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Double integating spheres: A method for assess-
ment of optical properties of biological tissues
Wigand Poppendieck
2004-12-20
LiTH-IMT/ERASMUS-R--04/27--SE
Institut für Biomedizinische Technik
Diplomarbeit
Double integrating spheres: A method for assessment of optical properties of biological
tissues
eingereicht von: cand. mach. Wigand Poppendieck
Matr. Nr. 1876408
am: Institut für Biomedizinische Technik
Prof. Dr. rer. nat. J. Nagel
Betreuer: Prof. Åke Öberg
Institutet för medicinsk teknik
Universität Linköping, Schweden
Ausgabedatum: 19. Januar 2004
Abgabedatum: 22. Dezember 2004
Universität Stuttgart
Double integrating spheres: A method for assessment of optical properties of biological tissues
Abstract The determination of the optical properties of biological tissue is an important issue in laser medicine. The optical properties define the tissue´s absorption and scattering behaviour, and can be expressed by quantities such as the albedo, the optical thickness and the anisotropy coefficient. During this project, a measurement system for the determination of the optical properties was built up. The system consists of a double integrating sphere set-up to perform the necessary reflection and transmission measurements, and a computer algorithm to calculate the optical properties from the measured data. The algorithm is called Inverse Adding Doubling method, and is based on a one-dimensional transport model. First measurements were conducted with the system, including measurements with phantom media (Intralipid-ink solutions) and with cartilage samples taken from the human knee joint. This work also includes an investigation about the preparation of tissue samples for optical measurements.
Doppelintegrationskugeln: Eine Methode zur Bestimmung der optischen Eigenschaften von biologischem Gewebe
Kurzzusammenfassung
Die Bestimmung der optischen Eigenschaften biologischen Gewebes ist ein wichtiger Aspekt in der Lasermedizin. Die optischen Eigenschaften definieren das Absorptions- und Streuverhalten des Gewebes, und können über Größen wie den Albedo, die optische Dicke und den Anisotropiekoeffizient ausgedrückt werden. Im Rahmen dieses Projekts wurde ein Meßsystem zur Bestimmung der optischen Eigenschaften aufgebaut. Das System besteht aus einem Doppelintegrationskugel-Aufbau, um die erforderlichen Reflexions- und Transmissionsmessungen durchzuführen, und einem Computeralgorithmus, um die optischen Eigenschaften aus den Meßdaten zu errechnen. Der Algorithmus wird als Inverse Adding Doubling-Methode bezeichnet, und basiert auf einem eindimensionalen Transportmodell. Erste Messungen wurden mit dem System ausgeführt, darunter Messungen mit �phantom media� (Intralipid-Tinte-Lösungen) und mit Knorpelproben aus dem menschlichen Kniegelenk. Diese Arbeit beinhaltet außerdem eine Untersuchung über die Präparierung von Gewebeproben für optische Messungen.
1
Table of contents
1 Introduction ..........................................................................................................4
2 Theoretical Background .......................................................................................6
2.1 General assumptions........................................................................................6
2.2 Optical properties .............................................................................................6
2.3 Phase functions ................................................................................................9
2.3.1 Approximation functions ............................................................................9
2.3.2 Function expansion .................................................................................12
2.4 Dimensionless quantities ................................................................................14
2.5 Reflection and transmission factors ................................................................15
3 Integrating Sphere Theory..................................................................................17
3.1 Derivation of the single integrating sphere formulas.......................................18
3.1.1 Reflectance sphere..................................................................................18
3.1.1.1 Light incident on the sphere wall (case A)..........................................19
3.1.1.2 Light incident on the sample (case B) ................................................20
3.1.2 Transmittance sphere..............................................................................21
3.1.2.1 Diffuse light incident upon the sample (case C) .................................21
3.1.2.2 Collimated light incident upon the sample (case D) ...........................22
3.2 Derivation of the double integrating sphere formulas .....................................23
3.2.1 Light incident upon the sphere wall (case E) ...........................................25
3.2.2 Light incident upon the sample (case F)..................................................27
3.3 Influence of baffles..........................................................................................30
3.3.1 Reflectance sphere with baffle ................................................................31
3.3.2 Transmittance sphere with baffle.............................................................32
3.3.3 Double integrating sphere with baffle ......................................................34
2
3.4 Reference measurements with a single integrating sphere ............................35
3.4.1 Introduction of the sphere constants........................................................35
3.4.2 Relation of the detected power to a reference power ..............................36
3.4.3 Determination of the sphere constants ....................................................39
4 Procedure to obtain the optical properties..........................................................42
4.1 Measurement set-up.......................................................................................42
4.2 Calculation of the optical properties out of the measurements .......................43
5 The Adding Doubling Method.............................................................................47
5.1 The reflection function and the transmission function.....................................48
5.2 Quadrature .....................................................................................................51
5.2.1 Radau quadrature....................................................................................52
5.2.2 Gaussian quadrature ...............................................................................56
5.3 The Redistribution function .............................................................................56
5.4 Layer initialization ...........................................................................................60
5.4.1 Infinitesimal generator initialization..........................................................63
5.4.2 Diamond initialization...............................................................................64
5.5 Derivation of the Adding Doubling equations..................................................66
5.6 Inclusion of boundary effects originating from the glass slides .......................69
5.7 Determination of the reflection and transmission factors ................................73
5.8 Example for an Adding Doubling calculation ..................................................74
6 The Inverse Adding-Doubling Method................................................................83
6.1 Uniqueness.....................................................................................................83
6.2 Case differentiation depending on the available measurements ....................85
6.3 Auxiliary calculations ......................................................................................87
6.3.1 Estimation of the reflection and transmission factors...............................87
6.3.2 The �quick guess�-method to obtain iteration starting values ..................88
3
6.4 Search algorithms...........................................................................................91
6.4.1 Determination of a, with τ and g fixed......................................................92
6.4.2 Determination of τ, with a and g fixed......................................................93
6.4.3 Determination of a and g, with τ fixed......................................................93
6.4.4 Determination of τ and g, with a fixed......................................................95
6.4.5 Determination of τ and g, with τs fixed .....................................................96
6.4.6 Determination of τ and g, with τa fixed.....................................................97
6.4.7 Determination of a and τ, with g fixed......................................................98
6.5 Example for an Inverse Adding Doubling calculation......................................99
7 Measurements with the integrating sphere set-up............................................104
7.1 Reference measurements ............................................................................104
7.2 Measurements with phantom media .............................................................106
7.3 Preparation of tissue samples ......................................................................114
7.4 Measurements with cartilage samples..........................................................117
7.5 Difficulties with the double integrating sphere set-up....................................120
8 Summary..........................................................................................................124
9 Prospects .........................................................................................................126
10 Appendix ..........................................................................................................128
10.1 Interdependence between ν, µ, µ´ and φ ......................................................128
10.2 Bracketing and parabolic interpolation..........................................................129
10.3 Golden section search and Brent´s method..................................................131
10.4 Two-dimensional minimization with the Amoeba algorithm ..........................131
10.5 The diffusion approximation of the radiative transport equation ...................134
11 Formula index ..................................................................................................141
12 References.......................................................................................................143
4
1 Introduction
From the proceeding development in laser medicine emerges an increasing interest
in techniques to determine the optical properties of various biological tissue. The
knowledge of these parameters is especially important with respect to dosimetry in
photodynamic therapy and diagnostic techniques, such as photodynamic tumour
treatment, transillumination imaging or fluorescence diagnostics [26].
The optical properties of a tissue are usually defined by three quantities: the
scattering coefficient µs [mm-1], the absorption coefficient µa [mm-1] and the single-
scattering phase function p(θ) [11]. The scattering coefficient and the absorption
coefficient can be transformed into two dimensionless quantities, the albedo a and
the optical thickness τ. The most important parameter characterizing the single-
scattering phase function is the anisotropy coefficient g.
The application of integrating spheres as a tool to measure these optical properties
has a long tradition and is an established technique. The first integrating sphere was
developed by R. Ulbricht, around 1900 [25]. Integrating spheres can be used to
measure parameters such as the reflection factor R and the transmission factor T,
depending on the wavelength of the incident laser beam.
The derivation of the three characteristic optical parameters (µs, µa, p(θ) or a, τ, g,
respectively) from the measured data is more complicated. We have three levels of
quantities: the measurement data, the reflection and transmission factors, and the
optical properties. The reflection and transmission factors are related to the
measurement data by the Integrating Sphere Theory. To connect them to the optical
properties, we have to introduce an algorithm based on a transport model. One
numerical algorithm to solve this problem is known as the Adding-Doubling (AD)
Method. By using the AD method and the Integrating Sphere equations, it is possible
to calculate expected measurement values, for given optical properties.
However, for the calculation of the optical properties out of given measurement
values, the algorithm has to be conducted in the other direction. As this cannot be
5
done analytically, an iterative method has to be applied: a set of optical properties is
guessed and used to calculate the expected measurement values with the AD
method. This set is then varied iteratively, until the calculated results match the
actual measurement values. This algorithm is called Inverse Adding Doubling (IAD)
Method [11].
The objective of this project is the set-up of a double integrating sphere system, that
can be used to measure the optical properties of various biological tissues. This also
includes the supply with an appropriate mathematical computation system, to
calculate the optical properties from the measurement data (IAD Method).
6
2 Theoretical Background
2.1 General assumptions
In order to simplify the calculations in the following chapters, the following
fundamental assumptions have to be made [10]:
- The light distribution is independent of time (exclusion of optical properties
changing with time, no irradiance times shorter than about 10-9 s)
- All media exhibit homogeneous optical properties
- The tissue geometry may be approximated by an infinite plane-parallel slab of
finite thickness (allowing generalization to layered tissues or infinitely thick
tissues)
- The tissue has a uniform index of refraction
- All boundaries are assumed to be smooth and specularly reflecting, according
to Fresnel´s law
- The polarization of light is ignored
2.2 Optical properties
Fig. 2.1 shows a model of the photon transport through tissue. In this model, there
are two possible types of interaction between the photons and the tissue: the photons
may either be scattered (1), resulting in a change of their movement direction, or be
absorbed by the tissue (2), i.e. their kinetic energy is transformed into heat energy.
Some of the photons may even pass the tissue without being affected at all (3).
In Fig. 2.1, the scattering and absorption events are symbolized by the blue and
orange spots in the tissue. Each time a photon hits one of these spots, it will be
scattered or absorbed, respectively.
7
Fig. 2.1: Model of photon transport through tissue
The scattering coefficient µs and the absorption coefficient µa of the tissue are
defined as the reciprocal of the average free path between two scattering events (ls)
or two absorption events (la), respectively. For example, if a tissue is highly
absorbing, we have a big number of absorption events, leading to a small average
free path la. This yields a high absorption coefficient µa. As a result of this
consideration, the higher the scattering or the absorption of a tissue is, the higher the
respective coefficient would become.
If a photon is scattered inside the tissue, it might be scattered in any direction. The
probability of the scattering direction is given by the so-called single-scattering phase
function p(s´,s). This function determines the probability that a photon coming from
the direction denoted by the unit vector s´ is scattered into the direction denoted by
the unit vector s, during a single scattering event.
The phase function p(s´,s) is usually constrained by assuming that it is only
dependent on the cosine ν of the angle θ between the two directions s´ and s (see
Fig. 2.1):
)()(cos)(),( νθ pppp ==⋅′=′ ssss (2.1)
As p denotes a probability distribution, the integral over all angles has to be unity:
8
1)(4
=∫π
ων dp (2.2)
dω is a differential solid angle. By using spherical coordinates, it can be expressed by
φθθω ddd ⋅⋅= sin (2.3)
with the colatitude (polar angle) θ and the longitude (azimuth) φ (see Fig. 2.2).
Fig. 2.2: Spherical coordinates
By substituting expression (2.3) in expression (2.2), we find
1sin)()(2
0 04
=⋅⋅= ∫ ∫∫= =
π
φ
π
θπ
φθθνων ddpdp (2.4)
With the use of
θνθθν
sincos dd −=⇒= (2.5)
expression (2.4) yields
1)()()(2
0
1
1
2
0
1
14
=⋅+=⋅−= ∫ ∫∫ ∫∫= −==
−
=
π
φ ν
π
φ νπ
φννφννων ddpddpdp (2.6)
As the phase function p(ν) is independent of φ, we finally obtain
9
1)(2)(1
14
== ∫∫−
ννπωνπ
dpdp (2.7)
The optical properties of turbid media are thus characterized by three quantities [11]:
- the scattering coefficient µs (reciprocal of the average distance that a photon will
travel within the medium before it is scattered)
- the absorption coefficient µa (reciprocal of the average distance that a photon will
travel within the medium before it is absorbed)
- the single-scattering phase function p(ν) (determining the probability that a photon
is scattered into a direction denoted by the cosine ν from the incoming direction,
during a single scattering event)
We further define the anisotropy coefficient g (-1≤g≤1), which is given by the average
cosine of the scattering angle:
∫∫−
==1
14
)(2)( νννπωννπ
dpdpg (2.8)
The functional form of p(ν) in tissue is usually not known. For mathematical
tractability, it has to be approximated by an appropriate function.
2.3 Phase functions
2.3.1 Approximation functions
The most simple phase function is a constant (isotropic) function
πν
41)( =isop (2.9)
The constant value results from the normalization condition (2.7). However, as
biological tissue does not exhibit isotropic scattering, the phase function (2.9) cannot
be used in this context.
10
Several researches have shown that the phase function in human dermis and aorta,
for a wave length of 633 nm, can be approximated by the use of a Henyey-
Greenstein function [11]:
232
2
)21(1
41)(
νπν
gggpHG −+
−⋅= (2.10)
The function depends only on the anisotropy coefficient g (see expression (2.8). The
Henyey-Greenstein phase function (2.10) meets the normalization requirement given
by expression (2.5):
1212
1)21(
121)(2
1
12
21
1232
21
1
=−+
−=
−+−
=−−−
∫∫ νν
νννπ
ggggd
gggdpHG (2.11)
The Henyey-Greenstein function is also compatible with expression (2.8):
ggg
gggggdpHG =
−+
++−+
−=
−−∫
1
12
22
2
21
1 21121
41)(2
νννννπ (2.12)
A series of Henyey-Greenstein functions is illustrated in Fig. 2.3. g varies between
values of -1 (scattering completety in the backwards direction θ=±180°) and 1
(scattering completely in the forwards direction θ=0°). For g=0, the sample exhibits
isotropic scattering. A typical value of the anisotropy coefficient for tissues in the red
region of the spectrum is g≈0,8 [11].
Fig. 2.3: Henyey-Greenstein functions
11
The Henyey-Greenstein approximation can be further adjusted by choosing a so-
called �modified Henyey-Greenstein function� [10]:
−+
−−+⋅=
232
2
)21(1)1(
41)(
νββ
πν
mm
mmHG gg
gp (2.13)
consisting of an isotropic term and a Henyey-Greenstein term. Phase function
measurements for dermis at 633 nm yield parameters of g=0.91 and β=0.1 [10]. For
β=0, expression (2.13) is reduced to the Henyey-Greenstein function (2.10). As a
lniear combination of the phase functions (2.9) and (2.10), the modified Henyey-
Greenstein function meets the normalization condition (2.7). The relation between g
and the �modified anisotropy coefficient� gm is given by
mHGmHG gdpdpg ⋅−=
⋅−+== ∫∫−−−
)1()()1(21
21)(2
1
1
1
1
21
1
βνννββννννπ (2.14)
Another approximation often used is given by the Eddington phase function [10]
[ ]νπ
ν gpE 3141)( +⋅= (2.15)
meeting the requirements (2.7) and (2.8):
ggdpgdp EE =
+==
+=
−−−−∫∫
1
1
321
1
1
1
21
1 21
21)(2;1
23
21)(2 νννννπννννπ (2.16)
The advantage of this approximation is the possibility to reduce the transport
equation (see chapter 10.5) to a diffusion equation, yielding useful results for the
radiative transport in media with small values of g.
However, for the approximation of the tissue phase function, the Eddington phase
function is not applicable, because the typical values of g are too big. An
approximation of the phase function without this restriction, but also allowing the
reduction of the transport equation to a diffusion equation, is the Delta-Eddington
approximation [5, 10]
[ ])31)(1()1(241)( ννδπ
νδ gffp E ′+−+−⋅= (2.17)
with the Dirac Delta function
12
1)(;0,00,
)( =
≠=∞
= ∫+∞
∞−
dxxxx
x δδ (2.18)
f is the fraction of light scattered forwards. For f=1, we obtain a delta function (all the
light is transmitted in the forwards direction), and for f=0, expression (2.17) is
reduced to the Eddington approximation (expression (2.15)).
g´ is an �anisotropy factor� that is usually not equal to the anisotropy coefficient. The
relation between g´ and g is given by
gffgffdpg E ′−+=
′+−+−==
−−∫ )1()
21()1()1(2
21)(2
1
1
21
1
νννσνννπ δ (2.19)
with the unity step function defined as
<≥
== ∫∞− 0,0
0,1)()(
xx
dttxx
δσ (2.20)
The normalization condition (2.7) is also met by the Delta-Eddington approximation:
1232
21)(2
1
1
21
1
=
++=
−−∫ ννννπ δ gfdp E (2.21)
2.3.2 Function expansion
The phase functions denoted in chapter 2.3.1 can be expanded as a sum of
Legendre polynomials [10]. The Legendre polynomials Pn are the solutions to the
Legendre differential equations
0)1(2)1( 2
22 =++−− n
nn PnndxdPx
dxPdx (2.22)
The first Legendre polynomials are given by [21]
)5105315231()()157063()(
)33035()()35()(
)13()()(
1)(
246161
6
3581
5
2481
4
321
3
221
2
1
0
−+−=+−=+−=
−=−=
==
xxxxPxxxxP
xxxPxxxP
xxPxxP
xP
(2.23)
The Henyey-Greenstein function can be expanded as [10]
13
[ ]...)(5)(3141)()12(
41)( 2
21
0+++⋅=+⋅= ∑
∞
=
ννπ
νπ
ν PggPPgnpn
nn
HG (2.24)
For the modified Henyey-Greenstein function, we find the following expansion [10]:
[ ]...)()1(5)()1(3141
)()12()1()(41)(
22
1
00
+−+−+⋅=
+⋅−+⋅⋅= ∑
∞
=
νβνβπ
νβνβπ
ν
PgPg
PgnPp
mm
nn
nmmHG
(2.25)
The Eddington phase function does not need to be expanded, as it is already a
polynomial. With P0(x) and P1(x) from expression (2.23), expression (2.15) can be
written as
[ ])(3)(41)( 10 ννπ
ν gPPpE +⋅= (2.26)
By using the Legendre expansion for the Dirac Delta function [17]
∑∞
=
+=−0
)()()12(21)(
nnn yPxPnyxδ (2.27)
the Delta-Eddington function (expression (2.17)) can be expanded as [10]
[ ][ ]...)(5)()1(3141
)()12())(31()1(41)(
21
01
++−′++⋅=
+⋅+′+⋅−⋅= ∑
∞
=
ννπ
ννπ
νδ
fPPfgf
PnfPgfpn
nE
(2.28)
For a modified Henyey-Greenstein function with given parameters gm and β, we can
approximate the corresponding Delta-Eddington function by comparing the
coefficients of the Legendre polynomials Pi(ν) (i=1...n) in the expressions (2.25) and
(2.28), yielding the following system of n equations:
[ ]
fngn
fgfg
gffg
nm
m
m
m
)12()1)(12(
7)1(75)1(5
)1(3)1(3
3
2
+=−+
=−=−
′−+=−
β
βββ
MMM
(2.29)
For β≠1 and gm≠0, the system cannot be solved for f. Thus, we neglect the equations
corresponding to values of i>2, and only look at the first two equations [10]. By
solving them for f and g´, we find the solutions
14
21
1
2 )1(,)1(m
mmm g
ggggf−
−=′−=
−β
β (2.30)
If we substitute the measurement values g=0.91 and β=0.1 (see chapter 2.3.1) in
expression (2.30), we obtain the values f=0.75 and g´=0.29. This shows that for
strongly forward scattering phase functions, the Delta-Eddington approximation
lumps a large portion (f=0.75) of the scattering into the Dirac Delta term, allowing the
anisotropy factor g´ of the Eddington term to fall. For the diffusion approximation of
the transport equation (see chapter 10.5), this decrease of the anisotropy factor leads
to an accuracy improvement, since the diffusion approximation is poor for strongly
anisotropic scattering, but relatively good for nearly isotropic scattering [10].
2.4 Dimensionless quantities
With the use of the physical sample thickness d (see Fig. 2.1), the coefficients µs and
µa (see chapter 2.2) can be expressed by two dimensionless quantities, the albedo a
and the optical thickness τ [11]:
τµ
µµµ d
a s
as
s =+
= (2.31)
( )asd µµτ +⋅= (2.32)
The albedo varies between 0 and 1: a=0 indicates that no scattering occurs in the
sample, while a=1 indicates the absence of absorption. The optical thickness is
defined as the product of the physical sample thickness and the sum of the scattering
and the absorption coefficient. For a sample with the optical thickness τ=1, there is a
probability of e-1=37% that light will travel through it without being scattered or
absorbed [10].
In the following chapters, the optical properties will be calculated in terms of three
dimensionless quantities:
- the albedo a
- the optical thickness τ
15
- the anisotropy coefficient g (see chapter 2.2)
By using the expressions (2.31) and (2.32), the scattering coefficient µs and the
absorption coefficient µa can be easily derived from a and τ, for a given physical
thickness d:
dasτµ ⋅= (2.33)
daa
τµ ⋅−= )1( (2.34)
With a given anisotropy coefficient g, an appropriate approximation for the phase
function p(ν) may be chosen. Approximation examples are given in chapter 2.3.
2.5 Reflection and transmission factors
To determine the optical properties, measurements with a double integrating sphere
set-up (see chapter 3) can be conducted. However, these measurements do not yield
directly a, τ and g. We can just obtain several measurement values that depend on
the reflectance and the transmittance of the sample.
The reflection factor and the transmission factor are defined relative to the irradiance
on the sample surface [11]. They vary between 0 and 1 and denote the fraction of the
total incident light that is reflected or transmitted, respectively, by the sample.
Depending on the character of the incident and the reflected or transmitted light, we
can distinguish three reflection factors and three transmission factors, respectively [8]
(see Table 2.1 and Fig. 2.4):
- If the light incident upon the sample is diffuse, we have only one reflection factor
and one transmission factor. The diffuse reflection (transmission) factor with
diffuse incident light, Rd (Td), denotes the fraction of light that is reflected
(transmitted) diffusely by the sample.
- If the light incident upon the sample is collimated, we have two reflection factors
and two transmission factors. The diffuse reflection (transmission) factor for
collimated incident light, Rcd (Tcd), denotes the fraction of light that is reflected
16
(transmitted) diffusely by the sample. The collimated (or specular) reflection
(transmission) factor for collimated incident light, Rc (Tc), denotes the fraction of
light that is reflected (transmitted) specularly.
Table 2.1: Reflection and transmission factors
Rd Diffuse reflection factor for diffuse incident light
Rc Collimated (specular) reflection factor for collimated incident light
Rcd Diffuse reflection factor for collimated incident light
Td Diffuse transmission factor for diffuse incident light
Tc Collimated (specular) transmission factor for collimated incident light
Tcd Diffuse transmission factor for collimated incident light
Fig. 2.4: Reflection and transmission factors
For collimated light incident upon the sample, the total amount of reflected
(transmitted) light is determined by Rct (Tct), given by the sum of the two
corresponding factors:
cdcctcdcct TTTRRR +=+= ; (2.35)
17
3 Integrating Sphere Theory
Integrating spheres are a means to measure the reflectance and the transmittance of
a sample. The general structure of a single integrating sphere set-up used to
measure the reflection factor is illustrated in Fig. 3.1.
Fig. 3.1: Set-up of a single integrating sphere system
The inner surface of the sphere is coated with BaSO4 [25]. A laser beam is incident
upon a small aperture in the sphere. The sample is attached to another aperture. A
third aperture is used for a photo detector. This photo detector measures the light
intensity incident upon it and transduces it into an electric voltage signal.
When measurements with biological tissue sample are conducted, the sample has to
be sandwiched within two thin glass plates to fix it in the proper position. This might
lead to inaccuracy during the measurement, as the glass surfaces influence the
reflective and transmittive behaviour of the sample.
18
3.1 Derivation of the single integrating sphere formulas
In the following, we assume that the efficiency of the detector is independent of the
angle of light incidence. Thus, the output voltage signal will be directly proportional to
the total light intensity incident upon the detector.
The total inner surface area of the sphere, including all holes and ports, is
24 RA ⋅= π (3.1)
with the sphere radius R. The area covered by the sample is denoted as s, the area
covered by the detector is denoted as δ. h is the area of the other holes (i.e. the
entrance port) in the sphere (see Fig. 3.1). By using these definitions, the actual
sphere wall area relative to the total sphere area A can be expressed by
++−=
++−=
Ah
As
AAhsA δδα 1)( (3.2)
The reflection factors of the sphere wall and of the detector will be denoted as m and
r, respectively.
3.1.1 Reflectance sphere
The total light power entering the sphere is given by P. This light beam can be
incident either on the sphere wall (Fig. 3.2A) or on the sample (Fig. 3.2B) [8]
Fig. 3.2: Light incident upon the sphere wall (A) and on the sample (B)
19
3.1.1.1 Light incident on the sphere wall (case A)
First, we regard case A. The total light power reflected by the sphere wall during the
first reflection is given by
mPP =1 (3.3)
The wall is considered to be a perfectly diffusing (Lambertian) surface; the power of
the reflected light is therefore distributed uniformly over the sphere wall. Thus, the
fraction of the reflected light power collected by the detector equals the fraction of the
detector area relative to the total sphere area:
11, PA
P δδ = (3.4)
Similar relations can be obtained for the light power collected by the sphere wall
(Pα,1), the sample (Ps,1) and the holes (Ph,1):
11,11,11, ;; PAhPP
AsPPP hs === αα (3.5)
The light leaving the sphere through the holes (Ph,1) is lost, the other fractions of light
remain inside the sphere. They are reflected from the detector (reflection factor r), the
sphere wall (reflection factor m), and the sample (reflection factor Rd, diffuse incident
light). The total reflected light of the second reflection is
FPPAsRPmP
ArP d 11112 =++= αδ (3.6)
with the abbreviation
AsRm
ArF d++= αδ (3.7)
Of this second reflection, the detector collects (cf. expression (3.4))
FPA
P 12,δ
δ = (3.8)
With the use of similar considerations, light from the third reflection is incident upon
the detector with a power of
20
FFPA
P 13,δ
δ = (3.9)
In general, the power of the light collected by the detector from the nth reflection is
given by
11,
−= nn FP
AP δδ (3.10)
The total power collected by the detector can be obtained by summing from n=1 to ∞:
∑∞
=
−⋅=1
11
n
nFPA
P δδ (3.11)
By using the equation for the geometric series (F<1), we find
11 )]()([11
11 P
ArAsRmAFP
AP
d δαδδ
δ ++−⋅=
−⋅= (3.12)
With the use of expression (3.3), expression (3.12) can be written as
PArAsRm
mA
Pd )]()([1 δα
δδ ++−
⋅= (3.13)
3.1.1.2 Light incident on the sample (case B)
In case B, the collimated entering light with the power P is incident upon the sample
(Fig. 3.2B). The reflected power consists of two parts: a collimated (specular) part
PRP cc = (3.14)
and a diffuse part
PRP cdcd = (3.15)
We assume that the collimated part of the reflected light is incident upon the sphere
wall and does not exit the sphere directly through the input hole (as may be the case
when the incoming light is perpendicular upon the sample). The reflection of the
collimated part at the sphere wall leads to a generation of diffuse light with the power
PmRP ccr = (3.16)
Thus, there are two sources of diffuse light, adding up to a power of
21
PRmRPPP cdccdcr )(1 +=+= (3.17)
This power source behaves in the same way as the power source P1 in (3.3). With
the considerations from above, the total power collected by the detector can be
obtained by using the expressions (3.12) and (3.17):
PArAsRm
RmRA
Pd
cdc
)]()([1 δαδ
δ ++−+
⋅= (3.18)
If the collimated part of the reflected light exits the integrating sphere directly through
the input hole, the term mRc disappears from the expression (3.18).
