effect on the brightness co-

Fig. 2. Normalized indicatrices of the brightness coefficient of reflected lightat τ′ = 3 and μ0 = 1: 1) k ⁄ ε′ = 0.2, 2) 0.1, and 3) 0.

Fig. 1. Normalized indicatrices of the brightness coefficient of reflected light(μ0 = 1) at k ⁄ ε′ = 0 (a) and 0.2 (b): 1) τ′ = 1, 2) 1.5, and 3) 10.

25

efficient at two extreme values of k ⁄ ε′, it is seen that for reduced small layer thicknesses the angular dependences

have the maximum ϑ of about 50 deg regardless of the parameter k ⁄ ε′. With increasing τ′ this maximum shifts some-

what to the region of smaller angles ϑ and thereafter disappears. All observed specific features of angular dependencesof the brightness coefficient are accounted for by different scattering multiplicities: small multiplicities increase bright-

ness with ϑ because of illumination of an extended region of the layer by a wide beam, and large ones decrease it asa consequence of the attenuation of radiation passing a long way in the medium.

To provide a clear picture, Fig. 2 presents angular dependences for the brightness coefficient of reflected lightcalculated from Eqs. (9) and (10) at fixed τ′ = 3 and a variable k ⁄ ε′. At the indicated optical thickness of the layer,the maximum of the brightness coefficient is more manifest for larger values of the parameter k ⁄ ε′ because of therelative increase in the contribution of low scattering multiplicities.

The data obtained are used for calculating the coefficients of reflection of light by the boundaries of the me-dium into its layer. Strictly speaking, to this end, not only the above angular light distributions should be used, but itis necessary to take into account additional radiation rereflections with other angular relationships between the bound-ary and the scattering medium. However, the calculations indicated that the contribution of these radiations can be ne-glected, since the power production of additional flows is low, and their angular distributions differ from the consi-dered ones only slightly. Table 1 supplies the coefficients of reflection of light from the boundaries for an infinitelythick layer r∞↑ and a layer of finite thickness r1↑ with τ′ = 1, and also the values of r↓ independent of k ⁄ ε′ and τ′.Here, arrow ↑ stands for the surface of the layer on which the external radiation is incident, and the arrow ↓ denotesthe opposite surface. In order to distinguish the turbid medium–air and turbid medium–plate (substrate)–air boundaries,the first is called the boundary and the second, the plate (substrate). Typical refractive indices are used, which arecharacteristic of the laboratory experiment.

From Table 1 it is seen that the refractive index has a noticeable effect on reflection coefficients. For opti-cally thin layers, r↑ is virtually independent of k ⁄ ε′. With increasing τ′ the values of r↑ decrease somewhat, especiallywhen the medium absorbs light weakly. These variations should be borne in mind when solving inverse problems. Onthe whole, however, the coefficients of reflection from the boundaries of media depend on optical properties of themedium only slightly.

1.2. Analysis of Asymptotic Equations of Transfer Radiation in the Medium with Account for ItsBoundaries. With account for the boundaries of a uniform plane-parallel layer, according to [6, 13], the asymptoticequations for the reflection and transmission coefficients of light for this layer are of the form

R = (1 − r1) sinh X

sinh (X + Y) + r1 , T =

(1 − r1) sinh Y

sinh (X + Y) , (11)

TABLE 1. Reflection Coefficients for Smooth Boundaries of a Plane-Parallel Light-Scattering Layer

k ⁄ ε′Boundary Plate (substrate)

n1 = 1.34 n1 = 1.52 n1 = 1.6 n1 = 1.34,n2 = 1.52

r∞� r∞� r∞� r∞�

0.2 0.47 0.60 0.64 0.60

0.1 0.46 0.59 0.63 0.59

0.05 0.45 0.58 0.63 0.58

0 0.43 0.56 0.61 0.57

r1� r1� r1� r1�

0.2 0.49 0.63 0.67 0.63

0.1 0.49 0.63 0.67 0.63

0.05 0.49 0.63 0.67 0.63

0 0.50 0.63 0.67 0.63

r� r� r� r�

0.40 0.53 0.58 0.54

26

where r1 is the reflection coefficient of the illuminated boundary (plate). For the above two cases it is calculated fromEqs. (3) and (1) or (3) and (4). The parameters X and Y take account of the type of the external radiating beam andthe conditions of reflection at the boundaries with illumination from within the medium. With the diffuse externalbeam, we have

