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Measuring and Modelling Light
Scattering in Paper
Niklas Johansson
Supervisor:
Prof. Per Edström, Mid Sweden University
Assistant supervisors:
Dr. Mattias Andersson and Dr. Magnus Neuman, Mid Sweden University
Department of Natural Sciences
Mid Sweden University
Doctoral Thesis No. 224
Örnsköldsvik, Sweden
2015
ISBN 978-91-88025-27-2
ISSN 1652-893X
Mid Sweden University
Department of Natural Sciences
SE-891 18 Örnsköldsvik
SWEDEN
Akademisk avhandling som med tillstånd av Mittuniversitetet framlägges till offentlig granskning för avläggande av teknologie doktorsexamen torsdagen den 4
juni 2015 klockan 10:00 i Mediacenter DPC, Mittuniversitetet, Järnvägsgatan 3,
Örnsköldsvik.
c
Niklas
Johansson, 2015
Printed by Hemströms offset och boktryck, Härnösand, Sweden, 2015
Abstract
This thesis is about measuring and modelling light reflected from paper by using goniophotometric measurements. Measuring bidirectional reflectance requires highly
accurate instruments, and a large part of the work in this thesis is about establishing
the requirements that must be fulfilled to ensure valid data. A spectral goniophotometer is used for measuring the light reflected from paper and methods are developed for analyzing the different components, i.e. the fluorescence, surface reflectance
and bulk reflectance, separately. A separation of the surface and bulk reflectance is
obtained by inkjet printing and analyzing the total reflectance in the absorption band
of the ink. The main principle of the method is to add dye to the paper until the bulk
scattered light is completely absorbed. The remaining reflectance is solely surface
reflectance, which is subtracted from the total reflectance of the undyed sample to
give the bulk reflectance. The results show that although the surface reflectance of a
matte paper is small in comparison with the bulk reflectance, it grows rapidly with
increasing viewing angle, and can have a large influence on the overall reflectance.
A method for quantitative fluorescence measurements is developed, and used
for analyzing the angular distribution of the fluoresced light. The long-standing
issue whether fluorescence from turbid (or amorphous) media is Lambertian or not,
is resolved by using both angle-resolved luminescence measurements and radiative
transfer based Monte Carlo simulations. It is concluded that the degree of anisotropy
of the fluoresced light is related to the average depth of emission, which in turn
depends on factors such as concentration of fluorophores, angle of incident light
and the absorption coefficient at the excitation wavelength.
All measurements are conducted with a commercially available benchtop sized
double-beam spectral goniophotometer designed for laboratory use. To obtain reliable results, its absolute measurement capability is evaluated in terms of measurement accuracy. The results show that the compact size of the instrument, combined
with the anisotropic nature of reflectance from paper, can introduce significant systematic errors of the same order as the overall measurement uncertainty. The errors
are related to the relatively large detection solid angle that is required when measuring diffusely reflecting materials. Situations where the errors are most severe,
oblique viewing angles and samples with high degree of anisotropic scattering, are
identified, and a geometrical correction is developed.
Estimating optical properties of a material from bidirectional measurements has
proved to be a challenging problem and the outcome is highly dependent on both
the quality and quantity of the measurements. This problem is analyzed in detail for
optically thick turbid media, targeting the case when a restricted set of detection angles are available. Simulations show that the measurements can be restricted to the
plane of incidence (in-plane), and even the forward direction only, without any significant reduction in the precision or stability of the estimation, as long as sufficiently
oblique angles are included.
i
Sammanfattning
Avhandlingen behandlar de teoretiska och praktiska aspekterna av att använda spektrala vinkelupplösta reflektansmätningar för optisk karakterisering av fiberbaserade
material såsom papper och kartong. En spektral goniofotometer används för att
mäta det reflekterade ljusets vinkelfördelning. En stor del av arbetet utgörs av att
utvärdera instrumentets noggrannhet, samt utreda hur de vinkelupplösta mätningarna skall utföras på bästa sätt för att erhålla en så fullständig karakterisering som
möjligt. Det reflekterade ljuset består av tre komponenter; ytreflektans, bulkreflektans samt fluorescens. En fullständig karakterisering förutsätter att dessa tre komponenter kan analyseras separat, vilket i detta arbete görs genom nyutvecklade metoder.
En metod har utvecklats för separation av ytreflektans och bulkreflektans. Metoden bygger på att analysera hur den totala reflektansen förändras vid ökande absorption i det reflekterande materialet. Absorptionen kontrolleras genom inkjettryckning där tryckfärg appliceras på substratet i sådan mängd att bulkreflektansen
helt släcks ut. Genom att kombinera mätningar på tryckt och otryckt substrat kan
de båda komponenterna separeras. Trots att ytreflektansen från ett matt papper är
liten i förhållande till bulkreflektansen, så visar resultaten att den ökar markant med
ökande betraktningsvinkel och kan därmed ha stor inverkan på den totala reflektansen. Bidraget från fluorescens kan kvantitativt analyseras genom att kombinera
mätningar utförda med respektive utan UV-filter. Vinkelupplösta mätningar och
Monte Carlo-simuleringar av fluorescensens vinkelfördelning visar att dess anisotropi är relaterad till det medeldjup vid vilket fluorescensen emitteras. Resultaten
förklarar observerade skillnader och motstridigheter i tidigare studier kring huruvida fluorescens kan anses vara Lamberskt fördelad.
Samtliga goniofotometriska mätningar är utförda med ett kompakt, kommersiellt tillgängligt, dubbelstråleinstrument. För att undersöka instrumentets lämplighet
för absoluta reflektansmätningar utförs en analys av dess mätnoggrannhet. Resultaten visar att instrumentets kompakta storlek i kombination med anisotrop reflektans från papper introducerar systematiska fel av samma storleksordning som
den totala mätnoggrannheten. Dessa fel uppstår på grund av den relativt stora
detektorapertur som måste användas vid mätningar av diffus reflektans, vilket är
karakteristiskt för papper och kartong. Resultaten visar även att felen är störst vid
flacka mätvinklar och för prover med hög grad av anisotropisk reflektans, och en
geometrisk korrektionsmetod för denna typ av systematiska fel föreslås.
Spektrala och vinkelupplösta mätningar medför per automatik stora mängder
mätdata. Genom att använda strålningstransportteori som en matematisk modell
för hur ljus sprids i papper kan mätdatat reduceras till en uppsättning beskrivande
materialparameterar. Att uppskatta dessa optiska parametrar utifrån vinkelupplösta
reflektansmätningar är i sig ett komplicerat problem, vilket dessutom är känsligt för
mätfel och val av mätvinklar. Detta inversa problem analyseras i detalj, och specifikt hur valet av mätvinklar kan reduceras utan att försämra förutsättningarna för
estimeringen. Simuleringar visar att mätningarna kan begränsas till infallsplanet,
eller till och med enbart framåtriktningen, så länge tillräckligt flacka mätvinklar är
inkluderade i mätsekvensen.
ii
Acknowledgements
First of all, I would like to thank my supervisors Per Edström, Mattias Andersson
and Magnus Neuman for their support and encouragement throughout the work. It
is always a pleasure to see true professionals at work, and even better to have the
favour to work with them. Apart from their supporting roles, I would also like to
thank them for creating a relaxed, easy-going atmosphere with lots of shared jokes
and laughters.
My colleagues in Örnsköldsvik and Härnösand are also thanked. Thomas Öhlund
and Ole Norberg are thanked for stimulating discussions and for keeping me company during several late hours at the office. Ludovic Gustafsson Coppel is thanked
for successful cooperation, and for being a reliable colleague and friend. I would
also like to thank Prof. Sverker Edvardsson for teaching interesting courses at the
end of my doctoral studies.
This work has been performed in close cooperation with several industrial partners; Metsä Board Husum, MoRe Research AB, Imerys Mineral Limited, Lorentzen
& Wettre AB and Knightec AB are thanked for the support and expertise they have
provided throughout the work. A special thanks goes to Henry Westin for valuable
input and for showing a genuine interest in my work. During my doctoral studies I
have also come in contact with several universities and research institutes, and special thanks are directed to Dr. Réjean Baribeau at the National Research Council of
Canada and Dr. Markku Hauta-Kasari at the InFotonics Center in Joensuu. I would
also like to thank Dr. Daniel Nyström for helping me win the Cactus Award in 2012.
The Knowledge Foundation (KK-stiftelsen), the Kempe foundations, Länsstyrelsen
Västernorrland and the Swedish Agency for Economic and Regional Growth
(Tillväxtverket) are gratefully acknowledged for their financial support.