3.1.2 Transmittance sphere
For the measurement of the transmission, the sample is place at the entrance port of
the integrating sphere, as illustrated in Fig. 3.3. As in chapter 3.1.1, there are two
possibilities: the light incident upon the sample (total power P) can be either diffuse
or collimated.
Fig. 3.3: Diffuse (C) and collimated (D) light transmitted through the sample
Similar to case B, the problem can be reduced to that of one or two diffuse sources.
By using the equations of case A, the light power collected by the detector can be
calculated. Instead of the reflection factors, now the transmission factors have to be
used.
3.1.2.1 Diffuse light incident upon the sample (case C)
In case C, there is only one source of diffuse light in the sphere (see Fig. 3.3C):
22
PTP d=1 (3.19)
The light power collected by the detector is then (expression 3.12):
PArAsTm
TA
Pd
d
)]()([1 δαδ
δ ++−⋅= (3.20)
3.1.2.2 Collimated light incident upon the sample (case D)
If the light incident upon the sample is collimated, the transmitted light consists (as in
case B) of a collimated part
PTP cc = (3.21)
and a diffuse part
PTP cdcd = (3.22)
The collimated part is reflected diffusely on the sphere wall. This leads to a diffuse
light source of the power
PmTP ccr = (3.23)
The total power of the two diffuse sources (expressions (3.22) and (3.23) is therefore
given by
PTmTPPP cdccdcr )(1 +=+= (3.24)
leading to a power collected by the detector (equation 3.12) of
PArAsTm
TmTA
Pd
cdc
)]()([1 δαδ
δ ++−+
⋅= (3.25)
In Table 3.1, the results for the 4 different cases (A-D) are summarized. As
announced in chapter 3.1.1.2, the term mRc in case B will become zero, if the
collimated reflected light exits the integrating sphere directly. The same would
happen to the term mTc in case D, if the collimated transmitted light (see Fig. 3.3D)
exited the sphere directly through a port on the left side.
23
Table 3.1: Results for the single integrating sphere set-up, different cases
Case Measurement Light on sample Relative detected power Pδ/P
A Reflection diffuse )]()([1 ArAsRm
mA d δαδ
++−⋅
B Reflection collimated )]()([1 ArAsRm
RmRA d
cdc
δαδ
++−+
⋅
C Transmission diffuse )]()([1 ArAsTm
TA d
d
δαδ
++−⋅
D Transmission collimated )]()([1 ArAsTm
TmTA d
cdc
δαδ
++−+
⋅
By looking at the different equations for the detected power, some similarities can be
observed. We can divide all equations into two main factors G and S:
DcaseDcase
CcaseCcase
BcaseBcase
AcaseAcase
SGP
SGP
SGP
SGP
⋅=
⋅=
⋅=
⋅=
,
,
,
,
δ
δ
δ
δ
(3.26)
The first factor (G=geometry) is given by the sphere geometry and the reflection
factors of the sphere (including Rd). It is identical for all cases (assuming that the
sphere parameters are identical):
)]()([11
ArAsTmAG
d δαδ
++−⋅= (3.27)
The second factor (S=sources) denotes the different sources of diffuse light for each
case. There can be either one (case A, case C) or two sources (case B, case D):
PTmTS
PTS
PRmRS
PmS
cdcDcase
dCcase
cdcBcase
Acase
⋅+=
⋅=
⋅+=
⋅=
][
][ (3.28)
3.2 Derivation of the double integrating sphere formulas
When subjecting biological tissue to laser light, the initial response is an increase in
temperature [8]. As the optical properties of the tissue vary depending on the
24
temperature, measurements for different temperatures should be conducted in order
to determine the optical properties as a function of the temperature. The
measurements of the reflectance and of the transmittance therefore have to be
carried out simultaneously, while the tissue is heated. This simultaneous
measurement can be achieved by the use of a double integrating sphere set-up (see
Fig. 3.4).
Fig. 3.4: Set-up of a double integrating sphere system
The sample is placed between the two integrating spheres, so that it is situated at the
exit port of the first sphere (reflectance sphere, used to measure the reflectance),
and on the entrance port of the second sphere (transmittance sphere, used to
measure the transmittance). These two ports should have identical sizes, so that all
the transmitted light will be collected within the spheres. The other parameters of the
spheres (radius, size of the holes) do not have to be identical. It is assumed that the
reflectance and the transmittance of the sample are homogeneous with respect to
25
which side of it the light is incident upon [8]. Again, there are two cases: the light may
be incident upon the sphere wall (Fig. 3.4E) or upon the sample (Fig. 3.4F).
As the respective parameters of the two spheres (m, r, A, δ, h) do not have to be
identical, they will be distinguished in the following by the index r (reflectance sphere)
and t (transmittance sphere), respectively. This is also applied to the abbreviation α
(expression (3.2)), but not to the sample size s, as it is considered to be identical for
both spheres.
3.2.1 Light incident upon the sphere wall (case E)
The incoming light (power P) is first reflected on the sphere wall, acting as a diffuse
light source. The part of the reflected light that is collected by the first detector is
given by expression (3.13):
PVm
AP
r
r
r
rr ⋅=
δδ 1 (3.29)
with the abbreviation
++−=
r
rr
rdrrr A
rAsRmV δα1 (3.30)
However, in the double integrating sphere system, this is only a part of the light
power totally collected by the first detector, because there is also some light travelling
between the spheres, owing to the sample´s transmittance.
The part of the reflected light incident upon the sample is (cf. expression (3.29)):
PVm
AsP
r
r
rs ⋅=1 (3.31)
A part Pt1 of this light is transmitted into the transmittance sphere, depending on the
diffuse transmittance factor Td:
PVm
AsTPTP
r
r
rdsdt ⋅== 11 (3.32)
The part of this transmitted light that is collected by the second detector in the
transmittance sphere can then be determined by using expression (3.12):
26
PVm
As
VT
AP
VAP
r
r
rt
d
t
tt
tt
tt ⋅⋅=⋅=
δδδ 11
1 (3.33)
where Vt is defined similar to Vr:
++−=
t
tt
tdttt A
rAsRmV
δα1 (3.34)
Another part of Pt1 is incident back upon the sample:
PVm
As
VT
AsP
VAsP
r
r
rt
d
tt
tts ⋅⋅⋅=⋅=′ 11
1 (3.35)
The part of Ps1´ that is transmitted back to the reflectance sphere is given by
PVm
As
VT
AsTPTP
r
r
rt
d
tdsdt ⋅⋅⋅=′=′ 11 (3.36)
A fraction of this light is again incident upon the first detector:
PVm
As
VT
As
VT
AP
VAP
r
r
rt
d
tr
d
r
rt
rr
rr ⋅⋅⋅⋅⋅=′=
δδδ 12
1 (3.37)
By using the abbreviations
t
d
tt
r
d
rr V
TAs
VT
As
⋅=⋅= ττ ; (3.38)
expression (3.37) can be rewritten as
PVm
AP tr
r
r
r
rr ⋅⋅⋅= ττ
δδ 2 (3.39)
The part of Pt1´ that is incident back upon the sample can be calculated by using the
abbreviations (3.38):
PVm
AsP tr
r
r
rs ⋅⋅⋅= ττ2 (3.40)
so that the power of the light travelling through the sample a third time is given by
PVm
AsTPTP tr
r
r
rdsdt ⋅⋅⋅== ττ22 (3.41)
Now the second portion of light incident upon the detector in the transmittance
sphere can be determined:
27
PVm
As
VT
AP
VAP tr
r
r
rt
d
t
tt
tt
tt ⋅⋅⋅⋅=⋅= ττ
δδδ 22
1 (3.42)
This exchange of light between the two spheres will continue ad infinitum. By
comparing the expressions (3.29) and (3.39), the nth portion of light collected by the
first detector is
PVm
AP n
trr
r
r
rrn ⋅⋅⋅= −1)( ττ
δδ (3.43)
The nth portion of light collected by the second detector can be calculated by
comparing the expressions (3.33) and (3.42):
PVm
As
VT
AP n
trr
r
rt
d
t
ttn ⋅⋅⋅⋅= −1)( ττ
δδ (3.44)
Along the lines of chapter 3.1.1.1, the total light power collected by the detectors can
be determined by using the equation for the geometric series (τrτt<1):
tr
ntr ττττ
−=∑
∞−
11)(
1
1 (3.45)
yielding the total light power collected by the detector in the reflectance sphere
PVm
AP
trr
r
r
rr ⋅
−⋅⋅=
ττδ
δ 11 (3.46)
and the total light power collected by the detector in the transmittance sphere
PVm
As
VT
AP
trr
r
rt
d
t
tt ⋅
−⋅⋅⋅=
ττδ
δ 11 (3.47)
The factors τr and τt can be physically explained as follows: if there is light of a
certain power in the reflectance sphere (transmittance sphere), the fracture τr (τt) of
this power is the amount of light that is transmitted into the transmittance sphere
(reflectance sphere) by travelling through the sample once.
3.2.2 Light incident upon the sample (case F)
If the entering light is incident upon the sample, the detected power can be calculated
similar to chapter 3.2.1. Now there are four initial sources of diffuse light: two in the
reflectance sphere (adding up to P1r) and two in the transmittance sphere (adding up
to P1t) The origin of these four sources can be gathered from Fig. 3.4F.
28
The light power generated from the sources is given by (3.17) for the reflectance
sphere and by (3.24) for the transmittance sphere:
PRRmP cdcrr )(1 += (3.48)
PTTmP cdctt )(1 += (3.49)
A part of the diffuse light generated in the reflectance sphere will be transmitted into
the transmittance sphere, acting as a third source of diffuse light there. Analogically,
a part of the diffuse light generated in the transmittance sphere travels back through
the sample and acts as a third source in the reflectance sphere.
According to the above explanation of the factors τr and τt, these third sources (P1r,t
and P1t,t) originating from the transmitted part of the diffuse light generated in the
respectively adjacent sphere, are given by
PTTmP cdctttr )(,1 += τ (3.50)
PRRmP cdcrrtt )(,1 += τ (3.51)
The total light power generated by the sources in the reflectance sphere and the
transmittance sphere, respectively, can thus be determined as
PTTmRRmPPP cdcttcdcrtrrFcaser )]([,11,1 +++=+= τ (3.52)
PRRmTTmPPP cdcrrcdcttttFcaset )]([,11,1 +++=+= τ (3.53)
In chapter 3.2.1, the diffuse light sources in the two spheres were given by
PmVT
AsPmPPmP r
r
d
rrrEcasetrEcaser ⋅=== τ,1,1 ; (3.54)
By substituting these terms in the expressions (3.46) and (3.47) by the terms for the
source powers given in the expressions (3.52) and (3.53), the total light powers
collected by the two detectors in case F can be deduced as follows:
PV
TTmRRmA
Ptrr
cdcttcdcr
r
rr ⋅
−+++
⋅=)1(
)(ττ
τδδ (3.55)
29
PV
RRmTTmA
Ptrt
cdcrrcdct
t
tt ⋅
−+++
⋅=)1(
)(ττ
τδδ (3.56)
The results for the double integrating sphere set-up are summarized in Table 3.2:
Table 3.2: Results for the double integrating sphere set-up, different cases
Case Light incident on Measurement Relative detected power
Reflection trr
r
r
rr
Vm
APP
ττδδ
−⋅⋅=1
1
E Sphere wall
Transmission tr
r
t
r
t
tt
Vm
APP
τττδδ
−⋅⋅=1
Reflection )1()(
trr
cdcttcdcr
r
rr
VTTmRRm
APP
τττδδ
−+++
=
F Sample
Transmission )1()(
trt
cdcrrcdct
t
tt
VRRmTTm
APP
τττδδ
−+++
=
Similar to chapter 3.1, the different equations for the total detected power can be
divided into factors. However, in contrast to expression (3.26), we now observe three
main factors (G, S and E):
ESGP
ESGPESGP
ESGP
FcasettFcaset
FcaserrFcaser
EcasettEcaset
EcaserrEcaser
⋅⋅=
⋅⋅=
⋅⋅=
⋅⋅=
,,
,,
,,
,,
δ
δ
δ
δ
(3.57)
As above, the first two factors (G and S) represent the sphere geometry and the light
sources for each case. Now, G depends on whether the detector is located in the
reflectance sphere (Gr) or in the transmittance sphere (Gt):
tt
tt
rr
rr VA
GVA
Gδδ
== ; (3.58)
If the incoming light is incident upon the sphere wall (case E), there is only one
diffuse light source in each sphere. However, for incoming light incident upon the
sample (case F), we have four diffuse light sources in each sphere, as the incident
light is divided into four parts (collimated reflected light, diffuse reflected light,
collimated transmitted light, diffuse transmitted light). This can be observed by
looking at the different factors S:
30
PRRmTTmS
PTTmRRmSPmS
PmS
cdcrrcdctFcaset
cdcttcdcrFcaser
rrEcaset
rEcaser
⋅+++=
⋅+++=
⋅=
⋅=
)]([)]([
,
,
,
,
τ
τ
τ (3.59)
The third, new factor (E=exchange) accounts for the multiple exchange of light
between the two spheres. It is identical for all four equations and originates from a
geometric series:
tr
ntrE
ττττ
−== ∑
∞−
11)(
1
1 (3.60)
3.3 Influence of baffles
In many applications of integrating spheres, it is not desired that light is reflected from
the sample directly upon the detector. This will introduce a false detector response
[19]. To achieve this, a baffle is placed within the sphere, between the sample and
the detector (see Fig. 3.5).
Fig. 3.5: Integrating sphere with baffle
The baffle is coated with the same material as the sphere and therefore has the
same reflectance factor m. If the baffle is assumed small, the light reflected from the
sample (case B) in the direction of the detector will be reflected by the baffle and
31
returned to the sphere. Therefore, the term rδ/A, caused by reflection from the
detector, will be neglected. This is reasonable, as this reflection is small compared to
the reflections from the sample and from the wall [8].
3.3.1 Reflectance sphere with baffle
For light incident upon the sphere wall (case A, chapter 3.1.1.1), the neglection of the
term rδ/A yields a new definition of F, compared to that in expression (3.7):
)( AsRmF d+= α (3.61)
From the first reflection (diffuse light source of the power P1, cf. expression (3.3)), the
detector collects the same amount of light as given in expression (3.4), but from the
second reflection, the collected power is reduced to
αδδδδ mP
AAsRP
AFP
AP d 1112, )( =⋅−⋅= (3.62)
due to the fact that the detector is collecting no light reflected by the sample.
As the light reflected from the sample is still present within the sphere, the total light
power in the sphere after two reflections is still given by
FPPAsRPmP d 1112 )( =+= α (3.63)
Thus, the power collected by the detector from the third reflection is
FPmA
mPA
P 123, αδαδδ == (3.64)
and in general the detected power from the nth reflection yields to
21,
−= nn FPm
AP αδδ (3.65)
The total detected power can be calculated along the lines of chapter 3.1.1.1:
PAsRm
AsRmA
PAsRm
AsRA
Pd
d
d
d ⋅+−−⋅
⋅=⋅+−
−⋅=
)]([1)](1[
)]([1)(1
1 αδ
αδ
δ (3.66)
32
For light incident upon the sample (case B, chapter 3.1.1.2), there are two sources of
diffuse light (expression (3.17)). The calculation of the detected power from the
diffuse source generated by the specularly reflected light (Pcr) can be conducted
similar as above:
crd
dcr P
AsRmAsR
AP ⋅
+−−
⋅=)]([1
)(1, α
δδ (3.67)
For the second diffuse source (Pcd) we have to take into account, that no light from
the first reflection is incident upon the detector. Thus, the term
cdcd PA
P ⋅=δ
δ ,1 (3.68)
has to be subtracted from the total power given in expression (3.66):
cdd
cdcdd
dcd P
AsRmm
AP
AP
AsRmAsR
AP ⋅
+−⋅=⋅−⋅
+−−
⋅=)]([1)]([1
)(1, α
αδδα
δδ (3.69)
The total collected power can be deduced by adding Pδ,cr and Pδ,cd. By using the
expressions (3.15) and (3.16), we obtain
PAsRm
mRAsRmRA
PPPd
cddccdcr ⋅
+−+−
⋅=+=)]([1
)](1[,, α
αδδδδ (3.70)
3.3.2 Transmittance sphere with baffle
If the light incident upon the sample is diffuse (case C, chapter 3.1.2.1), we have one
source of diffuse light in the sphere, with the power TdP (cf. expression (3.19). As this
source is located on the sample, this case corresponds to the light source of the
power Pcd, that has already been discussed in chapter 3.3.1. Thus, by replacing Pcd
by TdP in expression (3.69), the total detected power for this case is
PAsRm
mTA
Pd
d ⋅+−
⋅=)]([1 α
αδδ (3.71)
For collimated light incident upon the sample (case D, chapter 3.1.2.2), we have two
diffuse light sources. This case is identical to case B (chapter 3.3.1), except that the
powers of the light sources are now given by the expressions (3.22) and (3.23). By
utilizing these expressions in (3.67) and (3.69), we obtain the total detected power
from expression (3.70):
33
PAsRm
mTAsRmTA
Pd
cddc ⋅+−
+−⋅=
)]([1)](1[
ααδ
δ (3.72)
Consequently, we can summarize the influence of a baffle on the detected power:
If a diffuse light source is located on the sphere wall, we have to multiply the detected
power without baffle with a factor
)(1 AsRb dw −= (3.73)
to obtain the power detected with the use of a baffle. This accounts for the fact that
no light reflected by the sample can reach the detector directly.
Similar, if a diffuse light source is located on the sample, the factor to multiply the
detected power is given by
αmbs = (3.74)
In addition to the inhibited direct light exchange between sample and detector, this
factor also considers that there is no light at all from the first reflection reaching the
detector.
Table 3.3 summarizes the results for the single integrating sphere, including the
influence of a baffle.
Table 3.3: Results for the single integrating sphere set-up, baffle influence included
Case Measurement Light on sample Relative detected power Pδ/P
A Reflection diffuse )]([1
)](1[AsRm
AsRmA d
d
+−−⋅
⋅α
δ
B Reflection collimated )]([1
)](1[AsRm
mRAsRmRA d
cddc
+−+−
⋅α
αδ
C Transmission diffuse )]([1 AsRm
mTA d
d
+−⋅
ααδ
D Transmission collimated )]([1
)](1[AsRm
mTAsRmTA d
cddc
+−+−
⋅α
αδ
34
3.3.3 Double integrating sphere with baffle
By using the observations from the chapters 3.3.1 and 3.3.2, we can now easily
transform the equations for the double integrating sphere without baffle into
equations that account for the existance of a baffle. Out of the three factors G, S and
E, the factor E remains constant. The factor G changes only in the manner that the
term rδ/A is neglected, as announced above. For the adjustment of the different
factors S (expression (3.59)), we have to determine the location of every source part
for the different equations, and then multiply it with the appropriate factor (bwr, bwt, bsr,
bst):
ttstrrsrtdwtrdwr mbmbAsRbAsRb αα ==−=−= ;);(1);(1 (3.75)
For light incident upon the sphere wall (case E, chapter 3.2.1), we have one diffuse
light source in each sphere. In the reflectance sphere, the source is located on the
sphere wall. Therefore, we have to multiply the factor Sr,case E with bwr . In difference
to this, the source in the transmittance sphere is located on the sample, demanding a
multiplication of St,case E with the factor bst.
For light incident upon the sample (case F), there are four sources of diffuse light in
each sphere. In both spheres, three of these sources are located on the sample,
while one is located on the sphere wall.
In the reflectance sphere, the four sources are given by expression (3.59):
PTTmRPRmS cdcttcdcrFcaser ⋅+++⋅= )]([, τ (3.76)
The first source (mrRcP) is located on the sphere wall and therefore has to be
multiplied with bwr, while the three other sources (RcdP, τtmtTcP, τtTcdP) are located
on the sample and have to be multiplied with bsr.
The procedure for the transmittance sphere is similar. Here the four sources are
denoted by (expression (3.59)):
PRRmTPTmS cdcrrcdctFcaset ⋅+++⋅= )]([, τ (3.77)
35
While the first source (mtTcP) is located on the sphere wall, the other three sources
(TcdP, τrmrRcP, τrRcdP) are located on the sample. Multiplication with the appropriate
factor (bwt or bst) yields the detected power, including the influence of the baffle.
The results for the double integrating sphere set-up, including the influence of a
baffle, are summarized in Table 3.4. Case E represents the case for light incident
upon the sphere wall, case F represents the case for light incident upon the sample.
The factors τr and τt can be taken from expression (3.38).
Table 3.4: Results for the double integrating sphere set-up, baffle influence included
Case Measurement Relative detected power
Reflection trrdrr
rdr
r
rr
AsRmAsRm
APP
τταδδ
−⋅
+−−⋅
⋅=1
1)]([1
)](1[
E
Transmission tr
r
tdtt
ttr
t
tt
AsRmmm
APP
τττ
ααδδ
−⋅
+−⋅=
1)]([1
Reflection )1()]([1)()](1[
trrdrr
rrcdcttrrcdrdcr
r
rr
AsRmmTTmmRAsRRm
APP
ττααταδδ
−⋅+−+++−⋅
=
F
Transmission )1()]([1)()](1[
trtdtt
ttcdcrrttcdtdct
t
tt
AsRmmRRmmTAsRTm
APP
ττααταδδ
−⋅+−+++−⋅
=
3.4 Reference measurements with a single integrating sphere
3.4.1 Introduction of the sphere constants
We introduce the �sphere constants�
αδ
mm
Ab
−⋅=′11 (3.78)
αmAsb
−⋅=1
12 (3.79)
b1´ is written with a prime, because there will be a slightly different definition of b1
later, to relate the detected power to a reference power (see chapter 3.4.2). The
equations for the single integrating sphere set-up (Table 3.3) can then be written as
follows [9]:
36
PRb
AsRbP
d
dAcase
2
1, 1
)](1[−−′
=δ (3.80)
PRb
AsRRRbP
d
dccdBcase
2
1, 1
)])(1[(−
−+′=
αδ (3.81)
PRbTb
Pd
dCcase
2
1, 1−
′=
αδ (3.82)
PRb
TAsRTbP
d
cddcDcase
2
1, 1
))](1[(−
+−′=
αδ (3.83)
Along these lines, the sphere constants for the double integrating sphere set-up are
defined as:
tt
t
t
tt
rr
r
r
rr m
mA
bmm
Ab
αδ
αδ
−⋅=′
−⋅=′
1;
1 11 (3.84)
tttt
rrrr mA
sbmA
sbαα −
⋅=−
⋅=1
1;1
122 (3.85)
With these abbreviations, the equations for the double integrating sphere set-up
(Table 3.4) can be written as [9]:
PTbbRbRb
AsRRbbP
dtrdtdr
rddtrEcaser 2
2222
21, )1)(1(
)](1)[1(−−−
−−′=δ (3.86)
PTbbRbRb
TmbbPdtrdtdr
dtrrtEcaset 2
2222
21, )1)(1( −−−
′=
αδ (3.87)
PTbbRbRb
TmTTbRbRAsRRbPdtrdtdr
ctcddtrdtcdrrdcrFcaser 2
2222
221, )1)(1(
)}()1)()](1[{(−−−
++−+−′=
ααδ (3.88)
PTbbRbRb
RmRTbRbTAsRTbP
dtrdtdr
crcddrtdrcdttdctFcaset 2
2222
221, )1)(1(
)}()1)()](1[{(−−−
++−+−′=
ααδ (3.89)
3.4.2 Relation of the detected power to a reference power
To measure the reflectance and the transmittance of a sample, usually photodiodes
or photomultiplier tubes are used as detectors [9]. In theory, the voltage Vδ recorded
by a detector is proportional to the light power Pδ incident upon it:
δδ KPV = (3.90)
The constant of proportionality, K, depends on the detector characteristics.
