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Modified KubelkaMunk model for calculation of the reflectance of coatings with optically-

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 39 (2006) 35713581 doi:10.1088/0022-3727/39/16/008

Modified KubelkaMunk model for

calculation of the reflectance of coatingswith optically-rough surfaces

A B Murphy

CSIRO Industrial Physics, PO Box 218, Lindfield NSW 2070, Australia and CSIRO Energy

Transformed National Research Flagship, Australia

E-mail: tony.murphy@csiro.au

Received 16 June 2006, in final form 3 July 2006

Published 4 August 2006Online at stacks.iop.org/JPhysD/39/3571

AbstractThe KubelkaMunk two-flux radiative transfer model is strictly applicableonly to the case of diffuse illumination but is often applied in the case ofcollimated illumination. Here, the application of the KubelkaMunktwo-flux model to the collimated illumination of optically-rough surfaces isinvestigated. Expressions for the reflectance from such surfaces areobtained. A relatively simple treatment of reflection from surfaces ofarbitrary roughness is developed that takes into account the characteristics ofthe spectrophotometer used to measure reflectance. The modifiedKubelkaMunk model is tested in the case of an optically-rough rutile

titanium dioxide coating on a titanium substrate and found to give goodagreement with experiment, even for negligible scattering within thecoating. Itis expected that if the surface is sufficiently rough to ensure that the lighttransmitted into the coating is diffuse, the modified KubelkaMunk modelwill be applicable irrespective of the magnitude of the absorption andscattering coefficients of the coating material.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The propagation of light in layered media is well understood

and relatively easily treated mathematically as long aseach layer is homogeneous and the interfaces betweenmedia are smooth (e.g. [1]). However, when the layers

are inhomogeneous, or the interfaces are optically-rough,

treatment becomes more difficult. Analytical treatments ofpropagation of light in inhomogeneous media are typically

complex, and for this reason transport theories are oftenused. Such theories treat the transport of radiative energy

through the medium directly, using effective absorption and

scattering coefficients. While the development of transporttheories is less rigorous than that of analytical treatments,

they are nonetheless very useful, and have been applied

to a wide range of problems. The KubelkaMunk model

[2, 3] is by far the most widely used transport theory, havingbeen applied to examine materials as diverse as paints [ 4],pigmented plastics [5], decorative and protective coatings [6],

solar-absorbing pigments and paints [7], human tissue [8],leaves [9], crystalline materials [10], melting of solids [11],

powders [12] and fibres and wool [13]. In this model, it is

assumed that the optical properties of the coating are describedby two constants, the absorption and scattering coefficients.

Kubelka and Munks original treatment [2,3] took into accountonly transport within a layer; Saunderson [5, 14] extended the

treatment to allow reflection from the front and back surfacesof the layer to be considered.

In the KubelkaMunk model, it is assumed that the light

is diffuse within the layer. Strictly, this can only occurwhen the incident light is diffuse; however, the model isfrequently used for collimated illumination [1518]. Vargas

and Niklasson [19] have examined the case of collimatedillumination and shown that the KubelkaMunk model, withthe Saunderson extension, is of very limited applicability,

being accurate only for weakly-absorbing coatings containinghighly-scattering particles whose sizes are larger than awavelength. They developed a slightly modified method, in

0022-3727/06/163571+11$30.00 2006 IOP Publishing Ltd Printed in the UK 3571

mailto:%20tony.murphy@csiro.auhttp://stacks.iop.org/JPhysD/39/3571mailto:%20tony.murphy@csiro.auhttp://dx.doi.org/10.1088/0022-3727/39/16/008

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A B Murphy

which the reflection coefficient from the front of the coating

was the Fresnel coefficient (i.e. the reflection coefficient for

collimated reflection of collimated light) and found it to have

a wider range of applicability. While it was rigorously correct

only for optically-thick weakly- or non-absorbing coatings,

useful results were also obtained for absorbing coatings whosereflectance is very weak and for coatings containing highly-

scattering particles whose sizes are larger than a wavelength.

The KubelkaMunk model is a two-flux model; the two

fluxes are diffuse light travelling in the forward and reverse

directions. The differential equations treat absorption and

scattering of the light. In cases where it is not reasonable

to assume that the light is diffuse, four-flux models (in which

the fluxes are both collimated and diffuse light travelling in

forward and reverse directions) may be used [2022].

The reflection coefficients used in the Saunderson

extension were for diffuse reflection of diffuse light. As

noted above, Vargas and Niklasson [19] also used reflection

coefficients for collimated reflection of collimated light.However, when collimated illumination is used, it is possible,

depending on the optical roughness of the surface, for the

reflected light to be collimated, diffuse or partially collimated

and partially diffuse. In the case of optically-rough surfaces,

the reflected light is mainly diffuse. The transmitted light is

also mainly diffuse. This means treatment by the two-flux

method is likely to be valid under collimated illumination for

a wider range of coating parameters than is the case for an

optically-smooth surface.

Treatment of collimated illumination of a general surface

requires expressions for reflection coefficients valid for

both optically-smooth and optically-rough surfaces, and in

particular the separation of these reflection coefficients intospecular (collimated) and diffuse components. In keeping

with the simplicity of the two-flux model, it is appropriate

to use relatively simple expressions for these reflection

coefficients. It is important, in comparing predictions of the

model with experiment, to take into account the properties

of the measurement apparatus (e.g. a spectrophotometer with

integrating sphere attachment), in particular the means by

which the apparatus separates the reflected light into diffuse

and specular components. Note that reflection from optically-

rough surfaces is often referred to as scattering. Here the term

reflection is usedfor both optically-roughand optically-smooth

surfaces, and the term scattering is reserved for the description

of scattering of light within the coating.In this paper, I derive a slightly-modified two-flux model

for the case of collimated illumination of a coating on an

opaque substrate that allows general surfaces to be treated.