X = √⎯⎯3k

ε′ ⎛⎜⎝

⎜⎜τ′ +

4

3

r�

1 − r�

⎞⎟⎠

⎟⎟ , Y = √⎯⎯3k

ε′ 4

3

1

1 − r�

. (12)

With directional illumination at the angle ϑ0, we obtain

X = √⎯⎯3k

ε′ ⎡⎢⎣τ′ +

4

3

1

1 − r�

− 4

3 g (μ0)⎤⎥

⎦ , Y =

4

3 √⎯⎯3k

ε′ ⎡⎢⎣

⎢⎢

r�

1 − r�

+ g (μ0)⎤⎥⎦

⎥⎥ . (13)

Here, r↑ and r↓ are the reflection coefficients of the boundaries with illumination from within the layer on theside of incidence of external radiation and on the opposite side, respectively. It should be noted that these coefficientsare incorporated by Zege et al. [13] in the parameters X (Eq. (12)) and Y (Eq. (13)) in order to retain a unique formof representing reflection and transmission coefficients (11) as a ratio of two hyperbolic functions. Such representationis fairly well organized, but the range of its applicability is automatically limited by small values of r↑ and r↓ [13].Actually, however, the reflection coefficients r↑ and r↓ can be fairly large (see Table 1). To avoid introducing the in-dicated constraints, we revert to the initial equations [13] obtained by summing up the terms of infinite geometric pro-gressions determined by multiple light reflections between two boundaries (substrates) and the scattering medium. Inother words, in lieu of Eqs. (11)–(13) we write the reflection and transmission coefficients R and T for illumination ofthe medium normal to it in the form

TABLE 2. Comparison of R and T Calculated from Eqs. (14)–(16) (upper number) and Eqs. (11)–(13) (lower number)

τ′

r� = r� = 0.05 r� = r� = 0.2

k ⁄ ε′ k ⁄ ε′ k ⁄ ε′ k ⁄ ε′0.05 0.2 0.05 0.2 0.05 0.2 0.05 0.2

R T R T

10.246 0.175 0.676 0.573 0.275 0.189 0.63 0.5180.244 0.169 0.672 0.561 0.262 0.16 0.609 0.462

2 0.396 0.24 0.42 0.26 0.384 0.221 0.405 0.2380.393 0.235 0.417 0.254 0.37 0.196 0.391 0.212

5 0.495 0.257 0.123 0.025 0.461 0.229 0.122 0.0230.492 0.252 0.122 0.024 0.446 0.205 0.117 0.021

10 0.505 0.257 0.018 5.28⋅10–4 0.468 0.229 0.017 4.83⋅10–4

0.502 0.252 0.018 5.15⋅10–4 0.453 0.206 0.017 4.3⋅10–4

r� = r� = 0.4 r� = r� = 0.6

10.307 0.196 0.566 0.438 0.323 0.184 0.491 0.3410.263 0.121 0.504 0.306 0.208 0.056 0.345 0.132

2 0.361 0.188 0.378 0.202 0.322 0.15 0.334 0.1550.319 0.131 0.335 0.141 0.224 0.056 0.233 0.061

5 0.402 0.186 0.117 0.02 0.322 0.135 0.104 0.0150.402 0.134 0.103 0.0141 0.237 0.057 0.073 5.93⋅10–3

10 0.406 0.186 0.017 4.1⋅10–4 0.322 0.135 0.015 3.14⋅10–40.406 0.134 0.015 2.9⋅10–4 0.238 0.057 0.01 1.2⋅10–4