37
In practice, due to the background signal V0, which occurs in all measurements and
originates from noise in the system and the presence of stray light, expression (3.90)
has to be expanded as follows:
0VKPV += δδ (3.91)
Thus, the detected power can be calculated from
)(10VV
KP −= δδ (3.92)
The constant K can be eliminated by using relative quantities, requiring additional
reference measurements conducted with a single integrating sphere set-up. For a
reference plate in the position of the sample, reflecting light diffusely with a reflection
factor Rd,ref, the detected power is given by the equation
),,,,( , mhAsRPP refdref Φ⋅= (3.93)
where P is the total power entering the sphere, and Φ is a function that is constant for
a given sphere geometry and a given reference plate, but depends on whether the
light incident upon the reference plate is collimated or diffuse [9]. Along the lines of
expression (3.92), Pref can be determined by measurements:
)(10,refrefref VV
KP −= (3.94)
By substituting (3.94) in (3.93), we obtain
)(110,refref VV
KP −⋅
Φ= (3.95)
Vref is the voltage recorded by the detector in the reflectance sphere with the
reference plate in position [25], Vref,0 is the background measurement, with the beam
blocked before entering the sphere. The two measurements are conducted as shown
in Fig. 3.6 [25]. With the expressions (3.92) and (3.95), the ratio of the detected
power with the sample in position Pδ to the incident power P can be written as
0,
0
refref VVVV
PP
−−
⋅Φ= δδ (3.96)
38
Fig. 3.6: Reference measurements
eliminating the constant K. As for the single integrating sphere set-up,
′1~ b
PPδ (3.97)
(see expressions (3.80)-(3.83)), we can define
′⋅Φ
= 111 bb (3.98)
Along these lines, the sphere constants b1r and b1t are defined, eliminating the need
to measure the total power P for the determination of the reflection and transmission
factors. With expression (3.98) and the definition
PP
PP
VVVVV
refrefref Φ==
−−
= δδδ
0,
0% (3.99)
the expressions (3.80)-(3.83) and (3.86)-(3.89) can be written as
d
dAcase Rb
AsRbV
2
1
1)](1[%
−−
= (3.100)
d
dccdBcase Rb
AsRRRbV
2
1
1)])(1[(%
−−+
=α (3.101)
d
dCcase Rb
TbV
2
1
1%
−=
α (3.102)
d
cddcDcase Rb
TAsRTbV
2
1
1))](1[(%
−+−
=α (3.103)
and
22222
21, )1)(1(
)](1)[1(%dtrdtdr
rddtrEcaser TbbRbRb
AsRRbbV−−−
−−= (3.104)
39
22222
21, )1)(1(
%dtrdtdr
dtrrtEcaset TbbRbRb
TmbbV−−−
=α (3.105)
22222
221, )1)(1(
)}()1)()](1[{(%dtrdtdr
ctcddtrdtcdrrdcrFcaser TbbRbRb
TmTTbRbRAsRRbV−−−
++−+−=
αα (3.106)
22222
221, )1)(1(
)}()1)()](1[{(%dtrdtdr
crcddrtdrcdttdctFcaset TbbRbRb
RmRTbRbTAsRTbV−−−
++−+−=
αα (3.107)
Because of expression (3.98), it is necessary that all measurements are conducted
with the same incident light power P.
3.4.3 Determination of the sphere constants
To determine the sphere constants b1r, b1t, b2r and b2t, we have to conduct several
measurements with the single integrating sphere set-up. The measurements have to
be made with both spheres individually to obtain the respective values for b1 and b2.
By looking at the expressions (3.99) and (3.100), we find that for Rd=0, the constant
b1 can be calculated as
0,
0,11
01 %refref
RAcase VVVV
Vbd −
−==
= (3.108)
by conducting two additional voltage measurements V1 and V1,0 [25]. V1 is the power
detected for light initially incident upon the sphere wall (case A) with no sample
present (Rd=0). V1,0 is the respective background signal, depending on the presence
of stray light and noise in the detection system [25]. According to Prahl [25], this
measurement has to be conducted with the beam blocked before it enters the
sphere. The measurements necessary to obtain V1 and V1,0 are shown in Fig. 3.7.
Fig. 3.7: Measurements necessary to determine b1
40
With these two measurements and the two reference measurements described in
chapter 3.4.2, b1 can be calculated from expression (3.108).
The constant b2 can be calculated by the use of measured values for s, A, α and m
(expression (3.79)) but it can also be obtained out of a direct measurement, by the
use of expression (3.100). In order to do this, we have to conduct further
measurements with a reference plate with a given diffuse reflection factor Rd,ref
(preferably, but not necessarily the same plate as in chapter 3.4.2). Again, the
incoming light beam has to be incident upon the sphere wall. The measurements
necessary to determine b2 are shown in Fig. 3.8 [25].
If we solve expression (3.100) for b2, we find for Rd=Rd,ref:
−⋅−⋅=
= refdd RRAcase
refd
refd VAsRb
Rb
,%
)](1[11 ,1
,2 (3.109)
Fig. 3.8: Measurements necessary to determine b2
If we neglect the factor [1-Rd,ref(s/A)] (which is reasonable as the resulting error is
small compared to other sources of uncertainty [9]), b2 can be determined by the
above measurements and expression (3.108):
−−
−⋅=
−−−−
−⋅=
−⋅=
=
=
0,22
0,11
,
0,
0,22
0,
0,11
,
0
,2 1111
%
%11
,
VVVV
RVVVVVVVV
RV
V
Rb
refd
refref
refref
refdRRAcase
RAcase
refdrefdd
d
(3.110)
41
By conducting the described measurements successively for both the reflection
sphere and the transmission sphere, the sphere constants b1r, b1t, b2r and b2t can be
determined.
To be able to use the expressions (3.100)-(3.107) for the evaluation of
measurements, we also have to determine the parameters αr and αt. The easiest
way to determine them is the calculation with the use of the expressions (3.1) and
(3.2) [25]. The sphere radius R and the port radii necessary to calculate δ, s and h
should be specified for commercial spheres, or can easily be determined by
measurements.
An example for the determination of all the necessary parameters can be found in
chapter 7.1.
42
4 Procedure to obtain the optical properties
4.1 Measurement set-up
The set-up that was used for the determination of the optical properties of tissue
samples is schematically illustrated in Fig. 4.1. As the entering light is incident
perpendicularly upon the sample, this set-up corresponds to the above described
case F (chapter 3.2.2).
Fig. 4.1: Measurement set-up
Overall, three detectors were used, yielding three voltage values Vr, Vt and Vc. Two
detectors are placed within the two integrating spheres, the third detector is situated
behind the spheres. It is used to measure the power of the light that is specularly
transmitted through the sample. As this part of light (as well as the specularly
reflected light) leaves the sphere without being reflected again, the terms including Rc
and Tc in the expressions (3.106) and (3.107) have to be omitted.
By the use of the reference measurements described in chapter 3.4, we can
transform the absolute values Vr and Vt into relative values Vr% and Vt%. For the
calculation of Vc%, another reference measurement has to be made. The measured
power
)(10,cccc VV
KPTP −== (4.1)
has to be related to the total power P, to directly obtain the value for Tc. Therefore,
the reference voltage Vc,ref is the voltage measured with the laser beam directly
43
incident upon the detector, without the use of a sample. Vc,ref,0 is the voltage
measured with the laser beam blocked, and should be equal to Vc,0, as the only
difference between these two measurements is the presence of the sample, which
should not affect the measured voltage when the laser beam is blocked. Fig. 4.2
shows the measurements necessary to obtain Vc,ref and Vc,ref,0.
Fig. 4.2: Reference measurements to calculate Vc%
4.2 Calculation of the optical properties out of the measurements
The equations to obtain the relative values Vr%, Vt% and Vc% out of the
measurements are the following:
0,,,
0,
0,
0,
0,
0, %;%;%refcrefc
ccc
refref
ttt
refref
rrr VV
VVV
VVVV
VVVVV
V−−
=−−
=−−
= (4.2)
As explained in chapter 3.4.2, the values Vr,0, Vt,0 and Vc,0 represent the background
signals corresponding to the respective measurements Vr, Vt and Vc.
The correlation between the measurement data and the reflection and transmission
factors is (cf. expressions (3.106) and (3.107))
44
22222
221
)1)(1(])1([%
dtrdtdr
cddtrdtcdrrr TbbRbRb
TTbRbRbV−−−
+−=
αα (4.3)
22222
221
)1)(1(])1([%
dtrdtdr
cddrtdrcdttt TbbRbRb
RTbRbTbV−−−
+−=
αα (4.4)
cc TV =% (4.5)
By the use of the expressions (4.3), (4.4) and (4.5), it is possible to calculate the 3
values Vr%, Vt% and Vc% for given reflection and transmission factors, as a solution
of the �double integrating sphere function�
%)%,%,(),,,,,(: ctrccddccddDIS VVVTTTRRRf → (4.6)
However, it is not possible to perform the calculation the other way around (to
calculate the 6 different reflection and transmission factors by using only these 3
equations).
If we look at the 3 levels of quantities described in chapter 1 (measurement values,
reflection/transmission factors and optical properties), the function fDIS yields a (one-
way) connection between the reflection/transmission factors and the measurement
values (see Fig. 4.3).
As announced in chapter 1, the final aim is to obtain the optical properties, given by
µa, µs and p(ν) (or a, τ and g, respectively). Therefore, we first have to find a
connection between the optical properties and the reflection/transmission factors.
This can be done by the use of an appropriate model of radiative transfer.
One of these models is used in the Adding Doubling Method (AD Method, see
chapter 5). The AD algorithm provides a numerical solution of the function
),,,,,(),,(: ccddccddAD TTTRRRgaf →τ (4.7)
(see Fig. 4.3). With the expressions (4.6) and (4.7), we can deduce that, by the use
of the AD Method and the formulas for the double integrating spheres (yielding the
expressions (4.3), (4.4) and (4.5)), the measurement results for Vr%, Vt% and Vc%
can be predicted, if the optical properties a, τ and g of the sample are given, by a
concatenation of the functions fAD and fDIS:
45
%)%,%,(),,(: ctrADDIS VVVgaff →τo (4.8)
In our case, we want to perform the calculation in the other direction. As the functions
fDIS and fAD are not implicitely invertible, we have to solve this problem numerically.
An iterative solution of the problem is given by the Inverse Adding Doubling Method
(IAD Method, see chapter 6). The function provided by the IAD Method can be
written as
),,(%)%,%,(:)( 1 gaVVVfff ctrADDISIAD τ→= −o (4.9)
(see Fig. 4.3). Of course, this algorithm assumes implicitly, that the inverse function
of (fAD°fDIS) exists, i.e. that (fAD°fDIS) is bijective. This means that for any set of
measurements (Vr%, Vt%, Vc%), a unique set of optical properties (a, τ, g) exists.
The correctness of this assumption is shown in chapter 6.1.
Fig. 4.3: Relations between the 3 levels of quantities
46
Thus, the procedure to obtain the optical properties of a sample by the use of a
double integrating sphere set-up and an IAD Program is as follows:
1. Conduct the necessary reference measurements to calibrate the measurement
set-up.
2. Determine the three measurement values Vr%, Vt% and Vc% for the sample.
3. Calculate the optical properties a, τ and g with the IAD program, by using the
measurement data as the input of the program.
4. If desired, calculate µa, µs and approximate p(ν) from the returned values a, τ and
g.
47
5 The Adding Doubling Method
The Adding Doubling (AD) Method was introduced by van de Hulst [18] as a one-
dimensional numerical solution of the radiative transport equation. For given optical
properties (albedo a, optical thickness τ and anisotropy coefficient g), the reflection
factors (Rd, Rcd, Rd) and the transmission factors (Td, Tcd, Tc) can be calculated by
the use of this algorithm.
The AD algorithm consists of several steps:
- Before the actual �Adding Doubling equations� can be used, we have to calculate
the reflection function R(µ,µ´) and the transmission function T(µ,µ´) (see chapter
5.1) of a very thin slab. As the functions R and T are determined numerically, we
can obtain the respective function values only for certain angles (µ,µ´)=(µj,µk)
(j,k=1�n), so that we can write the functions as matrices Rjk and Tjk. Thus, we
first have to determine these �quadrature angles� (see chapter 5.2).
- To derive the reflection function and the transmission function from the radiative
transport equation, we need a (numerical) solution of the �redistribution function�,
which appears in the radiative transport equation. For the previously determined
quadrature angles, this solution can be found by the use of the �δ-M method� (see
chapter 5.3).
- Now the matrices Rjk and Tjk for the thin slab (�initial matrices�) can be determined
from the radiative transport equation, by the use of an initialization algorithm (see
chapter 5.4).
- The reflectance and the transmittance of a slab with the double thickness can be
derived from the results of the thin slab by using the Adding Doubling equations
(see chapter 5.5). This �doubling method� is repeated, until the desired slab
thickness has been reached.
- In general, a sample in an integrating sphere set-up is sandwiched by two glass
plates. These glass plates influence the reflection and transmission
48
measurements. This influence is included in the AD algorithm by the possibility to
add boundary layers with the �adding method� (see chapter 5.6). The adding
method is a generalization of the doubling method, for the combination of non-
identical slabs.
- After the final reflectance matrix and transmittance matrix of the sample have
been found, the different reflection and transmission factors Rd, Rcd, Rd, Td, Tcd
and Tc are derived from Fresnel´s equations and Beer´s Law (see chapter 5.7).
To show how all these steps are performed for numerically given values (a, τ g), an
example calculation is performed in chapter 5.8.
5.1 The reflection function and the transmission function
Before the actual process of adding and doubling can begin, we have to generate a
thin starting layer and calculate the reflection function R and the transmission
function T of this layer. The definition of the reflection function and the transmission
function follows the one given by van de Hulst [12, 18]:
The reflection function R(ϕ,ϕ´) is defined as the radiance conically reflected by a slab
in the direction denoted by the angle ϕ for light that is conically incident from the
direction denoted by the angle ϕ´ (see Fig. 5.1). The reflection is normalized to an
incident diffuse flux π (created by an isotropic diffuse source F0(r)=1, cf. chapter 10.5,
expression (10.7)). A similar definition is applied for the transmission function T(ϕ,ϕ´).
µ and µ´ are the cosines of the angles ϕ and ϕ´, respectively:
)cos();cos( ϕµϕµ ′=′= (5.1)
In the following, the reflection function and the transmission function will be written in
terms of the cosines µ and µ´:
),();,( µµµµ ′=′= TTRR (5.2)
49
Fig. 5.1: Light incident upon a slab at angle ϕ´, reflected and transmitted at angle ϕ
For an azimuth-independent incident light intensity L(µ´), the reflected intensity
distribution Lr(µ) and the transmitted intensity distribution Lt(µ) can be calculated,
according to van de Hulst [18], by the following integrals (0≤µ´≤1):
∫ ′′⋅′⋅′=1
0
)](2),([)( µµµµµµ dLRLr (5.3)
∫ ′′⋅′⋅′=1
0
)](2),([)( µµµµµµ dLTLt (5.4)
The factor 2µ´ is included in order to follow the definition given by van de Hulst.
As the Adding Doubling algorithm provides only a numerical solution, the function
values for R and T will be calculated only for certain angles µi (i=1�n). These n
angles (actually they are the cosines of the respective angles) are called �quadrature
angles� (see chapter 5.2). A fast and reasonably accurate calculation can already be
conducted by choosing only n=4 angles [12]; higher values improve the accuracy at
the expense of the calculation time.
With this simplification, the function values of R and T belonging to the quadrature
angles can be written in form of nxn-matrices:
),();,( kjjk
kjjk TTRR µµµµ == (5.5)
50
The intensities L in the directions of the quadrature angles may be written as vectors
with n components:
)();();( jtj
tjrj
rkk LLLLLL µµµ === (5.6)
In the expressions (5.5) and (5.6), the angle µk represents the direction of the
incident light (µ´=µk), while the angle µj belongs to the reflected or transmitted light
(µ=µj). Thus, the incoming radiance L has the index k, while the reflected radiance Lr
and the transmitted radiance Lt have the index j.
The integrals in the expressions (5.3) and (5.4) can be approximated by the following
sums:
∑ ⋅⋅⋅=j
kkk
jkjr LwRL µ2 (5.7)
∑ ⋅⋅⋅=j
kkk
jkjt LwTL µ2 (5.8)
with the quadrature angles µk and their corresponding weights wk (see chapter 5.2).
To simplify the notation, we define the �matrix star multiplication� A∗B as
∑ ⋅⋅⋅=∗j
klkk
jk BwABA µ2 (5.9)
It was shown by Grant and Hunt [4] that all the following operations including star
multiplications are mathematically valid. This becomes obvious when the Star
multiplication is considered as a normal matrix multiplication including a diagonal
matrix
jkkkjk wc δµ ⋅⋅= 2 (5.10)
with the Kronecker-delta symbol δjk. By the use of expression (5.10), we can write the
star multiplication as [10]
AcBBA =∗ (5.11)
By applying the star multiplication notation on the expressions (5.7) and (5.8), we find
51
kjkjr LRL ∗= (5.12)
kjkjt LTL ∗= (5.13)
The first step of the AD Method therefore is to find appropriate quadrature angles µi
and their corresponding weights wi.
5.2 Quadrature
In our context, the idea behind quadrature is to approximate the integral function
∫ ′′⋅′′′⋅′=′′1
0
]2),(),([),( µµµµµµµµ dBAC (5.14)
at certain values µj and µl (j,l=1�n) by the sum
∑ ⋅⋅⋅=j
klkk
jkjl BwAC µ2 (5.15)
Ajk and Bkl are matrices filled with the function values of A(µj,µk) and B(µk,µl). The
values µi (i=1�n) are called quadrature points, wi is the weight corresponding to the
quadrature point µi.
The range of the integral (0 to 1) originates from the fact that we have to integrate
over angles from 90° to 0°. In terms of cosines, this leads to a range of 0 to 1. Care
has to be taken, if the index of refraction of the slab ns is greater than the one of the
surrounding medium na. In this case, we have a critical angle ϕc with the cosine µc:
=
s
ac n
narcsinϕ (5.16)
For angles exceeding the critical angle, no light can leave the slab, as it is totally
reflected back into the slab. To maintain integration accuracy, the integral in
expression (5.14) has to be divided into two parts [12]:
∫∫
∫
′′⋅′′′⋅′+′′⋅′′′⋅′=
′′⋅′′′⋅′
1
0
1
0
]2),(),([]2),(),([
]2),(),([
c
c
dBAdBA
dBA
µ
µ
µµµµµµµµµµµµ
µµµµµµ
(5.17)
52
In order to obtain appropriate quadrature angles µi and weights wi (j=1�n), different
quadrature methods can be applied. The range of almost every integration in the
Adding Doubling method is from zero to one. Three different quadrature methods
spanning this range are Gaussian quadrature, Lobatto quadrature and Radau
quadrature [12]. These methods exhibit almost equal accuracies, but their choice is
determined by the boundary conditions [12]:
- In Gaussian quadrature, neither endpoint 0 or 1 is included in the quadrature
points.
- In Lobatto quadrature, both endpoints 0 and 1 are included in the quadrature
points.
- In Radau quadrature, one quadrature point may be specified. Usually, the
endpoint 1 is chosen, to avoid extrapolation or interpolation errors.
In the AD program, Gaussian quadrature is used for the first half of the integral
(range from 0 to µc), to avoid calculations at the endpoints (in particular, to avoid µ=0
to be used as a quadrature angle). For the range from µc to 1, Radau quadrature is
used, in order to specify µ=1 as a quadrature point. Both intervals (0 to µc and µc to
1) get the same number of quadrature angles. If there is no critical angle, Radau
quadrature is used for the whole range (0 to 1) [12].
5.2.1 Radau quadrature
Radau quadrature is a method for the approximation of the integral over an arbitrary
function f(x), for an integration range of -1 to 1. The general formula is [22, 23]
)1()()(1
1
1
1
−+≈ ∑∫−
=−
fwxfwdxxf n
n
iii (5.18)
The free abscissas xi (i=1�n-1) are the roots of [12]
0)(1)()( 111 =′−+= −−− xP
nxxPx nnnφ (5.19)
By using the following recurrence relation of the Legendre polynomials [6]
53
)()()()1( 112 xnPxnxPxPx nnn −=′− −− (5.20)
expression (5.19) can be written without derivatives:
01
)()(1
)()(1)()( 12
111 =
++
=−−−
+= −−−− x
xPxPx
xnPxnxPn
xxPx nnnnnnφ (5.21)
The roots of φn-1 can be found by Newton´s method [23]. In order to use this method,
we have to find the derivative
211
1 )1())()(())()()(1()(
xxPxPxPxPxx nnnn
n ++−′+′+
=′ −−−φ (5.22)
By using expression (5.20), we can eliminate the derivative terms in expression
(5.22), to find
211
1 )1)(1()()1()()12()()1()(
xxxPnxPnxnxxPxnxx nnn
n +−+−−−++−+
=′ +−−φ (5.23)
The term Pn+1(x) can be eliminated by using the relation [23]
)()()12()()1( 11 xnPxxPnxPn nnn −+ −+=+ (5.24)
Finally, we obtain the following equation for the derivative:
21
1 )1)(1()()1()()1()(
xxxPnxnxxPnxnxx nn
n +−−−+−+−++
=′ −−φ (5.25)
To find the roots of expression (5.21) by Newton´s method, we therefore only have to
determine Pn-1(x) and Pn(x). Problems emerge only for the cases x=-1 and x=1, as
divisions by zero occur (see expressions (5.21) and (5.25)). To avoid this, we have to
look at these cases separately.
If x=1, we can recall the fact that for all Legendre polynomials Pi(x), we have Pi(1)=1
[23]. Thus, from expression (5.21) immediately follows
1)1(1 =−nφ (5.26)
To obtain the value for the first derivative of φn-1, we look at expression (2.22), the
differential equation met by all Legendre polynomials. For x=1, we find
54
2)1()1(
2)1()1( +
=+
=′ nnPnnP nn (5.27)
Substitution in expression (5.22) yields
21
4
)11()2
)1(2
)1((2)1(
2
1−
=+−
++
−
=′−n
nnnn
nφ (5.28)
To obtain the values for x=-1, we have to use another fact about Legendre
polynomials [23]:
)()1()( xPxP nn
n −=− (5.29)
With Pn(x)=1, this leads to Pn(-1)=(-1)n. Further, if we substitute x=-1 in expression
(2.22), we find
2)1()1()1(
2)1()1( 1 +
−=−+
−=−′ + nnPnnP nnn (5.30)
By substitution in expression (5.19), we obtain
nnnn
nnnn ⋅−=
−−
−+−=− −−
−11
1 )1(2
)1()1(2)1()1(φ (5.31)
The first derivative was determined with the help of Mathematica [23]:
4)1()1()1(
21
1nnn
n−
−=−′ −−φ (5.32)
Now the function φn-1 and its derivative can be calculated for the whole interval [-1, 1],
by using the expressions (5.21), (5.25), (5.26), (5.28), (5.31), (5.32). The required
Legendre polynomials can be calculated iteratively from P0(x)=1, P1(x)=x and the
recurrence relation [6]
1)()()12()( 1
1 +−+
= −+ k
xkPxxPkxP kkk (5.33)
Before we can find the actual roots of φn-1, we have to �bracket� then, i.e. to find a
small intervals in which the roots are situated (see chapter 10.2). In the AD program,
this is done with the help of a subroutine zbrak [23].
55
This routine works as follows [27]: A given interval [x1,x2] (for bracketing the roots in
our case, we have x1=-1, x2=1) is divided into n equally spaced smaller subintervals.
For each of these intervals, the routine checks if there is a sign change (+ to � or � to
+) from one end of the interval to the other, which indicates the presence of a root in
this interval (the root is �bracketed� in this interval). zbrak returns all those intervals
bracketing a root.
The AD program then checks how many roots are bracketed. If less than all n-1 roots
are bracketed, the number of subintervals is halved, yielding 2n intervals, and zbrak
is called again. This algorithm is repeated, until all roots are bracketed.
Now all n-1 roots can be determined iteratively by using Newton´s formula
)()(
,1
,1,1,
kin
kinkiki x
xxx
−
−+ ′
−=φφ
(5.34)
with a starting value xi,0 within the corresponding bracket interval. The quadrature
points are sorted such that their value is continuously decreasing from x1 to xn-1. We
now have n quadrature points, including the previously specified point xn=-1 (see
expression (5.18)). To get the quadrature angles µi from the quadrature points xi, we
have to scale them to the appropriate interval [µc,1] (if the desired range is from 0 to
1, simply set µc=0 in the following equation) [12, 23]:
2)1(1 icc
ixµµ
µ−−+
= (5.35)
Note that the first quadrature angle µ1 is the angle with the smallest (cosine) value,
actually denoting the highest quadrature angle. The corresponding weights wi can be
calculated from xi and µc by the following equation:
)()1(21
1 ini
ci xPx
w−′−
−=
µ (5.36)
The derivative of the Legendre polynomial in expression (5.36) can be obtained
iteratively, from P´0(x)=0, P´1(x)=1 and the recurrence relation [6]
1)()()12()( 1
1 +−+
=′ −+ k
xkPxxPkxP kkk (5.37)
56
5.2.2 Gaussian quadrature
Gaussian quadrature (with a weighting function W(x)=1) is also defined for an
integration range of -1 to 1 [20]:
∑∫=−
≈n
iii xfwdxxf
1
1
1
)()( (5.38)
In this case, the free abscissas xi (i=1�n) are the roots of a simple Legendre
polynomial [12]:
0)( =xPn (5.39)
The subroutine gauleg used to find these roots is already provided in C. As in chapter
5.4.1, the quadrature points decrease in their value from x1 to xn. As the integration
range has to be changed to the interval [0, µc], we have to calculate the quadrature
angles µi from the quadrature points xi by using the equation
)1(2 i
ci x−=
µµ (5.40)
Again, the first quadrature angle µ1 denotes the highest quadrature angle (with the
smallest cosine value). The corresponding weights wi for the Gaussian quadrature
can be calculated from xi and µi as follows:
[ ]22 )()1( ini
ci xPx
w′−
=µ (5.41)
with the derivative of the Legendre polynomial determined by expression (5.37).
5.3 The Redistribution function
The phase function p(ν) denotes the amount of light that is scattered at an angle with
the cosine ν from the incident direction, during a single scattering event (see chapter
2.2). In difference to this, the redistribution function h(µ,µ´) determines the fraction of
light from a cone of an angle with the cosine µ´, that is scattered into a cone of an
angle with the cosine µ (cf. Fig. 5.1). This definition is similar to the definition of the
reflection function and the transmission function (see chapter 5.1). But while the
reflection function and the transmission function are defined for the whole slab and
57
include absorption and multiple scattering, the redistribution function is defined only
for a single scattering event.
h(µ,µ´) is calculated by averaging the phase function p(ν) over all possible azimuthal
angles φ (0≤φ<2π) for fixed values of µ and µ´. According to the definition given by
[12], this average is multiplied by 4π, in order to normalize h to a constant value h=1
for isotropic scattering.
( ) ( )∫∫ =⋅=′ππ
φνφνπ
πµµ2
0
2
0
2214),( dpdph (5.42)
The interdependence between the scattering angle θ and the angles ϕ and ϕ´
(defined similar to Fig. 5.1) is derived in chapter 10.1. We find the following equation
(in terms of the respective cosines):
)cos(11 22 φµµµµν ′−−+′= (5.43)
If we substitute expression (5.43) into expression (5.42), we find the following
expression for the redistribution function:
( )∫ ′−−+′=′π
φφµµµµµµ2
0
22 )cos(112),( dph (5.44)
The cosines µ and µ´ may also be negative, to account for light travelling in the
opposite direction.