This extends previous studies that examined only optically-

smooth surfaces. Further, I obtain simple expressions for

reflection coefficients at surfaces of arbitrary roughness that

allow the characteristics of the measurement apparatus to be

taken into account. This is done by developing expressions,

using a physical optics approach, for reflection coefficients

from a general surface that depend on the acceptance cone of

the measurement apparatus.

Therangeof surface characteristicsfor which themodified

two-flux model will be valid is then investigated. The model istested for the case of collimated illumination of an optically-

rough rutile TiO2 coating on a titanium substrate. This

Figure 1. Schematic of measurement geometry of the diffusereflectance attachment of the Cary 5 spectrophotometer. Thediagram on the left represents the geometry for diffuse reflectancemeasurements (D position), and that on the right shows thegeometry for total reflectance measurements (S position). In eachcase, the sample is represented by the rectangle on the right of theintegrating sphere.

provides a good test of the model, since scattering within

the coating is weak, and absorption can be either strong or

weak, depending on the wavelength. Rutile TiO2 coatings ontitanium substrates can be produced, for example, by flame-

oxidation or oven-oxidation of titanium and have been applied

in the photocatalytic splitting of water into hydrogen and

oxygen [2325]. Application of the modified KubelkaMunk

model developed here allows the interpretation of reflectance

measurements of these coatings, and inversion of the equations

derived allows the absorption coefficient and refractive index

to be determined from the measured reflectance [26]. The

band-gap of the coating can then be determined from the

wavelength-dependence of the absorption coefficient using

standard methods [27]; reduction of the band-gap of rutile

TiO2 is imperative to improve its efficiency in photocatalytic

water-splitting [23].In section 2, the measurement of reflectance using a

spectrophotometer with an integrating sphere is described.

This system is used for the measurements, which are presented

and discussed in section 5. The modified KubelkaMunk

model is derived in section 3 and the reflection coefficients

for a general surface are derived and analysed in section 4.

Conclusions are given in section 6.

2. Reflectance measurements

Measurements of reflectance are typically performed using a

spectrophotometer with an integrating sphere attachment. For

example, the measurements to be presented in section 5 wereperformed using the diffuse reflectance attachment of a Cary 5

UV-visible spectrophotometer. A schematic of the geometry is

given in figure 1. The incident light is collimated, and reflected

light is captured by an integrating sphere.

The diffuse reflectance attachment has two settings. In the

D position, the sample is oriented so that the incident light is

normal to the surface of the sample. Light reflected specularly,

i.e. normal to the surface, is not captured by the integrating

sphere, so only diffusely reflected light is measured. In the

S position, the sample is oriented so that the incident light

is at a small angle to the normal to the surface. In this case,

both specularly- and diffusely-reflected light are captured by

the integrating sphere.The sample measured in the experiment described in

section 5 was smaller than the approximately 8 by 12 mm

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Figure 2. Geometry, showing boundary conditions at z = 0 andz = h. Collimated light is denoted by solid arrows and diffuse lightby dotted arrows.

aperture at the back of the integrating sphere. A matt black

plate with a small (3 mm diameter) aperture is placed between

the sphere and the sample. Hence a 3 mm diameter spot on

the sample was illuminated. The entrance aperture of the

integrating sphere is oval in shape and 11.04 mm vertically

by 13.44 mm horizontally. The inner diameter of the sphere is

110 mm. Hence, in the D position, light reflected from the

centre of the sample at an angle greater than 2.87 verticallyand3.50 horizontally is captured. Taking into account the factthat a 3 mm diameter circular region is illuminated, it can be

calculated that light reflected at angles greater than 2.9 0.8vertically and 3.5

0.8 horizontally is captured in the D

position.

In the calculations of reflectance from a rough surface that

are presented in this paper, diffuse reflectance corresponds

to measurements made in the D position and collimated

reflectance to the difference between measurements made in

the S and D positions. A matt Teflon reference was used to

provide a nominal 100% reflectance measurement.

3. Modified KubelkaMunk model

Figure 2 shows the geometry considered, which is appropriate

to a coating on an opaque substrate. The coating lies between

the front plane at z = 0 and the back plane at z = h.The coating rests on an infinite substrate starting at z = h.The incident light travels in the positive z direction and is

collimated.

I assume that the light within the coating is diffuse, as is

required to apply a two-flux model. This is a reasonable a

priori assumption in the case of an optically-rough surface at

the air-coating interface. I allow the light reflected from the

front surface of the coating to have both collimated and diffuse

components. While at first glance this may seem inconsistent

with the previous assumption, it allows the validity of that

assumption to be checked. Further, there are cases in which the

reflected light maybe partiallycollimated while thelightwithin

the coating is diffuse, for example coatings with an optically-smooth front surface that are strongly scattering. Finally, since

the spectrophotometer allows both the collimated and diffuse

components of the reflectance to be measured, it is useful to

calculate both components.

I distinguish between reflectance, which refers to the

reflection of light from the coatingsubstrate system, and

reflection coefficients, which refer to reflection from a single

surface. Both are dimensionless ratios. I will use R todenote reflectance and r to denote a reflection coefficient. The

following reflection coefficients have to be considered:

r icc: reflection of collimated light as collimated light, r icd: reflection of collimated light as diffuse light, r idd: reflection of a diffuse light as diffuse light.