27

R = r1 + r2+3 (1 − r1) (1 − r�)

1 − r�r2+3∗

, T = (1 − r1) ⎛⎜⎝

⎜⎜

r2+3r�t2+3∗

1 − r�r2+3∗

+ t2+3

⎞⎟⎠

⎟⎟ , (14)

r2+3 = Rm + r�TmTm

∗

1 − r�Rm∗

, t2+3 = Tm (1 − r�)

1 − r�Rm∗ , (15)

r2+3∗

= Rm∗

+ r� (Tm

∗)2

1 − r�Rm∗

, t2+3∗

= Tm

∗ (1 − r�)

1 − r�Rm∗ . (16)

Here, the subscript "m" indicates the reflection and transmission coefficients of the scattering medium with no reflect-ing boundaries (they are calculated from Eqs. (11)–(13) at r↑ = r↓ =0), the subscript "2 + 3" per tains to a system con-sisting of the medium with the boundary (substrate) opposite to the illuminated side, coefficients with asteriskscorrespond to diffuse illumination, and those without them, to directional illumination.

We now compare the calculations of R and T from Eqs. (11)–(13) and (14)–(16). The results of comparisonare presented in Table 2. It covers the cases where the reflection coefficients of the boundaries r↑ and r↓ are not linkedwith optical properties of the light-scattering layer and are constants. The entire range of optical characteristics of themedium where asymptotic equations with no boundaries are operative is embraced. In each square, the upper numberrepresents calculation from Eqs. (14)–(16), and the lower number is calculation from Eqs. (11)–(13). To make the pic-ture clearer, r1 is assumed to be zero, i.e., the light transmitted by the illuminated boundary is normalized.

From deriving Eqs. (11)–(13) [13] it follows that, with weak absorption and small values of r↑ and r↓, the dif-ferences between these equations and Eqs. (14)–(16) are small. This is also clear from the data in Table 2. When r↑,r↓ < 0.2, both approaches give about the same results within the limits of a relative error of the order of 10%. Withincreasing absorption and coefficients r↑ and r↓, the differences become much more noticeable. Here the values of Rand T obtained from Eqs. (11)–(13) are always smaller than the corresponding values obtained from Eqs. (14)–(16). Itshould also be noted that, if the values of r↑ and r↓ are small, R of course monotonically increases with increase inthe optical thickness. However, when r↓ is large, at small τ′ the value of R is determined mainly by r↓. With increas-ing τ′ the light-scattering layer begins to strongly screen the substrate and in some cases R decreases.

Above it has been pointed out that in reconstructing parameters of the scattering medium from its reflectionand transmission coefficients it is important to accurately write the expressions for R and T. Let us examine the typicallaboratory situation where the considered object — water emulsion — is placed in a glass cell with two glass covers.For this situation, Fig. 3 illustrates calculations using, as previously, two approaches at various values of k ⁄ ε′ and τ′.However, the coefficients were normalized not to external radiation transmitted by the boundary between two media

Fig. 3. Coefficients of reflection R (a) and transmission T (b) vs. τ′ at μ0 = 1:1) kτ = 0.01, 2) 0.05, and 3) 0.2. Solid lines stand for calculation from Eqs.(14)–(16), and dashed lines denote calculation from Eqs. (11)–(13).

28

but instead to radiation incident on it (as in the experiment). It was assumed that light incident on the boundaries fromwithin is completely depolarized (a = 0.5). For the cell covers n1 = 1.52, and for the emulsion n2 = 1.34. Accordingto Eqs. (4), (5), and (9), r↓ = 0.54, and r↑, as shown by calculations at various k ⁄ ε′ and τ′, varies from 0.569 to 0.63.

From Fig. 3 it is seen that even at low absorption the discrepancies between the two approaches are verymarked (only for k ⁄ ε′ ≤ 0.001 are the results identical, as shown by calculations), which is linked with the high reflec-tion coefficients of the cell boundary. Since media usually absorb light, it is practically always expedient to solve in-verse problems using Eqs. (14)–(16) rather than Eqs. (11)–(13).

