Download Department of Physics, Chemistry and Biology Master’s Thesis Lars Karlsson

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Department of Physics, Chemistry and Biology
Master’s Thesis
Infrared studies of trenches etched in silicon
Lars Karlsson
LITH-IFM-EX--07/1857--SE
Department of Physics, Chemistry and Biology
Linköpings universitet
SE-581 83 Linköping, Sweden
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Master’s Thesis
LITH-IFM-EX--07/1857--SE
Infrared studies of trenches etched in silicon
Lars Karlsson
Supervisor:
Hans Arwin
ifm, Linköpings universitet
Examiner:
Kenneth Jährendahl
ifm, Linköpings universitet
Linköping, 23 November, 2007
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Avdelning, Institution
Division, Department
Datum
Date
Laboratory of Applied Optics
Department of Physics, Chemistry and Biology
Linköpings universitet
SE-581 83 Linköping, Sweden
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
2007-11-23
—
LITH-IFM-EX--07/1857--SE
Serietitel och serienummer ISSN
Title of series, numbering
Övrig rapport
—
URL för elektronisk version
http://urn.kb.se/resolve?urn:nbn:se:liu:diva-7326
Titel
Title
Optiska studier av etsade kiselstrukturer med spektroskopisk ellipsometri i det
infraröda våglängdsområdet
Infrared studies of trenches etched in silicon
Författare Lars Karlsson
Author
Sammanfattning
Abstract
Previous studies of protein adsorption on silicon have been restricted by the choice
of a simple structure or large surface for protein to adsorb on. The aim of this
project was to develop an optical model for more complex nanostructures in form
of trenches etched in silicon and then examine if a protein would adsorb to the
surface. The method used was infrared ellipsometry.
The experimental values from measurements on the sample were used to develop an optical model that represent the nanostructure. A three-layered biaxial
model proved to be accurate. One sample was then exposed to the protein albumin
and then measured upon again. The results before and after protein adsorption
were compared and a small optical signature was found were it could be expected
for this specific protein. This shows that it is possible to detect adsorption in a
complex nanostructure and to develop an accurate optical model for said structure.
...
Nyckelord
Keywords
Silicon trenches, Infrared ellipsometry, protein adsorption
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Abstract
Previous studies of protein adsorption on silicon have been restricted by the choice
of a simple structure or large surface for protein to adsorb on. The aim of this
project was to develop an optical model for more complex nanostructures in form
of trenches etched in silicon and then examine if a protein would adsorb to the
surface. The method used was infrared ellipsometry.
The experimental values from measurements on the sample were used to develop an optical model that represent the nanostructure. A three-layered biaxial
model proved to be accurate. One sample was then exposed to the protein albumin
and then measured upon again. The results before and after protein adsorption
were compared and a small optical signature was found were it could be expected
for this specific protein. This shows that it is possible to detect adsorption in a
complex nanostructure and to develop an accurate optical model for said structure.
...
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Acknowledgements
This master’s thesis would not be possible without the help and support of Hans
Arwin whose enthusiasm for science has made a great impact on me as a person.
Of course I have to thank my friends for sticking with me trough thick and thin
over the years, especially Pontus, Daniel, Markus, Roger, Jonatan and Pelle. I can
not think of a better gang to ride into the sunset with now that this adventure is
over.
