Examples
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© 2001 Wolfgang Theiss

SPRAY
Spectral ray-tracing simulations
by Wolfgang Theiss

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directly or indirectly by this document.
Printed: 01.12.2009, 22:52 in Aachen, Germany

SPRAY examples

Contents

I

Table of Contents
Foreword

0Part I Introduction

3

1 About...................................................................................................................................
this document
3
2 Overview................................................................................................................................... 3

Part II Lenses

5

1 Chromatic...................................................................................................................................
error
5
2 Edge...................................................................................................................................
effects
7

Part III IR reflectance unit

12

1 Overview................................................................................................................................... 12
2 Spot...................................................................................................................................
size vs. angle of incidence
13
3 Alignment...................................................................................................................................
sensitivity
16
4 ATR...................................................................................................................................
experiment
19Part IV Light scattering

23

1 Dipole...................................................................................................................................
scattering - blue skies and red sunsets
23
2 Paints................................................................................................................................... 27
Paints: Beyond.........................................................................................................................................................
Kubelka-Munk
27
Scatterers......................................................................................................................................................... 28
Large TiO2..................................................................................................................................................
spheres28
Small TiO2..................................................................................................................................................
spheres30
CdS clusters.................................................................................................................................................. 31
The test system......................................................................................................................................................... 33
Test the test.........................................................................................................................................................
system35
Large TiO2.........................................................................................................................................................
particles in resin37
Sm all TiO2.........................................................................................................................................................
particles in resin
39
Color by CdS.........................................................................................................................................................
clusters
40

Index

0

Part

I

SPRAY examples

Introduction

1

Introduction

1.1

About this document

3

Spectral Ray Tracing

Examplesã W.Theiss
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 2001
1.2

Overview
This document describes introductory examples demonstrating important SPRAY features. The
SPRAY manual should be consulted for details of the individual objects.
Chromatic error of a collecting lensCollecting lens: edge effects
An IR reflectance unit for variable angle of incidence
Dipole scattering - blue skies and red sunsets
Paints: Beyond Kubelka-Munk© 2001 Wolfgang Theiss

Part

II

SPRAY examples

2

Lenses2.1

Chromatic error

Lenses5

SPRAY is used to investigate the shift of the focal plane of a collecting lens with wavelength.
A circular light source (left) creates a parallel beam which is focussed by a collecting lens (right):

Sending some test rays the function of the lens is demonstrated:

In the focal plane of the lens an array detector (length: 3 mm, 30 pixels) is placed which records
position-dependent spectra along the symmetry axis of the system:© 2001 Wolfgang Theiss

SPRAY examples

Lenses6

If the optical constants of BK7 glass are used for the lens the following intensity distribution isobserved:

Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0
300
400
500
Wavelength
[nm]

1.0
0.8
0.6
0.4
0.2
0.0

600
700

5
800

25
20
15
10
Pixel #

0

Clearly blue light (400 nm) is focussed closer to the lens than red light (700 nm) which is due tothe dispersion of the glass. The refractive index of BK7 has the following spectral dependence:© 2001 Wolfgang Theiss

SPRAY examples

Lenses7

BK7

Refractive index

1.56
1.55
1.54
1.53
1.52
1.51
1.50

2.2

300

400

500

600
700
Wavelength [nm]800

Edge effects
The imaging quality of a simple lens decreases with the distance of incoming rays from the centerof the lens. The configuration previously used to demonstrate chromatic errors is used here toshow how the size of the focus depends on the diameter of the incoming beam.
Setting the beam diameter to 0.2 cm (i.e. the radius of the circular light source to 0.1 cm) a very
sharp focus is achieved. As shown before, the position of the focal plane depends on wavelength
due to the dispersion of the optical constants of the glass:© 2001 Wolfgang Theiss

SPRAY examples

Lenses8Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0
300
400

1.0
0.8
0.6
0.4
0.2
0.0
25
20

500
Wavelength
[nm]

15

600
10

700

5
800

Pixel no.

0

Increasing the beam diameter to 0.6 cm a much larger portion of the lens is used to collect light:© 2001 Wolfgang Theiss

SPRAY examples

Lenses9

However, now the focus is much broader:

Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0
300
400

1.0
0.8
0.6
0.4
0.2
0.0
25
20

500
Wavelength
[nm]

15

600
10

700

5
800

Pixel no.

0

Blocking the rays in the center of the beam with a circular stop of 0.4 cm diameter© 2001 Wolfgang Theiss

SPRAY examples

Lenses10

makes clear that the rays at the outer regions of the lens are focussed in a different plane as thosethat hit the lens in the center (see figure above):

Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0
300
400

1.0
0.8
0.6
0.4
0.2
0.0
25
20

500
Wavelength
[nm]

15

600
10

700

5
800

Pixel no.

0

© 2001 Wolfgang Theiss

Part

III

SPRAY examples

3

IR reflectance unit

3.1

Overview

IR reflectance unit

12This section demonstrates the simulation of an almost realistic reflectance unit. Devices like thisare often used in the sample compartment of infrared spectrometers in order to record
reflectance spectra of samples at various angles of incidence and different polarizations.
The SPRAY setup includes the following items: A circular light source emits divergent radiation
directed towards a parabolic mirror. The light source is placed in the focus of the paraboloid hence a parallel beam is created by the mirror. A second parabolic mirror creates a focus on arotatable plane mirror.
In real spectrometers a Michelson interferometer - consisting of a beam splitter and a fixed andmovable plane mirror - is usually placed between the two paraboloids. The interferometer isskipped in this setup because the performance of the reflectance unit is of interest here.
The plane mirror directs the radiation upwards onto an elliptical mirror. The center of therotatable mirror as well as the sample position are the focal points of the ellipsoid. A second
ellipsoid directs the beam reflected by the sample (here a perfect plane mirror is taken as
sample) towards a second rotatable mirror. Finally the beam is focussed by another ellipsoidal
mirror to the detector element which records the spectrum.
Here is a side view of the setup (with and without some test rays sent to visualize the function of
the elements):© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

13

Viewed from the top the scenery looks like this:

3.2

Spot size vs. angle of incidence
In some analytical problems the size of available samples is limited and you have to know how
large the illuminated sample spot is. Placing a screen slightly above the sample position is theeasiest way to visualize the illuminated sample area.
The following pictures show the intensity distribution on a quadratic screen (10 mm by 10 mm)© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

14

for different rotation angles of the rotatable mirrors. First almost normal incidence of light istaken:An almost circular spot of about 2 mm diameter is found:

For larger angles of incidence the spot broadens:© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

15

For extremely large angles of incidence the spot gets even larger than the screen:© 2001 Wolfgang Theiss

SPRAY examples

3.3

IR reflectance unit

16

Alignment sensitivity
Optical setups are not always perfectly aligned, and it may be of interest how variations of somegeometrical parameters influence the performance of a device. SPRAY simulations can help toinvestigate misalignment effects. This is shown in the following simple example.
Ideal alignment leads to a throughput of the reflectance unit of 100% (ideal mirrors are assumed):© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

17

If the sample is moved up from its ideal position by 5 mm no rays reach the detector any more:

In order to investigate how the detector signal decreases with sample position many SPRAY
simulations have to be performed. Using OLE automation the sample height can be automaticallyvaried by an Excel macro, the detector signal is recorded and the following display of thedetected fraction of rays vs. the z coordinate of the sample is generated:© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

18

For almost normal incidence of lightthe z-alignment of the sample is significantly less critical:© 2001 Wolfgang Theiss

SPRAY examples

3.4

IR reflectance unit

19

ATR experiment
Reflectance units like the one discussed here are often used in combination with ATR (attenuatedtotal reflection). If a multiple reflection ATR crystal made of silicon is placed at the sample
position the situation is this:Here is a side view of the setup:© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

20

Obviously the ATR crystal has to be moved up in order to work properly because the radiation
has to enter the crystal from the 45° edges. In order to find the best position we can let SPRAY
do the work, controlled by OLE automation commands from Excel's VisualBasic in this case.
Varying the height of the ATR element and recording the detector signals one gets the following
results:

Trying a z-value of 0.33 seems to be promising. In fact, the light throughput is much better now:© 2001 Wolfgang Theiss

SPRAY examples

IR reflectance unit

21

There are 5 total reflections at the bottom side of the ATR element on the average:© 2001 Wolfgang Theiss

Part

IV

SPRAY examples

Light scattering

4

Light scattering

4.1

Dipole scattering - blue skies and red sunsets

23This is a simple example of light scattering effects studied with SPRAY. A parallel light beam(sun light) is directed into a region with very small particles - these particles represent density
fluctuations in the earth's atmosphere which lead to dipole scattering. SPRAY determines how
much light is received behind the scatterers and the fraction of rays scattered to the side. In bothcases not only the spectra but also the corresponding colors are computed.
The required scattering properties are computed using an external small program that performs
Mie computations (light scattering by spheres). Although water spheres have been used togenerate the data, the results are quite universal since the spheres are so small that they havealmost only dipole properties. The scattering characteristics (viewed with the tool view_rt.exe)
are the following:
Intensity [10 -19 ]
30

30

25

25

20

20

15

15

10

10

5

5

0

0
25000
20000
15000

020
4060

80100
120140160
180Scattering angle
[°]

10000

Wavenumber[1/cm]

Setting the volume fraction of the scattering particles to 0.0001 one gets the following
probabilities for scattering (red curve) and absorption (blue) per cm distance:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

24

Absorption and scattering data

Probability [1/cm]

0.30
0.25
0.20
0.15
0.10
0.05
0.00

12000

16000

20000
24000
Wavenumber [1/cm]

Clearly blue light (wavenumber 25000 1/cm, or 400 nm wavelength) is scattered more efficiently
than red light (700 nm wavelength, corresponding to about 14000 1/cm). The consequences forthe propagation of light through a volume filled with dipoles are investigated in a simple SPRAY
configuration described in the following.
The parallel light beam created by a circular light source (left object in the picture below) is
directed to a large sphere filled with the scattering dipoles (volume fraction: 0.0001). The rays
transmitted straightahead through the scattering volume are recorded by the rectangular detector
displayed to the right. The large detector in front is used to detect the radiation scatteredsideways:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

25

Testing the configuration with some rays (at 500 nm wavelength) gives a qualitative view of theeffects:

A certain fraction of the rays remains unscattered and reaches the 'throughput' detector, whereas
the scattered fraction is distributed in all directions.
Doing the simulation with 41 spectral points in the range 400 ... 800 nm and 10000 rays per
spectral point, the following results are obtained. The transmitted radiation has the following
spectral distribution and color (which is computed by SPRAY as well):© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

26'Throughput detector'
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

400

500

600
700
Wavelength [nm]800

The missing rays in the blue are scattered to the side:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

27'Side detector'
Detected fraction

0.10
0.08
0.06
0.04
0.02
0.00

400

500

600
700
Wavelength [nm]800

The conclusions from this simple simulation are the following: Light travelling through a medium
filled with scattering dipoles looses blue light and appears to be red (like sunlight closely before
sunset when it has to travel the largest distance through the earth's atmosphere). Watching lightscattering dipoles from the side (that's what you do looking at the sky) gives a blue appearance.

4.2

Paints

4.2.1

Paints: Beyond Kubelka-Munk
This example shows how SPRAY can be used to predict colors of paints. In most cases paints
consist of a transparent host material (resin) with embedded particles that scatter and absorb
light. For the design of paints it is important to know how the amount of reflected light and its
spectral composition depend on the concentrations and types of inclusions. Problems of this kindare often approached applying the so-called Kubelka-Munk theory. This rather old concept is a

© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

28

two-flux radiation transfer model. In its simplest form it assumes diffuse illumination, isotropic
scattering and perfect interfaces air-resin and resin-substrate. The probabilities per distance forabsorption K(l) and scattering S(l) must be known - these are the input quantities of the model.
The Kubelka-Munk theory returns the diffuse reflection coefficient of the paint for eachwavelength.
Using SPRAY you can do the same - but with less approximations and much more realisitic
results. The required input quantities are the scattering and absorption properties of each particle
species, the optical constants of all materials involved and - of course - the geometry of the setup
like the thickness of the resin or the direction of the incident radiation. You can increase thecomplexity of the model as you like - approaching more and more the real paint.
This SPRAY application example investigates a very simple model paint. Spherical inclusions areembedded in a resin with a constant and real refractive index of 1.5 - the resin itself does notabsorb light. The following inclusions are used:· Large TiO2 spheres· Small TiO2 spheres· Very small CdS spheres
The scattering and absorption properties of the inclusions are summarized in their individual
sections (just double-click the links above).
The SPRAY configuration used to investigate the paint properties is described in the section 'The
test system'. Before the actual work is done some short initial tests confirm the consistency of the
setup.
The TiO2 spheres will be used to create diffuse light scattering almost independent ofwavelength. In a first exercise we will learn how much inclusions we have to take in order to geta white paint covering a black substrate. The required amount strongly depends on the particle
size. This is demonstrated comparing the case of large and small TiO2 spheres.
In a second step CdS clusters are added. They mainly function as absorbers in the blue and will
be responsible for the color of the paint.

