Double integrating spheres: A method for assessment of optical properties of biological tissues

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Double integating spheres: A method for assess-

ment of optical properties of biological tissues

Wigand Poppendieck

2004-12-20

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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

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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.

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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

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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).

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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.

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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 ) ( ) , ( p p θ pν p s′s = s′⋅s = = (2.1)

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1 ) ( 4 =

∫

π ω ν d p _(2.2)

dω is a differential solid angle. By using spherical coordinates, it can be expressed by

φ θ θ

ω d d

d = 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

1 sin ) ( ) ( 2 0 0 4 = ⋅ ⋅ =

∫ ∫

∫

= = π φ π θ π φ θ θ ν ω ν d p d d p _(2.4)

With the use of

θ ν θ θ ν sin cos ⇒ d =− d = (2.5) expression (2.4) yields 1 ) ( ) ( ) ( 2 0 1 1 2 0 1 1 4 = ⋅ + = ⋅ − =

∫ ∫

∫

= =− = − = π φ ν π φ ν π φ ν ν φ ν ν ω ν d p d d p d d p _(2.6)

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1 ) ( 2 ) ( 1 1 4 = =

_∫

∫

− ν ν π ω ν π d p d p (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 1 4 ) ( 2 ) (ν ν ω π ν ν ν π d p d p g (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

π ν 4 1 ) ( = iso p (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.

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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]: 2 3 2 2 ) 2 1 ( 1 4 1 ) ( ν π ν g g g p_HG − + − ⋅ = (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): 1 2 1 2 1 ) 2 1 ( 1 2 1 ) ( 2 1 1 2 2 1 1 2 3 2 2 1 1 = − + − = − + − = − − −

∫

ν ν ν ν ν π g g g g d g g g d p_HG (2.11)

The Henyey-Greenstein function is also compatible with expression (2.8):

g g g g g g g g d p_HG =         − + + + − + − = − −

∫

1 1 2 2 2 2 2 1 1 1 2 1 2 1 4 1 ) ( 2 ν ν ν ν ν π (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].

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The Henyey-Greenstein approximation can be further adjusted by choosing a so-called „modified Henyey-Greenstein function“ [10]:

        − + − − + ⋅ = 2 3 2 2 ) 2 1 ( 1 ) 1 ( 4 1 ) ( ν β β π ν m m m mHG g g g p (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

m HG mHG d p d g p g = − ⋅         ⋅ − + = =

∫

− − − ) 1 ( ) ( ) 1 ( 2 1 2 1 ) ( 2 1 1 1 1 2 1 1 β ν ν ν β βν ν ν ν π (2.14)

Another approximation often used is given by the Eddington phase function [10]

[

ν

]

π ν g p_E 1 3 4 1 ) ( = ⋅ + (2.15)

meeting the requirements (2.7) and (2.8):

g g d p g d p_E _E =     ₊ = =     + = − − − −

∫

1 1 3 2 1 1 1 1 2 1 1 2 1 2 1 ) ( 2 ; 1 2 3 2 1 ) ( 2π ν ν ν ν π ν ν ν ν ν (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]

[

2 (1 ) (1 )(1 3 )

]

4 1 ) ( δ ν ν π ν δ f f g p_E = ⋅ − + − + ′ (2.17)

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1 ) ( ; 0 , 0 0 , ) ( =    ≠ = ∞ = +∞

_∫

∞ − dx x x x x δ δ (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

g f f g f f d p g _E = + − ′     ₋ ₊ ₋ ₊ _′ = = − −

∫

) 1 ( ) 2 1 ( ) 1 ( ) 1 ( 2 2 1 ) ( 2 1 1 2 1 1 ν ν ν σ ν ν ν π _δ (2.19)

with the unity step function defined as

   < ≥ = =

∫

∞ − 0, 0 0 , 1 ) ( ) ( x x dt t x x δ σ (2.20)

The normalization condition (2.7) is also met by the Delta-Eddington approximation:

1 2 3 2 2 1 ) ( 2 1 1 2 1 1 =     ₊ ₊ = − −

∫

ν ν ν ν π p_δ_E d f g (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 2 2 ₋ ₊ ₊ ₌ − n n _n _n _P_n dx dP x dx P d x (2.22)

The first Legendre polynomials are given by [21]

) 5 105 315 231 ( ) ( ) 15 70 63 ( ) ( ) 3 30 35 ( ) ( ) 3 5 ( ) ( ) 1 3 ( ) ( ) ( 1 ) ( 2 4 6 161 6 3 5 81 5 2 4 81 4 3 2 1 3 2 2 1 2 1 0 − + − = + − = + − = − = − = = = x x x x P x x x x P x x x P x x x P x x P x x P x P (2.23)

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[

1 3 ( ) 5 ( ) ...