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Contents
1 Introduction
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
2 Theory
2.1 Maxwell’s equations, plane waves and polarization
2.1.1 Polarization . . . . . . . . . . . . . . . . . .
2.2 Ellipsometry . . . . . . . . . . . . . . . . . . . . .
2.2.1 Reflection ellipsometry . . . . . . . . . . . .
2.2.2 Generalized ellipsometry . . . . . . . . . . .
2.2.3 Infrared ellipsometry . . . . . . . . . . . . .
2.3 Optical modeling . . . . . . . . . . . . . . . . . . .
2.3.1 Reflectance . . . . . . . . . . . . . . . . . .
2.3.2 Reflection at an anisotropic surface . . . . .
2.4 Effective medium approximation . . . . . . . . . .
2.4.1 The Bruggeman effective medium . . . . . .
2.5 Euler angles . . . . . . . . . . . . . . . . . . . . . .
2.6 Capillary action . . . . . . . . . . . . . . . . . . . .
2.7 Proteins . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Albumin . . . . . . . . . . . . . . . . . . . .
2.7.2 Protein adsorption . . . . . . . . . . . . . .
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3 Experimental and instrumentation
3.1 Ellipsometer . . . . . . . . . . . . .
3.2 Silicon samples . . . . . . . . . . .
3.2.1 Euler angles . . . . . . . . .
3.3 Protein adsorption . . . . . . . . .
3.4 Measurements . . . . . . . . . . . .
4 Results and discussion
4.1 Experimental results . . . . .
4.2 Optical model devolepment .
4.2.1 Silicon trenches . . . .
4.3 Sample with protein . . . . .
4.4 Conclusions and suggestion of
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further work
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Contents
Bibliography
35
A Experimental results
37
B Experimental and model results
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Chapter 1
Introduction
1.1
Background
Recentley there has been a merging of bio- and nanotechnology. Many new applications in biosensors, bioelectronics and biomaterials are found. Protein absorption
onto silicon surfaces has previously been studied when trying to observe what is
adsorbed and how much is absorbed. Infrared ellipsomerty is a non destructive
analysis technique that can be used for identification and for obtaining structural
information. There is a large number of reference spectra availably for organic
molecules such as proteins.
Previously there have been studies with bioadsorption on flat surfaces and
monolayers [1] or in porous silicon [2]. In porous silicon it is possible to have a
large surface but exact structure of the internal surface is unknown and therefore
there have not been reliable optical models for the nanostructure in those studies.
For monolayers the surface structure is known but there is a very small surface for
the proteins to absorb to.
In this study we will use trenches etched in silicon which gives us a known
structure to build a model for and it has more surface than a flat silicon structure.
Simulations show an enhanced sensitivity for studies of bioadsorption in nanostructures in the form of trenches etched in silicon [3]. Such samples have been
prepared [4] but so far no acceptable optical model is available.
1.2
Objective
This project has the main objective to measure with IR-ellipsometry on silicon
trenches and to develop and represent nanostructure to describe their reflection
properties. A second objective is to expose these trenches to protein molecules
which adsorb on the "walls" in the structure. Finally it will be investigated if the
dielectric function of the adsorbed protein can be determined.
1
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Introduction
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Chapter 2
Theory
2.1
Maxwell’s equations, plane waves and polarization
Electromagnetism was summarized by Maxwell in 1873 in his four equations [5].
Maxwell´s equations describe electromagnetic fields and their behavior when they
interact with matter. The equations in the most common form are
∇·D
=
ρ
(2.1)
∇·B =
0
(2.2)
∂D
+J
(2.3)
∇×H =
∂t
∂B
(2.4)
∇×E = −
∂t
These fields are the electric displacement field D, the electric field E, the magnetic
flux density B and the magnetic field H. J is the current density and ρ in this case
is the charge density. For ellipsometry the most interesting field is the E-field. One
solution to Maxwell’s equations in the case of harmonic fields is the plane wave.
The electric field, the magnetic field and the propagation vector q are orthogonal
to each other and form a right-handed system. The amplitudes of the fields will
also be proportional to each other according to
E0 = cµH0
where c is the speed of light, is the electric permittivity and µ is the permeability.
The amplitudes E0 and H0 are defined by
E = E0 ei(q·r−ωt)
(2.6)
H = H0 ei(q·r−ωt)
(2.7)
where r is the coordinate vector.
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4
2.1.1
Theory
Polarization
When studying reflection at surfaces it is necessary to describe the orientation of
the fields in a propagating electromagnetic wave, i.e. the polarization. This is done
by decomposing the fields into two components, usually with the plane of incidence
as a reference. The plane of incidence is defined by the propagation vectors of the
incident, reflected and refracted waves when studying reflection and transmission
at surfaces. When no oblique reflection or incidence occurs, a plane of incidence
is chosen in such a way that it simplifies the description of the polarization. The
two components are the p-component, seen in fig. 2.1, for which the electric field
lies in the plane of incidence, and the s-component, also seen in fig. 2.1, where the
electric field is perpendicular to the plane of incidence.