For the case of an isotropic phase function (see expression (2.9)), the redistribution
function is a constant:
14
2),(2
0
==′ ∫π
πφµµ dh (5.45)
If the phase function p(ν) is approximated by a Henyey-Greenstein function (see
expression (2.10)), the redistribution function can be expressed in terms of a
complete elliptic integral of the second kind E(x) [12]:
+⋅
+−−
⋅=′γαγ
γαγαπµµ 2
)(12),(
2
Egh (5.46)
58
with the anisotropy coefficient g and
222 112;21 µµγµµα ′−−=′−+= ggg (5.47)
However, in the Adding Doubling program, another way is used to calculate the
redistribution function: the δ-M method [17]. It works especially well for highly
anisotropic phase functions [17]. M=n is the number of quadrature points. In the δ-M
method, the phase function is substituted by an expression consisting of a Dirac delta
function (cf. expression (2.18)) and M-1 Legendre polynomials Pk(ν) (cf. expressions
(2.22) and (2.23)). This substitution function p*(ν) is defined as [17]
+−+−= ∑
−
=
∗∗1
0)()12()1()1(2
41)(
M
kkk
MM Pkggp νχνδπ
ν (5.48)
where the coefficients are calculated by using the moments of the actual phase
function p(ν) [17]:
∫−
∗ =−−
=1
1
)()(2;1
νννπχχ
χ dPpgg
kkM
Mk
k (5.49)
If we use a Henyey-Greenstein approximation (expression (2.10)) for the phase
function, we find [12, 17]:
kk g=χ (5.50)
yielding
M
Mk
k ggg
−−
=∗
1χ (5.51)
The redistribution function can then be written as [12]
∑−
=
∗ ′+=′1
0
* )()()12(),(M
kkkk PPkh µµχµµ (5.52)
If we use the redistibution function only for the quadrature angles µj and µk (j,k=1�n)
and the corresponding negative angles �µj and �µk, we can write it as a (2x2)-matrix
of (nxn)-matrices:
59
= ++−+
+−−−
jkjk
jkjkjk hh
hhh (5.53)
with
),();,();,();,( kjjkkjjkkjjkkjjk hhhhhhhh µµµµµµµµ ++=−+=+−=−−= ++−++−−− (5.54)
Owing to the substitution of the phase function (p(ν) to p*(ν)), the albedo a and the
optical thickness τ have to be substituted by values a* and τ*, too [12, 17]. This has
to be done because in expression (5.48), a part (gM) of the scattered light is moved to
the unscattered component, denoted by the Delta function (the scattering coefficient
is reduced to µs*=(1-gM)µs, cf. chapter 10.5, expression (10.11)).
With the definition (2.31) for the albedo, we find the following expression for a*:
s
aM
M
asM
sM
as
s
g
gg
ga
µµµµ
µµµ
µ
+−
−=
+−−
=+
=∗
∗∗
1
1)1(
)1( (5.55)
Solving expression (2.31) for the absorption coefficient yields
( )11 −= −asa µµ (5.56)
By substituting expression (5.56) in expression (5.55), we find
M
M
M
M
agga
agga
−−
=−+−
−=
−∗
11
111
1 (5.57)
Expression (2.32) yields the following equation for τ*:
Msas
Mas dggdd µτµµµµτ −=+−⋅=+⋅= ∗∗ ))1(()( (5.58)
We can solve expression (2.31) for µs:
da
s τµ = (5.59)
Substitution of expression (5.59) in expression (5.58) leads to
ττττ )1( MM agdgda
−=−=∗ (5.60)
60
5.4 Layer initialization
The radiative transport equation (cf. chapter 10.5, expression (10.5)) is given by
∫ ′′+
=+∇⋅+ π
ωνµµ
µµµ 4
),()(),(),()(1 dLpLLas
s
as
srsrsrs (5.61)
Note that the fraction in front of the integral can be replaced by the albedo a (see
expression (2.31). If we assume the radiance L(r,s) to be independent of the
azimuthal angle φ and the radial position r, depending only on the position z (normal
to the slab surface) and the cosine µ of the angle ϕ, expression (5.61) can be written
as
∫ ∫−=′ =′
′′′=+∇⋅+
1
1
2
0
),()(),(),()(1
µ
π
φ
µφµνµµµµ
ddzLpazLzLas
s (5.62)
In cylindric coordinates, we can further write
dzzdL
dzzdL
zLdd
zLdzd
zLdrd
zL ),(
0
),(
0
cossin
),(
),(
),(
),()( µµµ
φϕϕ
µφ
µ
µ
µ =
⋅
=
⋅=∇⋅ ss (5.63)
By substituting expression (5.63) in expression (5.62), we obtain
∫ ∫−=′ =′
′′′=++
1
1
2
0
),()(),(),(1
µ
π
φ
µφµνµµµµµ
ddzLpazLdzzdL
as
(5.64)
The radiance can also be written in terms of the optical depth τ, as defined in
expression (2.32). If we consider the position z (normal to the surface) as the current
physical depth d, we find
as
dzµµ
ττ+
==)( (5.65)
yielding L(z(τ),µ)=L(τ,µ). Written in terms of differentials, expression (5.65) leads to
as
dzddddz
µµτ
ττ
+=
⋅= (5.66)
Substitution of expression (5.66) in expression (5.64) yields
61
∫ ∫−=′ =′
′′′=+1
1
2
0
),()(),(),(
µ
π
φ
µφµτνµττµτµ ddLpaL
ddL (5.67)
As the radiance L(τ,µ´), is assumed to be independent of φ´, we can pull it out of the
inner integral to find
∫ ∫−=′ =′
′
′′=+
1
1
2
0
)(),(),(),(
µ
π
φ
µφνµτµττµτµ ddpLaL
ddL (5.68)
With the definition of the redistribution function h(µ,µ´) (expression (5.42)), we can
finally write
∫−=′
′′′=+1
1
),(),(2
),(),(
µ
µµτµµµττµτµ dLhaL
ddL (5.69)
to find the time-independent, one-dimensional, azimuthally-averaged, radiative
transport equation [12].
As we are going to calculate the redistribution function with the δ-M method, we have
to substitute the albedo a and the optical thickness τ in expression (5.69) by a* and τ*
(see chapter 5.3, expressions (5.57) and (5.60), note that dτ=dτ*). If the integral is
approximated by the use of a quadrature with the quadrature angles µi and the
corresponding weights wi (i=1�M), the equation can be written in a discrete form, by
distinguishing between positive and negative quadrature angles and using
expression (5.54) [12]:
[ ]∑=
∗−+∗++∗
∗∗
∗
−++=++∂
+∂+
M
kkjkkjkkj
jj LhLhwaL
L
1),(),(
2),(
),(µτµτµτ
τµτ
µ (5.70)
[ ]∑=
∗−+∗++∗
∗∗
∗
++−=++∂
−∂−
M
kkjkkjkkj
jj LhLhwaL
L
1),(),(
2),(
),(µτµτµτ
τµτ
µ (5.71)
By defining the redistribution matrices
),1(),();,( nkjhh kjkj ≤≤−+=++= −+++ µµµµ hh (5.72)
the radiance vectors
)1(),()();,()( njLL jj ≤≤−== ∗∗−∗∗+ µττµττ LL (5.73)
62
and the diagonal matrices
[ ] [ ] ),1(; nkjw jkjjkj ≤≤== δδµ wM (5.74)
with the Kronecker-delta symbol δjk, the expressions (5.70) and (5.71) may be written
in the following matrix-vector forms [16]:
[ ])()(2
)()( ∗−−+∗+++∗
∗+∗
∗+
+=+∂
∂+ τττ
ττ wLhwLhLLM a (5.75)
[ ])()(2
)()( ∗+−+∗−++∗
∗−∗
∗−
+=+∂
∂− τττ
ττ wLhwLhLLM a (5.76)
By integrating the expressions (5.75) and (5.76) over a thin layer from τ0* to τ1*, we
get
[ ] [ ]1/01/01/001 2)()( −−++++∗
∗∗+∗+∗+ +∆=∆+−+ wLhwLhLLLM ττττ a (5.77)
[ ] [ ]1/01/01/001 2)()( +−+−++∗
∗∗−∗−∗− +∆=∆+−− wLhwLhLLLM ττττ a (5.78)
with ∆τ*=τ1*-τ0* and the �central intensities� L+0/1 and L-
0/1 given by
∫∗
∗
±∗
±
∆=
1
0
)(11/0
τ
τ
τττ
dLL (5.79)
In the AD program, there are two different types of initialization implemented, their
choice depending on the optical thickness τthinnest of the starting layer. The choice of
τthinnest depends on the smallest quadrature angle µ1. In the AD program, it is
determined by repeatedly halving τ, until a value τthinnest≤µ1 is achieved. For τ=∞, the
starting thickness is set to τthinnest=0.5µ1 [23].
The two initialization routines differ in choosing different approximations for the
central intensities L±0/1. The central intensities can be approximated by assuming that
they are related linearly to the boundary intensities L±(τ0*) and L±(τ1*) [16]. For very
thin starting layers (τthinnest<10-4), the infinitesimal generator initialization (see chapter
5.4.1) is used, otherwise the more complicated, but also more accurate diamond
initialization (see chapter 5.4.2) is used.
63
5.4.1 Infinitesimal generator initialization
If the starting matrices R and T are calculated by the infinitesimal generator
initialization, the following approximations for the integral (5.79) are used:
)();( 11/001/0∗−−∗++ == ττ LLLL (5.80)
The average downward irradiance is approximated by the downward irradiance at the
top, while the average upward irradiance is approximated by the upward irradiance at
the bottom of the thin layer.
By substituting expression (5.80) into the expressions (5.77) and (5.78), we obtain
[ ] [ ])()(2
)()()( 10001∗−−+∗+++∗
∗∗∗+∗+∗+ +∆=∆+−+ τττττττ wLhwLhLLLM a (5.81)
[ ] [ ])()(2
)()()( 01101∗+−+∗−++∗
∗∗∗−∗−∗− +∆=∆+−− τττττττ wLhwLhLLLM a (5.82)
Solving these equations for L+(τ1*) and L-(τ0*), respectively, yields
)()2
()(2
)( 01
11
1∗+++
∗−∗∗−−+−∗
∗∗+
−∆−+
∆= τττττ LwhIMILwhML aa (5.83)
)()2
()(2
)( 11
01
0∗−++
∗−∗∗+−+−∗
∗∗−
−∆−+
∆= τττττ LwhIMILwhML aa (5.84)
with the identity matrix I. The radiance L+(τ1*) leaving the slab at τ1* can therefore be
calculated as the sum of a reflected part of the radiance L-(τ1*) entering the slab at
τ1*, and a transmitted part of the radiance L+(τ0*) entering the slab at τ0* (see
expression (5.83)). Analogous conclusions can be drawn from expression (5.84) for
the radiance L-(τ0*) leaving the slab at τ0*. The expressions (5.83) and (5.84) can
therefore be written as
)(~)(~)();(~)(~)( 100011∗−∗+∗−∗+∗−∗+ +=+= ττττττ LTLRLLTLRL (5.85)
with the reflection and transmission matrices
)2
(~;2
~ 11 whIMITwhMR ++∗
−∗−+−∗∗
−∆−=∆=aa ττ (5.86)
64
By using the definitions of the matrices M and w (see expression (5.74)), we can
write the initial reflection and transmission matrices for the infinitesimal generator
initialization in index notation:
∆−+
∆⋅=
∆⋅=
∗−+
∗∗++
∗∗
jjkkjk
jjkkjk
jjk whaTwhaR
µτδ
µτ
µτ 1
2~;
2~ (5.87)
In order to follow the definition of the reflection and transmission function introduced
by van de Hulst [18] (see expressions (5.7) and (5.8)), we have to divide the
reflection and the transmission matrix by the factor 2µkwk, to find the matrices R and
T as defined in chapter 5.1 [23]:
∆−+
∆⋅=
∆⋅=
∗−+
∗∗++
∗∗
jjj
jkjk
kjjkjk
kjjk w
haThaRµτ
µδ
µµτ
µµτ 1
22;
2 (5.88)
Note that because of the definition of the Kronecker-delta symbol, we have
δjk/µkwk=δjk/µjwj.
5.4.2 Diamond initialization
For the diamond initialization, we assume that the integral (5.79) can be
approximated by an average of the radiances at the top and the bottom of the thin
layer:
2)()(;
2)()( 01
1/010
1/0
∗−∗−−
∗+∗++ +
=+
=ττττ LLLLLL (5.89)
Substitution of expression (5.89) in (5.77) and (5.78) yields:
[ ]
++
+∆=
∆+
+−+
∗−∗−−+
∗+∗+++∗
∗
∗∗+∗+
∗+∗+
2)()(
2)()(
2
2)()()()(
0110
1001
τττττ
τττ
ττ
LLwhLLwh
LLLLM
a (5.90)
[ ]
++
+∆=
∆+
+−−
∗+∗+−+
∗−∗−++∗
∗
∗∗−∗−
∗−∗−
2)()(
2)()(
2
2)()()()(
1001
0101
τττττ
τττ
ττ
LLwhLLwh
LLLLM
a (5.91)
If we solve expression (5.90) for L+(τ1*), we find:
65
[ ])()(4
)()2
(2
)()2
(2
101
01
11
∗−∗−−+−∗∗
∗+++∗
−∗
∗+++∗
−∗
+
∆⋅+
−∆
−=
−∆
+
τττττ
ττ
LLwhMLwhIMI
LwhIMI
aa
a
(5.92)
With the abbreviations
−∆
=∆⋅
= ++∗
−∗
−+−∗∗
)2
(2
~;4
~ 11 whIMτwhMρ aa ττ (5.93)
expression (5.92) can be written as
[ ] [ ] [ ])()(~)(~)(~1001∗−∗−∗+∗+ ++−=+ ττττ LLρLτILτI (5.94)
An analogous equation can be obtained by solving expression (5.91) for L-(τ0*):
[ ] [ ] [ ])()(~)(~)(~1010∗+∗+∗−∗− ++−=+ ττττ LLρLτILτI (5.95)
Substituting expression (5.95) in expression (5.94) yields
[ ][ ] [ ] )())~()~((~)(~)~(~~
)(~)~(~~
11
01
11
∗−−∗+−
∗+−
−+++++−=
+−+
ττ
τ
LτIτIIρLρτIρτI
LρτIρτI (5.96)
With the abbreviation
[ ] 11 ~)~(~~ −−+−+= ρτIρτIΓ (5.97)
and the simplifications
[ ] 111 )~(2)~()~()~()~()~( −−− +=−++⋅+=−++ τIτIτIτIτIτII (5.98)
[ ] 111 2~)~(~~2~)~(~~ −−− −=+−+−=++− ΓIρτIρτIIρτIρτI (5.99)
we find
[ ] [ ] )(2)()~(~2)( 011
1∗+∗−−∗+ −++= τττ LIΓLτIρΓL (5.100)
By substituting expression (5.94) in expression (5.95), an analogous equation can be
found for L-(τ0*):
66
[ ] [ ] )(2)()~(~2)( 101
0∗−∗+−∗− −++= τττ LIΓLτIρΓL (5.101)
Thus, the initial reflection and transmission matrices for the diamond initialization (cf.
expression (5.86)) are given by
IΓTτIρΓR −=+= − 2~;)~(~2~ 1 (5.102)
with the respective abbreviations defined as in the expressions (5.93) and (5.97). As
in chapter 5.4.1, the reflection and the transmission matrix from expression (5.102)
have to be divided by the factor 2µkwk, to get the matrices R and T as defined by van
de Hulst [18]:
jkkk
jkjkkk
jk Tw
TRw
R ~1;~1µµ
== (5.103)
5.5 Derivation of the Adding Doubling equations
To obtain the Adding Doubling equations, we look at two similar, adjacent slabs (see
Fig. 5.2). The radiances travelling from left to right across the boundaries 0, 1 and 2
are denoted as L0+(µ), L1
+(µ) and L2+(µ), respectively, while L0
-(µ), L1-(µ) and L2
-(µ)
denote the radiances travelling from right to left [12].
Fig. 5.2: Nomenclature for the derivation of the Adding Doubling equations
67
Rmn and Tmn are defined as the reflection and transmission operators for light incident
upon a boundary m and moving towards a boundary n (see chapter 5.1). They can
be calculated for certain quadrature angles (see chapter 5.2), as shown in the
chapters 5.3 and 5.4. For homogeneous tissues, we have Rmn=Rnm and Tmn=Tnm.
For the left slab, we obtain the exiting radiance L1+ as the sum of the transmitted part
of the incident radiance L0+ and the reflected part of the incident radiance L1
-.
Similarly, L0- is the sum of the reflected part of L0
+ and the transmitted part of L1-
[10, 12]. With the use of the diagonal matrix c from expression (5.11), we can write
the expressions (5.12) and (5.13) for this case as
−++ += 110
001
1 cLRcLTL (5.104)
−+− += 110
001
0 cLTcLRL (5.105)
Analogous equations can be obtained for the right slab:
−++ += 221
112
2 cLRcLTL (5.106)
−+− += 221
112
1 cLTcLRL (5.107)
If we combine the two layers (01) and (12) to one layer (02), we find the equations
−++ += 220
002
2 cLRcLTL (5.108)
−+− += 220
002
0 cLTcLRL (5.109)
If the reflection and transmission operators for the single layers (01) and (12) are
known, the expressions (5.104)-(5.109) can be used to obtain the reflection and
transmission operators for the combined layer (02) [12].
Substitution of expression (5.104) into expression (5.107) yields
−−+− ++= 221
110
00112
1 )( cLTcLRcLTcRL (5.110)
If we solve for L1-, we find:
68
)()( 221
0011211012
1−+−− +−= cLTcLcTRccRRIL (5.111)
Similarly, L1+ can be obtained by combining the expressions (5.104) and (5.107):
)()( 22110
00111210
1−+−+ +−= cLcTRcLTccRRIL (5.112)
Substitution of expression (5.112) into (5.106) and of expression (5.111) into (5.105)
yields
[ ] [ ] −−+−+ +−+−= 22121101121012
0011121012
2 )()( cLRcTRccRRIcTcLTccRRIcTL (5.113)
[ ] [ ] −−+−− −++−= 2211101210
00101121101210
0 )()( cLTccRRIcTcLRcTRccRRIcTL (5.114)
With the use of
1121011121011112101
11210111210
)()())(()()(
−−−−−−−
−−−
−=−=−=
−=−
cRRccRRcccccRRccccRRcccccRRIc
(5.115)
and a similar transformation of c(I-R12cR10c)-1, we can write:
[ ] [ ] −−−+−−+ +−+−= 221211011210112
00111210112
2 )()( cLRcTRcRRcTcLTcRRcTL (5.116)
[ ] [ ] −−−+−−− −++−= 22111012110
001011211012110
0 )()( cLTcRRcTcLRcTRcRRcTL (5.117)
By comparing the factors in the expressions (5.116) and (5.117) with T02, T20, R20
and R02 in the expressions (5.108) and (5.109), we find the basic Adding Doubling
equations. If we define E=c-1 (unity matrix of the star multiplication), we can write the
equations with the star multiplication notation:
01112101202 )( TRRETT −∗−= (5.118)
21110121020 )( TRRETT −∗−= (5.119)
212110112101220 )( RTRRRETR +∗∗−= − (5.120)
010112110121002 )( RTRRRETR +∗∗−= − (5.121)
For identical and homogeneous slabs (�doubling� case, R01=R10=R12=R21=R and
T01=T10=T12=T21=T), the expressions (5.118)-(5.121) can be simplified to
69
RTRRRETRR +∗∗−== −12002 )( (5.122)
TRRETTT 12002 )( −∗−== (5.123)
Beginning with the starting thickness τthinnest, for which the reflection and transmission
operators Rstart and Tstart have been determined in matrix form by the initialization
routine (see chapter 5.4), the Adding Doubling routine repeatedly doubles the slab
thickness by using the expressions (5.122) and (5.123), until the desired thickness τ
is achieved. The resulting operators R and T are taken as the reflection matrix and
the transmission matrix of the sample.
If τ=∞ (slab with infinite thickness), the diffuse transmission Td is calculated from T
after every doubling step (see chapter 5.7). The doubling process is repeated until
the change of Td between two doubling steps is less than 10-6.
5.6 Inclusion of boundary effects originating from the glass slides
The Adding Doubling program also provides a consideration of effects on the
reflection and transmission operators resulting from the influence of the glass slides
usually applied to fix the sample during the integrating sphere measurements.
In order to do this, we have to combine the boundary layers and the sample layer
(with the reflection and transmission matrix obtained as described in chapter 5.5) to a
combined �sandwich layer�. As the individual layers are no longer equal anymore, we
have the �adding case� instead of the �doubling case�, and the expressions (5.118)-
(5.121) have to be used.
The �sandwich set-up� of the sample and the glass slides with the different reflection
and transmission operators is illustrated in Fig. 5.3. If the reflection and transmission
operators R01, T01, R10, T10 (left glass slide) and R23, T23, R32, T32 (right glass slide)
are determined, they can successively be combined with the operators R12=R21 and
T12=T21 of the sample, by using the Adding Doubling formulas (5.118)-(5.121).
If we use two identical glass slides (R01=R32, R10=R23, T01=T32, T10=T23), the final
results for the reflection and transmission operators are independent of the main
70
direction of light (R03=R30=R and T03=T30=T). This independence was one of the
assumptions for the integrating sphere theory (see chapter 3.2).
In order to combine the glass layers and the sample layer, the reflection and
transmission operators of the glass slides have to be calculated. We have four
interfaces (air-glass, glass-sample, sample-glass and glass-air) and two main
directions of light (left to right, right to left). On each interface we observe reflection
and transmission, depending on the indices of refraction of the two adjacent media
and on the angles of incidence and reflection or transmission, respectively.
Fig. 5.3: Sample between two glass slides
The quadrature angles µi (i=1�n) used in the calculation of the reflection and
transmission operators of the sample denote the angles of light within the sample µs,i.
The angles in the glass slides or in the air may differ from the angles within the
sample, due to refraction indices ng or na that are not equal to ns (refraction index of
the sample). The cosines µg,i and µa,i can be calculated from µs,i=µi, by Snell´s law:
( ) ( )22/,/
/,/ 11sinsin iagsiagi
ag
siag nn
nn
µµϕϕ −−=⇒= (5.124)
71
However, care has to be taken for angles exceeding the critical angle (total reflection
for angles ϕi=arccos(µi) with sin(ϕi)>ng/a/ns). In the Adding Doubling program, these
angles are set to a value of µg/a,i =0.
For unpolarized light, the Fresnel reflection r(µ,µ´) for light incident at an angle ϕ´ with
the cosine µ´ in a medium with an index of refraction n, onto a medium with an index
of refraction n´ (yielding the light angle ϕ with the cosine µ), can be calculated by the
following equation [23]:
22
21
21),(
′′+′′−
+
′+′′−′
=′µµµµ
µµµµµµ
nnnn
nnnnr (5.125)
With expression (5.125) and the respective cosines from expression (5.124), the
Fresnel reflections r on the four different interfaces can all be expressed in terms of
only the quadrature angle µi. If the incident angle µ´ exceeds the critical angle, so
that the angle µ is set to zero instead of being calculated from expression (5.124), we
find r(µ,µ´)=1. This corresponds to the physical observation of total reflection for
angles exceeding the critical angle.
For a non-absorbing glass slide with the Fresnel reflections rin(µi) (reflection when the
light enters the glass) and rout(µi) (reflection when the light leaves the glass), the total
amount of reflected light (including multiple reflections in the glass) can be calculated
with the help of [23]
)()(1)()(2)()()(
ioutiin
ioutiinioutiinig rr
rrrrR
µµµµµµ
µ⋅−
⋅−+= (5.126)
As there is no absorption in the glass, the total transmission can be determined as
)()(1)()()()(1)(1)(
ioutiin
ioutiinioutiinigig rr
rrrrRT
µµµµµµ
µµ⋅−
⋅+−−=−= (5.127)
In the case of non-absorbing glass slides, Rg and Tg are independent of the light
direction: For the direction �air!glass!sample�, we find rin(µi)=r(µg,i,µa,i) and
rout(µi)=r(µs,i,µg,i). The direction �sample!glass!air� yields rin(µi)=r(µg,i,µs,i) and
rout(µi)=r(µa,i,µg,i). With r(µ,µ´)=r(µ´,µ) (expression (5.125)), Rg(rin,rout)=Rg(rout,rin)
72
(expression (5.126)) and Tg(rin,rout)=Tg(rout,rin) (expression (5.127)), we can conclude
R01=R10, T01=T10, R23=R32 and T23=T32 (see Fig. 5.3).
If the absorption of the glass is desired to be included, we have to set a value for the
absorbing optical thickness of the glass τg=µa,g·dg. With the cosine of the angle of
light inside the glass µg(µi) (determined by Snell´s law), the total reflections and
transmissions of the glass slide are given by [23]
[ ]
−⋅⋅−
−⋅⋅−+
=
)(2
exp)()(1
)(2
exp)()(21)()(
ig
gioutiin
ig
gioutiiniin
ig
rr
rrrR
µµτ
µµ
µµτ
µµµ
µ (5.128)
[ ] [ ]
−⋅⋅−
−⋅−⋅−
=
)(2
exp)()(1
)(exp)(1)(1
)(
ig
gioutiin
ig
gioutiin
ig
rr
rrT
µµτ
µµ
µµτ
µµ
µ (5.129)
During the calculation of the glass slide operators Rg and Tg with absorbing glass
slides, we have to distinguish between the two main directions of light. For light
travelling from left to right through the glass, we may obtain different operators,
compared to light travelling from right to left. However, the final reflection and
transmission matrices of the glass-sample combination will be independent of the
light direction even for absorbing glass slides, as long as the two glass slides are
identical.
Before we can write the final (diagonal) matrices for the glass layers, the values
Rg(µi) and Tg(µi) have to be divided by the factor 2µiwi, to adjust them to the definition
of the reflection and transmission function, given in chapter 5.1 [12]. The reflection
matrix and the transmission matrix can then be written as
jj
jgjkjk
jj
jgjkjk w
TT
wR
Rµµ
δµµ
δ2
)(;
2)(
== (5.130)
The layers given by the expression (5.130) can now be combined with the sample
layer, by the use of the Adding equations (5.118)-(5.121).
73
5.7 Determination of the reflection and transmission factors
The different reflection factors (Rd, Rcd, Rc) and transmission factors (Td, Tcd, Tc) can
be calculated from the final reflection operator R and the final transmission operator
T, obtained through the procedure described above.