Finally, I would like to thank my family; Maria, Lukas, Elias and Gustav for constantly reminding me of what’s most important in my life.
iii
Contents
Abstract
i
Sammanfattning
ii
Acknowledgements
iii
List of papers
vi
1
Introduction
1
1.1
2
Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Light scattering in paper
2
3
Goniophotometric measurements
4
3.1
Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
Measuring the BRDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.3
Measurement uncertainty in BRDF measurements . . . . . . . . . . . . 11
4
5
Radiative transfer theory
12
4.1
The radiative transfer equation . . . . . . . . . . . . . . . . . . . . . . . 14
4.2
The phase function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3
The forward and inverse problem . . . . . . . . . . . . . . . . . . . . . . 18
Summary of the papers and contribution of the
thesis
20
5.1
Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2
Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3
Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.4
Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.5
Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6
Discussion
26
7
Future work
27
8
Conclusions
28
iv
References
29
v
List of papers
The thesis consists of the following papers, herein referred to by their respective
Roman numerals:
I
N. Johansson, M. Neuman, M. Andersson and P. Edström, “The influence of finite sized detection solid angle on bidirectional reflectance distribution function measurements,” Applied Optics 53(6), p. 1212–1220
(2014). Selected for publication in Virtual Journal of Biomedical Optics, 9(4).
II
N. Johansson, M. Neuman, M. Andersson and P. Edström, “Separation
of surface and bulk reflectance by absorption of bulk scattered light,”
Applied Optics 52(19), p. 4749–4754 (2013). Selected for publication in
Virtual Journal of Biomedical Optics, 8(8).
III
N. Johansson and M. Andersson, “Angular variations of reflectance and
fluorescence from paper - the influence of fluorescent whitening agents
and fillers,” in Proc. 20th Color and Imaging Conference, Los Angeles,
USA, p. 236–241 (2012).
IV
L. G. Coppel, N. Johansson and M. Neuman, “Angular dependence of
fluorescence from turbid media,” manuscript submitted to Optical Express (2015).
V
N. Johansson, M. Neuman, M. Andersson and P. Edström, “The inverse
radiative transfer problem - considerations for optically thick media,”
manuscript submitted to Inverse Problems in Science and Engineering
(2015).
vi
1 Introduction
1 Introduction
To be able to accurately determine the reflectance characteristics of an object is of
great importance in many fields. Apart from determining color and appearance attributes of objects, the information can also be used for such diverse purposes as
restoration and conservation of art and for detection of cancer tumours. A complete
characterization involves measuring both the spectral and angular distribution of
the reflected light. The angular distribution is for most materials determined almost
exclusively by the surface characteristics of the object. Paper however, can be classified as a turbid medium, where the angular distribution of the reflected light is
governed to a large extent by the light scattering that takes place within the bulk of
the object, from here on referred to as bulk reflectance.
Within the paper industry, reflectance is measured according to well defined standards [1], and with instruments that are accurately calibrated against traceable reflectance standards. In the standardized measurement geometry, the detection angle
is fixed. The sample is diffusely illuminated and the reflected light is measured at
0◦ , i.e. in a direction perpendicular to the surface of the sample. It has however been
shown that the bulk reflectance from paper is anisotropic [2, 3], where the intensity
of reflected light depends on both the angle of incidence and the viewing angle. In
order to capture these variations, and obtain a more comprehensive characterization
of the reflectance from paper, angle resolved measurements are required.
This type of characterization requires measurements with a spectral goniophotometer, an instrument that can perform spectrally resolved measurements of light
reflected or transmitted in different directions for different angles of incidence, i.e
bidirectional measurements. Angle resolved reflectance and transmittance measurements are of interest in many scientific fields, particularly when studying light scattering in turbid media, such as paper [3–5] or biological tissue [6,7]. Goniophotometric measurements are also required when characterizing gonio-apparent materials,
i.e. materials that exhibit a shift in color depending on the illumination and viewing
angle [8, 9], such as coatings with pearlescent or metallic pigments used within the
automotive and packaging industry [10].
The intensity and angular distribution of the reflected light depends on several
factors, and specifically on the optical properties of the object. The bulk reflectance
can be modelled with radiative transfer (RT) theory, that describes how light propagates in turbid media. RT models can therefore be used to simulate the optical
response of a turbid medium, often in terms of its bidirectional reflectance distribution function (BRDF), for any illumination and viewing condition, provided that the
optical parameters of the medium are known. These can in turn be estimated from
bidirectional reflectance measurements, by solving the so called inverse RT problem.
Apart from paper technology applications, modelling light scattering in turbid media is also of great interest in computer graphics in order to obtain realistic rendering
of complex materials under different viewing conditions. It is also frequently used
in such diverse fields as optical tomography [11–13] and atmospheric science [14].
This thesis is about utilizing spectral goniophotometry for optical characteriza1
Measuring and modelling light scattering in paper
tion of paper. Paper I deals with the reliability and expected accuracy of this type of
measurements on paper, and contains both theoretical and experimental elements.
In paper II and III, measurement methods are developed for analyzing the surface
reflectance, bulk reflectance and fluorescence separately. The angular distribution of
the fluorescence is investigated in more detail in paper IV. Finally, paper V targets
the inverse problem, and specifically how it is affected by the choice of measurement
angles.
1.1 Aim of the thesis
The overall aim of the thesis is to investigate how bidirectional reflectance measurements can be used to increase the understanding of how light is reflected from paper,
and how the measurements should be performed to ensure valid data.
This aim is approached through three specific targets. The first is to investigate
the applicability of a compact benchtop sized goniophotometer for reflectance measurements of paper, in terms of accuracy in absolute measurements. This study focuses on quantifying measurement errors related to the relatively large detection
solid angle that is required when measuring diffusely reflecting materials. The second is to develop measurement methods for separating light radiated from a paper into its surface reflectance, bulk reflectance and fluorescence components, which
opens up new ways of analyzing how surface treatments, filler pigments and fluorescent whitening agents affect the optical response of paper. Finally, the third is
to establish how the measurement sequence should be designed in order to provide
proper data for the inverse problem.
2 Light scattering in paper
This section is intended to give a brief description of the different components involved in light scattering in paper. For a comprehensive description of paper optics
and details of how different fibres, additives and stages in the manufacturing process
affect the optical properties, the textbook by Pauler can be recommended [15].
Figure 1 shows a simplified illustration of the different sources of scattering and
luminescence in paper. The origin of the light scattered from paper can either be the
bulk of the paper or the surface of the paper, as illustrated in Figure 1. An important
component that is not shown explicitly in the figure is the pores of the paper, which
contribute significantly to the scattering. Surface reflectance is light reflected directly
at the interface between the surrounding medium (air) and the substrate. Since the
light has not interacted with the bulk, its spectral content remains unaltered. Bulk
reflectance consists of light that penetrates into the bulk of the paper, and after multiple scattering events inside the material, is eventually reflected back through the
top surface. The last component is fluorescence, caused by the interaction of light
and fluorophores added to the paper. Filler pigments are added mainly in order
to increase the light scattering in the paper, and thereby the opacity. Fluorescent
2
2 Light scattering in paper
Fluorescence
Surface refl.
Bulk refl.
FWA
Filler pigments
Figure 1: The different origins of light reflected or fluoresced from paper. The bulk reflectance
consists of light scattered from fibres, filler pigments and pores, whereas fluorescence is
caused by light interacting with fluorescent whitening agents (FWA).
whitening agents (FWA) are added to increase the whiteness of the paper. The fluorophores absorb light in the UV-region of the spectrum and emits in the blue part,
thereby compensating for the yellowness inherent in the plain paper.
The light detected during reflectance measurements normally contains all three
components, and it is often of interest to study them separately. The fluorescence can
be removed by using UV-blocking filters, but separating the total reflectance into a
surface and bulk component is a more difficult task. Separating them is of interest in
several scientific fields, such as optical coatings and paper industry [16, 17], conservation of art [18] and biomedical science [19]. Matte paper , that can be described as
a dielectric with anisotropic bulk scattering and often a rough surface, is an example
of a material were a separation is difficult to obtain with existing methods. Methods
solely based on changes in the polarization state of the reflected light are not applicable on this type of materials, which has been reported in several studies [20–23].
More refined ellipsometric techniques [24, 25], such as angle-resolved ellipsometry
perform better, but becomes increasingly complex when analyzing highly scattering
materials with significant surface roughness [26–29]. For these reasons, an alternative method has been developed and is presented in Paper II.
A major part of this thesis is about developing methods for measuring the angular distribution of the different reflectance and luminescence components. The modelling has however been focused on bulk scattering, described in detail in Section 4.
