murphy 2006
TRANSCRIPT
-
8/3/2019 Murphy 2006
1/12
Modified KubelkaMunk model for calculation of the reflectance of coatings with optically-
rough surfaces
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2006 J. Phys. D: Appl. Phys. 39 3571
(http://iopscience.iop.org/0022-3727/39/16/008)
Download details:
IP Address: 200.45.63.36
The article was downloaded on 26/09/2011 at 16:31
Please note that terms and conditions apply.
View the table of contents for this issue, or go to thejournal homepage for more
ome Search Collections Journals About Contact us My IOPscience
http://iopscience.iop.org/page/termshttp://iopscience.iop.org/0022-3727/39/16http://iopscience.iop.org/0022-3727http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/contacthttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/journalshttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/searchhttp://iopscience.iop.org/http://iopscience.iop.org/0022-3727http://iopscience.iop.org/0022-3727/39/16http://iopscience.iop.org/page/terms -
8/3/2019 Murphy 2006
2/12
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: [email protected]
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:%[email protected]://stacks.iop.org/JPhysD/39/3571mailto:%[email protected]://dx.doi.org/10.1088/0022-3727/39/16/008 -
8/3/2019 Murphy 2006
3/12
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
3572
-
8/3/2019 Murphy 2006
4/12
Modified KubelkaMunk model for calculation of the reflectance
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)
3573
-
8/3/2019 Murphy 2006
5/12
A B Murphy
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
3574
-
8/3/2019 Murphy 2006
6/12
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)
3575
-
8/3/2019 Murphy 2006
7/12
A B Murphy
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)
3576
-
8/3/2019 Murphy 2006
8/12
Modified KubelkaMunk model for calculation of the reflectance
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.
3577
-
8/3/2019 Murphy 2006
9/12
A B Murphy
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
3578
-
8/3/2019 Murphy 2006
10/12
Modified KubelkaMunk model for calculation of the reflectance
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.
3579
-
8/3/2019 Murphy 2006
11/12
A B Murphy
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)
References
[1] Heavens O S 1955 Optical Properties of Thin Solid Films(London: Butterworths)
[2] Kubelka P and Munk F 1931 Ein Beitrag zur Optik derFarbanstriche Z. Tech. Phys. 12 593
[3] Kubelka P 1948 New contributions to the optics of intensely
light-scattering materials. Part I J. Opt. Soc. Am. 38 448[4] Krewinghaus A B 1969 Infrared reflectance of paints Appl.
Opt. 8 807
3580
-
8/3/2019 Murphy 2006
12/12
Modified KubelkaMunk model for calculation of the reflectance
[5] Saunderson J L 1942 Calculation of the color of pigmentedplastics J. Opt. Soc. Am. 32 727
[6] Rich DC 1995 Computer-aided design and manufacturing ofthe color of decorative and protective coatings J. Coat.Technol. 67 53
[7] Orel Z C, Gunde M K and Orel B 1997 Application of the
KubelkaMunk theory for the determination of the opticalproperties of solar absorbing paints Prog. Org. Coat. 30 59
[8] van Gemert M J C, Welch A J, Star W M, Motamedi M andCheong W-F 1987 Tissue optics for a slab geometry in thediffusion approximation Lasers Mater. Sci. 2 295
[9] Yamada N and Fujimura S 1991 Nondestructive measurementof chlorophyll pigment content in plant leaves fromthree-color reflectance and transmittance Appl. Opt. 30 3964
[10] Sardar D K and Levy L B Jr 1996 Comparative evaluation ofabsorption coefficients of KCl:Eu2+ and CaF2:Eu
2+ using aspectrophotometer and an integrating sphere J. Appl. Phys.79 1759
[11] Marx R, Vennemann B and Uffelmann E 1984 Meltingtransition of two-dimensional butadiene iron tricarbonylstudied by optical spectroscopy Phys. Rev. B 29 5063
[12] Kavaly B and Hevesi I 1971 Investigations on diffuse
reflectance spectra of V2O5 powder Z. Naturforsch. a 26 245[13] Thiebaud F and Kneubuhl F K 1983 Infrared properties of
quartz fibers and wool Infrared Phys. 23 131[14] Vargas W E 2002 Inversion methods from KubelkaMunk
analysis J. Opt. A: Pure Appl. Opt. 4 452[15] Blevin W R and Brown W J 1962 Total reflectances of opaque
diffusers J. Opt. Soc. Am. 52 1250[16] Gunde M K, Logar J K, Orel Z C and Orel B 1995 Application
of the KubelkaMunk theory to thickness-dependent diffusereflectance of black paints in the mid-IR Appl. Spectrosc.49 623
[17] McNeil L E and French R H 2000 Multiple scattering fromrutile TiO2 particles Acta Mater. 48 4571
[18] McNeil L E and French R H 2001 Light scattering from redpigment particles: multiple scattering in a stronglyabsorbing system J. Appl. Phys. 89 283
[19] Vargas W E and Niklasson G A 1997 Applicability conditionsof the KubelkaMunk theory Appl. Opt. 36 5580
[20] Ishimaru A 1978 Wave Propagation and Scattering in RandomMedia vol 1 (New York: Academic) pp 191201
[21] Maheu B, Letoulouzan J N and Gouesbet G 1984 Four-fluxmodels to solve the scattering transfer equation in terms ofLorenzMie parameters Appl. Opt. 23 3353
[22] Vargas W E 1998 Generalized four-flux radiative transfermodel Appl. Opt. 37 2615
[23] Murphy A B, Barnes P R F, Randeniya L K, Plumb I C,Grey I E, Horne M D and Glasscock J A 2006Efficiency of solar water splitting using semiconductorelectrodes Int. J. Hydrogen Energy at press, doi:10.1016/j.ijhydene.2006.01.014
[24] Barnes P R F, Randeniya L K, Murphy A B, Gwan P B,
Plumb I C, Glasscock J A, Grey I E and Li C 2006 TiO2photoelectrodes for water splitting: carbon doping by flamepyrolysis? Dev. Chem. Eng. Miner. Process. 14 51
[25] Khan S U M, Al-Shahry M and Ingler W B Jr 2002 Efficientphotochemical water splitting by a chemically modifiedn-TiO2 Science 297 2243
[26] Murphy A B 2006 Determination of the optical properties ofan optically-rough coating on an opaque substrate fromdiffuse reflectance measurements CSIRO Industrial Physics
ReportCIP-2434[27] Singh J, Oh I-K and Kasap S O 2003 Optical absorption,
photoexcitation and excitons in solids: fundamentalconcepts Photo-excited Processes, Diagnostics and
Applications ed A Peled (Amsterdam: Kluwer) pp 2555
[28] Kortum G 1969 Reflectance Spectroscopy (New York:Springer)
[29] Wendlandt W W and Hecht H G 1966 ReflectanceSpectroscopy (New York: Interscience (Wiley Interscience))
[30] Bohren C F and Huffman D R 1983 Absorption and Scatteringof Light by Small Particles (New York: Wiley)
[31] Beckmann P and Spizzichino A 1963 The Scattering ofElectromagnetic Waves from Rough Surfaces (Oxford:Pergamon)
[32] He X D, Torrance K E, Sillion F X and Greenberg D P 1991 Acomprehensive physical model for light reflection Comput.Graph. 25 175
[33] Thorsos E I 1988 The validity of the Kirchhoff approximationfor rough surface scattering using a Gaussian roughnessspectrum J. Acoust. Soc. Am. 83 78
[34] Nayar S K, Ikeuchi K and Kanade T 1991 Surface reflection:physical and geometrical perspectives IEEE Trans. Pattern
Anal. Mach. Intell. 13 611[35] Ament W S 1953 Towards a theory of reflection by a rough
surface, Proc. IRE41 1426[36] Filinski I 1972 The effects of sample imperfections on optical
spectra Phys. Status Solidi. b 49 577
[37] Poruba A, Fejfar A, Remes Z, Springer J, Vanecek M, Koeka J,Meier J, Torres P and Shah A 2000 Optical absorption andlight scattering in a microcrystalline silicon thin films andsolar cells J. Appl. Phys. 88 148
[38] Boithias L 1987 Radio Wave Propagation (New York:McGraw-Hill) pp 4950
[39] Landron O, Feuerstein M J and Rappaport T S 1996 Acomparison of theoretical and empirical reflectioncoefficients for typical exterior wall surfaces in a mobileradio environment IEEE Trans. Antennas Propag.44 341
[40] Miller A R, Brown R M and Vegh E 1984 New derivation forthe rough-surface reflection coefficient and for distributionsof sea-wave elevations IEE Proc. H 131 114
[41] Beard C I 1961 Coherent and incoherent scattering of
microwaves from the ocean IRE Trans. Antennas Propag.9 470[42] Hristov T and Friehe C 2002 EM propagation over the ocean:
analysis of RED experiment data Proc. 12th Conf. onInteractions of the Sea and Atmosphere (Long Beach, CA)(Boston: American Meteorological Society) Paper 9.2
[43] Curiel F, Vargas W E and Barrera R G 2002 Visible spectraldependence of the scattering and absorption coefficients ofpigmented coatings from inversion of diffuse reflectancespectra Appl. Opt. 41 5969
[44] Dechamps M and Lehr P 1977 Sur loxydation du titane enatmosphere doxygene: Role de la couche oxydee etmechanisme doxydation J. Less-Common Met. 56 193
[45] Rasigni G, Varnier F, Rasigni M, Palmari J B and Llebaria A1982 Autocovariance functions, root-mean-square-roughness height, and autocovariance length for rough
deposits of copper, silver and gold Phys. Rev. B 25 2315[46] Cardona M and Harbeke G 1965 Optical properties and band
structure of wurtzite-type crystals and rutile Phys. Rev. A137 1467
[47] Devore J R 1951 Refractive indices of rutile and sphaleriteJ. Opt. Soc. Am. 41 416
[48] Ribarsky M W 1985 Titanium dioxide (TiO2) (rutile)Handbook of Optical Constants ed E D Palik (Orlando, FL:Academic) pp 795800
[49] Eagles D M 1964 Polar modes of lattice vibration and polaroncoupling constants in rutile (TiO2) Phys. Chem. Sol.25 1243
[50] Horn B K P and Sjoberg R W 1979 Calculating the reflectancemap Appl. Opt. 18 1770
3581
http://dx.doi.org/10.1016/S0300-9440(96)00659-5http://dx.doi.org/10.1016/S0300-9440(96)00659-5http://dx.doi.org/10.1063/1.360965http://dx.doi.org/10.1063/1.360965http://dx.doi.org/10.1103/PhysRevB.29.5063http://dx.doi.org/10.1103/PhysRevB.29.5063http://dx.doi.org/10.1016/0020-0891(83)90027-1http://dx.doi.org/10.1016/0020-0891(83)90027-1http://dx.doi.org/10.1088/1464-4258/4/4/314http://dx.doi.org/10.1088/1464-4258/4/4/314http://dx.doi.org/10.1366/0003702953964165http://dx.doi.org/10.1366/0003702953964165http://dx.doi.org/10.1016/S1359-6454(00)00243-3http://dx.doi.org/10.1016/S1359-6454(00)00243-3http://dx.doi.org/10.1063/1.1331344http://dx.doi.org/10.1063/1.1331344http://dx.doi.org/10.1126/science.1075035http://dx.doi.org/10.1126/science.1075035http://dx.doi.org/10.1121/1.396188http://dx.doi.org/10.1121/1.396188http://dx.doi.org/10.1109/34.85654http://dx.doi.org/10.1109/34.85654http://dx.doi.org/10.1063/1.373635http://dx.doi.org/10.1063/1.373635http://dx.doi.org/10.1109/8.486303http://dx.doi.org/10.1109/8.486303http://dx.doi.org/10.1109/TAP.1961.1145043http://dx.doi.org/10.1109/TAP.1961.1145043http://dx.doi.org/10.1016/0022-5088(77)90041-8http://dx.doi.org/10.1016/0022-5088(77)90041-8http://dx.doi.org/10.1103/PhysRevB.25.2315http://dx.doi.org/10.1103/PhysRevB.25.2315http://dx.doi.org/10.1103/PhysRev.137.A1467http://dx.doi.org/10.1103/PhysRev.137.A1467http://dx.doi.org/10.1016/0022-3697(64)90022-8http://dx.doi.org/10.1016/0022-3697(64)90022-8http://dx.doi.org/10.1016/0022-3697(64)90022-8http://dx.doi.org/10.1103/PhysRev.137.A1467http://dx.doi.org/10.1103/PhysRevB.25.2315http://dx.doi.org/10.1016/0022-5088(77)90041-8http://dx.doi.org/10.1109/TAP.1961.1145043http://dx.doi.org/10.1109/8.486303http://dx.doi.org/10.1063/1.373635http://dx.doi.org/10.1109/34.85654http://dx.doi.org/10.1121/1.396188http://dx.doi.org/10.1126/science.1075035http://dx.doi.org/10.1063/1.1331344http://dx.doi.org/10.1016/S1359-6454(00)00243-3http://dx.doi.org/10.1366/0003702953964165http://dx.doi.org/10.1088/1464-4258/4/4/314http://dx.doi.org/10.1016/0020-0891(83)90027-1http://dx.doi.org/10.1103/PhysRevB.29.5063http://dx.doi.org/10.1063/1.360965http://dx.doi.org/10.1016/S0300-9440(96)00659-5