The diffuse reflection factor Rd and the diffuse transmission factor Td can be
calculated from R and T, respectively, by adding all components of the nxn-matrices
Rjk or Tjk corresponding to reflection or transmission angles ϕj smaller than the critical
angle (i.e. all components µj with j>crit, where µcrit is the cosine of the smallest
quadrature angle exceeding the critical angle ϕc (see expression (5.16)). Because of
the definition of the reflection and transmission function (see chapter 5.1), each
component Rjk or Tjk has to be multiplied by the factor 4µjwjµkwk [23]. Owing to the n2-
law of radiance
22
22
1
1
nL
nL
= (5.131)
stating that the ratio between the radiance and the squared refraction index of the
respective medium is invariant along a light path [10], the result also has to be
multiplied by the factor ns2, with ns being the refraction index of the sample [23]:
∑=
=+=
⋅⋅=nkj
kcritjkkjjjksd wwRnR
,
1,1
2 4 µµ (5.132)
∑=
=+=
⋅⋅=nkj
kcritjkkjjjksd wwTnT
,
1,1
2 4 µµ (5.133)
The factors Rct=Rc+Rcd and Tct=Tc+Tcd can be calculated by adding the components
of Rjk or Tjk (multiplied by 2µkwk), that correspond to the incident angle µk=µn=1 (light
perpendicularly incident upon the sample) [23]:
∑=
⋅=n
jjjjnct wRR
12µ (5.134)
∑=
⋅=n
jjjjnct wTT
12µ (5.135)
To separate the specular reflection and transmission, we have to calculate Rc and Tc
by other means. For a given optical thickness τ, this can be done by the use of
Beer´s law, including the multiple (Fresnel) reflections on the various interfaces.
74
If Rg1, Rg2, Tg1 and Tg2 are the reflection and transmission factors of the glass slides
for specular light under the angle µ=1 (1: glass slide before the sample, 2: glass slide
after the sample, see expressions (5.126)-(5.129)), the specular reflection and
transmission of the sample can be calculated by the following equations [23]:
( )( )ττ2exp1
2exp
21
212
1 −−
−+=
gg
gggc RR
TRRR (5.136)
( )( )ττ
2exp1exp
21
21
−−
−=
gg
ggc RR
TTT (5.137)
Finally, we can determine Rcd and Tcd with the use of
cctcd RRR −= (5.138)
cctcd TTT −= (5.139)
by substituting the expressions (5.134)-(5.137).
5.8 Example for an Adding Doubling calculation
To show how the calculations in the previous chapters are conducted, the Adding
Doubling algorithm will be performed numerically in this chapter [12]. To keep the
complexity of the calculations small, we will use only n=4 integration points
(quadrature angles). The sample used in this example shall be characterized the
following optical properties: a=0.9, τ=1, g=0.9. The index of refraction of the sample
is given by ns=1.5, while the surrounding medium has an index of refraction of na=1.
The refraction indices denoted above lead to a critical angle of ϕc=41.8°, with a
cosine of µc=0.7454 (see expression (5.16)). We have to determine two quadrature
points between 0 and 0.7454 with Gaussian quadrature, and two quadrature points
between 0.7454 and 1 with Radau quadrature (see chapter 5.2).
As we have n=2 for each quadrature algorithm, the Gaussian quadrature points can
be calculated as the roots of
75
0)13(21 2
2 =−= xP (5.140)
(see expressions (5.39) and (2.23)). The solutions of this equation are x1=+0.5774
and x2=-0.5774. The corresponding quadrature angles for the interval [0, 0.7454] can
be calculated from expression (5.40): µ1=0.1575 and µ2=0.5878.
In Radau quadrature, x4=-1 is already fixed. We only have to calculate x3 as the root
of
012
1)(2
1)()( 11 =⋅−
+=′−+=
xxxPxxPxφ (5.141)
(see expressions (5.19) and (2.23)). We find x3=0.3333, and from expression (5.35),
we can calculate the quadrature angles for the interval [0.7454,1]: µ3=0.8302 and
µ4=1.0000.
The weights wi corresponding to the 4 quadrature angles µi are calculated with the
help of the expressions (5.41) and (5.36), respectively. In Table 5.1, the results for
the weights wi are summarized, together with the quadrature angles µi. The two
smaller cosines (µ1 and µ2) correspond to virtual angles in the surrounding medium,
as they exceed the critical angle [12].
Table 5.1: Results for the quadrature angles µi and the corresponding weights wi [12]
Quadrature angles µi Corresponding weights wi
µ1=0.1575 w1=0.3727
µ2=0.5878 w2=0.3727
µ3=0.8302 w3=0.1910
µ4=1.0000 w4=0.0637
In the next step, the redistribution function has to be calculated for the quadrature
angles, with the use of the δ-M method (see chapter 5.3). According to the
expressions (5.57) and (5.60), a=0.9 and τ=1 have to replaced by
4095.0)1(;7558.011
=−==−−
= ∗∗ ττ MM
M
agaggaa (5.142)
76
(g=0.9, M=4). The starting value τ0* has to be smaller than µ1 (see chapter 5.4). As
we want τ0*=2nτ* (so that we find the desired optical thickness after n doubling
steps), the largest possible value for τ0* is τ0*=0.1024. Just two doubling steps are
then needed to reach τ*=0.4095 (τ=1) [12].
From expression (5.51), we further obtain the values
2120.0,4475.0,7092.0,0000.1 3210 ==== ∗∗∗∗ χχχχ (5.143)
For the redistribution function, we find for M=4 (expression (5.52)):
)()(7)()(5)()(3)()(
)()()12(),(
33*322
*211
*100
*0
1
0
*
kjkjkjkj
M
ikijiikj
PPPPPPPP
PPih
µµχµµχµµχµµχ
µµχµµ
+++=
+= ∑−
=
∗
(5.144)
With the expressions (2.23) and (5.143), expression (5.144) can be written as
)35()35(4839.1
)13()13(2376.21277.21),(3
213
21
2212
21
kjjj
kjkjkjh
µµµµ
µµµµµµ
−⋅−⋅+
−⋅−⋅++=∗
(5.145)
or
5594.1)(6782.14663.5
0345.5)(5645.52741.9),(22
223333
++−+
++−=∗
kjkj
kjkjkjkjkjh
µµµµ
µµµµµµµµµµ (5.146)
If we substitute the quadrature angles from Table 5.1 in expression (5.146), we find
the following results for the redistribution matrix:
=
=
−
−
−−
−−−
−
−
−
++−+
+−−−
8491.62362.47369.10365.02362.41555.39575.16630.07369.19575.19435.13037.10365.06630.03037.16081.1
3739.01533.03452.00346.01533.01204.00863.02311.03452.00863.00580.06583.00346.02311.06583.03503.1
3739.01533.03452.00346.01533.01204.00863.02311.03452.00863.00580.06583.00346.02311.06583.03503.1
8491.62362.47369.10365.02362.41555.39575.16630.07369.19575.19435.13037.10365.06630.03037.16081.1
jkjk
jkjkjk hh
hhh
(5.147)
77
The reflection and transmission matrix for the starting layer are found by the use of
diamond initialization (see chapter 5.4.2), as the starting thickness is too high to allow
the use of the more simple infinitesimal generator initialization.
With the diagonal matrices
=
=
0637.000001910.000003727.000003727.0
100008302.000005878.000001575.0
; wM (5.148)
(see expression (5.74)) and the redistribution matrix (expression (5.147)), we can use
expression (5.93) to calculate
=∆⋅
=
−−
−
−+−∗∗
0005.00006.00025.00002.00002.00005.00007.00020.00007.00005.00007.00081.00003.00054.00301.00618.0
1
4~ whMρ τa (5.149)
and
=
−∆
=
−−−−−−−−
−−
++∗
−∗
0428.00157.00125.00003.00063.00476.00170.00058.00036.00123.00632.00160.0
0003.00156.00597.02514.0
)2
(2
~ 1 whIMτ aτ (5.150)
For the expressions (5.97) and (5.102), we further need the following matrix:
=+
−−
−
−
0004.00006.00023.00002.00002.00005.00008.00016.00007.00006.00011.00065.00001.00063.00312.00498.0
1)~(~ τIρ (5.151)
Now we can use expression (5.97) to obtain
[ ]
=+−+=−−
9591.00145.00115.00000.00058.09549.00157.00047.00033.00113.09416.00124.00000.00127.00464.08019.0
11 ~)~(~~ ρτIρτIΓ (5.152)
and then calculate
78
=+=
−−
−
−
0008.00011.00045.00001.00004.00011.00019.00038.00013.00014.00028.00135.00001.00101.00502.00805.0
1)~(~2~ τIρΓR (5.153)
and
=−=
9183.00289.00231.00000.00116.09098.00313.00094.00067.00227.08833.00248.00001.00253.00927.06039.0
2~ IΓT (5.154)
We finally find the (symmetric) initial reflection and transmission matrix by dividing the
matrices from the expressions (5.153) and (5.154) by 2µkwk (i.e. the first column has
to be divided by 2µ1w1, the second by 2µ2w2, and so on, see expression (5.103)).
The result for the reflection matrix is
==
−−
−
0063.00035.00103.00010.00035.00036.00043.00320.00103.00043.00064.01146.00010.00320.01146.06860.0
~2
1jk
kkjk R
wR
µ (5.155)
and for the transmission matrix we find
==
2121.70912.00526.00004.00912.08689.20714.00799.00526.00714.00159.22117.00004.00799.02117.01435.5
~2
1jk
kkjk T
wT
µ (5.156)
The negative values in the reflection matrix are caused by the small number of
quadrature angles [12]. The integrated quantities Rd and Rct are always positive, as
negative reflectances are physically impossible. If we want to look at the reflectance
at a particular angle for light incident at a particular angle, we have to use a lot more
quadrature points [12].
After we found the initial matrices Rjk and Tjk for a starting optical thickness of τ0=0.25
(τ0*=0.1024), we have to apply the Adding Doubling formulas (5.122) and (5.123) on
them, to reach the desired thickness of τ=1. We have to perform two doubling steps.
For the diagonal matrices cjk (see expression (5.10)) and Ejk=cjk-1, we find
79
=
=
8541.700001534.300002823.200005177.8
1273.000003171.000004382.000001174.0
; jkjk Ec (5.157)
With the definition of the star matrix multiplication (see chapter 5.1), we can then
calculate
=∗− −
1273.00000.00000.00000.00000.03171.00001.00001.00000.00001.04385.00005.00000.00001.00005.01183.0
1)( RRE (5.158)
and use this result to obtain
=+∗∗−==
−
−
0110.00068.00191.00016.00068.00075.00102.00564.00191.00102.00173.01879.00016.00564.01879.09521.0
12002 )( RTRRRETRR (5.159)
and
=∗−== −
6264.61685.00969.00083.01685.06144.21313.01298.00969.01313.07895.13229.00083.01297.03229.01513.3
12002 )( TRRETTT (5.160)
for τ=0.5 from the expressions (5.122) and (5.123).
In the second doubling step, we first calculate
=∗− −
1273.00000.00000.00000.00000.03172.00002.00003.00000.00002.04390.00012.00000.00003.00012.01191.0
1)( RRE (5.161)
again, to get the final reflection matrix
=+∗∗−==
−
−
0167.00134.00336.00094.00134.00161.00235.00857.00336.00235.00422.02613.00094.00857.02613.01069.1
12002 )( RTRRRETRR (5.162)
and the final transmission matrix
80
=∗−== −
6040.52877.01656.00319.02877.01814.22205.01781.01656.02205.04264.13873.00319.01781.03873.02367.1
12002 )( TRRETTT (5.163)
Next, we have to find the reflection matrix and the transmission matrix of the
boundary (glass) layers. We assume non-absorbing glass slides with a refraction
index of ng=1.5. With the use of expression (5.124) and the quadrature angles in
Table 5.1, we can find the cosines µg,i and µa,i in the glass slides and in the air:
=
=
=
0000.15486.000
;
0000.18302.05878.01575.0
4,
3,
2,
1,
4
3
2
1
4,
3,
2,
1,
a
a
a
a
g
g
g
g
µµµµ
µµµµ
µµµµ
(5.164)
The cosines µa,1 and µa,2 are set to zero, because the first two quadrature angles
ϕ1=arccos(µ1)=80.9° and ϕ2=arccos(µ2)=54.0° exceed the critical angle ϕc=41.8°.
The Fresnel reflections rag(µi)=rga(µi) (air-glass interface) and rgs(µi)=rsg(µi) (glass-
sample interface) are determined by using expression (5.125). With the refraction
indices na=1.0, ng=ns=1.5 and the cosines from expression (5.164), we find:
=
=
=
=
0000
),(),(),(),(
)()()()(
;
0400.00754.011
),(),(),(),(
)()()()(
4,4,
3,3,
2,2,
1,1,
4
3
2
1
4,4,
3,3,
2,2,
1,1,
4
3
2
1
gs
gs
gs
gs
gs
gs
gs
gs
ag
ag
ag
ag
ag
ag
ag
ag
rrrr
rrrr
rrrr
rrrr
µµµµµµµµ
µµµµ
µµµµµµµµ
µµµµ
(5.165)
By substituting rin(µi)=rag(µi) and rout(µi)=rgs(µi)=0 in the expressions (5.126) and
(5.127), we can determine the reflection and the transmission of the glass slide:
)(1)();()( iagigiagig rTrR µµµµ −== (5.166)
By dividing Rg(µi) and Tg(µi) by 2µiwi (Table 5.1, see expression (5.130), the reflection
matrix
====
3142.000002379.000002823.200005177.8
32231001 RRRR (5.167)
and the transmission matrix
81
====
5399.700009154.200000000.000000000.0
32231001 TTTT (5.168)
of the glass slides can be determined. Compared to the results given in [12], these
matrices differ by a factor 4µjwjµkwk, but the results for the final layer (expressions
(5.171) and (5.172)) correspond again with [12].
First, we add the (01)-layer to the sample layer. With R12=R21 from expression
(5.162), T12=T21 from expression (5.163) and the glass layers R10 and T01, we can
use the expressions (5.120) and (5.118) to find
=+∗∗−= −
1580.00547.01499.00495.00547.01583.01921.01696.01499.01921.09936.06053.00495.01696.06053.03975.1
212110112101220 )( RTRRRETR (5.169)
and
=∗−= −
3819.52687.00000.00000.02804.00221.20000.00000.01805.02240.00000.00000.00385.01827.00000.00000.0
01112101202 )( TRRETT (5.170)
The (23)-layer can be added in the same way. We find the final transport matrices
R=R03=R30 and T=T03=T30 for the entire layer:
=+∗∗−= −
4781.00728.00000.00000.00728.04070.00000.00000.00000.00000.02823.20000.00000.00000.00000.05177.8
323220123202330 )( RTRRRETR (5.171)
and
=∗−= −
1942.52919.00000.00000.02919.09197.10000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0
02123202303 )( TRRETT (5.172)
The only thing that is left to do is the determination of the reflection and transmission
factors, out of the final matrices R and T. With the expressions (5.132)-(5.135), we
find
82
1228.0)0162.04781.0)0404.00728.0(21006.04070.0(5.1 2 =⋅+⋅⋅+⋅=dR (5.173)
6769.0)0162.01942.5)0404.02919.0(21006.09197.1(5.1 2 =⋅+⋅⋅+⋅=dT (5.174)
and
0840.01273.04781.03171.00728.0 =⋅+⋅=ctR (5.175)
7539.01273.01942.53171.02919.0 =⋅+⋅=ctT (5.176)
The discrepancy between the results and multiplying the terms out occurs because of
rounding inaccuracy. From the expressions (5.136) and (5.137), we can calculate the
specular reflection and transmission, with Rg1=Rg2=0.04 and Tg1=Tg2=0.96:
0450.004.004.01
96.004.004.0 2
22
=⋅⋅−⋅⋅
+=−
−
eeRc (5.177)
3391.004.004.01
96.096.02
1
=⋅⋅−⋅⋅
=−
−
eeTc (5.178)
With the help of the expressions (5.138) and (5.139), the factors Rcd and Tcd can
easily be calculated. The six reflection and transmission factors as the final result of
the Adding Doubling calculation are summarized in Table 5.2
Table 5.2: Final results of the Adding Doubling calculation
Reflection factors Transmission factors
Rd=0.1228 Td=0.6769
Rc=0.0450 Tc=0.3391
Rcd=0.0390 Tcd=0.4148
83
6 The Inverse Adding-Doubling Method
The Inverse Adding Doubling (IAD) method is an algorithm that can be used to
calculate the optical properties of a sample, out of measurements with an integrating
sphere set-up. The algorithm consists of the following steps [11]:
1. Guess a set of optical properties (a, τ, g).
2. Calculate the reflection and transmission factors (Rd, Rcd, Rd, Td, Tcd, Tc) by using
the Adding Doubling Method (expression (4.7), see chapter 5).
3. Calculate the predicted measurement results (Vr%, Vt%, Vc%) by using the
integrating sphere formulas (expressions (4.3)-(4.5), see chapter 3)
4. Compare the calculated values with the actually measured values.
5. Repeat iteratively, until a match is made. The set of optical properties resulting in
the correct measurement results is taken as the optical properties of the sample.
6.1 Uniqueness
As announced in chapter 4.2, we have to show first that there is a unique set of
optical properties for each set of measurements. This is not immediately obvious. For
example, if the albedo of a sample is increased, the total reflection will increase,
while the total transmission decreases. However, the same observations can be
made for an increased optical sample thickness [11].
Prahl et al. [11] have shown the uniqueness of the results for two cases. In the first
case all three measurement values (Vr%, Vt%, Vc%) are known, in the second case
the measurement of the unscattered transmission (Vc%) is unavailable and a fixed
value for g has to be assumed.
In Fig. 6.1A, the first case (all three measurements available) is shown. For variable
Vr% and Vt% and a fixed value Vc%=0.1, the corresponding values for a and g are
plotted. In this case, the IAD algorithm calculates the optical thickness directly out of
84
Vc% (see chapter 6.3.1). Thus, τ remains constant for a constant value of Vc%. The
index of refraction is set to ns=1 (no critical angle, see chapter 5). From the diagram
can be deduced that there is a unique pair (a, g) for each set (Vr%, Vt%, Vc%=0.1).
For other values of Vc%, similar diagrams can be plotted [11].
If it is not possible to obtain an accurate measurement value for Vc% (e.g. if the
sample is too thick), additional assumptions have to be made. One possibility is to
set g to a fixed (assumed) value g0. Consequently, only a and τ are calculated from
Vr% and Vt%. To simplify the calculation, the algorithm calculates the so-called
�reduced optical properties� (a´, τ´), with isotropic scattering assumed (g´=0). The
results are plotted in Fig. 6.1B. In this example, the index of refraction is set to n=1.4.
As above, there is a unique pair (a´, τ´) for each set of measurements (Vr%, Vt%).
Fig. 6.1: Optical properties for different values of Vr% and Vt%
If a (fixed) anisotropy coefficient g=g0≠0 is desired to be assumed, a and τ can be
calculated from the reduced optical properties by using the �similarity relations� (cf.
chapter 5.3, expressions (5.57) and (5.60)). With g´=0, we find f=g0 from expression
(2.19) (µs´=(1-g0)µs, cf. chapter 10.5, expression (10.12)). a and τ can then be
calculated from [11, 18]
85
001 gagaa
′+−′
= (6.1)
0
0
1 gga
−′′
+′=τ
ττ (6.2)
6.2 Case differentiation depending on the available measurements
The ideal case for the IAD algorithm is that all three measurement values (Vr%, Vt%,
Vc%) are available. However, often the collimated transmission (Vc%) cannot be
measured, owing to the sample´s thickness. This might even be the case for very
small sample thicknesses [25]. Sometimes, also the measurement in the
transmission sphere (Vt%) is not available (e.g. for measurements with only one
integrating sphere). Thus, we have three general cases [25]:
1. One measurement available (Vr%)
If only one measurement is available, the optical thickness is assumed to be infinite
(τ=∞). The anisotropy coefficient g has to be set to a default value. Thus, only the
albedo a is varied, until the correct value for Vr% is obtained.
2. Two measurements available (Vr%, Vt%)
If two measurements are available, additional information can be submitted to the
program. If one of the three parameters (a, τ, g) is given by a default value, the two
others will be varied, until the calculated values for Vr% and Vt% match the measured
values. Another possibility is to fix either the scattering optical thickness τs=dµs or the
absorbing optical thickness τa=dµa (τ=τs+τa, cf. expression (2.32)).
If no additional information is given, the different reflection and transmission factors
have to be estimated (see chapter 6.3.1). In order to do this with only two
measurements given, strong assumptions have to be made [23]. By looking at the
estimated reflection and transmission factors, different cases can be distinguished
[24]:
- If the diffuse reflection Rcd is zero, there is no scattering (µs=0), yielding a=0. g
cannot be determined. Thus, only the optical thickness τ has to be calculated.
86
- If the diffuse transmission Tcd and the collimated transmission Rcd are zero, τ is
assumed to be infinite, and only a can be determined.
- If the diffuse transmission Tcd is zero, but not the collimated transmission Tc, it can
be deduced that there is no scattering in the sample (µs=0), and the albedo is set
to a=0. Only the optical thickness τ has to be determined.
- If the total reflection Rct and the total transmission Tct add up to one, the
absorption coefficient is µa=0. This implies a=1. Only the optical thickness τ can
be calculated.
- If none of these special cases occurs, we can only determine the reduced optical
properties a´ and τ´, for an assumed anisotropy coefficient g´=0.
3. Three measurements available (Vr%, Vt%, Vc%)
If all three measurements are available, we check for the same special cases as for
two measurements. If none of them occurs and Vc%≠0, all three parameters can be
determined (for Vc%=0, we actually have the case of two available measurements,
and only the reduced properties can be determined). The optical thickness τ is
calculated from Vc%(=Tc) with the help of Beer´s Law (see chapter 6.3.1), the albedo
a and the anisotropy coefficient g are then varied until the calculated values for Vr%
and Vt% match the measurements.
Overall, we require 7 search algorithms (see chapter 6.4), to cover all the different
cases:
- Determination of the albedo a, with τ and g fixed (see chapter 6.4.1)
- Determination of the optical thickness τ, with a and g fixed (see chapter 6.4.2)
- Determination of the albedo a and the anisotropy coefficient g, with τ fixed (see
chapter 6.4.3)
87
- Determination of the optical thickness τ and the anisotropy coefficient g, with a
fixed (see chapter 6.4.4)
- Determination of the optical thickness τ and the anisotropy coefficient g, with
τs=dµs fixed and a=τs/τ (see chapter 6.4.5)
- Determination of the optical thickness τ and the anisotropy coefficient g, with
τa=dµa fixed and a=(τ-τa)/τ (see chapter 6.4.6)
- Determination of the albedo a and the optical thickness τ, with g fixed (see
chapter 6.4.7)
6.3 Auxiliary calculations
6.3.1 Estimation of the reflection and transmission factors
In some cases, it is useful to estimate different reflection and transmission factors
(Rcd, Rc, Rct, Tcd, Tc, Tct) directly from the measurement values. Rct and Tct are
denoted as total reflection and total transmission (see expression (2.35)).
In the estimation, it is assumed that the detectors in the two spheres directly measure
the corresponding factor, neglecting the integrating sphere formulas. The light is
supposed to be incident upon the sample, thus we use Rcd and Tcd, instead of Rd and
Td. We only distinguish between two cases:
If the specularly reflected (transmitted) light leaves the spheres directly through an
exit port, the measured value Vr% (Vt%) equals the diffuse reflection (transmission)
factor:
%%; tcdrcd VTVR ≈≈ (6.3)
If the specularly reflected (transmitted) light remains inside the sphere, the measured
value Vr% (Vt%) equals the total reflection (transmission) factor:
88
%%; tctrct VTVR ≈≈ (6.4)
If all three measurements are available, we further have
%cc VT = (6.5)
For two measurements (Vr%, Vt%) available, we assume τ=∞ and Tc=0. If Vt% is not
available either, we have to set Tcd=0, too.
Now we know three values (Rcd or Rct, Tcd or Tct, and Tc), out of the total six. As two
more values can be calculated by the use of expression (2.35), we only have to
determine Rc, to obtain all six factors.
To calculate Rc, the optical thickness τ of the sample has to be determined (if not set
to infinite above). In order to do this, we first have to calculate the reflection and
transmission factors Rg1, Rg2, Tg1, Tg2 for the two glass slides fixing the sample. This
is done as described in chapter 5.6, by the use of Fresnel´s equations.
If we then solve expression (5.132) for τ, we find
++−=
221
22
2121 4
2lncgggggg
c
TRRTTTT
Tτ (6.6)
For a given optical thickness τ, we can determine the specular reflection factor Rc by
using expression (5.135).
6.3.2 The „quick guess“-method to obtain iteration starting values
For the routines using two-dimensional search algorithms (chapters 6.4.3, 6.4.4,
6.4.5, 6.4.6 and 6.4.7), initial values for the optical properties have to be determined
before the iteration process can start. In general, this is done by the use of a grid, to
scan the corresponding domain for appropriate values. However, if the starting value
found for the albedo is small (i.e. <0.2), there is little information about g available
[24]. Thus, the starting values generated by the use of the grid do not work so well,
and another method has to be used, to calculate starting values. This is the so-called
�quick guess�-method:
89
The reflection and transmission factors are estimated as described in chapter 6.3.1.
With the help of these factors, the reduced albedo a´ (cf. expression (6.1)) is
estimated as follows [24]:
[ ]
( )
−
−⋅−−
<∧<≤−⋅+
<∧<⋅−−
≥−
−
−−⋅−
=
=′
otherwise,1
411
4.01.005.0,05.045.0
4.005.0,1011
1.01
,1
1941
1,1
2
2
2
ct
ctcd
ctcdcd
ctcdcd
ct
cd
ct
ctcd
ct
TTR
TRR
TRR
TR
TTR
T
a (6.7)
To calculate the starting values from the reduced albedo, first the number of available
measurements has to be checked:
If only the reflection measurement is available, the optical thickness is set to infinite
(τ=∞), and the anisotropy coefficient is set either to g0=0 or to a given default value
g0. The starting value for the albedo is then calculated by expression (6.1).
For two measurements available, the reduced optical thickness τ´ has to be
estimated, similar to a´ [24]:
( )
≤<∧≥⋅
>∧≥⋅−⋅⋅
≤∧≥∞
<
=′
1.0001.0,lnln05.0ln
1.001.0,2ln)(5exp2
001.0,
01.0,(6.5))expression()(
ctcdct
ct
ctcdctcd
ctcd
cdct
TRRT
TRTR
TR
RTτ
τ (6.8)
The anisotropy coefficient is then set to either g0=0 or to a given default value g0. The
albedo and the optical thickness are calculated from a´ and τ´ with the use of
expression (6.1) and [24]
90
01 ag−′
=ττ (6.9)
If all three measurements are available, the calculation of the starting values
depends on which properties are desired to be determined [24]:
- If the algorithm tries to find only a, then τ is calculated from Vc%. g is set to either
zero or a given default value g0. a is calculated from expression (6.1).
- If the algorithm tries to find only τ, the starting value for τ is calculated from Vc%. a
is set to zero, and g is set to either zero or a given default value g0.
- If the algorithm tries to find a and τ (this is the case for Vc%=0 and Vt%>0), the
starting values are determined as follows: g is set either to zero or to a given
default value. If this default value is g0=1, the albedo is set to a=0, otherwise it is
calculated from expression (6.1). The optical thickness is determined by using the
expressions (6.8) and (6.9), it might also become infinite.