]

4 1 ) ( ) 1 2 ( 4 1 ) ( 2 ₂ 1 0 + + + ⋅ = + ⋅ =

∑

∞ = ν ν π ν π ν n g P gP g P p n n n HG (2.24)

For the modified Henyey-Greenstein function, we find the following expansion [10]:

[

1 3(1 ) ( ) 5(1 ) ( ) ...

]

4 1 ) ( ) 1 2 ( ) 1 ( ) ( 4 1 ) ( 2 2 1 0 0 + − + − + ⋅ =       _⋅ ₊ ₋ _⋅ ₊ ⋅ =

∑

∞ = ν β ν β π ν β ν β π ν P g P g P g n P p m m n n n m mHG (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 ( )

]

4 1 ) ( ₀ ν ₁ν π ν P gP p_E = ⋅ + (2.26)

By using the Legendre expansion for the Dirac Delta function [17]

∑

∞ = + = − 0 ) ( ) ( ) 1 2 ( 2 1 ) ( n n n x P y P n y x δ (2.27)

the Delta-Eddington function (expression (2.17)) can be expanded as [10]

[

]

[

1 3 (1 ) ( ) 5 ( ) ...

]

4 1 ) ( ) 1 2 ( )) ( 3 1 ( ) 1 ( 4 1 ) ( 2 1 0 1 + + − ′ + + ⋅ =       ₋ _⋅ ₊ _′ ₊ _⋅ ₊ ⋅ =

∑

∞ = ν ν π ν ν π ν δ fP P f g f P n f P g f p n n E (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:

[

]

f n g n f g f g g f f g n m m m m ) 1 2 ( ) 1 )( 1 2 ( 7 ) 1 ( 7 5 ) 1 ( 5 ) 1 ( 3 ) 1 ( 3 3 2 + = − + = − = − ′ − + = − β β β β M M M (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

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2 11 2 (1 ) , ) 1 ( m m m m g g g g g f − − = ′ − = −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]:

τ µ µ µ µ d a s a s s ₌ + = (2.31)

(

s a

)

d µ µ τ = ⋅ + (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

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- 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: d a s τ µ = ⋅ (2.33) d a a τ µ =(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

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(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: cd c ct cd c ct R R T T T R = + ; = + (2.35)

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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.

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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

2

4 R

A= 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

      ₊ ₊ − = + + − = A h A s A A h s A δ δ α ( ) 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]

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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

mP

P₁ = (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:

1 1

, _AP

P_δ = δ (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):

1 1 , 1 1 , 1 1 , ; ; P A h P P A s P P P_α =α _s = _h = (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

F P P A s R P m P A r P₂ = δ ₁+ α ₁+ _d ₁= ₁ (3.6)

with the abbreviation

A s R m A r F = δ + α + _d (3.7)

Of this second reflection, the detector collects (cf. expression (3.4))

F P A

P_δ_,₂ = δ ₁ (3.8)

With the use of similar considerations, light from the third reflection is incident upon the detector with a power of

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FF P A

P_δ_,₃ = δ ₁ (3.9)

In general, the power of the light collected by the detector from the nth reflection is given by 1 1 ,n = PFn− A P_δ δ (3.10)

The total power collected by the detector can be obtained by summing from n=1 to ∞:

∑

∞ = − ⋅ = 1 1 1 n n F P A P_δ δ (3.11)

By using the equation for the geometric series (F<1), we find

1 1 )] ( ) ( [ 1 1 1 1 P A r A s R m A F P A P d δ α δ δ δ = ⋅ ₋ = ⋅ ₋ ₊ ₊ (3.12)

With the use of expression (3.3), expression (3.12) can be written as

P A r A s R m m A P d( ) ( )] [ 1 α δ δ δ = ⋅ ₋ ₊ ₊ (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

P R

P_c = _c (3.14)

and a diffuse part

P R

P_cd = _cd (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

P mR

P_cr = _c (3.16)

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P R mR P P P₁ = _cr + _cd =( _c + _cd) (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):

P A r A s R m R mR A P d cd c )] ( ) ( [ 1 α δ δ 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)