Although not used in this thesis, it might still be instructive to give a brief review
3
Measuring and modelling light scattering in paper
of methods to model the surface reflectance as well. For perfectly flat surfaces, the
surface component can be modelled by Fresnel’s equations. As the roughness of the
surface increases, other approaches are needed. In the Torrance-Sparrow model [30],
the surface is being modelled as a composition of small, perfectly reflecting tilted
facets. By incorporating shadowing effects, where the incident light does not reach
parts of the surface, and masking effects, where a part of the reflected light does not
reach the viewer, the agreement between measured and simulated reflectance is improved. These improvements are most significant at grazing viewing angles where
effects such as off-specular maxima are present. The Torrance-Sparrow model has
been the subject of further development and several variants have been presented
throughout the years [31–34]. A prerequisite for all these models is that the wavelength of the incident light is small in comparison with the dimensions of the surface
roughness, i.e. that geometrical optics is applicable. As the dimensions approach
the wavelength of the incident light, models where the light is treated as an electromagnetic wave, i.e. physical optics, must be used. Most of these wave scattering
models [35,36] are based on the work by Beckmann and Spizzichino [37]. A comparison of different surface models and their pros and cons can be found in the work by
Nayar et al. [38] and Westin et al. [39]
3 Goniophotometric measurements
...nobody will object to an ardent experimentalist boasting of his measurements and rather
looking down on the ”paper and ink” physics of his theoretical friend, who on his own part is
proud of his lofty ideas and despises the dirty fingers of the other.
Max Born
This section gives a description of spectral goniophotometry as a method to assess
the spectral and angular distribution of the reflected light. An overview of different
types of instruments is given in Section 3.1, together with a description of the instrument used in this thesis. A detailed derivation of the bidirectional reflectance distribution function (BRDF), which is defined in radiometric quantities, from the measurement signals and instrument parameters, is given in Section 3.2. Bidirectional
reflectance, where both the incident and reflected light is contained within infinitesimally small solid angles, is defined as a ratio between two differential quantities.
It can therefore only be measured approximately, and the necessary approximations
are described in detail.
For practical purposes, BRDF measurements are usually performed relative to a
calibrated reflectance standard, i.e. relative measurements, as opposed to absolute
measurements where the BRDF is assessed directly in terms of reflectance or radiance measurements and no reference standard is required. Relative measurements
have the advantage that the reliability in reported values increases due to the traceability of the reference standard. The calibration procedure does however become
more complex due to the increased degrees of freedom in measurement geometry.
4
3 Goniophotometric measurements
Measurement errors are in most cases dependent on the geometry used and a single
calibration geometry is simply not enough to ensure reliable results, as opposed to
standardized methods with fixed geometry. The obvious disadvantage with absolute measurements is that the measurement parameters need to be known with high
accuracy since they are required in order to convert measured quantities into BRDF
values. Section 3.3 gives a detailed description of how to estimate the measurement
uncertainty for a double-beam instrument.
3.1 Instrumentation
Spectral goniophotometers can roughly be divided into two categories; scanning and
imaging devices, where scanning devices in general are more accurate but slower
than imaging devices. For scanning instruments [8, 40–48] the measurements at different viewing angles are taken one by one in a sequence. The most common configurations are to use broadband illumination and a wavelength dispersive detector,
e.g. spectroradiometer, or the opposite, broadband illumination passing through a
monochromator and a broadband detector. The high accuracy does however come at
the cost of longer measurement times. Scanning instruments can in turn be divided
into two categories; large accurate reference instruments at metrology institutes, and
commercially available compact benchtop sized instruments. The prevailing method
for the first category is to utilize a spectral radiometer to measure the reflected radiance directly, as well as the incident irradiance from the light source [40, 41]. For
the latter category, double-beam spectrophotometers are often used [49], where the
measurement signal consists of the ratio between the incident and reflected power,
and this is the type of instrument that has been used in this thesis.
For faster measurements, camera-based systems where multiple angles can be
detected simultaneously have become more common, particularly within the field of
computer graphics. In general, camera based systems have a lower accuracy in terms
of spectral resolution and dynamic range. The lower dynamic range of imaging
sensors can however be improved by making use of HDR cameras [50], which also
reduces the acquisition time. The dynamic range and spectral resolution can also
be improved by combining different detectors such as multispectral cameras and
spectroradiometers [51].
A schematic illustration of the instrument used in this thesis is given in Figure
2. The instrument is a Perkin Elmer Lambda 1050 spectrophotometer equipped with
an ARTA goniophotometer accessory from OMT Solutions BV [49]. The spectrophotometer is equipped with two light sources, a tungsten halogen light bulb for VIS
and a deuterium lamp for UV. Monochromatic light is obtained by letting the light
pass through double holographic grating monochromator, where one grating is for
UV/VIS and the other for NIR. The beam splitting is obtained by a chopper module, and the sample beam is then guided via a system of mirrors into the sample
compartment of the goniophotometer, whereas the reference beam is guided via an
optical fibre bundle directly to the detector.
5
Measuring and modelling light scattering in paper
spectrophotometer
goniometer
optical fiber bundle
detector
reference beam
monochromator
chopper module
Φr
sample beam
θi
Φi
θr
sample
rotational axis
light source
Figure 2: Schematic illustration of the double-beam spectral goniophotometer used in this thesis. The beam splitting enables simultaneous measurement of the incident (Φi ) and reflected
(Φr ) power. The goniometer module (illustrated as a top view) makes it possible to change
both the angle of incidence θi and detection angle θr , independently of each other.
The sample is placed on a motorized rotation stage, and the angle of the incident
light is varied by rotating the sample. The reflected light is detected with an integrating sphere detector equipped with a photomultiplier tube for UV/VIS and an
InGaAs detector for NIR. The sphere is mounted on a second motorized rotation
stage, allowing the detector to be positioned at any angle relative the normal of the
sample surface. The detection solid angle is varied by adjusting the width of the detector aperture, and the size of the illuminated spot is adjusted by a beam mask and
by altering the width of the exit slit of the monochromator.
3.2 Measuring the BRDF
Bidirectional reflectance measurements are often presented in terms of the BRDF,
which in turn is defined in terms of radiometric quantities. In this section the relationship between radiometric quantities and measured quantities in terms of reflectance is given. Although the procedure is fairly straightforward it might appear
confusing at a first glance, or to quote William L Wolfe [52]: ”Radiometry appears to be
a simple subject, and the basics are indeed uncomplicated. The devil is in the details.”. For a
more comprehensive review of radiometry, the textbooks by Wolfe [52] and Palmer
and Grant [53] can be recommended.
The defining equation for the BRDF is [54]
fr (θi , φi , θr , φr ) =
dLr (θi , φi , θr , φr )
,
dEi (θi , φi )
(1)
where dLr (θi , φi , θr , φr ) is the differential radiance reflected in the direction (θr , φr )
6
3 Goniophotometric measurements
Quantity
Radiant energy
Definition
-
Symbol
Q
Unit
[J]
Radiant flux, power
dQ
dt
Φ
[W]
Radiant intensity
dΦ
dω
I
[W · sr−1 ]
Irradiance
dΦ
dAi
E
[W · m−2 ]
Radiant exitance
dΦ
dAi
M
[W · m−2 ]
d2 Φ
dAi cos θdω
L
[W · sr−1 · m−2 ]
Radiance
Table 1: Basic radiometric quantities. The defining angles are described in Figure 3. The
difference between irradiance and radiant exitance is that the former is defined for radiant
flux incident on a surface dAi , and the latter for flux emitted or reflected from a surface. The
radiance is defined in terms of projected area dAi cos θ, which is illustrated in Figure 4.
and differential solid angle dωr , originating from the differential irradiance dEi impinging on area element dAi from a direction (θi , φi ), see Figure 3. The definitions
of the different radiometric quantities are given in Table 1. A measurement situation inevitably involves finite detector apertures and finite size of illuminated area,
and the measured BRDF is therefore an approximation of the differential form. A
detailed description of how the measured BRDF differs from the strictly theoretical
is given below.
Non-directional illumination
The BRDF is defined for directional illumination, which implies that the incident
light beam must be perfectly collimated. The solid angle ωi through which each surface element of the illuminated area receives flux is then infinitesimal. In practice this
is impossible to obtain and the light beam is either slightly diverging or converging.