4.2.2

Scatterers

4.2.2.1

Large TiO2 spheres

The scattering properties of TiO2 spheres are computed using the external Mie program. The
optical constants of TiO2 have been taken from the SPRAY database. In order to avoid somenumerical problems of the Mie software, a small imaginary part has been added to the dielectric
function which would be purely real otherwise.
The radius distribution for the Mie computation is this:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

29

Size distribution large TiO2 spheres

Probability

0.5
0.4
0.3
0.2
0.1
0.0
0

1

2

3
4
5
Sphere radius [10-6 m]

The spectral and angular distribution of the scattered intensity is dominated by forward andbackward scattering
Intensity [10 -11 ]
10
8
6
4
2
0
25000

10
8
6
4
2

20000

0
0 20

40 60

80 100

Scattering angle[°]

120 140160 18015000Wavenumber[1/cm]
10000

If these particles are embedded in the resin with a volume fraction of 1% one gets the following
absorption (blue curve) and scattering probabilities/distance (red):© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

30

Absorption and scattering probability

K(l), S(l) [1/cm]

200
175
150
125
100
75
50
25
0

12000

16000

20000
24000
Wavenumber [1/cm]

In summary: The large TiO2 particles scatter light mainly in the forward direction. There is also
pronounced backward scattering, but no absorption. All these properties are independent ofwavelength.

Small TiO2 spheres

Reducing the size of the large TiO2 particles by a factor of 10
Size distribution small TiO2 spheres
0.5
Probability

4.2.2.2

0.4
0.3
0.2
0.1
0.0
0.0

0.1

0.2
0.3
0.4
0.5
Sphere radius [10 -6 m]changes the angular distribution of the scattered intensity significantly. There is still preferredforward and backward scattering but now the difference to the other directions is not so large
any more:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

31

Intensity [10 -13 ]
10
8
6
4
2
0
25000

10
8
6
4
2

20000

0
0 20

40 60

80 100120 140

Scattering angle[°]

160 18015000 Wavenumber[1/cm]
10000

Filling up the resin with these small TiO2 spheres (1% volume fraction) gives these scattering
probabilities:

Absorption and scattering probability

K(l), S( l) [1/cm]

3000

2000

1000

0

12000

16000

20000
24000
Wavenumber [1/cm]

Note the much larger probability for scattering events compared to the large TiO2 spheres whichis the main difference between the two types of TiO2 scatterers discussed here.
4.2.2.3

CdS clusters

The optical constants of the CdS clusters are taken from the SPRAY database. The radius
distribution used in the Mie computation is shown below:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

32

Size distribution CdS clusters

Probability

0.5
0.4
0.3
0.2
0.1
0.0
0.00

0.01

0.02
0.03
0.04
Sphere radius [10 -6 m]

Due to the very small sizes the scattering is almost dipole scattering:
Intensity [10-17 ]

6
5
4
3
2
1
0

6
5
4
3
2
1
0
25000

0
40

20000

80
Scattering angle
[°]

120

15000
160

10000

Wavenumber[1/cm]The CdS particles in the resin (1% volume fraction) absorb and scatter with the following
probabilities:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

33

Absorption and scattering probability
K(l), S(l) [1/cm]

3000

2000

1000

0

4.2.3

12000

16000

20000
24000
Wavenumber [1/cm]

The test system
The paint test system consists of a circular light source (transparent) sending a parallel beam
towards the air-resin interface (rectangular, 2 mm * 2 mm). Below this interface there is anotherone representing the resin-substrate transition. A view of this configuration is given below:

Here is a side view of the arrangement:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

34

The space between top and bottom interface is filled with a composite scatterer (the TiO2 andCdS particles). The distance between the interfaces, i.e. the thickness of the paint, is 100
microns. The concentration of the TiO2 and CdS particles can be varied in the model, like allother parameters and dimensions.
Four perfect mirrors enclose the scattering volume in order to avoid radiation leaving the test
volume (reflecting boundary conditions). The next picture shows the four mirrors which areabout to be moved to their final positions:Finally we show the complete system with some test rays:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

35

Above the light source there is a huge rectangular detector (10 m * 10 m) collecting all rays thatleave the top surface in the upward direction. Since in the views shown above the observer
looks downwards the detector is not visible.

4.2.4

Test the test system
Before the actual work begins we should perform some simple tests of the SPRAY setup. This isa very important step since there is a good chance that some of the numerous parameters may
be wrong (although we did our best).
No scattering particles, perfect absorber at the bottom interface
Here the rays are partly reflected at the air-resin interface. Since the resin has a constant and realrefractive index of 1.5 the reflectance of the top interface is about 4%. All rays transmitted to thebottom interface are absorbed if we use a perfect absorber for that interface. Here is a side view
with some test rays:

The detector should have a constant signal of about 4%. Doing the simulation in the spectral
range 300 ... 1000 nm with 20 spectral points and 10000 rays/spectral point one gets thefollowing detector signal:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

36

Detected fraction

Rectangular Detector
0.048
0.046
0.044
0.042
0.040
0.038
0.036
0.034
0.032
0.030
300

400

500

600

700 800 900 1000
Wavelength [nm]This is the expected behaviour. The noise level can be estimated the following way: If 10000
rays are processed and 4% reach the detector, the average number of rays is 400. The statisticalvariation of the number of detected rays is the square root of 400, i.e. 20. The amplitude of thenoise is then given by (420-380)/10000 = 0.4% which is roughly what you see in the graphabove.
No scattering particles, perfect mirror at the bottom interface
In this case all rays are reflected at the bottom interface

and should reach the detector. We get this:

Detected fraction

Rectangular Detector
1.2
1.0
0.8
0.6
0.4
300

400

500

600

700 800 900
Wavelength [nm]1000

No scattering particles, gold mirror at the bottom interface© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

37

This case is used to test a case with spectral variation. The optical constants of gold are takenfrom the SPRAY database. The obtained spectrum (300 ... 1000 nm, 71 points, 1000 rays/
point) and the corresponding color are shown below:
Rectangular Detector

Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900
Wavelength [nm]1000This looks like gold, and we can continue with more serious investigations.

4.2.5

Large TiO2 particles in resin
Now we can start to look at light scattering. In this first investigation we want to find out how
many of the large TiO2 particles we have to add to the paint to make it optically dense, i.e.
independent of the substrate.
The bottom interface is set to be a perfect absorber again. As shown in the tests before, thedetector signal is 4% (due to the specular reflectance of the air-resin interface) if the volume
fraction of the TiO2 spheres is 0. We now slowly increase the volume fraction, compute spectrain the range 300 ... 1000 nm with 7 points (we do not need spectral resolution in this case) andcollect the results with the Collect program.

© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

38

Starting at a volume fraction f = 0.01 a side view shows what happens:

The rays travel through the whole resin layer. Due to the pronounced forward scattering many of
the rays that get scattered nevertheless reach the absorbing bottom. Consequently the detector
signal is not too high yet:

Rectangular Detector
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900
Wavelength [nm]1000

At f = 0.2 almost no particles reach the bottom interface:

The diffuse reflectance signal is much higher now:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

39

Rectangular Detector
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0
300

400

500

600

700 800 900 1000
Wavelength [nm]The next graph summarizes the results for various volume fractions:Diffuse reflectance
1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
300400
500
600
700

0.3
800

Wavelength
[nm]

4.2.6

0.0
0.4

0.2
900
1000

0.1
Volume fraction f

Small TiO2 particles in resin
To investigate the influence of the TiO2 particle size on the paint performance we repeat theprevious computations for the small TiO2 spheres. Here is side view for f = 0.02:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering40

It turns out that much smaller volume fractions are needed to obtain an optically thick paint:
Diffuse reflectance
1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0
300400
500
600
700
800
900
1000
Wavelength
[nm]

4.2.7

0.0
0.020
0.015
0.010
0.005
Volume fraction f

Color by CdS clusters
Finally we add some absorbers to the paint to get a real color. As shown above, the small CdS
inclusions absorb in the blue much more than in the red - hence we can expect a color like red,orange or yellow.
Mixing a paint with f = 0.02 of the small TiO2 particles and f = 0.001 of CdS clusters, thefollowing spectrum (71 points covering the range 300 ... 1000 nm with 1000 rays/point) is
obtained:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

41

Rectangular Detector
Detected fraction

1.0
0.8
0.6
0.4
0.2
0.0

300

400

500

600

700 800 900 1000
Wavelength [nm]To visulasize what happens we can compare the fate of some test rays for different wavelengths.
At 400 nm (blue light) most of the rays get absorbed on their way through the paint:

On the other hand, red light (700 nm) can travel much longer through the resin and has a muchlarger chance to get to the top again without absorption:© 2001 Wolfgang Theiss

SPRAY examples

Light scattering

42

The graph below summarizes the results of various simulations:
Variaton of CdS volume fraction

0.00000
0.00005
0.000100.000200.000500.00100
0.00200

Diffuse reflectance

1.0
0.8
0.6
0.4
0.2
0.0
300

400

500

600

700 800 900
Wavelength [nm]1000

© 2001 Wolfgang Theiss