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P T

P₁= _d (3.19)

The light power collected by the detector is then (expression 3.12):

P A r A s T m T A P d d )] ( ) ( [ 1 α δ δ 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

P T

P_c = _c (3.21)

and a diffuse part

P T

P_cd = _cd (3.22)

The collimated part is reflected diffusely on the sphere wall. This leads to a diffuse light source of the power

P mT

P_cr = _c (3.23)

The total power of the two diffuse sources (expressions (3.22) and (3.23) is therefore given by P T mT P P P₁ = _cr + _cd =( _c + _cd) (3.24)

leading to a power collected by the detector (equation 3.12) of

P A r A s T m T mT A P d cd c )] ( ) ( [ 1 α δ δ 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)

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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 m R s A r A m A α _d δ δ + + − ⋅ B Reflection collimated )] ( ) ( [ 1 m R s A r A R mR A _d cd c δ α δ + + − + ⋅ C Transmission diffuse )] ( ) ( [ 1 m T s A r A T A _d d δ α δ + + − ⋅ D Transmission collimated )] ( ) ( [ 1 m T s A r A T mT A _d cd c δ α δ + + − + ⋅

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:

D case D case C case C case B case B case A case A case S G P S G P S G P S G P ⋅ = ⋅ = ⋅ = ⋅ = , , , , 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):

)] ( ) ( [ 1 1 A r A s T m A G d δ α δ + + − ⋅ = _(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):

P T mT S P T S P R mR S P m S cd c D case d C case cd c B case A case ⋅ + = ⋅ = ⋅ + = ⋅ = ] [ ] [ (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

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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

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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): P V m A P r r r r r = ⋅ δ δ1 (3.29)

with the abbreviation

      + + − = r r r r d r r r A r A s R m V 1 α δ (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)):

P V m A s P r r r s1= ⋅ (3.31)

A part Pt1 of this light is transmitted into the transmittance sphere, depending on the

diffuse transmittance factor Td:

P V m A s T P T P r r r d s d t1 = 1= ⋅ (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):

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P V m A s V T A P V A P r r r t d t t t t t t t = ⋅ = ⋅ ⋅ δ δ δ1 1 1 (3.33) where Vt is defined similar to Vr:

      + + − = t t t t d t t t A r A s R m V 1 α δ (3.34)

Another part of Pt1 is incident back upon the sample:

P V m A s V T A s P V A s P r r r t d t t t t s = ⋅ = ⋅ ⋅ ⋅ ′ 1 1 1 (3.35) The part of Ps1´ that is transmitted back to the reflectance sphere is given by

P V m A s V T A s T P T P r r r t d t d s d t = ⋅ ⋅ ⋅ ′ = ′ 1 1 (3.36)

A fraction of this light is again incident upon the first detector:

P V m A s V T A s V T A P V A P r r r t d t r d r r t r r r r = ⋅ ⋅ ⋅ ⋅ ⋅ ′ = δ δ δ 2 1 1 (3.37) By using the abbreviations

t d t t r d r r _V T A s V T A s ⋅ = ⋅ = τ τ ; _(3.38)

expression (3.37) can be rewritten as

P V m A P _r _t r r r r r = ⋅ ⋅τ τ ⋅ δ δ 2 (3.39)

The part of Pt1´ that is incident back upon the sample can be calculated by using the

abbreviations (3.38): P V m A s P _r _t r r r s2 = ⋅ ⋅τ τ ⋅ (3.40)

so that the power of the light travelling through the sample a third time is given by

P V m A s T P T P _r _t r r r d s d t2 = 2 = ⋅ ⋅τ τ ⋅ (3.41)

Now the second portion of light incident upon the detector in the transmittance sphere can be determined:

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P V m A s V T A P V A P _r _t r r r t d t t t t t t t = ⋅ = ⋅ ⋅ ⋅τ τ ⋅ δ δ δ2 2 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 P V m A P n t r r r r r rn = ⋅ ⋅ ⋅ −1 ) (τ τ δ δ (3.43)

The nth portion of light collected by the second detector can be calculated by comparing the expressions (3.33) and (3.42):

P V m A s V T A P n t r r r r t d t t tn = ⋅ ⋅ ⋅ ⋅ −1 ) (τ τ δ δ (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):

t r n t rτ _τ _τ τ − =

∑

∞ − 1 1 ) ( 1 1 (3.45) yielding the total light power collected by the detector in the reflectance sphere

P V m A P t r r r r r r = ⋅ ⋅ ₋_τ _τ ⋅ δ δ ₁ 1 (3.46) and the total light power collected by the detector in the transmittance sphere

P V m A s V T A P t r r r r t d t t t = ⋅ ⋅ ⋅ ₋_τ _τ ⋅ δ δ ₁ 1 (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.