The differential reflected radiance dLr (θi , φi , θr , φr ) is therefore approximated by the
total radiance reflected by a surface element dAi , into the direction (θr , φr ), given by:
Z
dLr (θi , φi ; θr , φr ; Ei )
(2)
Lr (θr , φr ) =
ωi
where ωi is the illumination solid angle defined by the cross section of the beam
and the distance between light source and sample plane. The dependence on the
irradiance, via the BRDF, is indicated by the last argument. Substituting Eq. (1) into
7
Measuring and modelling light scattering in paper
Z
dωi
dωr
θi
θr
φi
Y
dAi
φr
X
Figure 3: Defining angles for bidirectional reflectance. Light incident from (θi , φi ) contained
within the differential solid angle dωi is reflected from a surface element dAi along the direction
(θr , φr ) and into the differential solid angle dωr .
Eq. (2) gives us
Z
Z
Lr = fr (θi , φi , θr , φr )dEi = fr (θi , φi , θr , φr ) dEi = fr (θi , φi , θr , φr )Ei ,
(3)
ωi
ωi
assuming that fr is constant over ωi . The BRDF is then given by
fr (θi , φi , θr , φr ) =
Lr (θi , φi , θr , φr )
,
Ei (θi , φi )
(4)
which is the defining equation for the radiance coefficient q = Lr /Ei , a more general
quantity defined for any kind of illumination geometry. We are simply approximating the BRDF with the measured radiance coefficient for non-directional illumination
defined by the illumination solid angle ωi .
Finite sized areas
The BRDF is defined as the ratio between the radiance reflected or emitted from a
surface element and the irradiance on the same surface element. A BRDF-measurement
involves measuring the flux reflected into a detection solid angle, which in the simplest case is directly defined by the detector aperture. The corresponding radiance
incident on the detector Ld is then equal to the reflected (or emitted) radiance Lr
according to the radiance invariance principle, see Figure 4.
For double-beam instruments, the BRDF is measured as the ratio between incident
8
3 Goniophotometric measurements
dAd
n̂d
n̂i
dAd cos θd
θd
θr
dωd
dωs
dAi
dAi cos θr
Figure 4: The radiance invariance principle states that the radiance emitted from surface
element dAi into solid angle dωd equals the radiance received by area dAd contained within
solid angle dωs , assuming a lossless medium between the two surfaces.
power Φi and reflected power Φr . In order to express the BRDF in terms of measured quantities, the specific measurement configuration regarding the relationship
between illuminated area Ai and detected area Af has to be taken into account. Figure 5 shows the two configurations most often encountered; Ai always smaller and
within Af , and the opposite, Af always smaller and within Ai . In both cases, the
irradiance is given by:
Φi cos θi
Φi
=
,
(5)
Ei =
Ai
As
where As = Ai cos θi is the size of the illuminated area at normal incidence and the
differential areas and fluxes have been replaced with their finite counterparts. For
the first case, illustrated in Figure 5(a) for θr = 0, the reflected radiance Lr is given
by:
Φr
Φr
Φr r 2
Lr = Ld =
=
,
(6)
=
Ad cos θd Ωf
Ad cos θd ω
Ad cos θd Ai cos θr
where we have replaced the solid angle Ωf , defined by the detectors field of view,
with the solid angle ω = Ai cos θr /r2 , since this is the effective solid angle through
which the detector receives flux. The angle θd is defined in Figure 4 and is equal
to zero in Figure 5. The BRDF is then, omitting the angular dependence for clarity,
given by
fr =
Φr
r2 As cos θi
Φr
r2
Lr
=
=
.
Ei
Φi Ad cos θd As cos θi cos θr
Φi Ad cos θr cos θd
(7)
9
Measuring and modelling light scattering in paper
Ad
Ad
As
ωd
As
Ωf
ω
Ai
Ai
Af
(a) Detected area larger than illuminated area
As
Ad
As
Ad
ωi
Ωf
Ai
Ωf
Af
Ai
Af
(b) Illuminated area larger than detected area
Figure 5: The two most common configurations in BRDF measurements are illustrated in (a),
where the entire illuminated area Ai is within the detectors field of view Ωf , and in (b) where
Ai is larger than the area Af covered by the detector.
For this configuration, the BRDF does not explicitly depend on the angle of incidence. For an increasing angle of incidence, the incident flux is distributed over a
larger area, and the irradiance decreases accordingly. This decrease is automatically
taken into account in the expression for the BRDF, provided that the entire illuminated area is within the detectors field of view for all viewing angles and angles of
incidence.
For the second configuration, illustrated in Figure 5(b), Lr is given by:
Lr = Ld =
Φr r 2
Φr
=
,
Ad cos θd Ωf
Ad cos θd Af cos θr
(8)
which gives us
fr =
Lr
Φr
r2
As
=
·
·
.
Ei
Φi Ad cos θr cos θd Af cos θi
(9)
The decrease in irradiance is compensated for by the ratio between illuminated and
10
3 Goniophotometric measurements
detected area in the last term. It should be noted that the detected area Af is also
dependent on the viewing angle as it becomes larger with increasing θr .
In this work, measurements are always conducted according to the first configuration displayed in Figure 5(a). The detector aperture is always perpendicular to
the position vector between the center of the sample surface and the center of the
detector aperture, and substituting θd = 0 into Eq. (7), we finally get the following
expression for the BRDF
fr =
r2
Φr
.
Φi cos θr Ad
(10)
Sometimes the cos θr term is dropped from the definition, which results in the so
called cosine-corrected BRDF. A more accurate expression, that takes the radiative
transfer between two finite surfaces into account, can be obtained by performing the
geometric correction described in Paper I.
3.3 Measurement uncertainty in BRDF measurements
In this section, the combined standard uncertainty of BRDF measurements with a
double-beam instrument is derived. The analysis is given for a setup were the entire
illuminated area is within the detectors field of view and where the measured quantity is reflectance.
The measurement signal V (λ) at certain wavelength λ is determined by the detectors
responsivity function ℜ(λ) according to V (λ) = Φ(λ)ℜ(λ). For a double-beam instrument where the same detector is used for measuring both the incident and reflected
flux, and where the wavelength of the incident light is equal to the wavelength of
the detected light, the detector responsivities are equal and the measured reflectance
equals ρ = Φr /Φi = Vr /Vi . Possible changes in responsivity due to changes in polarization state etc. are assumed to be negligible. From Eq. (10), the BRDF is then given
by
fr (θi , φi ; θr , φr ) =
Vr ℜi
r2
r2
·
·
=ρ·
.
Vi ℜr Ad · cos θr
Ad · cos θr
(11)
Following the procedure recommended by the Joint Committee for Guides in Metrology (JCGM) in [55], the combined standard uncertainty uc (fr ) of the BRDF is
2
u2c (fr ) =
∂fr
∂ρ
+
∂fr
∂ cos θr
u2 (ρ) +
2
∂fr
∂r
2
u2 (cos θr ) ,
u2 (r) +
∂fr
∂Ad
2
u2 (Ad )
(12)
11
Measuring and modelling light scattering in paper
where the partial derivatives are given by
∂fr
r2
=
∂ρ
Ad cos θr
,
ρr2
∂fr
=− 2
∂Ad
Ad cos θr
2ρr
∂fr
=
∂r
Ad cos θr
,
∂fr
ρr2
=−
2
∂ cos θr
Ad (cos θr )
The relative combined standard uncertainty uc (fr )/fr is then given by
uc (fr )
fr
2
=
u(ρ)
ρ
2
+
2u(r)
r
2
+
u(Ad )
Ad
2
+
u(cos θr )
cos θr
2
(13)
By introducing g = cos(θr ), the standard uncertainty of the cosine of the viewing
angle is given by
u(cos θr ) = u(g) = u(θr ) ·
dg (θr )
= u(θr ) · sin(θr ) .
dθr
(14)
Using Eq. (14) in Eq. (13), we can finally write the relative combined standard uncertainty as
uc (fr )
fr
2
=
u(ρ)
ρ
2
+
2u(r)
r
2
+
u(Ad )
Ad
2
2
+ (u(θr ) tan θr ) .
(15)
Note that the measurement uncertainty depends on tan θr . The error due to angular
misalignment increases rapidly as the viewing angle increases, and is one of the
reasons why BRDF measurements at grazing angles are often less accurate. The
uncertainty in reflectance u(ρ) depends in turn on components such as repeatability,
detector nonlinearity, stray light and so on. These components have been estimated
in paper I. For a more accurate error budget, the accuracy of the measurement signals
Vr and Vi must be estimated separately, and accuracy of neutral density filters etc.
must be taken into account [56].
4 Radiative transfer theory
Radiative transfer (RT) theory is a transport theory that can be used to describe how
light propagates in a scattering and absorbing homogeneous medium. In this work
we model a medium whose optical properties do not vary with spatial coordinates,
and we use the Henyey-Greenstein phase function to describe the single-scattering
process. This phase function relates the scattering distribution to the relative scattering angle only, i.e. to the angle between incoming and outgoing direction, and not to
the absolute directions of the incident and outgoing light, as illustrated in Figure 6.
12
4 Radiative transfer theory
Scattering particle
Ii
Is
Figure 6: The optical properties do not vary with spatial coordinates, and the scattering distribution depends on the relative scattering angle only. This means that the scattering distribution
looks the same everywhere in the medium and does not depend on absolute direction of incident light. The asymmetry factor g of the phase function (here Henyey-Greenstein) determines
the ellipticity of the scattering distribution.
Compared to other popular theories for light scattering, RT positions it self in the
mid range in terms of both explanatory power and complexity. Approximations of
RT theory are often encountered, such as Kubelka-Munk and diffusion theory (DT),
which are frequently used in paper technology and optical tomography respectively.
Although widely used and sufficiently accurate in many applications, they cannot
resolve the asymmetry factor and therefore fail in predicting anisotropic scattering.
Furthermore, these methods are in principle only valid for normal incidence and
when the scattering coefficient is much larger than the absorption coefficient. The
main benefit with these approximations is that they can be solved analytically, which
makes e.g. the inverse problem trivial.
RT treats the light as a bundle of rays and cannot explain polarization and interference effects since it does not take the wave nature of light into account. A rigorous
treatment of scattering of electromagnetic waves involves solving Maxwell’s equations for a certain configuration of particles, and a direct approach of obtaining an
analytical solution quickly becomes intractable as the number of particles increases
and multiple scattering needs to be taken into account. Several numerical methods
for solving Maxwell’s equations for more complex situations have been developed,
such as the T-matrix method [57] and the discrete dipole approximation (DDA) [58].
An overview of the most frequently used methods and DDA in particular is given
by Kahnert [59] and Yurkin and Hoekstra [60] respectively.
RT theory has until quite recently been regarded as a phenomenological approach
to describe scattering of electromagnetic waves. Substantial effort has been put into
13
Measuring and modelling light scattering in paper
deriving it from first principles, such as Preisendorfer’s attempts to connect this
phenomenological ”island” with the ”mainland” of physics [61]. By the work of
Mishchenko [62–65] this connection was finally established when he showed that RT
can be derived from Maxwells equations, provided that a set of specific conditions
are fulfilled. These conditions can be regarded as being fulfilled when modelling
light scattering in paper with RT theory [66].
Within the RT framework, light propagation is described by a single equation,
the radiative transfer equation, which is described in the next section.
4.1 The radiative transfer equation
Before we continue with the derivation of the radiative transfer equation (RTE), it
might be worthwhile to emphasize some differences in nomenclature between RT
and radiometry. Within RT, intensity is equivalent to the radiometric quantity radiance, whereas flux corresponds to the radiometric quantities irradiance and radiant
exitance. The intensity per unit frequency interval is often denoted specific intensity.
Depending on discipline, the term ”specific” might however refer to some other unit,
and extra care must therefore be taken to avoid confusion.
The radiative transfer in a plane-parallel geometry, i.e a slab geometry, is illustrated in Figure 7(a). The horizontal extension of the medium is much larger than
the vertical extension (i.e the thickness), and boundary effects at the sides of the
medium can therefore be ignored. The figure shows light with intensity I propagating in a plane-parallel medium of thickness d with scattering coefficient σs and
absorption coefficient σa . At depth z = z ′ , the reduction in intensity when travelling
a distance ds in the direction Ω is given by:
Z
σs ds
I(z, Ω′ )p(Ω′ , Ω)dΩ′ ,
(16)
dI(z, Ω) = −(σa + σs )I(z, Ω)ds +
4π 4π
where p(Ω′ , Ω) is the phase function that gives the probability of scattering from
direction Ω′ to the direction Ω.
The different components contributing to the change in intensity are illustrated
in Figure 7(b), where the radiative transfer within a small volume element is shown.
The first component on the right hand side of Equation (16) is the reduction in intensity I(z, Ω) due to absorption and ”out-scattering”, i.e scattering away from the
direction of propagation (red arrows). The integral term sums up the positive ”inscattering” contributions (blue arrows) from light travelling in all possible directions
Ω′ (including Ω) that is being scattered into the the direction of propagation. By using dz = ds · u, where u = cos θ, the RTE can then finally be written as
Z
σs
dI(z, Ω)
= −(σa + σs )I(z, Ω) +
I(z, Ω′ )p(Ω′ , Ω)dΩ′ .
(17)
u
dz
4π 4π
An alternative representation is obtained by introducing a new coordinate, the dimensionless optical depth τ (z), defined as
Z ∞
τ (z) =
(σa + σs )dz ′ ,
(18)
z
14
4 Radiative transfer theory
Ω
z=d
I
τ (z)
θ
ds
dz
I(z, Ω′ )
I(z, Ω)
z
z=0
ds
τ
(b) RT in a small volume element
(a) Plane-parallel geometry
Figure 7: For radiative transfer in a plane-parallel geometry (a), the intensity depends only on
the depth z, or alternatively the optical thickness τ . The different scattering terms in the RTE
are illustrated in (b)
and illustrated in Figure 7(a). The upper bound of the integral is set to infinity mostly
for historical reasons, reflecting its origin in studies of semi-infinite atmospheric layers. For our case, with a well-defined layer, the physical thickness d is a more natural
upper bound. The optical depth is then zero at the top of the substrate (z = d) and
equal to the optical thickness τ = τ (0) = (σa + σs )d at the bottom of the substrate.
Using dτ = −(σa + σs )dz, where the minus sign comes from the fact that τ and z
increase in opposite directions, the RTE takes the form [67]
u
dI(τ, u, φ)
σs
1
= I(τ, u, φ) −
dτ
4π σs + σa
Z2π Z1
p(u′ , φ′ ; u, φ)I(τ, u′ , φ′ )du′ dφ′ ,
(19)
0 −1
The RTE is then expressed in the dimensionless quantities optical thickness τ and
single scattering albedo a, defined as
a=
σs
.
σs + σa
(20)
The mean free path, i.e. the average distance between an absorption or scattering
event is given by le = 1/σe , where σe = σs + σa is the extinction coefficient.
For most cases, the RT problem is also expressed in terms of a third parameter,
namely the asymmetry factor g of the phase function, which is described in the next
section.
4.2 The phase function
In RT, the angular distribution of the scattered light is described by introducing
a phase function p(u′ , φ′ ; u, φ), giving the probability of scattering from direction
(u′ , φ′ ) to the direction (u, φ). In most cases the actual phase function is difficult
15
Measuring and modelling light scattering in paper
100
10
p(Θ)
u
s
p
g = 0.95
g = 0.8
g = 0.45
g=0
90
1.5
60
120
1
150
30
0.5
1
180
0
0.1
210
0.01
330
240
0.001
-180
-90
0
Θ [◦ ]
90
(a) Henyey-Greenstein phase function
300
270
180
(b) Rayleigh phase function
Figure 8: (a) Henyey-Greenstein phase function for different values of the asymmetry factor.
(b) The Rayleigh phase function is shown for s-polarized (E-field perpendicular to the plane of
incidence), p-polarized (E-field parallel to plane of incidence) and unpolarized incident radiation. The plane of incidence is equal to the plane of the paper.
to assess, and approximations must be used. A common approximation is the the
Henyey-Greenstein (H-G) phase function [68]
p(cos Θ) =
1
1 − g2
.
4π (1 + g 2 − 2g cos Θ)3/2
(21)
This function depends only on the scattering angle Θ, which in terms of the incident
and scattered directions (θi , φi ) and (θr , φr ), as defined in Figure 3, is given by
cos Θ = cos θr cos θi + sin θr sin θi cos(φr − φi ) .
(22)
It is therefore often used when modelling isotropic media, see Figure 6, where the
probability for scattering in a certain direction only depends on the angle between
incoming and outgoing direction, i.e. the scattering angle. The shape of the phase
function is controlled by the asymmetry factor g = hcos Θi, where hcos Θi is the average of the cosine of the scattering angle. The asymmetry factor ranges from -1
(complete backward scattering) to 1 (complete forward scattering), where g = 0 is
isotropic scattering. The H-G phase function for different values of g is shown in
Figure 8(a) for a single plane of observation, i.e cos(φr − φi ) = −1. Within computer
graphics the Schlick phase function is sometimes used [69]. This is a slight modification of H-G function where the exponent 3/2 in the denominator of Eq. (21) is
replaced with 2 to speed up computations in real-time rendering.
Scattering from small particles, much smaller then the wavelength of the light,
can be described by the Rayleigh phase function
pR (Θ) =
3
(1 + cos2 Θ) ,
4
(23)
shown in Figure 8(b). The Rayleigh phase function is nearly isotropic and since it is
symmetric around Θ = 90o , its asymmetry factor is equal to zero. As the size of the
16
4 Radiative transfer theory
Isca
φ
θ
Iinc
Figure 9: Scattering from a long cylinder becomes more and more concentrated to the surface
of a cone, as the ratio between its length and radius increases.
particles increases, the scattering becomes more concentrated to the forward direction and the Rayleigh approximation becomes less valid. Scattering from particles
comparable to or larger than the wavelength of the incident light requires a more rigorous treatment by solving Maxwell’s equations. This can only be done analytically
for very simple geometries, such as perfectly spherical particles or long cylinders by
using Mie theory [70].
In this thesis, the H-G phase function is used to model the angular distribution
of light scattering in paper, treating the paper as a homogeneous medium. The size
of filler and coating pigments in paper are of the same order as the wavelength of
the light, and scattering from these particles is thus in the Mie regime. The forward
directed scattering from particles of this size is taken into account by the asymmetry
factor and the H-G phase function is therefore regarded as a valid approximation
for this source of scattering. Interference effects are assumed negligible due to multiple scattering and the highly randomized structure of paper. In order to take the
elongated shape of the fibres into account, more advanced models are required. One
approach is to use a statistical description of the paper structure in combination with
Monte Carlo simulations [71]. The fibres are then approximated by long cylinders,
see Figure 9, whose phase function can be calculated by Mie theory [70, 72, 73]. The
phase function and the horizontal alignment of the fibres affect for instance the lateral light scattering in a paper sheet [74].
Fluorescence emission is assumed to be isotropic. Although the fluorescence
emission from a single fluorophore molecule depends on the molecule’s orientation,
it is assumed that net emission is isotropic due to randomly oriented molecules. The
angular distribution of the fluorescence leaving the paper is however not necessarily
isotropic, but is related to the average depth of emission as shown in Paper IV and
illustrated in Figure 10.
17
Measuring and modelling light scattering in paper
ΦL
ΦL
*
ΦL
d
*
*
βL
βL
βL
θr
θr
θr
Figure 10: The anisotropy of the fluorescence emission is related to the average depth of
emission d within the material. A Lambertian distribution of the fluorescence ΦL [Wsr−1 ],
i.e. a constant luminescent radiance βL , is obtained at a specific average depth (mid). For
a large value of the absorption coefficient at the excitation wavelength (high concentration
of fluorophores), the emission takes place near the surface. Since the fluorescence process
can be regarded as being isotropic, the resulting fluorescence ΦL exiting the material is also
isotropic (top left), which gives non-Lambertian distribution of the luminescent radiance βL
(bottom left). For large value of d, there is a higher probability that the fluorescence emitted
at oblique angles will be scattered or absorbed before exiting the material, which also results
in non-Lambertian distribution (right). It should be noted that interactions with the top surface
would also change the shape of the distribution.
4.3 The forward and inverse problem
The forward problem consists of solving Eq. (19) for a given set optical parameters and directions (θ, φ). Since this equation lacks an analytical solution, one has to
rely on numerical methods. A common method is using discrete ordinates, where
the integral over directions is approximated by numerical quadrature [67, 75]. Several numerical implementations exist, such as DISORT [76] and DORT2002 [77, 78],
where the latter is adapted to light scattering in paper and used extensively in Paper
V. Another way to solve the RTE is to use spherical harmonics, where the RTE is
transformed to a set of partial differential equations by expanding the intensity into
a series of spherical harmonics [79]. For an overview of numerical methods for solving the RTE, the textbook by Modest is recommended [80]. Monte Carlo methods
are a different approach to solve the RTE. The Open PaperOpt simulation tool [71] is
a Monte Carlo implementation of radiative transfer in paper, and is used for most of
the simulations in Paper IV.
The inverse problem, also referred to as the parameter estimation problem, is to determine the optical parameters x = (σa , σs , g) given a set of reflectance and/or transmittance data. It is a common problem that is also encountered in e.g. high temperature applications [81,82], biomedicine [83,84] and optical tomography [12,13]. Apart
from the methods developed by Edström [85] and Joshi [86], methods that retrieve
18
4 Radiative transfer theory
(a) 40 g/m2
(b) Opaque
Figure 11: The main difficulty in the parameter estimation problem consists of decoupling
the scattering coefficient σs and the absorption coefficient σa , as shown in (a) for a sample
with basis weight 40 g/m2 and total transmittance of 10%. The local flatness of the objective
function makes the minimum (σs , σa ) = (150, 10) difficult to find (black circle). The flatness
increases with increasing optical thickness, until the sample becomes opaque and there is
no longer any unique minimum in this parameter subspace. The parameters σs and σa can
therefore not be decoupled for an opaque sample. The reflectance from an opaque sample
depends only on the albedo a and asymmetry factor g, which gives an easier estimation
problem as shown in (b), where the objective function is shown in the (a, g)-parameter space.
all three parameters from only reflectance measurements are uncommon.
In this thesis, the data when solving the parameter estimation problem consists
of a set of BRDF values in chosen directions. The parameter estimation problem then
amounts to minimizing the differences between model predictions (simulations) and
the given data, often by minimizing an objective function. In this work, the simulations are made with the DORT2002 software. The main difficulty of the inverse
problem is illustrated in Figure 11(a), where the objective function is shown for a
sample with basis weight 40 g/m2 in the (σs , σa )-parameter space. The local elongated flatness near the solution at (σs , σa ) = (150, 10) makes the problem difficult to
solve, and it is therefore difficult to decouple these two parameters. As the optical
thickness of the sample increases, the problem with local flatness becomes worse For
an opaque sample, there is no longer any unique solution and the σa and σs parameters cannot be decoupled. If we instead look at the problem of estimating the single
scattering albedo a and asymmetry factor g of the opaque sample, the problem is
much easier, see Figure 11(b). This problem is investigated in detail in Paper V, and
particularly how sensitive it is to the choice of measurement angles for the input
data.
19
Measuring and modelling light scattering in paper
5 Summary of the papers and contribution of the
thesis
The overall contribution of this thesis is the use of angle resolved reflectance measurements in combination with light scattering models as a comprehensive method
for optical characterization of turbid media. Paper I focuses on the absolute measurement capability of a commercially available benchtop-sized spectral goniophotometer, and its applicability for BRDF measurements on paper is investigated in
detail. Special attention is given to the influence of anisotropic scattering over finite
sized detector aperture, and the magnitude of the error introduced. In Paper II and
III, goniophotometric measurements are used to investigate various aspects of light
scattering from paper. In Paper II, a method is proposed for separating the total
reflectance into a bulk and surface component, and successfully applied to a matte
paper. Paper III deals with the angular distribution of fluorescence and reflectance of
matte paper, and the influence of filler pigment and FWA content on the distribution
of the total radiance factor. In Paper IV, the angular distribution of fluorescence is
investigated in more detail with the aim to determine how FWA content, angle of incidence and the optical parameters of the sample affect the distribution of fluoresced
light. The inverse problem is analyzed in detail in Paper V, with the objective to establish necessary requirements regarding choice of detection angles for a successful
estimation of the optical parameters.
5.1 Paper I
The influence of finite sized detection solid angle on bidirectional reflectance
distribution function measurements
This paper deals with limitations and often overlooked sources of error introduced in
compact double-beam instruments. The applicability of this kind of instrument for
BRDF measurements on paper is evaluated, and the accuracy that can be expected
in absolute reflectance measurements is assessed. The main focus is on errors related
to the relatively large detection solid angle due to the compact design of benchtop
instruments. Two error sources are analyzed. Firstly, the geometrical error due to a
simplified description of the radiative transfer between illuminated area and detector aperture, and secondly the convolution error due to anisotropic scattering over
the detector aperture. The results show that the former error source is almost independent of viewing angle, and introduces relative errors in measured BRDF as large
as 3% and of the same magnitude as the estimated overall measurement uncertainty.
Systematic errors due to convolution errors depend on the anisotropy and total reflectance level of the sample. For samples with high absorption and large degree
of anisotropic scattering, the relative errors can be as high as 3% when measuring at
oblique viewing angles. The results also emphasize the difficulty in avoiding systematic errors when performing absolute measurements. Comparative measurements
with an accurate reference instrument at the National Research Council of Canada
(NRCC) show deviations up to 10%, mainly due to systematic errors that need to be
20
5 Summary of the papers and contribution of the thesis
identified and compensated for.
Commercially available goniophotometers are emerging on the market, and are
an attractive alternative for laboratories and research facilities due the relatively low
cost, compact size and user-friendliness. The main contribution of this paper is the
quantification of errors that cannot be regarded as negligible for this category of
instruments. To the author’s knowledge, this is the first time this type of errors
are quantified and compared to the overall measurement uncertainty. Furthermore,
measurement situations are identified where the errors are particularly significant,
and these often overlooked sources of error must be taken into account in order to
avoid systematic errors. The results are of interest for anyone using or planning to
use such an instrument, regardless of scientific field.
Paper I was written in cooperation with Prof. Per Edström, Dr. Magnus Neuman
and Dr. Mattias Andersson. The contribution of the author of this thesis was to
perform measurements, to do calculations and main part of reasoning and writing.
5.2 Paper II
Separation of surface and bulk reflectance by absorption of bulk scattered light
Paper II presents a new method for separating light reflected from the surface of a
material, surface reflectance, from light reflected through multiple scattering within
the material, bulk reflectance. The method is applied to a matte photo paper and is
realized by inkjet printing, using overlapping prints of water based dye. By adding
dye to the sample, the absorption is successively increased until the bulk scattered
light is completely absorbed. By goniophotometric measurements, it is shown that
the bulk reflectance is efficiently cancelled at wavelengths within the absorption
band of the dye, and the remaining reflected light is therefore surface reflectance.
It is shown that the surface topography of the sample is unaltered, and the remaining surface reflectance is therefore regarded as identical with the surface reflectance
of the undyed sample. The bulk reflectance is assessed by subtracting the surface
reflectance from the total reflectance of the undyed sample. By using dyes with different colors, the surface reflectance at different parts of the spectra is assessed. The
method is therefore also applicable to materials where the surface reflectance is not
spectrally neutral. The results show that the surface component constitutes as small
part of the total reflectance, but increases rapidly at grazing viewing angles.
This is to the author’s knowledge the first time this method is proposed, and the
findings in this paper contribute to several scientific fields. The results are directly
applicable to light scattering in paper and print, but are also of interest in fields such
as biomedical optics and computer graphics. The method is successfully applied
to matte paper, which is a dielectric, highly randomized material with significant
anisotropic bulk scattering and rough surface. The simplicity of the method makes
it possible to study a range of materials that are difficult to analyze with existing
methods. A reliable separation method opens up for new ways of analyzing e.g.
biological tissues and optical coatings, and is also a valuable tool in the development
21
Measuring and modelling light scattering in paper
of more comprehensive reflectance models.
Paper II was written in cooperation with Dr. Magnus Neuman, Prof. Per Edström
and Dr. Mattias Andersson. The contribution of the author of this thesis was to
perform measurements, to do calculations and main part of reasoning and writing.
5.3 Paper III
Angular variations of reflectance and fluorescence from paper - the influence
of fluorescent whitening agents and fillers
In this paper, the angular distribution of reflectance and fluorescence is investigated
for a set of paper samples containing different amounts of filler and FWA. By using
incident light of wavelengths within the excitation band of the FWA, the angular
distribution of the total radiance factor, i.e. reflected and luminescent, is assessed by
goniophotometric measurements. Since we are using a broadband detector and not
a bispectrometer, the total fluorescence emitted for a certain excitation wavelength is
included in the measurement. By equipping the detector with a UV-blocking filter,
the total fluorescence is extracted by excluding the reflected radiance from the measurements. The reflected radiance is in turn assessed by subtracting the measured
fluorescence from the total radiance. The results show that the angular distribution
of the fluorescence can be regarded as Lambertian in comparison to the reflectance.
The findings may however depend on the optical properties of the sample and choice
of angle of incidence, which is investigated further in Paper IV. It is also observed
that the total radiance becomes more isotropic with increasing amount of FWA, as
well as with increasing amount of filler pigments. The addition of filler does however reduce the effect of FWA, since the average penetration depth decreases and
less light interacts with the FWA.
The findings are directly applicable to paper manufacturing where FWA is added
to increase the whiteness. A Lambertian fluorescence distribution makes the desired
effect independent of viewing angle, and the results are therefore a good starting
point when analyzing more complex distributions of FWA in paper. A Lambertian
distribution is a common assumption in e.g. computer graphics when rendering materials exhibiting fluorescence, but to the authors knowledge this is the first time it
is verified experimentally for paper substrates. The results are also of interest in epaper technology, where the overall appearance should resemble traditional paper
as closely as possible.
Paper III was written in cooperation with Dr. Mattias Andersson. The contribution of the author of this thesis was to perform measurements, to do calculations and
main part of reasoning and writing.
22
5 Summary of the papers and contribution of the thesis
5.4 Paper IV
Angular dependence of fluorescence from turbid media
The angular distribution of fluorescence from a set of papers containing different
amounts of FWA is investigated by both Monte Carlo light scattering simulations
and goniophotometric measurements. The scattering and absorption coefficients together with the quantum efficiency are estimated from bispectral measurements. The
problematic estimation of the parameter values at the excitation wavelength, due to
the large absorption by the FWA, is made by using a method suggested by Coppel et
al [87], that relies on differences in luminescent radiance factor between an opaque
and translucent (single sheet) sample. Both measurements and simulations show
that the angular distribution of fluorescence depends on the concentration of FWA,
where increasing concentration results in a more Lambertian distribution. Furthermore, it is shown that higher values of the single scattering albedo and larger angles
of incidence also makes the distribution more Lambertian. It is deduced that all of
these effects can be attributed to the average depth of fluorescence emission. If the
emission takes place closer to the surface, there is a higher probability that light exits
at oblique angles due to the shorter optical path. The simulations also show that
the angular distribution is independent of wavelength for all emission wavelength
above 420 nm.
The findings shed new light on the underlying mechanisms behind the angular
distribution of fluorescence, and explain the different and sometimes contradictory
results reported in the literature. The results show that the angular distribution is
strongly correlated to the mean depth of the fluorescence process. This means that
the angular distribution of fluorescence must be taken into account when determining the quantum efficiency of fluorescing turbid media. When the optical parameters
are known, the depth of the fluorophores may be deduced from the angular distribution, and the findings can therefore find applications in fluorescence spectroscopy
and dye tracing in medical applications.
Paper IV was written in cooperation with Dr. Ludvic Gustafsson Coppel and
Dr. Magnus Neuman. The contribution of the author of this thesis was to perform
measurements and part of the analysis, reasoning and writing.
5.5 Paper V
The inverse radiative transfer problem - considerations for optically thick media
This paper analyses the inverse problem, i.e. estimating the optical parameters from
bidirectional measurements, in more detail and highlights the main reasons why an
estimation of all three parameters is difficult. It specifically targets the the problem
of estimating the optical parameters of optically thick turbid media when a restricted
set of detection angles are available. In most situations encountered, restrictions in
detection angles are necessary or inevitable for various reasons. It might be due to
23
Measuring and modelling light scattering in paper
mechanical limitations of the measurement equipment, or that an unobstructed view
of the sample is not possible. A common reason is that the measurements are often
time consuming and that the overall measurement time needs to be reduced.
As a first step, the characteristics of the inverse problem is illustrated by contour
plots of the objective function
1X
F (x) =
{Mi (x) − bi }2 ,
(24)
2 i
using DORT2002 software to generate both model predictions Mi (x) and measurement data points bi for different sets of parameters x. The true parameter values
are thereby known, which allows us to evaluate the accuracy of the parameter estimation. A visual evaluation of the contour plots reveals that only the asymmetry
factor and the scattering albedo can be estimated from bidirectional reflectance measurements on an opaque sample, and that measurements on a translucent sample
are needed in order to decouple the scattering and absorption coefficient. It is also
shown that measurements on a translucent sample is in principle enough for a full
three parameter estimation, but the estimation problem becomes increasingly difficult as the optical thickness of the sample increases. The rest of the paper focuses
on studying the two-parameter estimation problem of an opaque sample. The reason for this is mainly that this problem provides better conditions for estimating
the asymmetry parameter, which for the full problem is very sensitive to both measurement noise and choice of initial value. Knowledge of both the albedo and the
asymmetry factor reduces the difficulty of estimating σa and σs .
The remaining analyses are focused on how the curvature of the objective function is affected by restrictions in available detection angles. A parameter estimation
is performed for different sets of detection angles. Four test cases with different values of asymmetry factor and albedo are analyzed, and noise is added to the measurement points in order to simulate more realistic measurement conditions. The results
show that the set of detection angles can be confined to the plane of incidence, and
even to forward directions only as long as sufficiently oblique angles are included in
the measurement sequence. The absolute errors in estimated asymmetry factor and
relative error in estimated albedo are in most cases below 0.02 and 0.2% respectively,
when using moderate noise levels. Test cases with high values of g showed in general the largest errors. The relative errors in albedo where significantly larger for test
cases with low albedo.
These findings are of direct importance to the development of fast on-line measurement systems for e.g. paper technology and optical imaging applications, as
well as in the development of extended parameter estimation methods. Increasing
the overall reflectance of a material is a common product design task in many applications, such as paper manufacturing, coating technology and thin film technology.
Solving the parameter estimation problem makes it possible to monitor how different stages in the manufacturing process affect the optical parameters. Knowledge of
how to obtain optimal conditions for a successful parameter estimation is therefore
highly valuable.
Paper V was written in cooperation with Dr. Magnus Neuman, Prof. Per Edström
24
5 Summary of the papers and contribution of the thesis
and Dr. Mattias Andersson. The contribution of the author of this thesis was to
perform simulations, to do calculations and main part of reasoning and writing.
25
Measuring and modelling light scattering in paper
6 Discussion
In paper I, it is shown that a thorough understanding of the measurement equipment
is required in order to conduct reliable goniophotometric measurements. BRDF measurements are not particularly error-prone, as long as one is aware of the limitations
imposed by the instrument. For reliable measurements, the impact of the approximations made when converting measured quantities to BRDF values must be fully
understood and properly handled. For accurate reference instruments at e.g. metrology institutes, these approximations are truly negligible, but can be significant for
commercial, benchtop sized instruments. For the instrument used in this thesis, the
errors in the approximations are in some instances of the same order as the estimated
overall measurement uncertainty, which is shown in Paper I. It should however be
pointed out that it is not reasonable to expect the same high accuracy as that of a
reference instrument. The accuracy can always be improved by conducting relative
measurements against a calibrated reference target, which also adds extra reliability
in terms of traceability. The findings in Paper I emphasize the importance of giving a
detailed description of the measurement parameters used when reporting measured
BRDF values. An aspect that has not been included in the analysis is sample uniformity. The geometrical error identified in Paper I can be reduced by reducing the size
of detector aperture and illuminated area. For paper samples, the optical properties
vary along the lateral dimensions of the sample, and a sufficiently large area must
be illuminated in order to obtain a representative measurement value. Reducing the
area is therefore not a recommended alternative to using the geometrical correction.
In paper II, a method for separating surface and bulk reflectance is proposed. It
is intended to be used for materials where the dye is efficiently absorbed into the
bulk of the material, such as matte paper grades. For paper grades were the surface
is not as porous, splitting the sample into two halves or applying the dye on the
backside of the sample are alternative methods. It should also be pointed out that
any interaction between the bulk scattered light and the surface is included in the
measured BRDF of the bulk reflectance and effects such as internal reflection might
alter the measured BRDF. To what extent the bulk reflectance is affected depends
on how well-defined the air-interface is, i.e. the smoothness of the surface, and it is
therefore assumed that the effect is small for this case with matte samples. This effect
is also present in the fluorescence measurements conducted in Paper IV.
In paper III, it is demonstrated that the fluorescence can be assessed qualitatively,
by placing a UV-blocking filter in front of the broadband detector. Quantitative fluorescence measurements in terms of Donaldson radiance factors can however only
be performed with a bispectrophotometer, where the wavelength of the incident and
detected light can be controlled independently of each other by two monochromators. The results in Paper III can therefore not be used for photometric purposes.
The angular distribution is however most likely correct, since the findings in Paper
IV show that the luminescent BRDF does not depend on emission wavelength. The
angular distribution of the fluorescence does however depend on the average depth
of emission, and thereby on the distribution of the FWA in the sample. The paper
samples used in the study are made on a small scale experimental paper machine,
26
7 Future work
where the FWA is evenly distributed in the sample, which differs from the distribution found in most commercial paper grades. It would therefore be interesting to
extend the study to commercial paper grades as well.
The comparison between measurements and simulations in Paper IV can be improved with better estimations of the asymmetry factor. Estimating the asymmetry
factor has proved to be difficult with earlier parameter estimation methods. It is
however shown in Paper V that the single scattering albedo and asymmetry factor
can be estimated directly from bidirectional measurements on an opaque sample. It
would therefore be interesting to repeat the experiments in Paper IV with asymmetry factors estimated in this way. A natural extension of the work presented in Paper
V is to develop a method for the inverse problem of estimating the scattering and
absorption coefficient when the values of the asymmetry factor and the albedo are
known a priori. One must however bear in mind that the asymmetry factor is only a
parameter in the RT model, and a simplification of the complex scattering that takes
place in the paper structure. For further improvement of RT models for light scattering in paper, the continued work should be focused on developing alternative phase
functions and more comprehensive surface models.
7 Future work
A natural extension of this work is to further improve both the experimental methods and the numerical models for studying light scattering in turbid media. Extending the existing bulk scattering models with models for surface scattering is one
possible improvement. Although this source of scattering is small for matte papers,
it becomes more significant for e.g. coated papers with smoother surfaces, and a
combined model is necessary in order to improve the agreement between measurements and simulations. A smoother surface introduces a well-defined air-substrate
interface that increases both the direct top surface reflectance, but also the internal
reflectance of bulk scattered light and fluorescence.
The experimental methods can be improved by utilizing the out-of-plane measurement capability of the instrument, which allows a more comprehensive characterization of materials with textured surfaces. A new field where this type of analyses
is of particular interest is solar cell technology. The efficiency of solar cells can be
increased by reducing front-surface reflections, which depend on parameters such
as the texture of the surface. As solar cells are getting thinner, increasing the efficiency by increasing the path length of the light within the active layer becomes
more important. Optimizing the texture of the backing layer, by reflecting light back
towards the active layer at oblique angles is one way to achieve higher efficiency. Another field where the reflectance of the backing/substrate is of important is in printed
electronics. The efficiency of the sintering process, where the deposited metal nano
particles are melted together into conductive layer, can most likely be increased by
taking the optical properties of the backing (i.e paper) into account.
RT models for light scattering can be developed further by e.g. considering other
27
Measuring and modelling light scattering in paper
phase functions than Henyey-Greenstein. By advancing to models based on scattering of electromagnetic waves, such as the discrete dipole approximation or Tmatrix methods, a more comprehensive description of light scattering is achieved,
This would enable simulations of e.g. polarization effects that the scalar RTE does
not allow.
8 Conclusions
This thesis shows that bidirectional reflectance measurements is a very powerful tool
for optical characterization of paper. The results are directly applicable in industrial
applications for monitoring how the optical response of the product is affected by
various factors such as filler content, FWA distribution and surface treatment. Measuring bidirectional reflectance is however not a trivial task and puts high demands
on the equipment. It is shown that the accuracy of measurements with a compact
goniophotometer to a large extent depends on the optical properties of the sample
and the measurement geometry. In addition, a compact design of the instrument
increases the risk of introducing systematic errors that are difficult to detect. These
findings are not limited to paper substrates, but hold for any material with similar
properties, such as fabrics, composites and biological tissue and are thus of importance in several fields.
A method for separating the total reflectance into a surface and a bulk reflectance
component has been developed and successfully applied to matte paper. By using
reflectance measurements separated into a surface and bulk component, more comprehensive reflectance models can be developed. The method targets materials that
are difficult to analyse with existing methods, namely dielectric, highly randomized
materials with significant anisotropic bulk scattering and a rough surface. The proposed method is thus of interest in several scientific fields, such as biomedical optics
and computer graphics.
It is also shown that fluorescence from paper can be studied in a qualitative manner with spectral goniophotometric measurements. Filler pigments and FWA alter
the angular distribution of the total radiance and the findings are therefore relevant
when trying to optimizing the usage and performance of additives, as well as for
improving current reflectance models. This thesis provides deeper insight into the
angular distribution of fluorescence and identifies the underlying mechanisms. It is
found that the degree of anisotropy depends to a large extent on the average depth of
fluorescence emission, that in turn depends on several factors such as concentration
of FWA, asymmetry factor, single scattering albedo and angle of incidence, which all
have to be taken into account when determining the fluorescence efficiency.
The problem of estimating optical properties of optically thick turbid media from
bidirectional measurements is closely related to the choice of measurement angles.
Simulations show that restricting the detection angles to the plane of incidence (inplane), and even the forward direction only, does not affect precision or stability of
28
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optical imaging, where a restricted set of angles might be necessary.
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