- If the algorithm tries to find a and g (for Vc%>0 and Vt%>0), the optical thickness τ
is calculated from Vc%. If τ=∞ or τ=0, we set a=a´ and g=g0 (zero or default
value). Otherwise τ´ is estimated from expression (6.8), and the equation to
calculate a is [24]:
ττ 11 −′⋅′+=aa (6.10)
If a<0.1, the anisotropy coefficient is set to g=0, otherwise it is calculated from [24]
ag τ
τ ′−=
1 (6.11)
In all three cases (one, two or three measurements available), we have to check if
the calculated values are in their defined range. If we have a negative value for a or
g, respectively, the value is set to zero. For g≥1, we set g=0.5 [24].
For each possible case, we have now found starting values for a, τ and g. These
values are used in the corresponding search algorithm, instead of the ones found
with the �grid method� (see chapter 6.4).
91
6.4 Search algorithms
The objective of the IAD algoritm is to determine the optical properties of a sample,
given by the albedo a, the optical thickness τ and the anisotropy coefficient g.
However, depending on the input data, one or two of these properties are already set
to a default value or can be calculated by other means (e.g. the direct calculation of τ
out of Vc%, see chapter 6.3.1). To account for this, the IAD program provides
different search algorithms to determine the one or two remaining optical properties
by an iterative use of the Adding Doubling Method.
The two-dimensional iterations in the search algorithms use an N-dimensional
minimization algorithm that is based on the downhill simplex method introduced by
Nelder and Mead [13]. The implementation of this algorithm that is used in the IAD
program is called amoeba (see chapter 10.4), and it varies the parameters from -∞ to
+∞. The same range is applied in the one-dimensional iterative minimization
algorithm, which uses Brent´s method (see chapter 10.3).
As the optical properties have fixed ranges (0≤a≤1, 0≤τ≤∞, -1≤g≤1), they have to be
transformed into a computation space, in which the can be varied from -∞ to +∞. The
transformation functions used for the three parameters are [24]:
)1(12aa
aacomp −−
= (6.12)
( )ττ ln=comp (6.13)
gggcomp −
=1
(6.14)
The inverse relations that transform the parameters back are given as [24]:
comp
compcomp
aaa
a2
42 2 +++−= (6.15)
( )compττ exp= (6.16)
comp
comp
gg
g+
=1
(6.17)
92
As explained in chapter 6.2, different search algorithms are applied to calculate the
optical properties. Each of them yields values for a, τ and g, as long as they can be
determined for the respective input data. By measuring the physical thickness of the
sample, the scattering coefficient and the absorption coefficient can be calculated
from a, τ and the expressions (2.33) and (2.34). The Henyey-Greenstein
approximation for the single-scattering phase function p(ν) can be derived from g and
expression (2.10).
6.4.1 Determination of a, with τ and g fixed
There are two possibilities that this search algorithm is used by the program (see
chapter 6.2). Either there is only one measurement value available, or the estimated
transmission factors Tc and Td (see chapter 6.3.1) are both zero. In both cases the
optical thickness will be set to τ =∞, and the anisotropy coefficient will be either set to
a given default value, or to zero.
First, the algorithm estimates the reflection and transmission factors, as described in
chapter 6.3.1. If the diffuse reflection Rd is zero, we have the non-scattering case.
The albedo is therefore set to a=0, and the algorithm terminates.
Otherwise, the algorithm transforms the two values a1=0.3 and a2=0.5 into the
computation space (expression (6.12)). Then the subroutine mnbrak is called. This
routine returns a point triplet (a1,comp,a2,comp,a3,comp) bracketing (see chapter 10.2) a
minimum of the function
600
600
10%%),,(%
10%%),,(%
)(−− +
−+
+
−=
meast
meastcalct
measr
measrcalcrcompa V
VgaVV
VgaVaM
ττ (6.18)
Two initial points (a1,comp,a2,comp) have to be passed to the routine; they do now
necessarily have to equal the returned points (a1,comp,a2,comp).
To evaluate the function Ma, acomp is transformed back to a. Then values for Vr% and
Vt% are calculated out of (a, τ=τ0, g=g0), with the Adding Doubling Method (see
expression (4.8) and chapter 5). After that, the relative deviation from the
measurement values is determined.
93
With three bracketing points a1,comp, a2,comp and a3,comp found, the algorithm uses
Brent´s method (see chapter 10.3) to find the minimum of the function Ma(acomp). The
corresponding value acomp is then transformed back to a, yielding the desired value
for the albedo (expression (6.15)).
6.4.2 Determination of τ, with a and g fixed
This routine is called if the albedo is set either to a=0 or to a=1. The albedo is set to
zero, if either Rd=0, or Td=0 and Tc≠0. It is set to one, if there is no absorption
(Rt+Tt=1). The search algorithm might also be called if the albedo is set to a default
value.
If a=0 (no scattering), the optical thickness τ is calculated from Tc by the use of
Beer´s law (see chapter 6.3.1). If Tc=0, the optical thickness is set to τ=∞.
If a≠0, we have to conduct an iterative calculation to find τ. The procedure is similar
to the one described in chapter 6.4.1. The optical thickness and the anisotropy
coefficent are set to their default values a0 and g0. Then the values τ1=1 and τ2=2 are
transformed into the computation space (expression (6.13)). With the resulting values
τ1,comp and τ2,comp used as initial values, the subroutine mnbrak (see chapter 10.2) is
called to calculate a point triplet (τ1,comp,τ2,comp,τ3,comp), such that the three points
bracket a minimum of the function
600
600
10%%),,(%
10%%),,(%
)(−− +
−+
+
−=
meast
meastcalct
measr
measrcalcrcomp V
VgaVV
VgaVM
ττττ (6.19)
With the use of Brent´s method (see chapter 10.3), the value τcomp minimizing the
function Mτ is determined. Finally, this value is transformed back to obtain τ (see
expression (6.16)).
6.4.3 Determination of a and g, with τ fixed
There are two possibilities that this routine is called by the main program: either if the
optical thickness is given as a default value, or if we have the �optimum case� that all
three measurement values Vr%, Vt% and Vc% exist and are non-zero. In the latter
case, the optical thickness is calculated from Vc%, as described in chapter 6.3.1.
94
To obtain appropriate starting values for the iteration, the algorithm first creates a grid
by filling an mxn-matrix with (a,τ,g). With τ=τ0 (either default value or calculated from
Vc%), we have n different values for a, and m different values for g, yielding mn
combinations.
For each of these combinations, values for Vr% and Vt% are calculated by the use of
the AD Method (chapter 5). The calculated values (Vr%ij, Vt%ij) are also saved in the
grid. Now each node of the grid matrix contains five values: aij, τij, gij, Vr%ij and Vt%ij.
The grid matrix is given by
=
=
−
−−
=
=
−−
−=
=
),,(%%
),,(%%
11)1(299.0
111
0
2
ijijijcalctijt
ijijijcalcrijr
ij
ij
ij
ijag
gaVV
gaVV
mig
nja
G
τ
τ
ττ
(6.20)
For the default values m=n=11, we obtain the albedos {1, 0.99, 0.96, 0.91, 0.84,
0.75, 0.64, 0.51, 0.36, 0.19, 0) and the anisotropy coefficients {-0.99, -0.792, -0.594, -
0.396, -0.198, 0, 0.198, 0.396, 0.594, 0.792, 0.99}.
Next, the grid is scanned for the node that contains the best suited optical properties.
For each node the absolute deviation ∆ij from the measurement is calculated:
meastijtmeasrijrij VVVV %%%% −+−=∆ (6.21)
The node with the smallest deviation is saved, and the corresponding optical
properties are used as starting values for the iteration.
Before starting the iteration, the program checks if the albedo starting value is
smaller than 0.2. In that case, the �quick guess�-method is used to find the starting
values (see chapter 6.3.2).
95
After the starting values astart and gstart are found, the three nodes p1, p2 and p3 of the
(a, g)-simplex are initialized by the use of the expressions (6.12) and (6.14):
+
=
+=
=
compstart
compstart
compstart
compstart
compstart
compstart
ga
pg
ap
ga
p)05.09.0(
)(;
)()05.09.0(
;)()(
321 (6.22)
The subroutine amoeba (see chapter 10.4) is then called, returning the minimum of
the function
60
60
10%%),,(%
10%%),,(%
),(−− +
−+
+
−=
meast
meastcalct
measr
measrcalcrcompcompag V
VgaVV
VgaVgaM
ττ (6.23)
Finally, the values for acomp and gcomp that correspond to the minimum are
transformed back by using the expressions (6.15) and (6.17), to obtain the desired
optical properties a and g.
6.4.4 Determination of τ and g, with a fixed
There is only one possibility that this algorithm is used by the main program: a default
value for the albedo a=a0 has to be provided by the user, so that the two other
unknowns can be determined by this routine.
Similar to the procedure in chapter 6.4.3, an mxn-matrix is created, resulting from (by
default) m=n=11 different values for both the optical thickness τ and the anisotropy
coefficient g:
=
=
−
−−
=
=
=
=
−
),,(%%
),,(%%
11
)1(299.0
2 6
0
ijijijcalctijt
ijijijcalcrijr
ij
iij
ij
ijg
gaVV
gaVV
njg
aa
G
τ
τ
τ
τ (6.24)
For the default matrix size, the generated values for g are identical to the ones in
chapter 6.4.3. τ is set to the following values: {0.03125, 0.0625, 0.125, 0.25, 0.5, 1, 2,
4, 8, 16, 32}.
96
Along the lines of chapter 6.4.3, the algorithm looks for the grid node with the
smallest absolute deviation (see expression (6.21)), to obtain the starting values. As
above, if the (fixed) albedo a0 is smaller than 0.2, the �quick guess�-method (see
chapter 6.3.2) has to be used to find appropriate starting values for τ and g.
With the starting values found, the three nodes p1, p2 and p3 of the (τ, g)-simplex can
be initialized (expressions (6.13) and (6.14)):
+
=
=
=
compstart
compstart
compstart
compstart
compstart
compstart
gp
gp
gp
)05.09.0()(
;)()2(
;)()(
321
τττ (6.25)
The subroutine amoeba (see chapter 10.4) returns the minimum of the function
60
60
10%%),,(%
10%%),,(%
),(−− +
−+
+
−=
meast
meastcalct
measr
measrcalcrcompcompg V
VgaVV
VgaVgM
ττττ (6.26)
By back-transforming the results (expressions (6.16) and (6.17)), the desired optical
properties τ and g can then be obtained.
6.4.5 Determination of τ and g, with τs fixed
This algorithm is used to determine the optical properties, if the �scattering optical
thickness� τs= dµs is given as a default value. It is quite similar to to algorithm
described in chapter 6.4.4.
First the grid is created:
=
=
−
−−
=
+=
=
=
−
),,(%%
),,(%%
11
)1(299.0
2 6
ijijijcalctijt
ijijijcalcrijr
ij
isij
ij
sij
ijg
gaVV
gaVV
njg
a
Ga
τ
τ
ττ
ττ
τ (6.27)
97
Now we vary the �absorption optical thickness� τa= dµa instead of τ, and then add τs
to obtain τij. The albedo aij belonging to each pair (τij, gij) is calculated by the use of
expression (2.31).
By scanning the grid for the best set of optical properties (expression (6.21), or
chapter 6.3.2, if the albedo is smaller than 0.2), starting values for the iteration are
generated. With the expressions (6.13) and (6.14), the (τa, g)-simplex is initialized:
+−
=
−=
−=
compstart
compsstart
compstart
compsstart
compstart
compsstart
gp
gp
gp
)05.09.0()(
;)(
)22(;
)()(
321
ττττττ (6.28)
By using this simplex, the minimum of the function
66 10%
%),,(%
10%
%),,(%),(
−− +
−+
+
−=
meast
measts
calct
measr
measrs
calcr
compcompg V
VgV
V
VgVgM
a
τττ
τττ
ττ (6.29)
is determined by amoeba (see chapter 10.4), to find τ and g (expressions (6.16) and
(6.17)). With τs given, the albedo a can then be calculated with expression (2.31).
6.4.6 Determination of τ and g, with τa fixed
If the �absorption optical thickness� µa is provided as a default value, this routine is
called. The algorithm is the same as the one described in chapter 6.4.5, except that
now τa is given instead of τs. Thus, the calculation of the albedo is different.
We have the grid
=
=
−
−−
=
+=
−=
=
−
),,(%%
),,(%%
11
)1(299.0
2 6
ijijijcalctijt
ijijijcalcrijr
ij
iaij
ij
aijij
ijg
gaVV
gaVV
njg
a
Gs
τ
τ
ττ
τττ
τ (6.30)
98
and the initial (τa, g)-simplex
+−
=
−=
−=
compstart
compastart
compstart
compastart
compstart
compastart
gp
gp
gp
)05.09.0()(
;)(
)22(;
)()(
321
ττττττ (6.31)
The function to minimize is
66 10%
%),,(%
10%
%),,(%),(
−
−
−
−
+
−+
+
−=
meast
meastcalct
measr
measrcalcrcompcompg V
VgV
V
VgVgM
aa
s
τττ τ
τττττ
τ (6.32)
6.4.7 Determination of a and τ, with g fixed
This algorithm is called if either a default value g0 for the anisotropy coefficient is
provided, or if there is not enough information given to calculate all optical properties,
and therefore only the reduced optical properties a´ and τ´ for an assumed value
g0=g´=0 can be calculated. The latter is the case if there are only two measurements
(or three measurements with Vc%=0) available without any �special case� (see
chapter 6.2) or default values given.
The procedure is once more similar to the ones described above. The grid used to
find the starting values is defined as
=
=
=
=
−−
−==
=
−
),,(%%
),,(%%
2
111
0
6
2
ijijijcalctijt
ijijijcalcrijr
ij
iij
ijij
ijag
gaVV
gaVV
gg
njaa
G
τ
τ
τ
(6.33)
After the starting values are found (expression (6.22) or chapter 6.3.2), we can
calculate the initial (a, τ)-simplex by using the expressions (6.12) and (6.13):
=
+=
=
compstart
compstart
compstart
compstart
compstart
compstart ap
ap
ap
)2()(
;)(
)05.09.0(;
)()(
321 τττ (6.34)
The function minimized by the amoeba-subroutine (see chapter 10.4) is given by
99
60
60
10%%),,(%
10%%),,(%
),(−− +
−+
+
−=
meast
meastcalct
measr
measrcalcrcompcompa V
VgaVV
VgaVaM
ττττ (6.35)
The back-transformation is done by the use of the expressions (6.15) and (6.16). If
only the reduced optical properties a´ and τ´ could be calculated by the algorithm, the
�similarity relations� (expressions (6.1) and (6.2)) can be used to calculate the optical
properties for an assumed anisotropy coefficient g0.
6.5 Example for an Inverse Adding Doubling calculation
In this chapter, the Inverse Adding Doubling algorithm is performed on a set of
�measurement values� (Vr%, Vt%, Vc%), to illustrate the necessary calculations. We
take the example from chapter 5.8, where we performed an Adding Doubling
calculation for the optical properties a=0.9, τ=1, g=0.9.
As the integrating sphere calculations are included in the IAD method, we have to set
the sphere parameters in the integrating sphere equations. We take the values that
were determined for the spheres used during this project (see chapter 7.1, Table
7.2). With αr=αt=0.9824 (see chapter 7.1), we can substitute the reflection and
transmission factors from Table 5.2 in the expressions (4.3)-(4.5), to get the
�measurement values�
33911.0%;43401.0%;05672.0% === ctr VVV (6.36)
If we use these values as the �input� for the IAD calculations, we should get the
�original� optical properties a=0.9, τ=1, g=0.9 as a result.
As all three measurement values are given, all optical properties can be determined.
The optical thickness τ is calculated from the specular transmission Tc=Vc%, with
expression (6.6). The values Rg1=Rg2=0.04 and Tg1=Tg2=0.96 (ng=ns=1.5) are the
same as in chapter 5.8 (expression (5.166), with µi=µ4=1). We find
00000.133911.004.004.0496.096.096.096.0
33911.02ln222
=
⋅⋅⋅+⋅+⋅
⋅−=τ (6.37)
The albedo a and the anisotropy coefficient g are determined iteratively from Vr% and
Vt%, by using the algorithm described in chapter 6.4.3. First, we create a grid Gagij
100
(see expression (6.20)), to obtain starting values for the iteration. We use m=n=11,
leading to 121 grid points, as combinations of 11 albedos aj={1, 0.99, 0.96, 0.91,
0.84, 0.75, 0.64, 0.51, 0.36, 0.19, 0) and 11 anisotropy coefficients gi={-0.99, -0.792,
-0.594, -0.396, -0.198, 0, 0.198, 0.396, 0.594, 0.792, 0.99}. With τij=τ0=1.0000, we
can perform the Adding Doubling calculation (with 4 quadrature points, along the
lines of chapter 5.8) for all the 121 sets of optical properties. The corresponding
values Vr%ij and Vt%ij can then be determined out of the resulting reflection and
transmission factors, by using the expressions (4.3) and (4.4). We obtain
=
000.0004.0009.0015.0023.0031.0040.0049.0056.0062.0068.0000.0003.0008.0015.0025.0038.0057.0083.0117.0153.0170.0000.0004.0010.0020.0034.0054.0083.0123.0172.0219.0239.0000.0005.0014.0026.0044.0070.0107.0155.0212.0264.0285.0000.0008.0019.0035.0057.0087.0128.0181.0243.0297.0320.0000.0012.0026.0046.0071.0105.0149.0205.0269.0324.0347.0000.0018.0038.0062.0091.0127.0174.0231.0295.0351.0374.0000.0028.0056.0087.0121.0160.0207.0264.0326.0382.0405.0000.0043.0085.0126.0167.0211.0259.0313.0372.0424.0446.0000.0066.0128.0186.0241.0292.0342.0393.0444.0490.0511.0000.0101.0193.0280.0360.0434.0499.0557.0597.0623.0635.0
% ijrV (6.38)
and
=
000.0072.0150.0230.0311.0388.0457.0516.0562.0591.0605.0000.0051.0104.0159.0214.0269.0324.0379.0434.0483.0504.0000.0035.0071.0110.0150.0195.0244.0301.0361.0414.0436.0000.0023.0048.0076.0109.0147.0194.0250.0313.0367.0390.0000.0015.0032.0054.0081.0116.0160.0216.0279.0334.0356.0000.0009.0022.0040.0063.0095.0137.0191.0252.0306.0329.0000.0006.0016.0031.0051.0079.0118.0169.0228.0281.0303.0000.0004.0012.0024.0041.0066.0101.0147.0201.0251.0272.0000.0002.0008.0017.0032.0052.0080.0119.0166.0211.0231.0000.0000.0001.0007.0018.0033.0053.0080.0114.0150.0167.0000.0000.0000.0000.0000.0000.0009.0013.0027.0037.0044.0
% ijtV (6.39)
In each row, the anisotropy coefficient gi is held constant, while the albedo aj varies
from 1 to 0. As the absorption increases with decreasing albedo (and constant optical
thickness), the reflection and the transmission decrease from left to right. For aj=11=0,
we have no scattering at all. Thus, Rcd and Tcd become zero, yielding
Vr%i,11=Vt%i,11=0.
In each column, we have a constant albedo aj, and anisotropy coefficients gi varying
from -0.99 to 0.99. For gi=1=-0.99, almost all light is scattered backwards, we observe
101
high values Vr%1,j for the reflection, and small values Vt%1,j for the transmission. With
increasing g, more and more light is scattered into the forward direction. Therefore,
the value for the reflection decreases, while we get higher values for the
transmission.
With expression (6.21), the absolute deviation ∆ij from the target values (expression
(6.36)) can be calculated:
=∆
491.0415.0332.0245.0157.0072.0040.0090.0129.0163.0182.0491.0437.0378.0317.0252.0184.0110.0081.0061.0145.0184.0491.0452.0409.0362.0307.0242.0216.0200.0189.0182.0184.0491.0463.0429.0388.0338.0301.0290.0282.0277.0274.0273.0491.0468.0440.0402.0353.0349.0345.0343.0341.0341.0341.0491.0470.0442.0405.0385.0388.0390.0392.0394.0395.0396.0491.0467.0436.0408.0417.0425.0433.0439.0444.0447.0448.0491.0459.0422.0440.0456.0471.0484.0494.0502.0508.0509.0491.0446.0454.0486.0513.0536.0556.0571.0583.0590.0593.0491.0446.0505.0556.0600.0637.0666.0690.0707.0717.0721.0491.0488.0585.0671.0745.0814.0867.0922.0948.0963.0969.0
ij (6.40)
The smallest deviation (∆11,5=0.040) is found for (i=11, j=5), yielding the starting
values astart=a5=0.84 and gstart=g11=0.99 (astart>0.2, cf. chapter 6.4.3). Now the (a, g)-
simplex for the amoeba-subroutine (see chapter 10.4) can be initialized (see
expressions (6.12), (6.14) and (6.22)):
=
=
=
94915.1505952.5
;00000.99
91394.3;
00000.9905952.5
321 ppp (6.41)
With this starting simplex, amoeba (see chapter 10.4) is called, to calculate the
minimum of the function (6.42). The development of the 3 simplex points during the
iteration, and their converging, is illustrated in Fig. 6.2 and Fig. 6.3. In Fig. 6.4 and
Fig. 6.5, the same iteration is shown, but with the simplex points (acomp, gcomp)
transferred back to (a, g), with the expressions (6.15) and (6.17). The 3 colours (blue,
red, green) refer to the 3 simplex points used during the iteration.
For a tolerance of 10-5 (i.e. termination for acomp and gcomp with Mag(acomp,gcomp)<10-5,
see expression (6.42)), the routine terminates after 84 iteration steps, returning the
values acomp=8.88888 and gcomp=8.99744. With the expressions (6.15) and (6.17), the
values can be transformed back, and we find the results
102
90000.088888.82
488888.888888.82 2
=⋅
+++−=a (6.42)
9,089997.099744.81
99744.8≈=
+=g (6.43)
As announced before, the IAD results for the optical properties (a=0.9, τ=1, g=0.9)
match the �input� values of the Adding Doubling calculation in chapter 5.8, as we
performed the IAD calculation with the results from the AD calculation.
Fig. 6.2: Development of gcomp during the iteration
Fig. 6.3: Development of gcomp during the iteration
103
Fig. 6.4: Development of a during the iteration
Fig. 6.5: Development of g during the iteration
104
7 Measurements with the integrating sphere set-up
All the measurements in this chapter are conducted with two integrating spheres with
an inner diameter of D=150 mm. The diameter of the two ports (the sample port and
the entry/exit port) is ds=dh=25 mm, while the diameter of the detector area is
dδ=18 mm. The inner surface of the spheres is coated with barium sulfate (BaSO4).
Both spheres contain a baffle (see chapter 3.3). The laser wavelength for all
measurements in this chapter is λ=632.8 nm.
While the detector used for the collimated measurement transduces the light intensity
into electrical voltage [V], the two detectors inside the spheres transduce the light
intensity into electrical power [W]. Nevertheless, although two different physical
quantities are measured, all measurement values are denoted with the letter V, in
correspondence with the chapters 3 and 4.
7.1 Reference measurements
Before any measurements can be evaluated, the reference measurements described
in the chapters 3.4 and 4.1 have to be conducted. With the single integrating sphere
set-up, six measurements have to be done for each sphere, to obtain the sphere
constants b1 and b2 (Vref, Vref,0, V1, V1,0, V2, V2,0, see Fig. 3.6, Fig. 3.7 and Fig. 3.8).
Two more reference measurements with the double integrating sphere set-up are
necessary (Vc,ref, Vc,ref,0, see Fig. 4.2). This leads to an overall of 14 required
measurements.
The measurements with the double integrating sphere set-up yielded the results
Vc,ref=13.83 V and Vc,ref,0=0.01 V. This leads to a reference value for the collimated
measurements of ∆Vc,ref= Vc,ref-Vc,ref,0=13.82 V.
The reference plate used for the reference measurements with the single integrating
sphere set-up consists of Spectralon with a diffuse reflection factor of Rref=0.97918,
for the used laser wavelength. Spectralon is a thermoplastic resin with a very high
diffuse reflectance, that is often used for the construction of optical components.
105
Each measurement was conducted 10 times, to account for uncertainties in the
measurement. Table 7.1 shows the results for ∆Vref=Vref-Vref,0, ∆V1=V1-V1,0 and
∆V2=V2-V2,0. The standard deviation σ is also given.
Table 7.1: Results for the reference measurements
Reflection sphere Transmission sphereMeas. No. ∆Vref,r ∆V1,r ∆V2,r ∆Vref,t ∆V1,t ∆V2,t 1 726 776 814 780 827 8702 723 776 815 789 830 8753 721 781 820 786 833 8784 723 768 808 774 824 8685 722 775 815 782 824 8686 724 770 807 783 822 8647 727 773 811 781 825 8708 723 773 812 784 822 8669 721 769 807 786 821 865
10 723 772 812 786 824 869Ø 723.3 773.3 812.1 783.1 825.2 869.3σ 1.94651 3.88873 4.12176 4.20185 3.79473 4.34741
From these values, the sphere constants b1,r, b2,r, b1,t and b2,t can be calculated with
the help of the expressions (3.108) and (3.110). If we use the average values of the
above measurements, we get the values summarized in Table 7.2:
Table 7.2: Results for the sphere constants
b1,r b2,r b1,t b1,t 1.06913 0.04879 1.05376 0.05181
The sphere constant b2 can also be calculated from the geometry and the reflection
properties of the sphere, with the use of expression (3.79). For this calculation, the
reflectance of the sphere wall is estimated to be around m=0.8.
The sphere area A is given by A=πD2=70686 mm2. For the calculation of s, δ and h,
we have to notice that they are considered as a fraction of the sphere area. Thus,
instead of simply calculating a circle area, they should be calculated with the help of
the following formula for the surface Ac of a sphere cap [7]:
RHAc ⋅= π2 (7.1)
R is the sphere radius, H is the height of the sphere cap, which can be derived from
106
22 rRRH −−= (7.2)
with the hole radius r (see Fig. 7.1). If we substitute expression (7.2) in expression
(7.1) and use the diameters D=2R and d=2r instead of the radii, we find:
−−=
22 11
2 DdDAc
π (7.3)
Fig. 7.1: Surface of a sphere cap
With the respective values ds=dh=25 mm and dδ=18 mm, we find s=h=494.3 mm2 and
δ=255.4 mm2. Substitution in the expressions (3.2) and (3.79) yields α=0.9824 and
b2=0.0327. The calculated value for b2 is 33% below the measued value for the
reflection sphere, and 37% below the value for the transmission sphere. One
possible reason for this might be, that the sphere wall reflectivity is higher as
assumed. A value of m=0.87-0.88 would lead to the measured results.