Superscript i represents the surface from which reflection

occurs as follows:

i = f represents reflection from the front surface of thecoating (at z = 0),

i

=b represents reflection from the back surface of the

coating (at z = 0), i = s represents reflection from the front surface of the

substrate (at z = h).Since the incident light is collimated, but the light within the

coating is diffuse, the following reflection coefficients have to

be calculated: rfcc, r

f

cd, rbdd and r

sdd.

Let the incident collimated light intensity be denoted

by Ic0, and the forward- and backward-directed diffuse light

intensities in the coating by Id(z) and Jd(z), respectively. The

boundary conditions at z = 0 and z = h are, respectively,

Id(0) = (1 r fcc rfcd)Ic0 + r bddJd(0) (1)

and

Jd(h) = r sddId(h). (2)The KubelkaMunk model uses an effective scattering

coefficient S and an effective absorption coefficient K

to describe the optical properties of the coating. The

effective scattering coefficient is related to the usual scattering

coefficient s by S= 2(1 )s, where the forward scatteringratio is defined as the ratio of the energy scattered by a

particle in the forward hemisphere to the total scattered energy.

For Rayleigh scattering, = 1/2, while for Mie scattering,1/2 < < 1. The effective absorption coefficient is related

to the usual absorption coefficient k by K=

k, where the

average crossing parameter is defined such that the average

path length travelled by diffuse light crossing a length dz is

dz. For collimated light, = 1, while for semi-isotropic(i.e. isotropic in the direction of propagation) diffuse light,

= 2 [28]. It is usual in applying the KubelkaMunk modelto write = 1/2 and = 2, so that S= s and K = 2k.

The differential equations describing the energy balance

between diffuse light in the forward (positive z) and backward

(negative z) directions are

dId

dz= (S+ K)Id + SJd, (3)

dJd

dz= (S+ K)Jd SId. (4)

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The general solution to these equations is

Id(z) = C1 exp(Sbz) + C2 exp(Sbz), (5)

Jd(z) = (a b)C1 exp(Sbz) + (a + b)C2 exp(Sbz), (6)

where a = (S+ K)/S and b = a2 1 and C1 and C2 areconstants. Using boundary conditions (1) and (2) gives

Id(z) = {(1 r fcc r fcd)Ic0[b cosh(Sbh Sbz)+(a r sdd) sinh(Sbh Sbz)]}{b(1 r bddrsdd) cosh(Sbh)+(a r bdd r sdd + ar bddr sdd) sinh(Sbh)}1, (7)

Jd(z) = {(1 r fcc r fcd)Ic0[br sdd cosh(Sbh Sbz)+(1 ar sdd) sinh(Sbh Sbz)]}{b(1 r bddrsdd) cosh(Sbh)+(a r bdd r sdd + ar bddr sdd) sinh(Sbh)}1. (8)

From (8), we obtain

Jd(0) =(1

r

fcc

r

f

cd)RKMIc0

1 r bddRKM , (9)

where

RKM =1 r sdd[a b coth(bSh)]

a + b coth(bSh) rsdd. (10)

The collimated reflectance from the coating and substrate

system is just the collimated reflected component of the

incident radiative flux normalized to the incident radiative flux

Rcc = r fcc . (11)

The diffuse reflectance from the coating and substrate system

is the sum of the diffuse reflected component of the incident

radiative flux and the transmitted diffuse backward flux atz = 0, normalized to the incident radiative flux

Rcd =r

f

cdIc0 + (1 r bdd)Jd(0)Ic0

. (12)

Using (9) gives

Rcd = r fcd +(1 rfcd rfcc)(1 r bdd)RKM

1 r bddRKM, (13)

with RKM given by (10). This result is similar to that first

obtained by Saunderson [5, 14] for diffuse reflectance in the

case of diffuse illumination

Rdd = r fdd +(1 r fdd)(1 r bdd)RKM

1 r bddRKM. (14)

4. Reflection coefficients

4.1. Reflection of collimated incident light from an

optically-rough surface

In this section, I consider the calculation of reflection

coefficients for collimated light incident on a surface that

may be optically-rough or optically-smooth, or intermediate.

The derivation of reflection coefficients for collimated light

incident on an optically-smooth surface, i.e. the Fresnelcoefficients, is treated in standard optics text-books (e.g.

[29, 30]). Expressions for Fresnel coefficients are given in

appendix A. Optically-rough surfaces are usually defined by

the Rayleigh criterion, which states that surfaces can be treated

as optically-smooth if the heights of surface irregularities are

less than /(8cos i ), where i is the angle of incidence. The

factor 8 is sometimes replaced by 16 or 32 [31].

Before continuing, it is useful to consider nomenclature.Reflection from rough surfaces is often described as scattering,

since a proportion of the reflected light is scattered at angles

of reflection other than that equal to the angle of incidence.

Here I use the term reflection to include such scattering,

i.e. to describe reflection at any angle. I define the reflection

coefficient as the ratio of the intensity of the reflected light to

the intensity of the incident light. (The reflection coefficient is

sometimes defined in terms of the electric field amplitude.)

Further, in this section, I only consider reflection from

the initial interaction of the incident light with the surface.

Reflection arising from scattering from discontinuities beneath

the surface or reflections from subsurface interfaces is taken

into account using the KubelkaMunk model that was derivedin section 3.