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The light power generated from the sources is given by (3.17) for the reflectance sphere and by (3.24) for the transmittance sphere:

P R R m P₁_r =( _r _c + _cd) (3.48) P T T m P₁_t =( _t _c + _cd) (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

P T T m P₁_r_,_t =τ_t( _t _c + _cd) (3.50) P R R m P₁_t_,_t =τ_r( _r _c + _cd) (3.51) The total light power generated by the sources in the reflectance sphere and the transmittance sphere, respectively, can thus be determined as

P T T m R R m P P P₁_r_,_case_F = ₁_r + ₁_r_,_t =[ _r _c + _cd +τ_t( _t _c + _cd)] (3.52) P R R m T T m P P P₁_t_,_case_F = ₁_t + ₁_t_,_t =[ _t _c + _cd +τ_r( _r _c + _cd)] _(3.53) In chapter 3.2.1, the diffuse light sources in the two spheres were given by

P m V T A s P m P P m P _r r d r r r E case t r E case r, = 1, =τ = ⋅ 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:

P V T T m R R m A P t r r cd c t t cd c r r r r ₋ ⋅ + + + ⋅ = ) 1 ( ) ( τ τ τ δ δ (3.55)

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P V R R m T T m A P t r t cd c r r cd c t t t t ₋ ⋅ + + + ⋅ = ) 1 ( ) ( τ τ τ δ δ (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 t r r r r r r V m A P P τ τ δ δ − ⋅ ⋅ = 1 1 E Sphere wall Transmission t r r t r t t t V m A P P τ τ τ δ δ − ⋅ ⋅ = 1 Reflection ) 1 ( ) ( t r r cd c t t cd c r r r r V T T m R R m A P P τ τ τ δ δ − + + + = F Sample Transmission ) 1 ( ) ( t r t cd c r r cd c t t t t V R R m T T m A P P τ τ τ δ δ − + + + =

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):

E S G P E S G P E S G P E S G P F case t t F case t F case r r F case r E case t t E case t E case r r E case r ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = , , , , , , , , 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):

t t t t r r r r V A G V A G = δ ; = δ _(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:

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P R R m T T m S P T T m R R m S P m S P m S cd c r r cd c t F case t cd c t t cd c r F case r r r E case t r E case r ⋅ + + + = ⋅ + + + = ⋅ = ⋅ = )] ( [ )] ( [ , , , , 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: t r n t r E τ τ τ τ − = =

∑

∞ − 1 1 ) ( 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

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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):

) ( As R m

F = α + _d (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

α δ δ δ δ Pm A A s R P A F P A P_,₂ = ₁⋅ − ₁⋅ _d( )= ₁ (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

F P P A s R P m P₂ = α ₁+ _d( ) ₁ = ₁ (3.63)

Thus, the power collected by the detector from the third reflection is

F P m A m P A P_δ_,₃ =δ ₂ α = δ α ₁ (3.64)

and in general the detected power from the nth reflection yields to

2 1 , − = n n m PF A P_δ δ α (3.65)

The total detected power can be calculated along the lines of chapter 3.1.1.1:

P A s R m A s R m A P A s R m A s R A P d d d d _⋅ + − − ⋅ ⋅ = ⋅ + − − ⋅ = )] ( [ 1 )] ( 1 [ )] ( [ 1 ) ( 1 1 _α δ α δ δ (3.66)

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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: cr d d cr _m _R _s _A P A s R A P ⋅ + − − ⋅ = )] ( [ 1 ) ( 1 , _α δ δ (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

cd cd _A P

P_δ₁_, =δ ⋅ (3.68)

has to be subtracted from the total power given in expression (3.66):

cd d cd cd d d cd _m _R _s _A P m A P A P A s R m A s R A P ⋅ + − ⋅ = ⋅ − ⋅ + − − ⋅ = )] ( [ 1 )] ( [ 1 ) ( 1 , _α α δ δ α δ δ (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

P A s R m m R A s R mR A P P P d cd d c cd cr ₋ ₊ ⋅ + − ⋅ = + = )] ( [ 1 )] ( 1 [ , , _α α δ δ δ δ (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