7.2 Measurements with phantom media
In order to test the double integrating sphere set-up, it is desired to do measurements
on samples whose optical properties are already known in advance. Furthermore,
these measurements should be conducted with different samples, covering a broad
spectrum of different scattering and absorption coefficients. To provide such
samples, phantom media were produced, consisting of water, Intralipid and black ink,
in various concentrations.
107
Intralipid is an intravenous nutrient, which consists of an emulsion of phospholipid
micelles and water [3]. It is turbid and shows almost no absorption for wavelengths
within the visible spectrum. Intralipid exhibits a relatively inert chemical nature, and is
widely available at low cost. Thus, it is sometimes used as a phantom medium in light
dosimetry studies, to simulate tissue with a low absorption coefficient (a≈1) [3].
Black ink was added to the solutions, to simulate absorbing media. The optical
properties of ink are mainly characterized by its absorption coefficient, the scattering
coefficient is comparatively low (a≈0) [3].
For the measurements with the double integrating sphere set-up, a �matrix� of
phantom media was prepared (see Fig. 7.2). Five different concentrations of Intralipid
were used, yielding expected scattering coefficients of µs=(2, 4, 8, 16, 32) [mm-1]
(�vertical axis� in Fig. 7.2) . Black ink was added in seven different concentrations,
leading to absorption coefficients of µa=(0.003, 0.01, 0.03, 0.1, 0.3, 1, 3) [mm-1]
(�horizontal axis� in Fig. 7.2).
Fig. 7.2: Phantom media in different concentrations (target values in mm-1)
108
The respective values µs and µa for pure Intralipid and pure ink were determined by
measurements of the collimated transmission, with a different set-up. The �target
values� for the different samples were calculated from the respective concentrations
in each solution. Overall, we obtain 35 combinations of Intralipid-ink solutions.
Table 7.3 shows the results of the double integrating sphere measurements for the
35 samples. The optimum case (all 3 measurements Vr%, Vt% and Vc% available)
was achieved for all combinations with expected scattering coefficients of 2, 4 and 8,
except for the combination with an expected scattering coefficient of 8 and an
expected absorption coefficient of 3. For this combination, and for all combinations
with expected scattering coefficients of 16 and 32, the collimated measurement Vc%
was not available, because the optical thickness was too high, so that not enough
light could reach the respective detector.
For the combinations with only two measurements available, further assumptions
have to be made, so that the optical properties can be calculated (see chapter 6.2).
Here, we give a default value for the anisotropy coefficient, so that the remaining two
optical properties (µs and µa) can be calculated from the two measurement values
Vr% and Vt%. The default anisotropy coefficient of the phantom media is g0=0.75.
Fig. 7.3 and Fig. 7.4 show the results of the Inverse Adding Doubling calculations,
which were performed with n=32 quadrature angles. The refraction index of the
sample was supposed to be ns=nwater=1.33 [15], the refraction index of the (non-
absorbing) glass plates was set to ng=1.5. The samples were d=1 mm thick.
The diagrams in Fig. 7.3 show the results for constant concentrations of Intralipid,
with the ink concentration being varied, so that the scattering coefficient is expected
to remain constant. In the diagrams in Fig. 7.4, we vary the Intralipid concentration
and keep the ink concentration constant, so that we should observe a constant
absorption coefficient. The �target values� (calculated from the respective
concentrations) are shown by the red lines. The squares and triangles represent the
IAD calculations. However, the abscissa values are not the IAD results, but the
respective �target values� (µa=(0.003, 0.01, 0.03, 0.1, 0.3, 1, 3) [mm-1] in Fig. 7.3,
µs=(2, 4, 8, 16, 32) [mm-1] in Fig. 7.4), to identify the respective samples.
109
Table 7.3: Results for the phantom media measurements Vr%=0.658 Vr%=0.650 Vr%=0.619 Vr%=0.534 Vr%=0.406 Vr%=0.238 Vr%=0.111Vt%=0.225 Vt%=0.211 Vt%=0.189 Vt%=0.128 Vt%=0.056 Vt%=0.008 Vt%=0.001
µ s=3
2
Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000Vr%=0.557 Vr%=0.547 Vr%=0.516 Vr%=0.438 Vr%=0.315 Vr%=0.196 Vr%=0.066Vt%=0.328 Vt%=0.317 Vt%=0.299 Vt%=0.220 Vt%=0.119 Vt%=0.041 Vt%=0.001
µ s=1
6
Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000 Vc%=0.000Vr%=0.438 Vr%=0.426 Vr%=0.402 Vr%=0.332 Vr%=0.224 Vr%=0.100 Vr%=0.035Vt%=0.442 Vt%=0.435 Vt%=0.409 Vt%=0.332 Vt%=0.211 Vt%=0.064 Vt%=0.008
µ s=8
Vc%=0.001 Vc%=0.002 Vc%=0.002 Vc%=0.002 Vc%=0.001 Vc%=0.001 Vc%=0.000Vr%=0.308 Vr%=0.293 Vr%=0.282 Vr%=0.228 Vr%=0.148 Vr%=0.059 Vr%=0.018Vt%=0.530 Vt%=0.525 Vt%=0.498 Vt%=0.429 Vt%=0.304 Vt%=0.129 Vt%=0.014
µ s=4
Vc%=0.046 Vc%=0.051 Vc%=0.048 Vc%=0.043 Vc%=0.038 Vc%=0.021 Vc%=0.003Vr%=0.189 Vr%=0.187 Vr%=0.180 Vr%=0.142 Vr%=0.093 Vr%=0.030 Vr%=0.010Vt%=0.511 Vt%=0.502 Vt%=0.487 Vt%=0.430 Vt%=0.329 Vt%=0.146 Vt%=0.020
µ s=2
Vc%=0.240 Vc%=0.234 Vc%=0.219 Vc%=0.214 Vc%=0.179 Vc%=0.092 Vc%=0.015 µa=0.003 µa=0.01 µa=0.03 µa=0.1 µa=0.3 µa=1 µa=3
Fig. 7.3: Scattering coefficient µs in mm-1 for different Intralipid-ink solutions
110
Fig. 7.4: Absorption coefficient µa in mm-1 for different Intralipid-ink solutions
The blue squares result from the calculations with three measurements, and could
therefore not be obtained for all the samples. The green triangles show the
calculations with two measurements. These calculations were conducted for all the
samples, even if the third measurement was available, to get a comparison.
111
For some of the calculations with two measurements, the amoeba algorithm (see
chapters 6.4.7 and 10.4) did not provide a tolerably accurate result. Thus, the
respective points were left out in the diagrams.
If we look at the diagrams in Fig. 7.3, we can see that the scattering coefficients
calculated from the measurements are quite similar for the solutions with a constant
concentration of Intralipid. This result can be observed for all five Intralipid
concentrations, except for the calculations from two measurements, with the target
value µs=2 mm-1. Here, the IAD calculations with g0=0.75 did not yield results for 5 of
the 7 samples, so that we get only 2 points.
For the absorption coefficients in Fig. 7.4, we get a similar result, although the
variation of µa for the samples with a constant ink concentration is higher. Especially
the samples with the target value µs=2 mm-1 (most left point in the diagrams) often
yield a significantly lower absorption coefficient.
For both the scattering coefficients and the absorption coefficients, we find that the
values calculated out of the measurements do not match the target values. While the
scattering coefficients are at least in the same magnitude, the absorption coefficients
exceed the target values by a factor of up to 20, for small ink concentrations. With
increasing target values of µa, the measurement results come closer to the target
values.
For the samples with three measurement values available, the calculation results
from all three measurements differ from those using only two measurements. This
indicates that the anisotropy coefficient calculated from all three measurements (in
this case we get all three optical properties as a result) does not equal the default
value g0=0.75.
Fig. 7.5 shows the anisotropy coefficients that were calculated for the samples with
Vc%≠0. The values for µs and µa in the diagram correspond to the target values, to
identify the respective samples. We can see that the calculated anisotropy
coefficients are lower that g0=0.75, we find values of g≈0.6-0.7.
112
Fig. 7.5: Anisotropy coefficient g for different Intralipid-ink solutions
There are several possibilities for the differences in the optical properties that were
measured with the double integrating sphere set-up and calculated with the Inverse
Adding Doubling method, compared to the �target values�, calculated from the values
for pure Intralipid and pure ink:
- Concentration inaccuracies in the preparation of the samples.
- Inaccuracies with the integrating sphere set-up, during the reference
measurements and the determination of the sphere parameters as well as during
the measurements with the samples. Some difficulties related to the integrating
sphere measurements are described in chapter 7.5.
- Inaccuracies due to the assumption of Intralipid as a pure scatterer and ink as a
pure absorber.
- Inaccuracies due to the assumption that the scattering (absorption) coefficient is
proportional to the concentration of Intralipid (ink), as suggested in [3]. This might
especially be the case for µa, where the results for higher concentrations are
significantly better than the results for low concentrations.
113
In addition to the calculations with n=32 quadrature angles from the diagrams, the
optical properties were also calculated by using n=4, n=8 and n=16 quadrature
angles, to investigate the accuracy of the IAD algorithm. Table 7.4 and Table 7.5
show the maximum deviation of the results for n=4, n=8, n=16, in % related to the
results for n=32.
Table 7.4: Maximum deviation for n=4, n=8, n=16, related to n=32 (3 measurements)
Max. deviation for n=4 n=8 n=16
a 0.497% 0.221% 0.028%
g 0.956% 0.289% 0.017%
µs 0.497% 0.221% 0.028%
µa 2.368% 0.557% 0.116%
Table 7.5: Maximum deviation for n=4, n=8, n=16, related to n=32 (2 measurements)
Max. deviation for n=4 n=8 n=16
a 1.440% (0.160%) 0.134% 0.012% (0.008%)
τ 6.586% (3.357%) 0.920% 0.056% (0.038%)
µs 5.051% (3.357%) 0.920% 0.044% (0.038%)
µa 93.363% (2.157%) 0.553% 0.746% (0.084%)
In Table 7.4, the optical thickness τ is not included, as it is calculated directly out of
Vc% (expression (6.6)) for the 3 measurement case. Thus, we get the same results
for τ, for all values of n. In Table 7.5, the anisotropy coefficient g is left out, because
we use a default value g0=0.75. However, we get deviations for τ, as it depends on n
now (see chapter 6.4.7).
For n=8, there was one sample (target values µs=2, µa=0.3) with the amoeba iteration
not finding results for a and τ, for the 2 measurement calculation. The iteration
converges for n=4 and n=16. However, the results for this sample differ strongly from
the result for n=32, especially the results for n=4 (see Table 7.5). For all 4 examined
values, this sample was the one that exhibited the maximum deviation. If we exclude
this sample for the 2 measurement calculation, the maximum deviations decrease
significantly (numbers in brackets in Table 7.5).
114
The devation between the results for n=16 and n=32 is relatively small: it is less than
0.1% for a, τ, g and µs. For µa, we get the highest deviations (up to 0.746% for the
�problem point� described above).
If we go down to n=8 points, the maximum deviations increase, but are still less than
1% for all five values.
For the calculations with n=4, there is a strong deviation in the results for the
�problem point�: the absorption coefficient µa calculated with n=4 is almost twice as
high as the one for n=32 (0.0180 compared to 0.0093). If we leave out this point, we
still get deviations up to 3.357%. Only the results for the albedo a seem to be
relatively stable (less than 0.5% deviation).
For all the investigated values for n, we still have the problem that the 2
measurement IAD algorithm does not find a result for all the samples. Out of 35
samples in total, this problem occurs with 5-6 samples, almost all of them with the
target value µs=2 mm-1. This number can be reduced by increasing the number of
grid points (see chapter 6.4.7) and by reducing the tolerance from 10-5, but still there
remains doubt in these results.
7.3 Preparation of tissue samples
After the integrating sphere measurements with phantom media (see chapter 7.2),
measurements with prostate samples were performed. The samples were dried and
attached to glass plates (see Fig. 7.6). Samples with five different thicknesses
(d=30 µm, 40 µm, 50 µm, 60 µm, 100 µm) were used.
However, it was not possible to obtain useful results out of the measurements with
the prostate samples. The collimated measurement Vc% was not available, even for
the thinnest sample. In the two other measurements (Vr% and Vt%), the results of
several measurements on the same sample differed very strongly, up to a factor 2
between the smallest and the highest value. One reason for this variety of the results
was the inhomogenity of the tissue: there were darker spots and small holes all over
the sample (see Fig. 7.6). Therefore, the results of the reflection and the transmission
115
measurement were strongly dependent on the exact position of the sample (e.g.
whether the laser beam was incident upon a darker spot or not). Small movements of
the sample holder led to big changes in the detected light power.
Fig. 7.6: Prostate samples on glass plates
Thus, before trying to get better samples for the integrating sphere measurements,
an investigation was performed about the optimal preparation of biological tissue
samples for optical measurements. The results of this investigation were the following
[1, 11, 14, 25]:
- Freshness: The samples should be fresh and in excellent condition, in order to
gain results similar to in vivo results (so far, no work is done on the change in the
optical properties during tissue removal). Significant changes in the optical
properties were observed for samples 24 h post mortem, at 4°C. Short-time
storage in saline solution leads to considerable changes, due to the loss of
absorbers, mainly haemoglobin.
116
- Boundaries: Glass slides should be attached to both sides of the sample, to
control the Fresnel reflection at the boundaries (to avoid inaccuracies owing to
rough tissue surfaces). We have to assure that no bubbles are formed between
the glass and the tissue, by using water or another fluid. A problem of the usage
of glass slides is the boundary mismatch (ng≠ns, see chapter 5.6), but the errors
caused by this are small, compared to those caused by other experimental
difficulties.
- Hydration: The tissue should be stored in air tight containers, sandwiched
between moist towels (soaked with isotonic saline) at cool (but above freezing)
temperature.
- Sample thickness: The tissue slices should be uniformly thick: between 100 µm
and 1 mm for samples with a low total attenuation coefficient (µs+µa~10 mm-1),
and between 15 µm and 150 µm for samples with a high total attenuation
coefficient.
- Sample cutting: Cutting non-frozen tissue in slices thinner than 500 µm is difficult.
Beek et al. used a vibratome and tissue glued to a table with cyanoacrylate to cut
slices thicker than 500 µm, and a hand-held microtone knife in combination with a
custom-built stand supporting the tissue on all sides to cut slices thinner than
500 µm. Prahl cutted dermis samples with a cold dermatome to obtain a thickness
of 200-400 µm, and with a freezing microtome to obtain a thickness of ~20 µm.
Roggan et al. examined the effect of different preparation techniques on the
optical properties. According to them, microtomes are not applicable for sample
thicknesses of 100-500 µm, as the maximum thickness is limited to about 40 µm.
They used a dermatome to get samples of about 200 µm thickness from porcine
liver. Fixation was achieved by using a vacuum table.
- Sample diameter: The sample should be big enough to cover the whole sphere
port. The distance from the edge of the irradiating beam to the edge of the port
should be much larger than the lateral propagation distance (µs´+µa), to avoid light
loss on the sides of the sample (which would result in a too high absorption
coefficient).
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- Variability: Enough measurements should be conducted, to account for variability
(1) from spot to spot on one sample, and (2) from sample to sample.
- Blood: An important question is whether blood should be removed from the tissue
before the mesaurements or not. Neither of the two possibilities is a good
approximation of the in vivo situation. The varying haemoglobin concentrations
are assumed to be the main reason for the wide range of optical properties
measured for the same type of tissue, but from different individuals. Therefore, a
defined blood removal might become an interesting procedure to obtain samples
with reproducible optical properties. For samples without blood, the optical
properties of the tissue itself are measured. In order to use these parameters for
clinical purposes, the optical properties of blood have to be measured separately
and then be combined with the blood-free parameters.
- Freezing: It is necessary to freeze the samples, in order to obtain very thin
sections. We have to consider the possibility of a change in the optical properties
due to the freezing process. For muscle tissue, investigations have shown a
change in the optical properties, while for other tissues (like dermis and aorta),
the influence of freezing is suspected to be not that strong.
7.4 Measurements with cartilage samples
After the investigation on tissue preparation, samples of cartilage were prepared for
integrating sphere measurements. The samples had a physical thickness of about
200 µm, and they were cut with the use of a vibrating microtome. The cartilage was
of human origin, it was taken from the knee of patients that underwent surgery in
order to get a joint prosthesis. The samples were stored in a refrigerator, in
physiological saline (0.9% NaCl).
The main problem during the cutting was that there were very few plain surfaces in
the cartilage. All surfaces had a curvature, so that it was very difficult to get
homogeneous samples with a diameter big enough to cover the whole sample port of
the integrating spheres.
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Overall, 10 similar samples were examined (named �sample 1� to �sample 10� in the
following). For each sample, 20 integrating sphere measurements were conducted:
after the first 10 measurements, the samples were rotated 90° from the original
position, perpendicular to the beam direction (the cartilage samples have a laminar
structure). In theory, this rotation should not influence the measurement results, as
we use a one-dimensional, azimuth-independent model with unpolarized light. One
aspect of the measurements is to examine, if the actual results also show this
independence.
During the 10 measurements for each position, the sample was slightly moved in the
plane perpendicular to the beam. The change in the measurement values caused by
this was less strong than for the measurements with the prostate samples (see
chapter 7.3).
For almost all of the samples, it was only possible to achieve two measurement
signals (Vr% and Vt%), the collimated transmission signal Vc% could not be detected,
because the samples were too thick. The only exception was sample 8, which was
thinner than the others. Thus, the IAD calculations for all the samples were
performed with two measurements and a default value g=0.87 for the anisotropy
coefficient. This default value was chosen, as it is the average value for the
anisotropy coefficient that was calculated for sample 8. Here it was possible to do a
three measurement calculation (in addition to the two measurement calculation), as
all three measurement signals could be detected. During all these calculations, n=32
quadrature angles were used.
Fig. 7.7 shows the results for the albedo a and the optical thickness τ, for all 10
samples. Each colour represents one sample, the two shapes symbolize the two
sample positions (see above). The points on the low right (a=1) belong to
calculations that did not terminate: if the maximum number of iterations is exceeded,
the IAD program sets the albedo to a=1. Out of the overall 200 calculations, this
problem emerged for 15 measurements. 11 of these 15 measurements belonged to
sample 8. For this sample, we get better results from the three measurement
calculation (see below). In the following, the 15 non-terminating calculations will be
neglected.
119
Fig. 7.7: Albedo a and optical thickness τ for human knee cartilage samples (g=0.87)
The calculations result in an albedo value of a=0.96-0.99 for almost all samples, only
sample 8 and sample 9 seem to have a lower albedo (average: a=0.965). The
calculated optical thickness τ is between 2 and 10 (average: τ=4.728).
Up to a certain degree, the points with the same colour form clusters in the a-τ-plane
(e.g. sample 6 in Fig. 7.7). With this observation, it is possible to draw some
conclusions about the sample structure. For example, most of the calculations for
sample 3 led to a relatively small optical thickness, compared to the other sample.
For sample 10, we find the opposite result. According to these results, it is probable
that e.g. sample 10 was either physically thicker than sample 3, or had an optically
thicker structure.
If we look at the development of the optical properties during the 90° rotation of the
sample, the results from Fig. 7.7 do not show a significant change. The average
deviation in the albedo a is 0.008. For the optical thickness τ, we find an average
deviation of 0.719. These values are about as high as the standard deviation during
120
the 10 measurements of a sample in the same position (0.006 for the albedo, 0.844
for the optical thickness).
The three measurement calculations from the 20 measurements on sample 8 yielded
the following average results and standard deviations for the optical properties:
( ) ( ) ( )031.0871.0;900.0463.5;021.0944.0 ====== σστσ ga (7.4)
While the values for a and g can directly be compared to results from the literature,
the optical thickness τ depends on the physical thickness of the sample. Thus, we
should calculate the scattering coefficient µs and the absorption coefficient µa, in
order to obtain values comparable to other results. This can be done with the help of
the expressions (2.33) and (2.34), with the physical thickness d of the samples. This
thickness is not exactly equal for all the 10 samples, but as the microtome was set to
a cutting thickness of 200 µm, we will use this value. For the 185 useable values from
the 200 two measurement calculations, we find the following average results and
standard deviations (in mm-1)
( ) ( )24.074.0;8.89.22 ==== σµσµ as (7.5)
The results for the albedo a and the anisotropy coefficient g are within the typical
magnitude for biological tissues. The same goes for the scattering coefficient µs and
the absorption coefficient µa [1, 2, 14].
7.5 Difficulties with the double integrating sphere set-up
When conducting measurements with a double integrating sphere set-up, we have to
consider different problems originating from the measurement set-up. In addition to
this, inaccuracies in the determination of the reflection and the transmission occur,
because of approximations and simplifications in the integrating sphere theory (see
chapter 3). Some important aspects, that have an influence on the results for the
optical properties determined from the measurements in the chapters 7.1-7.4, are
summarized in the following:
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- Reference measurements: As the measurement values Vr%, Vt% and Vc% are
relative quantities related to reference measurements, it is important that these
reference measurements are performed accurately enough. In the set-up used for
the measurements described in the chapters 7.1-7.4, the sensitivity of the Vc%
detector could be adjusted by the use of a coloured glass plate (working as a
damper), that was fixed between the spheres and the detector. During these
measurements, the detector had to be very sensitive, in order to obtain signals for
as many samples as possible. Thus, the value Vc,ref turned out to be relatively
high (13.83 V, see chapter 7.1), as for increased damping no signal was available
for some of the phantom media (see chapter 7.2). However, for values exceeding
12 V, the detector is supposed to be not working in the linear range anymore
(because of saturation effects). This might yield inaccuracies for Vc,ref, as the
�real� reference value might be higher than the actually measured value. In this
case, the values obtained for Vc% would be too high.
- Interference: When light is incident upon the glass slide, it is reflected at the air-
glass interface, and also reflected at the glass-sample interface. These two
reflections can interfere with each other. The influence on the measurement can
be up to 10%. A way to avoid this is to take a box of glass slides, to observe their
reflection profiles on a wall, and to discard all the slides that exhibit definite
interference effects. For very thick glass plates, interference can be avoided by
slightly rotating them, to displace the second reflection. However, thick glass
plates are not ideal (displacement of the spheres, causing loss of diffuse light).
The best possibility to avoid interference is the usage of optically flat plates, which
however are expensive [25].
- Diffuse reflection: In order to make a measurement of the collimated transmission
possible, very thin samples have to be used. However, for thin samples, the
diffuse reflection decreases, and might become considerably less than 1%,
especially for low albedos [14]. Even if the radiation field in the sphere is enforced
by the multiple reflections between the spheres, the signal reaching the detector
is very small, because the detector area is very small related to the sphere area.
122
- Sphere wall reflectance: The integrating sphere formulas depend on the
reflectance m of the sphere wall (see chapter 3). If we calculate the sphere
constants by reference measurements (see chapter 3.4.3), m does not have to be
determined explicitly, but as it is unstable owing to dust and partial contamination,
the reference measurements should be repeated from time to time, to check if the
results are still valid [14].
- Collimated transmission: The measurement of the collimated transmission
Vc%=Tc is the most difficult measurement. The relation between Vc% and the
sample thickness d (or the concentration, see chapter 7.2) has to be linear, and it
should pass through the correct value for d=0. This value can be calculated by
using the indices of refraction. Without the inclusion of multiple reflections, we
obtain this value by multiplying the transmission factors for the various surfaces.
For example, for water (ns=1.33) between two (non-absorbing) glass slides
(ng=1.5), we find rag(µ=1)=rga(µ=1)=0.04 and rgs(µ=1)=rsg(µ=1)=0.0036, Thus, the
collimated transmission is Tc=(1-rag)(1-rgs)(1-rsg)(1-rga)=0.915. Another important
aspect is the inclusion of diffusely transmitted light in Vc%. For a typical beam
divergence of 0.2 rad, about 0.4% of the diffusely transmitted light hits the Vc%
detector. Especially if Vt% is much higher than Vc%, this has a considerable
influence on the Vc% measurement. In theory, this problem can be avoided by
increasing the distance between the spheres and the Vc% detector, however this
is limited by the beam quality [14, 25].
- Polarization: Both the integrating sphere theory and the Adding Doubling model
assume unpolarized light. So far, very little work is done on the effects of light
polarization.
- Side losses: As announced in chapter 7.3, photons are lost on the side
boundaries of the sample. A possibility to minimize this effect is to keep the ratio
between the beam spot size and the sample diameter as small as possible.
However, typical set-ups exhibit side losses of up to 15% [14], and the side losses
can only be compensated if we use a highly complex transport model. The side
losses may be a reason for the high absorption coefficients obtained for the
phantom media measurements in chapter 7.2.
123
- Model limitations: The transport model used in the Adding Doubling method is a
one-dimensional model. It assumes infinite lateral expansion, and systematic
errors (side losses, diffuse photons reaching the Vc%-detector, interferences [14])
are neglected. To account for every aspect of the doulbe integrating sphere set-
up, we have to use more complex models (e.g. Monte Carlo simulation [10, 14]).
124
8 Summary
The objective of this project was to set up a system for the measurement of the
reflection and transmission factors and the calculation of the optical properties
(µs, µa, p(ν)) or (a, τ, g) (see chapter 2) of biological tissue. This system consists of a
double integrating sphere measurement set-up and a computer algorithm to perform
the necessary calculations.
The double integrating sphere set-up consists of two spheres with a sample between
them. A laser beam is incident upon the sample, and two detectors installed in the
spheres are used to simultaneously measure the fractions of light that are reflected
(reflection sphere) and transmitted (transmission sphere) by the sample. A third
detector is installed behind the transmission sphere, to measure the collimated
transmission. Overall, we get three measurement values (Vr%, Vt%, Vc%). With the
help of the integrating sphere theory (see chapter 3), these measurement values can
be related to the reflection and transmission factors (Rd, Rc, Rcd, Td, Tc, Tcd) of the
sample.
To finally calculate the optical properties from the measurements, we also have to
find a mathematical connection between the reflection and transmission factors and
the optical properties (see chapter 4). This connection is provided by the Adding
Doubling algorithm (see chapter 5), which is based on a one-dimensional transport
model. For given optical properties, the reflection and transmission factors can be
calculated by the use of this model. The algorithm first calculates the optical
properties for a very thin sample and then repeatedly doubles the thickness, until the
desired thickness is reached. The influence of the glass plates usually used to keep
the sample in position is also considered by the possibility to include boundary
layers.
However, we have to do the calculation in the other direction. As the AD method
cannot be �inverted� analytically, we have to solve the problem iteratively. This is
done with the use of the Inverse Adding Doubling method (see chapter 6). Here, a
starting set of optical properties is guessed and then iteratively varied, until the
�expected measurement results� (calculated from the guessed optical properties, with
125
the AD method and the integrating sphere theory) match the actually measured
values. The IAD program used here is based on the implementation of Prahl
[23, 24, 25], and allows the adjustment of parameters such as the accuracy (by
setting the number of quadrature angles), and to set default values for a, τ, g, µs or µa
(if not enough information is provided from the measurements).