Two important approaches to the calculation of reflection

properties of rough surfaces are that based on physical

optics (the BeckmannSpizzichino model) and that based

on geometrical optics (the TorranceSparrow model). The

geometrical optics approach is mathematically simpler but is

only valid when the wavelength of the incident light is much

smaller than the dimensions of the surface irregularities. Since

this is not always the case for the surfaces of interest, I follow

the more general physical optics approach.

Beckmann and Spizzichino [31] derived expressions

describing the reflection of light from rough surfaces.

However, they gave results for only one polarization, anddid not consider shadowing effects. I use the results of

He et al [32], who gave expressions for the bidirectional

reflectance distribution function (BRDF) for incident light of

both polarizations and for unpolarized light and also took into

account shadowing. The assumptions made in the derivation

are:

The height distribution on the surface is assumed to beGaussian and spatially isotropic. Under such conditions,

the probability that a point on the surface falls within the

height range z to z + dz is p(z)dz, where

p(z)=

1

2 0exp(

z2/22

0). (15)

The mean height is z = 0 and 0 is the rms roughnessof the surface. To specify the surface fully, a horizontal

length measure is also required. The measure used is the

autocorrelation length , which isa measure ofthe spacing

between surface peaks. The rms slope of the surface is

proportional to 0/.

The electric field at a given point on the surface is set tothe value that would exist if the surface were replaced by

its local tangent plane (the tangent plane or Kirchoff

approximation). Thorsos showed that this approximation

is accurate for / 1 when an appropriate shadowing

treatment is used [33]. The assumption is made, in evaluating an integral thatarises in the derivation, that the surface is either optically

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Modified KubelkaMunk model for calculation of the reflectance

very rough (i.e. (2/)2 1, where is an effectivesurface roughness, defined below) or that the surface has

gentle slopes (i.e. / 1). Multiple reflections from the surface are ignored. This

contribution is negligible for a surface with gentle slopes.

He et al [32] gave expressions for the BRDF as the sum

of a specular component, a directional diffuse component,

and a uniform diffuse component. The latter corresponds to

the subsurface scattering and multiple subsurface reflections

described by the KubelkaMunk model. These have been

accounted for in section 3 and are ignored here. The

directional diffuse component corresponds to what is here

called diffuse reflection. The BRDF for unpolarized light is

given by

= s + d, (16)where the specular component is

s =r

F exp(g)Z

cos i d i, (17)

and the diffuse component is

d =rF(

)GZD cos i cos r

. (18)

Here rF() is the Fresnel reflection coefficient evaluated at the

bisecting angle

= cos1(|k r k i |/2), (19)

where k

i

and k

r

are, respectively, the unit vectors in the

direction of the incident and reflected light, is a delta

function that is unity in the cone of specular reflection and

zero elsewhere, i and r are, respectively, the polar angles

of incidence and reflection and r is the azimuthal angle of

reflection. It is assumed that the azimuthal angle of incidence

is i = 0. The geometric factor G is given by

G = 4(1 + cos i cos r sin i sin r cos r )2

(cos i + cos r )2. (20)

The surface roughness function g is given by

g = [(2/)(cos i + cos r )]2. (21)

The distribution function D is given by

D = 22

42

m=1

gm exp(g)m!m

exp

v2xy 2

4m

, (22)

where

vxy =2

sin2 i 2sin i sin r cos r + sin2 r

1/2. (23)

Calculation ofD can cause numerical problems because of the

large numbers involved for large values ofm. Nayar et al [34]

give useful approximate expressions for D. For g 1 (i.e. asmooth surface)

D = 22

42exp(g)g exp

v2xy 24

, (24)

and for g 1 (i.e. a rough surface)

D = 22

421

gexp

v2xy 2

4g

. (25)

The effective roughness was introduced by He et al toallow averaging to occur over only the illuminated (non-

shadowed) parts of the surface. Particularly for grazing angles

of incidence or reflection, it can be considerably smaller than

the rms roughness 0. They are related by

= 0(1 + z20/20 )1/2, (26)

where z0 is the root of the equation

2z = 0

4(Ki + Kr ) exp

z

2

220

, (27)

and

Ki = tan i erfc( cot i /20), (28)Kr = tan r erfc( cot r /20). (29)

The shadowing function Z is given by

Z = Zi (i )Zr (r ), (30)

where

Zi (i ) =

1 12

erfc( cot i /20)

(cot i ) + 1, (31)

Zr (r ) =[1 1

2erfc(cotr /20)]

(cot r ) + 1, (32)

(cot ) = 12

20

cot erfc

cot

20

. (33)

The BRDF expressions (17) and (18) are now fully defined.

These expressions, together with the expression for the

reflection coefficient r in terms of the BRDF

r =

/20

cos i cos r sin r dr dr (34)

obtained in appendix B, can be used to obtain expressions for

the collimatedcollimated and collimateddiffuse reflection

coefficients, rfcc and r

f

cd, respectively, from a rough surface.