2. Rough Boundary. We assume that the surface roughnesses are much larger than the wavelength of inci-dent light; therefore, methods of geometric optics can be used. The surface is considered as a set of a large numberof microsites distributed along the angles of inclination by a certain law. Let S be the area of a macrosurface onwhich the rough boundary can be projected. Then, the total area of microsites, the normals to which lie in the solidangle dω near the direction specified by polar δ and azimuth ζ angles, is

dσ = Sf (δ, ζ) dω . (17)

From this equation, only the distribution function f(δ, ζ) of microsites can be determined. If azimuthal symmetry ofthe distribution is the case, then

dσ = 2πSf (δ) sin dδ . (18)

This approach is used to solve various problems. In [14], a convenient equation is proposed

Φ (δ) = f (δ)

f (0) =

1

1 + b tan2 δ

, (19)

where b is the empirical constant. Taking account of the normalization condition ∫ S

cos δdσ = S, one can easily show that

f (δ) = (1 − b)2

π (1 + b tan2 δ) (1 − b + b ln b)

. (20)

Expression (20) is used for calculating the coefficient of reflection from the rough surface in various situations.2.1. Coefficient of Reflection from the Boundary between Two Media with Normal Illumination. We first

consider the simplest case of normal light incidence. The reflection coefficient is

R = 2π ∫ 0

π ⁄ 4

r (δ) f (δ) sin δ cos δdδ . (21)

Here, r(δ) is the reflection coefficient of a microsite, the normal to which is oriented to the direction of illuminationat the angle δ. The integral is taken up to π ⁄ 4, since more inclined surfaces do not participate in reflection becausethey reflect light to the front hemisphere rather than to the back one. The general picture of reflection for incident un-polarized light is presented in Table 3. Consideration is given to various degrees of roughness characterized by the pa-rameter b and to the relative refractive indices n at the boundary between two media. Where a square of Table 3contains two numbers, the upper one gives the reflection coefficient R1 for light incident from a less dense medium to

a denser one and the lower number gives R2 for illumination in the opposite direction. To provide better insight into

the regularities observed at various b, Table 3 gives a column of the values of Σ = 2π ∫ 0

π ⁄ 4

f(δ) sin δ cos δdδ charac-

terizing the portion of the total area of projections of the microsites participating in light reflection.For a smooth surface, R1 and R2 are of course the same. With increasing roughness (with decreasing b)

R1 < R2, since, in reflection, sites inclined to incident light begin to participate, which with illumination from an opti-

29

cally denser medium reflect more light than from a less dense one. The differences between R1 and R2 are more no-ticeable the higher the reflective index. With decrease in the parameter b the coefficient R1 monotonically decreases.This is linked to the decrease in the portion of sites Σ participating in reflection and with the weak dependence oftheir coefficient r(δ) on δ when δ ≤ π ⁄ 4. The same is the case with R2 for small n ≤ 1.1. However, at high refractiveindices, r(δ) increases rapidly with δ. Two opposite effects manifest themselves: with increase in the degree of rough-ness of the surface, the number of the reflecting sites diminishes, while the coefficient of reflection from them in-creases. Therefore, at intermediate values of n, with decreasing b the quantity R2 passes through the maximum, and atlarge n the second effect is dominant and R2 rapidly increases. We remaind the reader that this calculation is done dis-regarding screening and rereflection at the boundary between two media.