P A s R m m T A P d d _⋅ + − ⋅ = )] ( [ 1 α α δ δ (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):

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P A s R m m T A s R mT A P d cd d c _⋅ + − + − ⋅ = )] ( [ 1 )] ( 1 [ α α δ 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 R s A

b_w = − _d (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

α

m

b_s = (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 [ A s R m A s R m A _d d + − − ⋅ ⋅ α δ B Reflection collimated )] ( [ 1 )] ( 1 [ A s R m m R A s R mR A _d cd d c + − + − ⋅ α α δ C Transmission diffuse )] ( [ 1 m R s A m T A _d d + − ⋅ α α δ D Transmission collimated )] ( [ 1 )] ( 1 [ A s R m m T A s R mT A _d cd d c + − + − ⋅ α α δ

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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): t t st r r sr t d wt r d wr R s A b R s A b m b m b =1− ( ); =1− ( ); = α ; = α (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):

P T T m R P R m S_r_,_case_F = _r _c ⋅ +[ _cd +τ_t( _t _c + _cd)]⋅ (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)): P R R m T P T m S_t_,_case_F = _t _c ⋅ +[ _cd +τ_r( _r _c + _cd)]⋅ _(3.77)

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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 t r r d r r r d r r r r A s R m A s R m A P P τ τ α δ δ − ⋅ + − − ⋅ ⋅ = 1 1 )] ( [ 1 )] ( 1 [ E Transmission t r r t d t t t t r t t t A s R m m m A P P τ τ τ α α δ δ − ⋅ + − ⋅ = 1 )] ( [ 1 Reflection ) 1 ( )] ( [ 1 ) ( )] ( 1 [ t r r d r r r r cd c t t r r cd r d c r r r r A s R m m T T m m R A s R R m A P P τ τ α α τ α δ δ − ⋅ + − + + + − ⋅ = F Transmission ) 1 ( )] ( [ 1 ) ( )] ( 1 [ t r t d t t t t cd c r r t t cd t d c t t t t A s R m m R R m m T A s R T m A P P τ τ α α τ α δ δ − ⋅ + − + + + − ⋅ =

3.4 Reference measurements with a single integrating sphere

3.4.1 Introduction of the sphere constants We introduce the „sphere constants“

α δ m m A b − ⋅ = ′ 1 1 (3.78) α m A s b − ⋅ = 1 1 2 (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]:

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P R b A s R b P d d A case 2 1 , 1 )] ( 1 [ − − ′ = δ (3.80) P R b A s R R R b P d d c cd B case 2 1 , 1 )]) ( 1 [ ( − − + ′ = α δ (3.81) P R b T b P d d C case 2 1 , ₁₋ ′ = α δ (3.82) P R b T A s R T b P d cd d c D case 2 1 , 1 ) )] ( 1 [ ( − + − ′ = α δ (3.83)

Along these lines, the sphere constants for the double integrating sphere set-up are defined as: t t t t t t r r r r r r m m A b m m A b α δ α δ − ⋅ = ′ − ⋅ = ′ 1 ; 1 1 1 (3.84) t t t t r r r r m A s b m A s b α α = ⋅ − − ⋅ = 1 1 ; 1 1 2 2 (3.85)

With these abbreviations, the equations for the double integrating sphere set-up (Table 3.4) can be written as [9]:

P T b b R b R b A s R R b b P d t r d t d r r d d t r E case r 2 2 2 2 2 2 1 , ) 1 )( 1 ( )] ( 1 )[ 1 ( − − − − − ′ = δ (3.86) P T b b R b R b T m b b P d t r d t d r d t r r t E case t 2 2 2 2 2 2 1 , ) 1 )( 1 ( − − − ′ = α δ (3.87) P T b b R b R b T m T T b R b R A s R R b P d t r d t d r c t cd d t r d t cd r r d c r F case r ₂ 2 2 2 2 2 2 1 , ) 1 )( 1 ( )} ( ) 1 )( )] ( 1 [ {( − − − + + − + − ′ = α α δ (3.88) P T b b R b R b R m R T b R b T A s R T b P d t r d t d r c r cd d r t d r cd t t d c t F case t 2 2 2 2 2 2 2 1 , ) 1 )( 1 ( )} ( ) 1 )( )] ( 1 [ {( − − − + + − + − ′ = α α δ (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:

δ δ KP

V = (3.90)