The total reflection coefficient is split into specular and diffuse

componentsr = rs + rd. (35)

Using (16), (18) and (34), the diffuse component ofr is given

by

rd =

/20

d cos i cos r sin r dr dr . (36)

For specular reflection, r = i and r = , which implythat = i . Further, for a perfectly-reflecting mirror-likesurface, we expect rF(0) = 1 and rs = 1. In calculating thespecular component of the BRDF for rough surfaces, rF(

) ismultiplied by exp(g)Z, so the specular reflection coefficientis modified accordingly to give

rs = rF(i ) exp(g)Z. (37)

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0.01 0.1 1 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reflectioncoefficie

ntsr,rs,rd

Wavelength-normalised rms roughness 0

/

/0

= 5: diffuse, total

/0

= 10: diffuse, total

/0

= 20: diffuse, total

specular (all values of /0)

Figure 3. Dependence of total, specular and diffuse reflectioncoefficients r, rs and rd, respectively, at normal incidence on rmsroughness normalized to wavelength. Results are given for different

values of inverse rms surface slope. The Fresnel reflectioncoefficient at normal incidence is assumed to be 100%. The regionsin which the total and diffuse reflection coefficients may beinaccurate are indicated by cross-hatching. The specular reflectioncoefficient is independent of rms surface slope.

Figure 3 shows the reflection coefficients rd, rs and r as

a function of wavelength-normalized surface roughness 0/

for i = 0, for different values of the inverse rms surfaceslope /0. It is assumed that the Fresnel reflection coefficient

for normal incidence rF(0) is 100% for the purposes of the

figure, to allow the effect of surface roughness to be shown

more clearly. Note that the specular reflection coefficient

rs is independent of /0. The diffuse reflection coefficientis lower for smaller values of /0; this is because of two

effects. The first is that on the scale of the surface roughness

the average angle of incidence increases as the surface slope

increases, decreasing the average reflection coefficient. The

second is a result of shadowing of the diffusely-reflected light;

i.e. Z < 1. Note that the equation for calculation of diffuse

reflection coefficient is only accurate for / > 1; hence for

low /0, the results are not reliable for low 0/. The regions

that may therefore be inaccurate are marked on the graph by

cross-hatching.

Figure 4 shows the angular dependence of the integrand

(1/)rF()GZD sin r in expression (36) for diffuse

reflection coefficient rd, for the caseofi = 0. Thevalue of theintegrand depends strongly on /0 and relatively weakly on0; in particular, for 0 1 m, the integrand is independentof0. While the integrand is zero for r = 0, since reflectionat this angle is classed as specular, a significant fraction of the

diffusely-reflected light is reflected within a few degrees of

r = 0, particularly for larger values of /0.In the measurement of reflectance using a spectropho-

tometer withan integrating sphere, the incident collimatedlight

has angle of incidence i = 0, and reflected light within anacceptance cone centred around r = 0 is measured as specu-larly reflected light (see section 2). If this acceptance cone has

half-angular width , then the reflection coefficients required

in the modifed KubelkaMunk model ((11) and (13)) are

r fcc = rs + rd|r (38)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

(1/)rF(')GZDs

inr

Angle of reflection r (degrees)

= 500 nm

0

= 1 m, /0

= 2

0

= 1 m, /0

= 5

0 = 1 m, /0 = 1 0

0= 1 m, /

0= 2 0

0

= 1 m, /0

= 5 0

0

= 50 nm, /0

= 1 0

0

= 100 nm, /0

= 1 0

Figure 4. Dependence of integrand in diffuse coefficient expressionon the angle of reflection, for different values of rms roughness andinverse rms surface slope, for normal incidence. The Fresnelreflection coefficient at normal incidence is assumed to be 100%.

The wavelength is 500 nm. Values for rms roughness greater than1 m are equal to the 1 m values shown.

and

rf

cd = rd|r > . (39)Using (36) and (37), we obtain (noting that the following

apply for normal incidence: rF(i ) = rF(0), the BRDF isindependent of azimuthal angle r , and Z = 1 for specularreflection)

rfcc = rF(0) exp(g) +1

0

rF(r /2)GZD sin r dr (40)

and

rf

cd =1

/2

rF(r /2)GZD sin r dr . (41)

In calculating r(0) and r(r /2) with (A.4), the refractive index

N1 is that of air, and the refractive index N2 is that of the

coating.

The total reflection coefficient at i = 0 is given byr(0) = rs + rd = rfcc + r fcd = rF(0) exp(g)

+1

/20

rF(r /2)GZD sin r dr . (42)

Figure 5 shows the total reflection coefficient r and its

components rf

cc and rf

cd for half-angular width = 3.2, whichis the average value for the diffuse reflectance attachment of

the Cary 5 spectrophotometer. For optically-rough surfaces

(e.g. 0 = 1 m), the diffuse component rfcd dominates. Foroptically-smooth surfaces (e.g. 0 = 10nm), the specularcomponent r

fcc dominates. For surfaces of intermediate

roughness, both components are important.

4.2. Comparison with other expressions

It is common (e.g. [3537]) when dealing with the reflection

from optically-roughsurfaces to use just thespecular reflection

coefficient

rs = rF(0) exp

4 0 cos i

2

, (43)

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300 400 500 600 700 800

0.001

0.010

0.100

1.000

(b) /0 = 20

0 = 10 nm: r & rf

cc, rf

cd

0

= 100 nm: r, rf

cc, r

f

cd

0

= 1 m: r & rf

cd(r

f

cc~ 0)

Wavelength (nm)

0.001

0.010

0.100

1.000

Reflectioncoefficientsr,rfcc,rf cd

(a) /0

= 10

Figure 5. Reflection coefficient r and its specular and diffusecomponents rfcc and r

f

cd, respectively, for three different rms surfaceroughnesses and for two values of inverse rms surface slope, fornormal incidence. The Fresnel reflection coefficient at normalincidence is assumed to be 100%.

which is equivalent to (37) for the case of normal incidence.

Boithias [38] suggested a modification to

rs = rF(0) exp

4 0 cos i

2

I0

1

2 4 0 cos i

22

, (44)

where I0 is the modified Bessel function of order zero; this

expression has also been used by, for example, Landron et al

[39]. Miller et al [40] derived this modification, and claimed

that it gave a better fit to experimental results of Beard [ 41]

for coherent reflection from sea waves. However, Hristov

and Friehe [42] have recently claimed that the Bessel function

factor is unnecessary.

Presumably in these cases it is assumed that light that is not

reflected specularly is reflected diffusely, so the total reflection

coefficient is r(0). Hence, ifrs is given by (43), we have

rd

=r(0)1 exp

4 0 cos i

2

. (45)There are a number of shortcomings inherent in using ( 43)

and (45), compared with using the expressions (40) and (41)

derived here for rfcc and r

f

cd, respectively. First, the division of

the reflected light into specular and diffuse components does

not take into account the component of the diffuse reflection

that is reflected at or close to the specular angle of reflection

(r i , r = i + ). Further, shadowing is neglected,and the effective angle of incidence on the scale of the surface

roughness increases when the average surface slope increases;

these affect the calculation of the total reflection coefficient.

Finally, deviations of from 0 are neglected.

Figure 6 compares expressions (43) and (44) for thespecular reflectance with the value of r

fcc given by (40), for

half-angular width = 3.2, which isthe average value for the

0.001

0.010

0.100

1.000

Specularreflectionco

efficient

Wavelength-normalised rms roughness 0/

rfcc, /0 = 5

rfcc

, /0

= 10

rfcc, /0 = 20

rfcc

, /0

= 50

rs, equation (43)

rs, equation (44)

Figure 6. Comparison of reflection coefficient rfcc forspectrophotometer of half-angular width = 3.2, with specularreflection coefficient rs calculated by expressions (43) and (44), as afunction of rms roughness normalized to the wavelength and fordifferent values of inverse rms surface slope.

diffuse reflectanceattachmentof the Cary5 spectrophotometer.

Expression (43) is a good approximation in the case of /0 10. For /0 10, significant deviationsoccur for / 0.1,due to the significant proportion of the diffuse component that

is reflected into the acceptance cone of the spectrophotometer,

and therefore measured as specular reflection.

Figure 6 also shows that expression (44) is a better

approximation than (43) for a limited range of parameters

( /0 10 and 0/ 0.1) but is worse for other parameters;this may explain the controversy about its accuracy relativeto (43).

4.3. Reflection of diffuse incident light from an

optically-rough surface

The diffusediffuse reflection coefficient is calculated using

an angular average over all angles of incidence of the Fresnel

reflection coefficient rF(i ), where i is the angle of incidence

of the light [43]:

rdd =2

/20

rF(i )di . (46)

We use the same values of diffusediffuse reflection coefficient

for reflection from a rough surface and a smooth surface. This

is because in both cases the reflection coefficient is an average

value over all angles of incidence, and the total reflected light

is equal to a good approximation if multiple surface reflections

at a rough surface are neglected. For the case of a coating on

a substrate,

rbdd = rdd(N1 = Nc, N2 = Nair), (47)

rsdd = rdd(N1 = Nc, N2 = Ns ), (48)where Nc, Ns and Nair are the complex refractive indices of the

coating, substrate and air, respectively. Note that the complexrefractive index is N= n + i , where n is the refractive indexand = k/4 is the extinction coefficient.

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300 400 500 600 700 8000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Refractiveindex

Wavelength (nm)

n(TiO2)

(TiO2)

n(Ti)

(Ti)

Figure 7. Real and imaginary parts of the refractive indices of rutileand titanium.

It is worth noting that the expressions (40), (41), (47)and (48) for r

fcc, r

f

cd, rbdd and r

sdd, respectively, are required

in four-flux models as well as two-flux models. The other

reflection coefficients that are required in four-flux models, r bcc,

rbcd, rscc, r

scd, and for diffuse illumination, r

f

dd, (the notation used

is defined in section 3) are easily calculated using analogous

expressions.

5. Experiment and discussion

A rutile titanium dioxide coating was formed by oxidizing a

piece of titanium sheet in oxygen at 1 bar at a temperature

of 850 C for 10 min. The titanium sheet was etched inKrolls solution for 10 s, prior to oxidation, to provide a roughsubstrate. Such oxidesemiconductor coatings on titanium have

been used to investigate the photocatalytic splitting of water

into hydrogen and oxygen [23, 24]. The thickness of the rutile

coating is estimated to be 2000 200 nm using the oxidationrate data given by Dechamps and Lehr [44]. Using an atomic-

force microscope to measure the surface profile and standard

analysis methods [45], the rms roughness 0 of the surface was

measured to be 571 nm and the autocorrelation length to be

6.48 m, given an inverse rms surface slope of /0 = 11.3.(For the substrate, 0 was 520 nm and was 9.71 m; these

data are not required by the model.)

The reflection coefficients r

f

cc and r

f

cd were calculatedusing (40) and (41) for acceptance cone half-angular width

= 3.2, which is the average value for the geometry of theCary 5 diffuse reflectance attachment, as discussedin section 2.

The complex refractive indices N = n + i of TiO2 and Tiused in the calculation are shown in figure 7; is known

as the extinction coefficient. The real part of the refractive

index of rutile was taken from Cardona and Harbeke [ 46] and

Devore [47], as reported by Ribarsky [48] and the imaginary

part from the work of Eagles [49]. The Ti data were taken from

Ribarsky [48].

Figure 8 shows calculated values of the reflection

coefficients rfcc, r

f

cd, rsdd and r

bdd. As expected for the relatively

rough surface, rf

cd rf

cc. This indicates that the transmittedlight will be predominately diffuse, so the modified Kubelka

Munk model should be applicable.

300 400 500 600 700 800

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reflectioncoefficient

Wavelength (nm)

rfcd

rf

cc

rbdd

rsdd

Figure 8. Reflection coefficients for a rutile TiO2 coating on a Tisubstrate, with 0 = 570 nm and = 6.5 m, and for = 3.2.

300 400 500 600 700 8000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

DiffusereflectanceR

cd

Wavelength (nm)

Calculated ( Nc

+ 10%, Nc

- 10%)

Measured

Figure 9. Measured and calculated value of the diffuse reflectancefrom the rutile TiO2 coating on a Ti substrate, with thickness2000 nm, negligible scattering coefficient, absorption coefficientcalculated from the data shown in figure 7 and other parameters asfor figure 8. The dashed and dotted line, respectively, show theeffect of increasing and decreasing the refractive index by 10%.

Figure 9 shows measured values of the reflectance Rcdand the values calculated using the modified KubelkaMunk

model. The scattering coefficient S is set to a negligible value

(to avoid dividing by zero, it has to be non-zero). The influenceon the calculated reflectance of altering the real and imaginary

components of the refractive index by 10% is shown in thegraph. The literature values of refractive index vary by at

least 10%, so this is a useful estimate of the uncertainty in

the calculated value. The influence of altering the coating

thickness by the same percentage is smaller.

The agreement between the measured and calculated

values is good for wavelengths above about 300 nm. The

reflectance curve has the same form as the refractive index

of rutile TiO2, except for a pronounced dip at around 400 nm.

This dip corresponds to the rapid decrease of the absorption

coefficient K = 2k = 8/ at the band-gap wavelength ofrutile TiO2. The absorption coefficient is greater than 10

7

m1

forwavelengths below350 nm, decreasing to less than 100 m1

for wavelengths above 450 nm. Clearly, the KubelkaMunk

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model is able to calculate the reflectance accurately in the

case of collimated illumination of a rough surface, even for

a coating with negligible scattering coefficient Sand for a very

wide range of absorption coefficient K . At wavelengths below

300 nm, the agreement between the measured and calculated

reflectance is not as good; agreement can be obtained if thereal and imaginary parts of the refractive index are increased

by about 25%. At these wavelengths at which absorption is

strong, measurement of the refractive index is difficult, and the

literature values differ by up to 50% [46], so the discrepancy

between the prediction of the model and measurement is likely

to be related to uncertainties in the refractive index data.

The main requirement for the KubelkaMunk model to

be valid is that the light fluxes within the coating are diffuse.

This will of course be the case if illumination is diffuse. If

illumination is collimated, then there has to be a mechanism

for the light flux to become diffuse. One way this can be

provided is by strong scattering within the coating. The

mechanism considered here is that provided by an optically-rough surface; the roughness of the surface means that the

incident light is scattered, so that both the reflected and

transmitted light are diffuse. The angular distribution of the

reflected light provides an indication of the diffuseness of the

transmitted light. Figure 4 indicates the reflected and hence

transmitted light will be mainly diffuse for optically-rough

surfaces (0 /8) with inverse surface slope /0 10.In the current case, 0 and /0 10, so the coatingis near the edge of the range of applicability of the modified

KubelkaMunk model. Nevertheless, the model predicts the

reflectance of the coating and substrate well.

For optically-rough surfaces rfcc rfcd, and thus can be

neglected in expression (13) for the reflectance. However,it remains important to use (41) for the diffuse reflection

coefficient rf

cd, rather than simply the Fresnel coefficient rF.

This is apparent from figure 3; for /0 10 and 0/ 0.1,the diffuse reflection coefficient is significantly smaller than

the Fresnel coefficient (which was assumed to be 100% for the

purposes of the figure).

It should be noted that it is possible for both the surface of

the coating and the interface between the coating and substrate

to be optically-rough, but for the reflected light to exhibit

interference effects. This can occur when the coating is of

approximately constant thickness and follows the contours

of the substrate and when scattering and absorption within

the coating are weak. This did not occur for the currentcoating; the surface of the substrate had an autocorrelation

length about 50% greater than that of the coating and had a

very different appearance on the scale of thesurface roughness.

However, it can occur in thin coatings (of the order of 200 nm

or less) formed by oxidation of the substrate or by deposition

techniques. Transport models such as the KubelkaMunk

model do not take into account optical phase and are therefore

not applicable when interference fringes occur.

6. Conclusions

The KubelkaMunk two-flux model is strictly only applicable

to the case of diffuse illumination. However, it has frequentlybeen used, as extended by Saunderson to allow treatment of

reflection from interfaces, to calculate diffuse reflectance of

coatings under collimated illumination. Previous work has

shown that useful results can be obtained for only specific

cases: optically-thick weakly- or non-absorbing coatings,

absorbing coatings whose reflectance is very weak and for

coatings containing highly-scattering particles whose sizes

are larger than a wavelength. The influence of the surfacemorphology of the coating has not been considered.

I have extended the KubelkaMunk model to the case

of collimated illumination of optically-rough surfaces, by

modifying the Saunderson extension to allow treatment of

reflection of collimated light from optically-rough, optically-

smooth and intermediate surfaces. Further, I have introduced

an expression for the reflection coefficient that allows the

separation of reflectance into diffuse and collimated (specular)

components, taking into account the characteristics of the

integrating sphere used to measure the reflectance. The

expression for the reflectance has been compared with other

simple treatments, which have been found to be inaccurate

for some classes of rough surfaces, including those for whichthe modified KubelkaMunk model is applicable. Analysis

of the angular distribution of the reflected radiation indicates

that the light in the coating will be diffuse, and hence the

modified KubelkaMunk model is applicable, for optically-

rough surfaces with inverse surface slope /0 10.

The modified KubelkaMunk model has been tested in

the case of an optically-rough rutile titanium dioxide coating

on a titanium substrate and found to give good agreement with

measurements for wavelength ranges in which absorption of

the coating is both strong and weak, even with neglibible

scattering. Hence, the modifications extend the range in

which the KubelkaMunk model can be applied to collimated

illumination to a wide range of optically-rough coatings. It isexpected that if the surface is sufficiently rough to ensure that

the light transmitted into the coating is diffuse, the modified

KubelkaMunk model will be applicable irrespective of the

magnitude of the absorption and scattering coefficients of the

coating.

Acknowledgments

I thank Dr Piers Barnes for measuring the rms roughness

and autocorrelation length of the rutile TiO2 coating and the

titanium substrate and Dr Ian Plumb and Dr Barnes for helpful

comments.

Appendix A. Fresnel reflection coefficients

The Fresnel reflection coefficient for unpolarized light is given

by rF(i ) = 12 [r(i ) + r(i )], where r(i ) and r(i )are, respectively, the reflection coefficients of light polarized

with electric field parallel and perpendicular to the plane of

incidence, and i is the angle of incidence of the light [29, 30].

We consider light passing from medium 1 to medium 2, where

the complex refractive index of medium l is

Nl = nl + il , (A.1)

where nl is the real part of the refractive index (usually referred

to as the refractive index) and l is the extinction coefficient.

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It can be shown that [29]

r(i ) =cos2 i + u v cos icos2 i + u + v cos i

, (A.2)

r(i ) = r(i )u

v sin i

tan i

+ sin2 i

tan2 i

u + v sin i tan i + sin2 i tan

2 i. (A.3)

We then obtain the Fresnel reflection coefficient by averaging

(A.2) and (A.3):

rF(i ) = 12 [r(i ) + r(i )]

= r(i )u + sin2 i tan

2 i

u + v sin i tan i + sin2 i tan

2 i, (A.4)

where

u = {n21(n22 22 ) [(n22 22 )2 + 4n22 22 ]sin2 i}2+ 4(n22n

21)

2

12

(n22 22 )2 + 4n22 22

1

, (A.5)

v = 2n21(n22 22 ) [(n22 22 )2 + 4n22 22 ]sin2 i+ ({n21(n22 22 ) [(n22 22 )2 + 4n22 22 ]sin2 i}2

+ 4(n22n21)

2)1/2

(n22 22 )2 + 4n22 2211/2

. (A.6)

Appendix B. Calculation of reflection coefficientsfrom BRDF

The geometry of the reflectance problem was discussed in

detail by Horn and Sjoberg [50]. I present some of the relevant

definitions and results and combine these with the results of

He etal [32] to obtain expressions for the reflection coefficient

r. Here represents the polar angle (relative to the surface

normal), and represents the azimuthal angle, of a direction.The subscript i denotes quantities associated with the incident

radiant flux, and the subscript r denotes quantities associated

with the reflected radiant flux.

The irradiance Ii is the incident flux density, while the

radiant exitance Mr is the reflected flux density. The incident

radiance Li is the incident flux per unit surface area per unit

projected solid angle, and the reflected radiance Lr is the flux

reflected per unit surface area per unit projected solid angle.

The projected solid angle is related to the actual solid angle

by

= cos . (B.1)The irradiance and the incident radiance are related by

Ii =

i

Li di . (B.2)

Similarly, the radiant exitance and reflected radiance are related

by

Mr =

r

Lr dr . (B.3)

We will make use of the geometric relations

Xd =

/20

X cos sin d d (B.4)

and

Xd =

/20

X sin d d. (B.5)

The reflection properties of a rough surface are usually

specified using the BRDF, defined as

(i , i , r , r ) =dLr (i , i , r , r )

dIi (i , i ). (B.6)

The BRDF gives information about how bright a surface will

appear viewed from a given direction when illuminated from

another given direction. We do not require such directional

information; rather we require the reflection coefficient, given

by

r = Mr /Ii . (B.7)We therefore need to obtain an expression for r in terms of the

BRDF. Using (B.3) and (B.4),

Mr =

r

Lr dr =

/20

Lr cos r sin r dr dr .

(B.8)

From the definition of the BRDF (B.6), and (B.4),

Lr =

i

Li di =

/20

Li cos i sin i di di . (B.9)

We use a collimated source; the irradiance for such a source

in the direction (0, 0) will be proportional to the product of

the delta functions (i 0)(i 0). It must also satisfy(using (B.2) and (B.5))

Ii =

i

Li di =

/20

Li sin i di di . (B.10)

This can be accomplished if

Li = Ii (i 0)(i 0)/ sin 0. (B.11)

Substituting (B.11) into (B.9) gives

Lr =

/20

Ii(i 0)(i 0)

sin 0cos i sin i di di

= Ii cos i . (B.12)Substituting this expression into (B.8) gives

Mr = Ii

/20

cos i cos r sin r dr dr . (B.13)

Using (B.7), we obtain the required expression for the

reflection coefficient in terms of the BRDF for a collimatedsource

r =

/20

cos i cos r sin r dr dr . (B.14)

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