To test the set-up, measurements were conducted with different Intralipid-ink
solutions (phantom media to simulate biological tissue) and cartilage samples (see
chapter 7). This part also includes an investigation on tissue preparation for optical
measurements, and a summary of difficulties and problems that can occur during the
measurements and the calculations with the system.
126
9 Prospects
With the double-integrating sphere set-up and the IAD algorithm, it is possible to
conduct measurements on biological tissue and then calculate the optical properties
of the tissue. However, before the system can be used for practical purposes, more
test measurements should be done, in order to check the accuracy of the integrating
sphere measurements.
By doing measurements on samples with already known optical properties, the
accuracy of the system can be examined. Measurements like this were already
performed in chapter 7.2. However, the results (especially those for the absorption
coefficients) did not exactly match the expected values so far. Thus, the system has
to be checked for sources of inaccuracy.
It is important that the sphere factors (b1 and b2) are determined accurately. Further
advice to increase the accuracy of the set-up is given in chapter 7.5, e.g. usage of
optically flat glass slides, avoidance of sphere wall contamination, avoidance of the
contribution of diffusely transmitted light in the Vc% measurement.
In order to do further measurements with biological samples, it is very important that
these samples are properly prepared for optical measurements. A guideline for tissue
preparation is given in chapter 7.3. The results of these measurements should be
compared to results for similar tissue from the literature.
The IAD program can also be improved, as the iteration procedure does not
terminate for all input values. A first step towards this problem could be an
examination of other two-dimensional optimization algorithms (apart from amoeba),
that might be more reliable, maybe at the cost of the calculation time.
To simplify the handling of the system, the measurement-calculation interface should
be improved. So far, the procedure was doing the measurements and writing the
results on a piece of paper, then preparing an input file for the IAD program with the
measurement results, and finally postprocessing the output file of the program,
containing the optical properties. An automation of this procedure is desired.
127
Therefore, in the future, the 3 detectors should be directly connected to a computer
providing the IAD program, so that the optical properties can be immediately
calculated (almost in real time for a small number of quadrature angles). This might
require the installation of an A/D converter, depending on the detectors used for the
measurements.
An important advantage of the double integrating sphere system is the possibility of a
simultaneous measurement of all the quantities necessary to obtain the optical
properties. This facilitates e.g. measurements with a sample being gradually heated
up. Other types of measurements that could take advantage of the simultaneous
acquisition of all necessary data are measurements with liquid samples, at different
flow velocities. For example, the effect of the blood flow velocity on the optical
properties can be examined with the double integrating sphere system. In order to do
this, a special sample holder has to be constructed, to facilitate measurements on
flowing liquids.
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10 Appendix
10.1 Interdependence between ν, µ, µ´ and φ
The interdependence between the scattering angle θ (angle between the direction of
the incident light s´ and the direction of the scattered light s) and the angles ϕ´ and ϕ
(angles between s´ or s, respectively, and the normal to the layer surface, ez) can be
determined by the use of the coordinate system introduced in Fig. 10.1. z is the axis
normal to the sample surface, and x and y are defined in a way, that the direction
vector s´ of the incident light lies in the xz-plane [12].
The cosine of the scattering angle is given by the scalar product of the unit vectors s´ and s, associated with the incident and scattered light directions:
Fig. 10.1: Interdependence between the scattering angle and the angles ϕ´ and ϕ
νθ ==⋅′ )cos(ss (10.1)
From Fig. 10.1 can be further deduced, that
′
′−=
′
′
=′
µ
µ
ϕ
ϕ0
1
)cos(0
)sin( 2
s (10.2)
With the azimuthal angle φ from the positive x-axis to the projection of s onto the xy-
plane, we find
129
⋅−⋅−
=
⋅⋅
=µ
φµφµ
ϕφϕφϕ
)sin(1)cos(1
)cos()sin()sin()cos()sin(
2
2
s (10.3)
By substituting the expressions (5.16) and (5.17) into expression (5.15), we obtain
)cos(11 22 φµµµµν −′−+′= (10.4)
10.2 Bracketing and parabolic interpolation
The root f(x0)=0 of a function f(x) is bracketed by a pair of points a<b, if we have
opposite signs of the function values at the two points, i.e. either f(a)>0 and f(b)<0, or
f(a)<0 and f(b)>0. If f is a continuous function, there has to be a root f(x0)=0 with
a<x0<b. One possibility to find this root is to choose a point c in the interval (a,b),
preferably the midpoint. Either f(a) and f(c), or f(b) and f(c) must have opposite signs.
Thus, we now have the minimum bracketed by the points a<c or c<b, respectively.
This procedure can be repeated, until the interval size is tolerably small [13].
However, in order to find the quadrature points for Radau quadrature (see chapter
5.2.1), Newton´s method is used for root finding.
To find a minimum of a function f(x), we need a bracketing triplet of points a<b<c ,
such that we have f(b)<f(a) and f(b)<f(c). In this case, the (nonsingular) function f has
to have a minimum in the interval (a,c) (see chapter 10.3). In the IAD program (see
chapter 6), the subroutine mnbrak is used to get starting points for the one-
dimensional minimum search with Brent´s method (see chapter 10.3).
mnbrak is called with two arbitrary initial points (a, f(a)) and (b, f(b)) given, and works
as follows: We check for f(a)>f(b). If this is not the case, a and b are switched, so that
we can go downhill in the direction from a to b [13]. The special case f(a)=f(b) is not
considered in the IAD program [24], the bracketing algorithm might fail for this case.
The first guess for the third point c is the point c=b+1.618(b-a). The odd factor is
chosen, such that b divides the interval [a, c] into fractions of the so-called �golden
mean�. This is done to optimize the searching algorithm following the bracketing [13].
If f(b)<f(c), we have already bracketed the minimum, and the algorithm terminates.
130
Otherwise, we have to continue searching. The next point is generated by parabolic
interpolation. The formula to find the minimum d of a parabola through the 3 points
(a, f(a)), (b, f(b)) and (c, f(c)) is [13]:
))()()(())()()(())()(()())()(()(
21 22
afbfcbcfbfabafbfcbcfbfabbd
−−−−−−−−−−
⋅−= (10.5)
If d is between b and c (see Fig. 10.2A), we can check if the triplets (b, d, c) or
(a, b, d) bracket the minimum, and in this case terminate the algorithm. Otherwise,
we have to repeat the interpolation algorithm with the new triplet (b, d, c).
If d is not between b and c, but further in the downward direction (d>b for a<b or d<b
for a>b, see Fig. 10.2B), we start over with the triplet (b, c, d). If d is further away
from c than the �limit� dlim=b+100(c-b), we set d=dlim.
If d is on the other side of b (either between a and b, or in the uphill direction), the
parabola through the three points is open to the negative side, i.e. d locates a
maximum. In this case, the point d for the next triplet (b, c, d) is found once more by
�golden section search�: d=c+1.618(c-b) [13, 24].
Fig. 10.2: Parabolic interpolation
In Fig. 10.2, the parabolic interpolations are successful, we find the bracketing points
(b, d, c). This is not always the case, but after a sufficient number of iteration steps, a
minimum is bracketed for every function with a minimum in the interval [-∞, +∞].
131
10.3 Golden section search and Brent´s method
There are several methods to find a minimum bracketed by three points a, b and c
(see chapter 10.2). One possibility is the �golden section search�: we choose a new
point d with either a<d<b or b<d<c. Without loss of generality, we choose a<d<b.
Then we evaluate f(d). If f(d)<f(b), the new bracketing triplet is a<d<b (see Fig.
10.3A). If f(d)>f(b), we have the new triplet d<b<c (see Fig. 10.3B). This procedure
can be repeated, until the distance between the two outer points is tolerably small.
Fig. 10.3: Algorithm to find a minimum bracketed by three points a<b<c
This algorithm is called �golden section search�, because the �new point� d is chosen
in a way that it divides the interval [a, b] into fractions with the lengths 0.618(b-a) and
0.382(b-a). This assures the fastest convergence for the worst possible case [13].
In the IAD program, we use a different algorithm called �Brent´s method� [13]. It
combines the golden section search with parabolic interpolation (see chapter 10.2):
We try a parabolic interpolation through the three points with the least function values
found so far. If the parabolic step falls within the bounding interval and implies a
movement from the best current value that is not too big [13], we accept the parabolic
interpolation. Otherwise, we choose the next point based on golden section search.
10.4 Two-dimensional minimization with the Amoeba algorithm
The subroutine amoeba is based on the downhill simplex method by Nelder and
Mead. It is used in the IAD program for a two-dimensional minimum search (see
chapters 6.4.3-6.4.7), without the necessity of function derivations [13].
132
The amoeba-algorithm can be used for an arbitrary number N of dimensions.
However, in the IAD program, it is used only for the two-dimensional case: the
minimization of a function M(p) with p=(x,y) (e.g. x=acomp and y=gcomp in chapter
6.4.3). Thus, we only consider the case N=2.
An N-dimensional simplex is a geometrical figure consisting of N+1 points and all
their interconnecting lines and areas. A two-dimensional simplex is a triangle. This
triangle lies in the (x,y)-plane. We have to pass an initial simplex (p1=(x1,y1),
p2=(x2,y2), p3=(x3,y3) to the amoeba-routine. This simplex should already be near the
desired minimum, which is assured by the grid algorithm (or the quick search
method) described in chapter 6. During the minimization algorithm, the simplex will
be moved and contracted around the minimum point.
At the beginning of each iteration step, we check the function values at the three
simplex points. The point with the highest function value (the worst point) will be
called �high point� phi. The point with the second-highest function value (�next-highest
point�) is referred to as pnhi. The point with the lowest function value (the best point) is
called �low point� plo (see Fig. 10.4).
The first try to �improve� the simplex is a reflection away from the high point: The new
point ptry1 is generated by a point reflection of phi around the middle of the line
between plo and pnhi (see Fig. 10.4A). We then check the function value M(ptry1). Now
we have several possibilities:
- If we find M(ptry1)≤M(plo) (the new point is at least equally good than plo), we seem
to move in the right direction. Therefore, we expand the simplex by a factor 2 into
this direction (see Fig. 10.4B), and go on to the next iteration step with the new
simplex (ptry2, plo, pnhi).
- If we find M(plo)<M(ptry1)<M(pnhi) (the new point is worse than plo, but better than
pnhi), we keep the point and use the new simplex (plo, pnhi, ptry1) in the next step.
- If we find M(ptry1)≥M(pnhi) (the new point is not better than plo and pnhi), we
abandon it and try instead a contraction along one dimension from the high point,
133
to get the new point ptry3 (see Fig. 10.4C). Then we check again: If M(ptry3)<M(phi)
(the new point is an improvement, compared to phi), we keep ptry3 and use the
new simplex (plo, pnhi, ptry3) in the next step. If M(ptry3)≥M(phi) (still no improvement
of the simplex found), we use a contraction along all dimensions towards the low
point, yielding the new points ptry4 and ptry5 (see Fig. 10.4D). The next step is then
taken with the simplex (plo, ptry4, ptry5).
Fig. 10.4: Illustration of the amoeba algorithm
In general, the algorithm terminates, if the difference between M(phi) and M(plo) is
tolerably small. However, in our IAD case, we know that the function value at the
minimum is zero (see definiton of M in chapter 6.4), therefore we check for
M(plo)<Mtol (with a default value Mtol=10-5) as the termination criterion. The point plo is
then returned to the main program as the minimum of the function M. The algorithm
also terminates after a maximum number of iteration steps.
134
10.5 The diffusion approximation of the radiative transport equation
The radiative transport equation is given by [10]
∫ ′′′++−=∇⋅π
ωµµµ4
),(),(),()(),()( dLpLL sas srsssrsrs (10.6)
L(r,s) is the radiance (in W·mm-2·sr-1) at the position denoted by the vector r, in the
direction of the unit vector s. The integration variable dω´ is a differential solid angle
in the s´ direction. µs, µa and the phase function p(s´,s) are defined in chapter 2.2.
The left hand side of expression (10.6) denotes the rate of change of the intensity at
the point indicated by the vector r, in the direction s. This change equals the intensity
lost through scattering and absorption plus the intensity gained due to light scattering
from all other directions into the direction s (right hand side of expression (10.5)).
As announced in chapter 2.1, the scattering medium is an infinite slab with finite
thickness and infinite breadth. The inward normal to the left side of the slab is defined
as the positive z-direction. The left surface of the slab is illuminated with a
monodirectional flux πF0(r), depending only on the radial component r of the position
vector r (in cylindric coordinates), as the left surface is defined as z=0 (see Fig. 10.5).
The flux is assumed to be independent of the azimuthal angle φ.
The incident radiance on the left surface is then given by [10]
)()(21))()(()(
21),( 000 µµδδππ
−=⋅−⋅⋅= rFsrFrLinc z0z eess (10.7)
ez is a unit vector in the z direction, and s0 is a unit vector in the direction of the
incident flux (in Fig. 10.5, we have s0=ez). µ (µ0) is the cosine of the angle between s
(s0) and ez. The factor π in expression (10.7) is included for historic reasons, so that
an isotropic diffuse source F0(r) will cause a flux πF0(r) at a surface (µ≥0 as the flux is
directed forward) [10]:
)()()()( 0
2
0
1
00
)0(20 rFddrFdrF πφµµω
π
µπ
==⋅⋅ ∫∫∫≥
zes (10.8)
135
Fig. 10.5: Incident flux πF0(r)
For a collimated irradiance E0(r), we find E0(r)= πF0(r).
If we substitute the Delta-Eddington approximation of the phase function (expression
(2.19)) into the transport equation (10.6), we obtain
[ ]∫
∫
′⋅′′+−′+
′⋅′−′⋅++−=∇⋅
π
π
ωπµ
ωδπµ
µµ
4
4
)(31)1)(,(4
)1(),(2
),()(),()(
dgfL
dLfLL
s
sas
sssr
sssrsrsrs (10.9)
The Delta function in the Delta-Eddington approximation can be written as [5]:
)()(2)1( φφδµµδπδ ′−′−⋅=⋅′− ss (10.10)
The cosines µ and µ´ and the azimuthal angles φ and φ´ determine the direction of
the unit vectors s and s´, respectively. With the use of expression (10.10) and the
properties of the Dirac Delta function (expression (2.20)), the first integral in
expression (10.9) can be solved:
),(2)()(),(2)1(),(2
0
1
14
srsrsssr ′⋅=′−′−′⋅=′⋅′−′ ∫ ∫∫= −=
LddLdL πφµφφδµµδπωδπ
φ νπ
(10.11)
Substituting expression (10.11) in expression (10.9) yields
136
[ ]∫ ′⋅′′+′′
++′−=∇⋅π
ωπµ
µµ4
)(31),(4
),()(),()( dgLLL sas sssrsrsrs (10.12)
with the �reduced scattering coefficient�
ss f µµ )1( −=′ (10.13)
The scattering coefficient is reduced because light scattered into the forward peak
(Dirac Delta term in expression (2.19)) is undistinguishable from unscattered light.
The fraction of light that is not scattered into the forward peak (�Eddington� term, see
expression (2.19)) is given by (1-f). Thus, the scattering coefficient has to be reduced
by this factor [10].
The radiance L(r,s) can be divided into a collimated and a diffuse part [10]:
),(),(),( srsrsr dc LLL += (10.14)
The collimated part includes both the light scattered into a direction parallel to the
incident beam, and the unscattered light.
As the light scattered forward is included in the collimated radiance, the attenuation
of the beam is given by the transport coefficient µt´=µs´+µa, instead of the total
attenuation coefficient µt=µs+µa. Thus, the collimated radiance is described by the
differential equation
),(),()(),()( srsrsrs ctcasc LLL ′−=+′−=∇⋅ µµµ (10.15)
with the boundary condition
)()(21)1(),()1(),( 000
µµδ −⋅−=−==
rFrrLrL sincszc ssr (10.16)
as the collimated light entering the slab is denoted by the incident radiance
(expression (10.7)), reduced by a factor (1-rs) to account for the loss of light due to
the specular reflection from the surface. rs is the specular reflection coefficient, which
can be determined by the Fresnel reflection formula for an angle of incidence
cos(θ0)=s0·ez [10]:
137
+−
++−
=)(tan)(tan
)(sin)(sin
21
02
02
02
02
t
t
t
tsr θθ
θθθθθθ (10.17)
The transmitted angle θt can be calculated from the incident angle θ0 and the indices
of refraction nair and ntissue, by the use of Snell´s law:
0sinsin θθtissue
airt n
n= (10.18)
The expressions (10.15) and (10.16) yield the collimated radiance
)()(exp)()(21)1(),( 00
000 µµδ
µµ
µµδ −⋅=
′−⋅−⋅−= rL
zrFrL c
tsc sr (10.19)
with the abbreviation
′−⋅⋅−=
000 exp)(
21)1()(
µµ z
rFrrL tsc (10.20)
In the expressions (10.19) and (10.20), z/µ0 is the distance that the light (incident at
an angle with the cosine µ0) travels in the tissue until it reaches a depth of z.
By substituting the expressions (10.14) and (10.19) in expression (10.12), we find:
[ ] [ ]∫∫ ′⋅′′+−′′
+′⋅′′+′′
+
+′−+′−=∇⋅+∇⋅
ππ
ωµµδπµ
ωπµ
µµµµ
400
4
)(31)()(4
)(31),(4
),()(),()(),()(),()(
dgrLdgL
LLLL
cs
ds
dascasdc
sssssr
srsrsrssrs (10.21)
With the help of expression (10.15), this equation can be simplified:
[ ] [ ]∫∫ ′⋅′′+−′′
+′⋅′′+′′
+
+′−=∇⋅
ππ
ωµµδπµ
ωπµ
µµ
400
4
)(31)()(4
)(31),(4
),()(),()(
dgrLdgL
LL
cs
ds
dasd
sssssr
srsrs (10.22)
s´·s is given by (see chapter 10.1)
)cos(11 22 φφµµµµ −′−′−−′=⋅′ ss (10.23)
The second integral in expression (10.22) can be solved by using the following
property of the Dirac Delta function [10]:
138
∫∫ ′′=′−′′′π
π
φφµωµµδφµ2
00
40 ),()(),( dfdf (10.24)
With the use of the expressions (10.23) and (10.24), the second integral in
expression (10.22) can be written as
[ ]
[ ]
[ ][ ]µµπ
φφφµµµµ
ωφφµµµµµµδ
ωµµδ
π
π
π
00
2
0
22000
4
2200
400
31)(2
))cos(11(31)(
))cos(11(31)()(
)(31)()(
grL
dgrL
dgrL
dgrL
c
c
c
c
′+⋅=
′−′−−−′+⋅=
′−′−′−−′′+⋅−′=
′⋅′′+⋅−′
∫
∫
∫ ss
(10.25)
Substitution in expression (10.22) finally yields
[ ]
[ ]µµµµ
ππµ
ωπµ
µµπ
00
0
4
31exp)()1(4
)(31),(4
),()(),()(
gz
rFr
dgLLL
ts
s
ds
dasd
′+⋅
′−⋅−
′+
′⋅′′+′′
++′−=∇⋅ ∫ sssrsrsrs
(10.26)
The diffuse radiance Ld(r,s) can be expressed as [10]
srFrsr d ⋅+= )(43
)(41
),(π
ϕπ ddL (10.27)
with an isotropic fluence contribution ϕd(r) and an anisotropic flux (per unit area)
contribution Fd(r). The factors 1/4π and 3/4π result from normalization, the right hand
side of expression (10.27) represents the first two terms of the Taylor expansion for
Ld(r,s) [10].
Substitution of expression (10.27) in (10.26) and multiplication with the factor 4π lead
to the following equation:
[ ]
[ ][ ]
[ ]µµµµ
πµ
ωϕπµ
ϕµµϕ
π
00
0
4
31exp)()1(
)(31)(3)(4
)(3)()())()((3)()(
gzrFr
dg
tss
ds
dasd
′+⋅
′−⋅−′+
′⋅′′+′⋅+′
+
⋅++′−=⋅∇⋅+∇⋅
∫ sssrFr
srFrsrFsrs
d
dd
(10.28)
139
By using the solid angle integration identities [10]
0))()(();(3
4))((;0444
=⋅⋅⋅⋅=⋅⋅=⋅ ∫∫∫πππ
ωπωω ddd CsBsAsBΑBsAsAs (10.29)
for all vectors A, B, C, the integral in expression (10.28) can be solved:
[ ][ ]
[ ]
))((1200)(4
)()(9)()(3)(3)(
)(31)(3)(
4
4
srFr
srFssrsssrFr
sssrFr
d
dd
d
⋅′+++=
′′⋅⋅⋅′′+⋅′′+′⋅+=
′⋅′′+′⋅+
∫
∫
g
dgg
dg
d
dd
d
ππϕ
ωϕϕ
ωϕ
π
π
(10.30)
Substitution in expression (10.28) yields
[ ]µµµµ
πµ
µµϕµϕ
00
0 31exp)()1(
))()()1((3)())()((3)()(
gzrFr
g
tss
asdad
′+⋅
′−⋅−′+
⋅+′′−−−=⋅∇⋅+∇⋅ srFrsrFsrs dd
(10.31)
By integrating expression (10.31) over all directions s (with the use of expression
(10.29)), we find
[ ]∫−
′+⋅
′−⋅−′+
−−=⋅∇+
1
10
00 31exp)()1(2
0)(4)((40
µµµµµ
πµπ
ϕπµπ
dgz
rFr tss
da rrFd
(10.32)
resulting in the following equation for the diffuse flux [10]:
′−⋅−′+−=⋅∇
00 exp)()1()()(
µµ
πµϕµz
rFr tssda rrFd (10.33)
To obtain an equation for the diffuse fluence ϕd(r), we multiply expression (10.31)
with s and integrate again over all directions s [10]:
[ ]∫ ′+⋅
′−⋅−′+
+′′−−=+⋅∇
π
ωµµµµ
πµ
µµπϕπ
40
00 31exp)()1(
)())1((40)(3
4
dgz
rFr
g
tss
asd
s
rFr d
(10.34)
By using the identity µ=s·ez, we can solve the integral and finally find
140
zd erFr
′−⋅−′′+′−=
000 exp)()1(3)(3)(
µµ
πµµµϕz
rFrg tsstrd (10.35)
with the reduced transport coefficient
astr g µµµ +′′−=′ )1( (10.36)
Taking the divergence of expression (10.35) and solving for the change in the diffuse
flux Fd(r) leads to
′−⋅−′
′′′−⋅∇′−=⋅∇
00
2 exp)()1()(3
1)(µµ
πµ
µµϕ
µ
zrFr
g ts
tr
tsd
tr
rrFd (10.37)
In expression (10.37), the collimated light term results from
′−′−=
′−⋅∇
000
expexpµµ
µµ
µµ zz ttt
ze (10.38)
By equating the expressions (10.33) and (10.37), we find a Helmholtz equation [10]:
′−⋅−′′+′′−=′−⋅∇
00
2 exp)()1)((3)(3)(µµ
πµµµϕµµϕz
rFrg tsttrsdatrd rr (10.39)
This equation is the diffusion equation for light transport in tissue.
141
11 Formula index
θ scattering angle rad
ν cosine of the scattering angle θ -
τ optical thickness -
ω solid angle sr
φ azimuthal angle rad
δ detector area mm2
α sphere wall area, relative to total sphere area -
ϕ angle of reflected/transmitted light, respective to surface normal rad
ϕ´ angle of incident light, respective to surface normal rad
µ´, µ cosines of ϕ´ and ϕ -
µa absorption coefficient mm-1
µi quadrature angle -
µs scattering coefficient mm-1
µt attenuation coefficient mm-1
µt´ transport coefficient mm-1 a albedo - A sphere area mm2
a*,a´,τ*,τ´ reduced optical properties - b1, b2 sphere constants - d physical thickness mm f factor in the Delta-Eddington approximation - g anisotropy coefficient - h hole area mm2 h redistribution function - L radiance, light intensity W/mm2 la, ls average free path between two absorption or scattering events mm m reflection factor of the sphere wall - M quadrature angles in vector notation - n number of quadrature angles - na, ns, ng refraction indices of atmosphere, sample, glass - P light power W
p(ν) scattering phase function - Pi(x) Legendre polynomial -
142
r position vector - R sphere radius mm r reflection factor of the detector -
R(µ´,µ) reflection function - Rd,cd,c,ct reflection factors - rin, rout Fresnel reflection factors - Rjk, Tjk reflection and transmission function in matrix notation - s direction vector of scattered light - s sample area mm2 s´ direction vector of incident light -
T(µ´,µ) transmission function - Td,cd,c,ct transmission factors - Vi voltage measurement V Vr,t,c% relative measurements with the three detectors - wi quadrature weights - β factor in the modified Henyey-Greenstein function -
143
12 References
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optical properties of tissues between 630 and 1064 nm. Physics in Medicine and
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M.J.C.: Two integrating spheres with an intervening scattering sample. Journal of
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van Gemert, M.J.C.: Double-integrating-sphere system for measuring the optical
144
properties of tissue. Applied Optics: Optical Technology and Biomedical Optics
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[19] http://www.labsphere.com/uploadDocs/A%20Guide%20to%20Integrating%20Sp
here%20Theory%20and%20Applications_kb100.pdf
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PnRootFinder.html
Linköpings tekniska högskola Institutionen för medicinsk teknik
Rapportnr: LiTH-IMT/ERASMUS-R--04/27--SE Datum: 2004-12-20
Svensk titel
Engelsk titel
Double integrating spheres: A method for assessment of optical properties of biological tissues
Författare
Wigand Poppendieck
Uppdragsgivare: IMT, IBMT Stuttgart
Rapporttyp: Diplomarbete
Rapportspråk: Engelska
Sammanfattning (högst 150 ord). Abstract (150 words)
The determination of the optical properties of biological tissue is an important issue in laser medicine. The optical properties define the tissue´s absorption and scattering behaviour, and can be expressed by quantities such as the albedo, the optical thickness and the anisotropy coefficient. During this project, a measurement system for the determination of the optical properties was built up. The system consists of a double integrating sphere set-up to perform the necessary reflection and transmission measurements, and a computer algorithm to calculate the optical properties from the measured data. This algorithm is called Inverse Adding Doubling method, and is based on a one-dimensional transport model. First measurements were conducted with the system, including measurements with phantom media (Intralipid-ink solutions) and with cartilage samples taken from the human knee joint.This work also includes an investigation about the preparation of tissue samples for optical measurements.
Nyckelord (högst 8) Keyword (8 words) Double integrating sphere, optical properties, absorption, scattering, Inverse Adding Doubling Bibliotekets anteckningar:
