SPRAY special:

Physical models
for color prediction
M.Theiss Hard- and Software for Optical Spectroscopy
Dr.-Bernhard-Klein-Str. 110, D-52078 Aachen
Phone: (49) 241 5661390
Fax: (49) 241 9529100
E-mail: theiss@mtheiss.com
Web: www.mtheiss.com© 2002 Wolfgang Theiss

SPRAY
Spectral ray-tracing simulations
by Wolfgang Theiss

All rights reserved. No parts of this work may be reproduced in any form or by any means - graphic, electronic, or
mechanical, including photocopying, recording, taping, or information storage and retrieval systems - without thewritten permission of the publisher.
Products that are referred to in this document may be either trademarks and/or registered trademarks of therespective owners. The publisher and the author make no claim to these trademarks.
While every precaution has been taken in the preparation of this document, the publisher and the author assume no
responsibility for errors or omissions, or for damages resulting from the use of information contained in thisdocument or from the use of programs and source code that may accompany it. In no event shall the publisher andthe author be liable for any loss of profit or any other commercial damage caused or alleged to have been caused
directly or indirectly by this document.
Printed: 01.12.2009, 21:28 in Aachen, Germany

SPRAY for color prediction

Contents

I

Table of Contents
Foreword

2

Part I Introduction

3

1 About...................................................................................................................................
this document
3

Part II Simple demo systems

6

1 A simple...................................................................................................................................
model of paper
6
2 A paint...................................................................................................................................
model
12
3 A model...................................................................................................................................
of digital prints
18
4 Investigating...................................................................................................................................
pigment properties
22

Part III Optical constants27

1 Overview................................................................................................................................... 27
2 Optical...................................................................................................................................
constants of typical materials
27
3 Demo...................................................................................................................................
pigments30
P1......................................................................................................................................................... 30
P2......................................................................................................................................................... 31
P3......................................................................................................................................................... 33
Carbon particles......................................................................................................................................................... 33

4 Host...................................................................................................................................
material of paper model
34

Part IV Light scattering and absorption

35

1 Overview................................................................................................................................... 35
2 Demo...................................................................................................................................
pigments
36
P1......................................................................................................................................................... 36
P2......................................................................................................................................................... 37
P3......................................................................................................................................................... 39
Carbon particles......................................................................................................................................................... 39

3 Light...................................................................................................................................
scattering in paper
40

Index

44

Foreword
Our company has specialized in software for the analysis and prediction of optical spectra. The specularreflectance and transmittance of almost arbitrary layer stacks can be treated with our main products SCOUT
and CODE. These programs are widely used to analyze spectroscopic measurements (determine
thicknesses, optical constants or compositional parameters) and to design layer stacks for opticalapplications, such as coatings for architectural glass or solar absorbers.

We now present our SPRAY ray-tracing software to people dealing with paints, digital prints or other imaging
techniques.
SPRAY predicts the optical performance of systems based on physical models. This text should give you an
impression on how you can solve problems with SPRAY. We are using quite simple models and fictional
pigments. We hope, however, that it will be clear how the simple models could be extended in order to beprepared for realistic work.
Besides developing software we also offer our experience in optical problem solving. We determine opticalconstants, develop analytical methods, optimize products and train people in using our software products.
At the end of this text you'll find some information about other products of M.Theiss Hard- and Software.
Best regards
Wolfgang Theiss© 2002 Wolfgang Theiss

SPRAY for color prediction

Introduction
3

1

Introduction

1.1

About this document

Spectral Ray-Tracing

Phyical models for color prediction
M. Theiss – Hard- and Software
Dr.-Bernhard-Klein-Str. 110, D-52078 Aachen, Germany
Phone: + 49 241 5661390 Fax: + 49 241 9529100
e-mail: theiss@mtheiss.com web: www.mtheiss.com
September 2002
This text shows how our SPRAY ray-tracing software can be used to predict colors of paints,
digital prints or similar imaging techniques. Other SPRAY applications are described in www.
mtheiss.com/spray_ex/index.html. The SPRAY technical manual is available in www.mtheiss.
com/docs/spray/index.html, a tutorial for beginners can be found in www.mtheiss.com/sprayt1/
index.html.
SPRAY history: Beyond Kubelka-Munk ...
In order to understand reflectance spectra of diffusely reflecting powders we started (many yearsago) with the simple Kubelka-Munk two-flux concept. We computed the absorption andscattering coefficients from the optical constants and size parameters of our particles using the
Mie theory (scattering of spherical particles). To be close to our experimental setup with a laser
light source, we had to replace the diffuse illumination in the theory by a sharp illuminating beam.
Then we wanted to describe the angle distribution of the emitted radiation for powders or
arbibrary thickness. So we deleloped a multi-flux theory. Finally, in the attempt to analyze
infrared diffuse reflectance spectra we found that we had to determine the intensity and angle
distribution of the incoming radiation in our spectrometer. At this point we gave up extending the© 2002 Wolfgang Theiss

SPRAY for color prediction

Introduction
4

Kubelka-Munk concept, and developed the SPRAY software that computes optical spectra by
ray-tracing.
SPRAY features
Like many ray-tracing products, SPRAY has light sources, mirrors, lenses and various basic
shapes which can be used to build up the scenery. Very powerful optical constant models andalmost arbitrary layer stacks covering the surface of the various geometric objects are highlightsof the software.

Meanwhile SPRAY has a number of features that turn it into an excellent tool for color
prediction. Based on physical modeling, the transfer of light from the light source through lightscattering, absorbing and fluorescent media and across partially reflecting interfaces is simulated.
SPRAY performs fully three-dimensional ray-tracing with almost no compromises.

© 2002 Wolfgang Theiss

SPRAY for color prediction

Introduction
5

Color prediction demo systems
In order to show you the possibilities of SPRAY with respect to color prediction this text
discusses some simple demo systems in the next section.
Warning: We are no experts in paints and digital prints (yet)!
However, having studied some pieces of literature, we have thefeeling that SPRAY could be a useful tool for people composing
images. If you agree, feel free to suggest more advanced setups.
We will be happy to learn how paints and prints really work.
Paper: We discuss a simple model for paper based on scattering inclusions in a homogeneous
host. You can tune the light propagation in the paper by the size distribution and the volume
fraction of the inclusions, and the complex index of refraction of the host material.
Paints: Here we describe the optical performance of several pigment types in a homogeneousbinder layer on paper. We show that SPRAY can be controlled as OLE automation server from
Excel's VisualBasic (or any other OLE automation client) in order to create automatically chartsof spectra and color coordinates vs. pigment volume fractions.
Prints: The setup of a simple test system is explained which can be used to simulate lightpropagation through ink dots on paper.
Pigment research: This section gives a few examples of pigment investigations you can do withSPRAY. Size variation and coatings with single and multiple layers are treated.

Background information
The following background information is given, just in case you want to know some details.
Optical constants: The heart of any physical model for color prediction are the optical constantsof the materials in the system. SPRAY has very powerful optical constant models and a large
database. The section about optical constants discusses some typical materials, and gives the
optical constants of all materials used in the examples above.
Light scattering: SPRAY has an integrated Mie program that computes the scattering andabsorption characteristics of spherical particles. You can define a size distribution, and theparticles may be coated with a thin film layer stack. The properties of all light scatterers used in© 2002 Wolfgang Theiss

SPRAY for color prediction

Introduction
6

the examples of this text are discussed in this section.

SPRAY features not covered by this text:· You can work with user-defined curved surfaces· SPRAY can handle fluorescent materials· Parallel computing: Lengthy computations can be distributed on several PCs· In combination with OLE automation you can generate video sequences with SPRAY

2

Simple demo systems

2.1

A simple model of paper
Paper is the basis of many prints. Its large diffuse reflectance is almost independent ofwavelength. Hence it is used as a white background, onto which pigments with strongly
wavelength-dependent absorption are deposited where color is wanted.
The propagation of light through paper is very important for the color of a printed area. For asuccessful color prediction a correct description of the underlying paper is required. A physicalmodel should reproduce available measured optical properties of the relevant types of paper in aquantitative way.
In this SPRAY demo, we use a simple paper model which is able to reproduce some basic
features of paper. Instead of taking into account all the specific paper ingredients and theirmicrostructure, we describe paper as a two-phase composite: Spherical air inclusions areembedded in a host material with optical constants similar to those of glass (see details below inthe section of optical constants). Here is a sketch of the setup:

The size distribution of the voids determines the angle-dependence of the average single
scattering event (which is computed by the Mie program integrated into SPRAY), whereas thevolume fraction of the voids controls the strengths of the scattering, i.e. the scattering coefficient.
The host material weakly absorbs in the UV which is controlled by an oscillator term in the
optical constant model.
In the following the properties of two types of (model) paper are compared: One with large
inclusions and another one with small ones.

© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
7

The graph below shows a SPRAY visualization of some test rays for the case of large inclusions.
A collimated incident beam is directed onto the paper from the top (normal incidence of light).
The path of 20 test rays is displayed in the following graph for a volume fraction of 0.3:

Starting many rays and placing large detectors above and below the paper one can compute thediffuse reflectance and transmittance of the model paper:
Diffuse reflectance
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900
Wavelength [nm]1000

© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
8

Diffuse transmittance
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900
Wavelength [nm]1000

In order to visualize intensity distributions one can place screens (virtual CCD cameras,
represented by the two blue bars in the sketch below) where detailed information is wanted:

The top screen looks like this© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
9

whereas the bottom screen shows a broader distribution:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
10

The intensity distributions as well as the diffuse reflectance and transmittance spectra could becompared to measured data in order to estimate how good the model describes the real paper.
The smaller spheres with the same volume fraction have a larger scattering coefficient and abroader scattering distribution. This leads to a more concentrated radiation distribution:

The diffuse reflectance is significantly higher compared to that of the larger inclusions:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
11

Diffuse reflectance
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900
Wavelength [nm]1000

The distribution 'measured' with the top screen shows a more pronounced confinement of therays:

This paper type (small inclusions) is used for the following simulations of paints and prints.

© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
12

2.2

A paint model
The most simple model of a paint is probably given by a binder layer with embedded pigments
deposited on the simple paper model discussed above:

For simplicity we use for the binder the same optical constants as for the host material of thepaper model. Four types of absorbing and scattering pigments are used: Three fictional ones
absorbing in the blue (named P1), green (P2) and red (P3) part of the spectrum, and carbon
particles which have a broad absorption over the whole visible spectrum. All particles are
assumed to be spheres which makes it possible to compute their scattering and absorption
features using the Mie program integrated in SPRAY.
Performance of the individual pigments
For each pigment type we have computed the diffuse reflectance of the paint, varying the volume
fraction f of the pigments in the binder. Pigment P1 absorbs in the blue:

The corresponding color coordinates are theses:© 2002 Wolfgang Theiss

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Simple demo systems
13

For the P2 pigment the following spectra and colors are obtained:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
14

Here are spectra and color coordinates of P3 absorbing in the red:© 2002 Wolfgang Theiss

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Simple demo systems
15

Finally the carbon particles can be used to realize different gray scales:© 2002 Wolfgang Theiss

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Simple demo systems
16

Mixing pigments
A scattering medium in SPRAY may contain several types of pigments. In that case theindividual probabilities for absorption and scattering events are summed up. The angle
distribution used for the re-direction of scattered rays is computed as an average of thescattering patterns of the constituents - weighted with the corresponding volume fractions.

The following screenshot shows an Excel solution for color prediction of mixed pigment systems.
You type in the volume fraction of the individual pigments and a small VisualBasic macro
computes the color of the mixture. SPRAY works as OLE server in the background anddelivers optical spectra and color coordinates.

© 2002 Wolfgang Theiss

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Simple demo systems
17

The spectra computed for the 7 pigment compositions are these:

The corresponding colors are:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
18

2.3

A model of digital prints
Colors in digital prints are made by placing dots of the available pigment types close to eachother. Viewing the image from the distance with a spatial resolution that does not resolve theindividual dots, the color impression is given by averaging the diffuse reflectance of several dots.

The following steps show how a simple SPRAY model for the simulation of digital prints on
paper can be developed. We will consider a circular area of 100 mm radius, fill in some printed
dots and compute the total diffuse reflectanc of the system.
We start with two circles of 100 mm radius that define bottom and top of the underlying paper.
On the left picture below you see the scenery from a raised observation point whereas the rightimage shows a side view:© 2002 Wolfgang Theiss

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Simple demo systems
19

The dots are modeled by flat ellipsoids which are filled with pigments. The volume fraction of thepigments inside the dots is 5%, the dot diameter and height are 40 mm and 8 mm, respectively:

The dots are embedded in a homogeneous layer with a thickness of 15 mm. A circulartransparent light source is set directly above the top surface. It illuminates the sample from the
top with Lambertian characteristics. Around the scenery we have placed a cylindrical ideal
mirror which introduces reflecting boundary conditions:

Now it's time for light. In the UV at 300 nm wavelength the penetration depth of the radiation in
the paper is very short. Here are some test rays:© 2002 Wolfgang Theiss

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Simple demo systems
20

At 600 nm wavelength the rays travel much longer distances before they are emitted orabsorbed:

The next graph shows the spectrum for 3 dots with pigment P1 and 2 dots with pigment P2:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
21

The corresponding coordinates and a rough impression of the color are given below:

Taking 0 to 5 dots of pigment tpye P2 gives the following spectra and color coordinates:© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
22

2.4

Investigating pigment properties
This section shows how pigment research can be assisted by SPRAY. Knowing the opticalconstants of the involved pigment materials can save a lot of experiments and time. Instead of
realizing many samples and doing many measurements you can predict the optical performanceof a system with SPRAY. You should, of course, check the established relations and numberswith selected real cases.
Here we investigate the following questions:· Does the color of a paint change with pigment size?
· How thick must the coating of a particle be in order to hide the interior completely?

© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
23· How depends the reflectance spectrum of a multiply coated plate on the viewing angle?
Pigment size variation
What happens to the color of a paint if you change the particle size? Here we show how thepredicted color of the 'P1 paint' (see above) depends on the size of the pigment particles.
Increasing the size from 200 to 500 nm (average radius) leads to a noticeable change in visualappearance:

The corresponding color changes are significant:Turning carbon into gold ...
With SPRAY you can investigate single and multiple coatings of spherical particles. Sometimes
pigments are coated on purpose in order to achieve a certain appearance or other function,© 2002 Wolfgang Theiss

SPRAY for color prediction

Simple demo systems
24

sometimes 'natural' coatings like surface oxides exist and have to be taken into account.
The demo question for coated spheres is this: How thick must a gold layer be on carbon
particles (average size about 2 mm) in order to make the pigments look like solid gold nuggets ofthe same size?
Well, with SPRAY you can just try. The next graphs show the reflectance spectra (for thesimple paint model discussed above) of various test 'samples' including pure carbon and puregold. The thickness of the gold coating is indicated in the legend:

The coating should have a thickness of at least 50 nm. But even at 60 nm thickness the color
coordinates are still different from those of a 'solid gold paint'. The latter are indicated by thegray lines in the graph below:© 2002 Wolfgang Theiss

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Simple demo systems
25

Inspecting advanced coatings
Finally we show an example of complex coatings. Any object in SPRAY may be covered withan almost arbitrary layer stack. Here a glass plate is covered with a 12 layer coating. The coating
is made repeating the basic stack [TiO2 (10 nm) / SiO2 (50 nm) / Ag (10 nm)] 4 times:

The plate has been illuminated by diffuse radiation. A special detector (with multiple angle
segments) records how many rays are leaving the plate in which direction. The following plotsummarizes the results: From 0 to 90 degrees the reflected radiation is displayed (0 degree is thesurface normal), the range 90 ... 180 deg covers the transmitted rays:© 2002 Wolfgang Theiss

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Simple demo systems
26

Signal

0.20
0.15
0.10
0.05
0.00
400
500
Wavelength 600
[nm]
700

25

50

75

0.20
0.15
0.10
0.05
0.00
175
150
125
100
Observation angle[deg]

The observed features are interference effects which depend on the angle of observation, ofcourse.
Changing the basic stack to [TiO2 (20 nm) / SiO2 (50 nm) / Ag (10 nm)], i.e. doubling the TiO2
thickness, leads to a significant shift of the spectra:© 2002 Wolfgang Theiss

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Simple demo systems
27

Signal

0.20
0.15
0.10
0.05
0.00
400
500
Wavelength 600
[nm]
700

3

Optical constants

3.1

Overview

25

50

75

0.20
0.15
0.10
0.05
0.00
175
150
125
100
Observation angle[deg]

The term 'Optical constants' is used for the so-called dielectric function e = e1+ i e2 or its
square root, the complex refractive index n + i k. The dielectric function represents a material in
Maxwell's equations and determines the solutions for electromagnetic waves.
Light propagation through any system depends on the optical constants of all involved materials
and geometry (shapes, sizes, distances). Hence optical constants must be the basis of any
physical model. SPRAY is equipped with a large database of optical constants. There are fixeddata sets taken from literature sources, and flexible models that can be adapted to individual
cases.
If you are not familiar with optical constants, the next section about typical materials could be
useful for you.
The optical constants of the demo pigments used in this text are given in separate sections below.

3.2

Optical constants of typical materials
Here are optical constants of some typical materials. All graphs show the real part in blue andthe imaginary part in red.
Noble metals© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
28

A noble metal like silver is characterized by a large negative real part of the dielectric function
and an imaginary part which increases towards large wavelengths:

The complex square root of the dielectric function, the refractive index, has an almost vanishingreal part and a much larger imaginary part in the case of metals:

Dielectrics
Insulators like glass have optical constants which are very different from those of metals. The
imaginary part is almost zero, the real part almost constant. In most cases the refractive index in
the blue is a little larger than in the red - this effect is called 'normal dispersion'. Here is therefractive index of a typical glass:© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
29

In glasses significant absorption occurs in the mid infrared region (by vibrational modes) and in
the far UV (by optical interband transitions). Due to the large electronic bandgap in these typesof material the visible spectral range is almost absorption-free.
Semiconductors
Semiconductors have smaller separations of energy bands (band gap) than insulators. In manycases visible light has enough energy to excite electronic interband transitions which leads tolarge absorption above the bandgap like in the case of CdS shown here:© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
30

Since CdS absorbs in the blue, CdS particles appear to be yellow if illuminated with white light.
There are almost no homogeneous, crystalline materials which have sharp, isolated absorption
bands in the visible spectral range. If you need absorption in the green or yellow you have to
make use of electronic transitions in organic molecules (assisted by some incorporated metal
atoms). The demo pigments P1, P2 and P3 (shown in the following) are fictional examples withmore or less typical optical constants.

3.3

Demo pigments

3.3.1

P1
The optical constants of the demo pigments P1, P2 and P3 are composed of a constant and an
oscillator contribution. The latter creates the absorption band responsible for the pigment's color.
The optical constants of real pigments may be obtained by a fitting procedure which adjusts theparameters of a suitable dielectric function model. Realistic models sometimes require several
oscillator terms or interband transition models.
The dielectric function and the refractive index of the demo pigment P1 are the following:© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
31

P1 absorbs in the blue part of the visible spectrum.
3.3.2

P2
Demo pigment P2 absorbs in the center of the visible spectral range. Here are its opticalconstants:© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
32© 2002 Wolfgang Theiss

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Optical constants
33

3.3.3

P3
P3 absorbs in the red:

3.3.4

Carbon particles
Carbon particles absorb in a broad spectral range. They can be used as 'black pigments'.
The optical constants of carbon have been taken from the SPRAY database. The dielectric
function and the refractive index are these:© 2002 Wolfgang Theiss

SPRAY for color prediction

Optical constants
34

3.4

Host material of paper model
The host material used in the simple paper model has optical constants similar to glass. The realpart of the refractive index shows a weak dispersion, the absorption increases towards the UV.
The following graphs show the complex refractive index and the absorption coefficient:© 2002 Wolfgang Theiss

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Optical constants
35

4

Light scattering and absorption

4.1

Overview
If the optical constants of embedded pigments and host material differ from each other, lightwaves travelling through the medium are scattered and absorbed. In order to perform ray-tracing
computations one must know the probability/distance for scattering and absorption events. In
addition, the angle distribution of the scattered light must be taken into account when the new
direction of a scattered ray is computed.
In the case of spheres embedded in a homogeneous host material the problem is solved by theso-called Mie theory. The SPRAY software contains a module that performs Mie computations
for user-defined size distributions of spheres. The spheres may have multiple coatings.
In the following the scattering and absorption properties of the demo pigments are given:· P1© 2002 Wolfgang Theiss

SPRAY for color prediction

Light scattering and absorption
36

· P2
· P3
· Carbon· Scattering voids in paper

4.2

Demo pigments

4.2.1

P1
The optical constants of pigment P1 have been given above. The following radius distribution
with 3 size classes has been assumed:
Size distribution

fraction

0.5
0.4
0.3
0.2
0.1
0.0

0.0

0.1

0.2
0.3
Radius [10 -6 m]

0.4

The Mie program computes the scattering and absorption data in a way such that the volume
fraction of the pigments can be varied afterwards without repeating the calculation again. For avolume fraction of 1% the following absorption (blue) and scattering (red) probabilities arefound:© 2002 Wolfgang Theiss

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Light scattering and absorption
37

Sorry, all scattering properties are displayed in SPRAY using wavenumbers. Wavenumbers areinverse wavelengths measured in cm (i.e. 1/cm). 10000 1/cm correspond to 1000 nm
wavelength (near infrared), 25000 1/cm are 400 nm (blue end of the visible).
Note the rich structure in the scattering probability which depends on the sphere size distribution.
The rays scattered by pigment P1 get new directions according to this distribution:

Intensity [10

5
4
3
2
1
0
0
30
60
90
Scattering angle
[°]

120
150
180

-13 ]

5
4
3
2
1
0
35000
30000
25000
20000
Wavenumber
15000[1/cm]
10000

The scattering distribution depends very much on the ratio of the light wavelength and the particle
radius. In the infrared (below 15000 1/cm) the ratio is large which leads to a broad angle
distribution. In the UV (above 25000 1/cm) the ratio is smaller. Here the distribution isdominated by a strong and sharp forward peak and (weak) backward scattering.
Note that in the whole spectral range from the near infrared to the near UV the distribution isvery different from isotropic scattering. Isotropic scattering (which is often used as a first guess)hardly occurs in real systems.
4.2.2

P2
The same radius distribution as for P1 was used here. The optical constants of pigment type P2
were given above.
The Mie results for the probabilities and the scattering distribution are shown in the graphs
below:© 2002 Wolfgang Theiss

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Light scattering and absorption
38

Intensity [10 -13 ]

5
4
3
2
1
0
0
30
60
90
Scattering angle
[°]

120
150
180

5
4
3
2
1
0
35000
30000
25000
20000
15000 Wavenumber[1/cm]
10000© 2002 Wolfgang Theiss

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Light scattering and absorption
39

4.2.3

P3
Working again with the radius distribution already used for pigment P1 and the optical constants
with absorption in the red one obtains the following scattering characteristics for pigment P3:

Intensity [10

5
4
3
2
1
0
0
30
60
90
120

Scattering angle
[°]

4.2.4

150
180

-13 ]

5
4
3
2
1
0
35000
30000
25000
20000
Wavenumber
15000[1/cm]
10000

Carbon particles
For the carbon particles the probabilities for absorption and scattering have similar values over
the whole visible spectral range:© 2002 Wolfgang Theiss

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Light scattering and absorption
40

The scattering characteristics does not show any backward contribution:
Intensity [10 -13 ]

3
3

2

2

1

1
0
0
40
80
Scattering angle
[°]

4.3

120
160

0
35000
30000
25000
20000
Wavenumber
15000 [1/cm]
10000

Light scattering in paper
The simple paper model discussed above is based on spherical voids embedded in a
homogeneous host material. The optical constants of the host material have been shown above.
Two different size distributions of the spherical voids are considered. The first one is centeredaround 1 mm radius:© 2002 Wolfgang Theiss

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Light scattering and absorption
41

Size distribution

fraction

0.6

0.4

0.2

0.0

0.0

0.5

1.0
1.5
Radius [10 -6 m]

2.0

The voids do not absorb, of course, and have a more or less constant scattering coefficient,given by the red curve in the graph below for a volume fraction of 0.3:

The angle dependence of the average scattering event is - due to the quite large size of the voids- forward oriented:© 2002 Wolfgang Theiss

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Light scattering and absorption
42

Intensity [10 -11 ]
5
4
3
2
1
0
35000
30000
25000
20000
Wavenumber
15000 [1/cm]

5
4
3
2
1
0
0

30

60

90

120

Scattering angle
[°]

150

180

10000

Reducing the size of the spherical inclusions to this distribution
Size distribution

fraction

0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.0

0.5

1.0
1.5
Radius [10 -6 m]

2.0

leads to larger scattering coefficients (for the same volume fraction)© 2002 Wolfgang Theiss

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Light scattering and absorption
43

and a broader angle distribution of the single scattering event:

Intensity [10 -13 ]

10
8
6
4
2
0
30000

10
8
6
4
2
0

25000
0

30

20000
60

90

Scattering angle
[°]

120

Wavenumber
15000 [1/cm]
150

180

10000

© 2002 Wolfgang Theiss

SPRAY for color prediction

44

Index

digital prints
18
dispersion
27, 34
distances
27
dots
18

-A-

-E-

absorption
27, 30, 34, 35
advanced coatings
22Ag
22
angle dependence
40
angle distribution
3, 35
angle of observation
22
angle-dependence
6electromagnetic waves
27
electronic bandgap
27
ellipsoids
18
energy bands
27
Excel
12

-Bbackward scattering
bandgap27
beam
3
binder12
black pigment33
blue
12, 30, 36
boundary conditions-Ffluorescent materials
forward peak
36

36

18

-Ccarbon
22, 33, 39
carbon particles
12
CCD camera
6
CdS
27
coating22
coatings
3, 35
color
12, 18color coordinates
3
color impression
18
colors3
coordinates
12
curved surfaces
3-Ddatabase
3, 27, 33
detector6
dielectric function27
dielectric function model
30
Dielectrics
27
diffuse illumination3
diffuse reflectance
6, 12, 183-Ggeometry
27
glass
27
gold
22
gray scale
12
green
12

-Hhost3, 40-Iinclusions
3, 40
index of refraction
3
infrared3
ink dots3
Insulators
27
intensity distribution
6
interband transitions
27
interference effects
22
isotropic scattering
36-KKubelka-Munk

3-LLambertian18© 2002 Wolfgang TheissIndexlayer stack
3
lens
3
light propagation
light source3light waves
35

3

pigments
12, 27, 30, 35
plate
22
powders3
prints3, 18
probability
36

-M-

-R-

Maxwell's equations
27
microstructure6
Mie program
12, 36
Mie theory
3, 35
mirror3, 18
Mixing12
mixture12
model
6, 12, 18
multi-flux theory3
multiple coatings
22, 35

radiation
18
radius22
radius distribution
37
ray-tracing3
red
12, 33, 36
reflectance
6, 12, 18, 22
reflectance spectra
3
refractive index
27-Nnear infrared
36
noble metal
27
normal dispersion

27-Oobservation point18
OLE automation
3
optical constants
3, 27
optical performance22
oscillator
6, 30
oxides22-Ppaint
12
Paints3
paper
3, 6, 18
paper model
40
Parallel computing
3
particles
22, 33peak
36
penetration depth
18
physical model
6, 27
physical modeling3
pigment
3, 18, 31, 33, 36, 37, 39
pigment research
22pigment size
22

45

-Sscattering
35
scattering probability
36
screens
6
Semiconductors
27
shapes3, 27
single scattering
6
SiO2
22
size classes
36
size distribution
3, 35, 40
size variation
22
sizes
27
spatial resolution
18
spectra12
spectrometer3
spheres35
spherical particles
22
surface oxides
22-Ttechnical manual3
thickness
18, 22TiO222transmittance6
two-flux concept
3
two-phase composite
typical materials
27

6© 2002 Wolfgang Theiss

SPRAY for color prediction

46

-UUV

6, 18, 27, 36-VV34
vibrational modes
27
video sequences
3
viewing angle
22
VisualBasic macro
12
voids
6, 40
volume fraction
3, 6, 18, 36, 40-Wwavelengths
36
wavenumbers
36
waves
35
white
6

© 2002 Wolfgang Theiss