2.2. Coefficient of Reflection from the Boundary between Two Media with Totally Diffuse Illumination.The reflection coefficients were calculated using equations from [15], which were corrected for misprints. It should benoted that, unlike the equations for reflection coefficients for normally directed illumination, these equations are lessexact, since the disregarded screening and rereflection effects are here more substantial, especially at low refractive in-dices for light incidence from an optically denser medium. Indeed, in this case, at a small total value of R2, grazingrays experience complete internal reflection and, on getting on the neighboring roughness, are rereflected from it. Thisfact, however, is disregarded in calculation; therefore, the reflection coefficients are understated. When the total reflec-tion is high (when n is large), in the overall balance this is not very significant, which is not the case with a lowreflectivity of the boundary. The calculated reflection coefficients are shown in Table 4. At various b and n, in eachsquare the upper number corresponds to R1 and the lower number to R2. The analysis indicated that determining R2according to [15] leads to appreciable errors, especially at low n. In connection with this, knowing R1, we calculatedR2 using the exact equation (7).

From Table 4 it naturally follows that, with any R1 and R2 pair, the first number is always many timessmaller than the second one. With increasing roughness these coefficients decrease. This especially pertains to R1 atsmall n. For R2, such transformations are small, since in a wide range of angles there is complete internal reflection.The degree of variability of R1 and R2 is also seen from the equation

δR2 = δR1 R1

n2 − 1 + R1

, (22)

TABLE 3. Reflection Coefficients of a Rough Boundary at Normal Light Incidence

b Σn

1.02 1.1 1.2 1.34 1.53 1.7

1.1 0.768.7⋅10–5 1.9⋅10–3 6.9⋅10–3 0.017 0.035 0.0538.9⋅10–5 2.1⋅10–3 8.5⋅10–3 0.027 0.14 0.24

4 0.85 9.4⋅10–5 2.1⋅10–3 7.6⋅10–3 0.019 0.039 0.0599.5⋅10–5 2.3⋅10–3 8.9⋅10–3 0.027 0.12 0.21

16 0.91 9.7⋅10–5 2.2⋅10–3 7.9⋅10–3 0.02 0.041 0.0629.8⋅10–5 2.3⋅10–3 8.8⋅10–3 0.025 0.093 0.17

64 0.94 9.8⋅10–5 2.2⋅10–3 8.1⋅10–3 0.02 0.042 0.0649.8⋅10–5 2.3⋅10–3 8.7⋅10–3 0.024 0.077 0.14

100 0.95 9.8⋅10–5 2.2⋅10–3 8.1⋅10–3 0.02 0.042 0.0649.8⋅10–5 2.3⋅10–3 8.6⋅10–3 0.024 0.074 0.13

500 0.96 9.8⋅10–5 2.2⋅10–3 8.1⋅10–3 0.021 0.043 0.0659.8⋅10–5 2.3⋅10–3 8.5⋅10–3 0.023 0.065 0.11

5000 0.97 9.8⋅10–5 2.3⋅10–3 8.2⋅10–3 0.021 0.043 0.0669.8⋅10–5 2.3⋅10–3 8.5⋅10–3 0.022 0.059 0.098

Smooth surface 1 9.8⋅10–5 2.3⋅10–3 8.3⋅10–3 0.021 0.044 0.0679.8⋅10–5 2.3⋅10–3 8.3⋅10–3 0.021 0.044 0.067

30

ensuing from Eq. (7). Here, δR2 = ΔR2 ⁄ R2 and δR2 = ΔR1

⁄ R1 are the relative variations in R2 and R1 as a result of

the roughness change. From the above equation it also follows that δR2 < δR1 is always the case. Indeed, when

n → 1, expression (6) becomes R1 = (n − 1) ⁄ 3 [4] and, according to Eq. (22), δR2

δR1 =

1

3n + 4 →

17

. With different re-

fractive indices, the inequality δR2 < δR1 is evident.

Conclusions. We indicate possible applications of the results obtained only in the area of biomedical optics. Inspectrophotometric diagnostics of blood in a cell based on reflection and transmission coefficients, it is always neces-sary to take account of the presence of two plane-parallel plates. As shown, their reflection coefficients are about 0.5–0.6; therefore, R and T should be calculated using Eqs. (14)–(16) for the sums of geometric progressions rather than themore commonly used Eqs. (11)–(13). The parameter k ⁄ ε′, which is the subject of measurement in such diagnostics, in-fluences slightly the reflection coefficients r↓ and r↑, which are mainly determined by refractive indices of the consid-ered scattering medium and plate material. Therefore, r↓ and r↑ can be assumed known and constant, which reduces theproblem of reconstructing the medium’s characteristics to the use of asymptotic equations [5, 6] disregarding theboundaries with relevant parameters dependent on r↓ and r↑, according to Eqs. (14)–(16). Another aspect of applicationof the results obtained is noninvasive diagnostics of tissue from the spectra of its diffuse reflection coefficient, bright-ness coefficient, or from light fluxes going off from the tissue surface at some distance from the irradiation place [16].Here, the reflection coefficient of the rough surface of skin enters, as an unknown parameter, into the algorithm fordetermining the sought structural and biophysical characteristics. It can be evaluated by specifying the degree of rough-ness of skin, for example, according to the model of [17]. This is the subject of future investigations.

This work was carried out with financial support from the Belarusian Republic Foundation for Basic Researchand the State Commission for Science and Technology of Belarus under contract No. F09GKNT-0.004.

NOTATION

a, portion of the light intensity with oscillations of the electric vector in the incidence plane; b, roughnessparameter; B, light brightness; f, distribution function of microsites with respect to angles; g, auxiliary function; k, ab-sorption index; l, thickness of the medium layer; n, relative refractive index of two media; r, reflection coefficient in

TABLE 4. Reflection Coefficients of a Rough Boundary with Totally Diffuse Illumination

bn

1.02 1.1 1.2 1.34 1.53 1.7

1.11.5⋅10–3 0.011 0.025 0.045 0.072 0.098

0.04 0.18 0.32 0.47 0.6 0.69

4 1.7⋅10–3 0.012 0.026 0.047 0.074 0.0990.04 0.18 0.32 0.47 0.61 0.69

16 2.1⋅10–3 0.14 0.029 0.05 0.078 0.10.041 0.19 0.33 0.47 0.61 0.69

64 2.6⋅10–3 0.016 0.032 0.053 0.081 0.110.041 0.19 0.33 0.47 0.61 0.69

100 2.7⋅10–3 0.016 0.033 0.054 0.082 0.110.041 0.19 0.33 0.47 0.61 0.69

500 3.3⋅10–3 0.018 0.035 0.057 0.085 0.110.042 0.19 0.33 0.48 0.61 0.69

5000 4⋅10–3 0.02 0.038 0.06 0.088 0.110.043 0.19 0.33 0.48 0.61 0.69

Smooth surface 6.1⋅10–3 0.025 0.044 0.068 0.096 0.120.045 0.19 0.34 0.48 0.61 0.7

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directional illumination; R, reflection coefficient in diffuse illumination; S, macrosurface area; t, brightness coefficientof transmitted light; T, transmission coefficient; x, y, X, Y, auxiliary functions; γ, scattering angle; δ, polar angle of thenormal to a microsite; ε, attenuation index; ζ, azimuth angle of the normal to a microsite; ϑ, polar angle of observa-tion (illumination); μ = cos (ϑ), auxiliary function; σ, microsite area; Σ, portion of the total area of projections of mi-crosites; τ, optical thickness; ϕ, angle of radiation incidence; Φ, normalized distribution function of microsites withrespect to angles; ψ, angle of refraction of radiation; ω, solid angle. Subscripts: 0, illumination of the medium normalto its surface; 1, light incidence from an optically less dense medium on a denser one; 2, light incidence from an op-tically denser medium on a less optically dense one; 2 + 3, medium together with an unluminated boundary; ∞, opti-cally infinite thick layer; ↓, illuminated side of the layer; ↑, unilluminated side of the layer; ′, reduced index; c,substrate (plate); m, medium with no reflecting boundaries; p, oscillations of the light vector in the incidence plane; s,oscillations of the light vector normal to the incidence plane; τ, layer of finite optical thickness. Superscripts: (0), non-absorbing layer; *, diffuse illumination.

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