Hip
N0
N1
Eip
Erp
θ0
θ0
Hrp
His
N0
N1
θ1
Htp
Etp
Ers
Eis
θ0
θ0
Hrs
θ1
Hts
Ets
Figure 2.1: p- and s-parts of the reflected, transmitted and incident fields in
reflection from a single surface with complex refractive index N0 of the ambient
medium and N1 of the substrate. The angle of incidence θ0 equals the angle of
reflection, and θ1 is the angle of refraction.
2.2
Ellipsometry
Ellipsometry has been around for quite some time and has a lot of standard applications today. Paul Drude provided a theoretical basis for ellipsometry in the
late 1800 [6]. He also performed experiments and measurement that determined
optical properties of metals [7]. Since the mid 1970s there have been great progress
in the field of ellipsometry thanks to the availability of faster computers.
Ellipsometry is defined as the measurement of the state of the change in polarization of a polarized light wave [8]. For the measurements carried out in this
project IR-ellipsometry was used. The interaction between the sample and the
light is in reflectionmode which is defined as reflection ellipsometry. When the
change of the polarization after it is reflected from a sample is analyzed, one can
get information about layers that are thinner than the wavelength of the light.
The change in polarization of the light beam depends on the surface and the thin
film properties. With the help of ellipsometry one can determine optical properties
of materials in terms of spectral dependence of refractive indices and determine
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2.2 Ellipsometry
5
microstructural parameters such as crystal orientation, layer thickness and porosity.
The technique has many standard applications in the semiconductor industry
today but lately it has become interesting for researchers in such fields as biology
and medicine. In these new areas ellipsometry faces the challenge to measure on
unstable surfaces and to develop microscopic and imaging ellipsometry [9].
2.2.1
Reflection ellipsometry
Reflection ellipsometry is a technique based on measuring polarization changes
occurring upon reflection at oblique incidence of a polarized monochromatic plane
wave. Figure 2.2 shows the principle of ellipsometry.
E
s-plane
p-plane
p-plane
θ
E
s-plane
Plane of incidence
Sample
Figure 2.2: The principle of ellipsometry: In oblique reflection the plane of incidence defines the angle of incidence and the complex amplitudes of the p- and spolarized complexed-valued electric field components before and after reflection,
respectively.
The basic quantity measured with an ellipsometer is the ratio
ρ=
χr
χi
(2.8)
where χr and χi are the complex-number representations of the states of polarization of the reflected and incident beams. In a cartesian coordinate system with the
p- and s-direction parallel and perpendicular to the plane of incidence the quantity
χ is found to be
Ep
χ=
(2.9)
Es
where Ep and Es are the complex-valued representations of the electric fields in
the p- and s-directions. For light reflected from optically isotropic samples no
coupling will occur between the orthogonal p- and s-polarizations. The complex-
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6
Theory
valued amplitude reflection coefficients Rp and Rs for light polarized in the p- and
s-direction then become
Rp =
Epr
Epi
(2.10)
Rs =
Esr
Esi
(2.11)
χi =
Epi
Esi
(2.12)
χr =
Epr
Esr
(2.13)
Epr Esi
Rp
=
= tan Ψei(δp −δs )
Esr Epi
Rs
(2.14)
Equation (2.8) becomes
ρ=
and the difference in phase is
∆ = δp − δs
(2.15)
Ψ and ∆ are introduced as parameters in a polar description of ρ and are called
ellipsometric angles and
Rp tan Ψ = (2.16)
Rs
This gives us the definitions to the two parameters Ψ and ∆ [8].
2.2.2
Generalized ellipsometry
We can see that Eq.(2.14) is valid for optically isotropic samples and for anisotropic
samples having diagonal Jones matrices. In the general case of anisotropic samples
Ψ and ∆ depend on the polarization state of the incident beam. This can be
described with a non-diagonal Jones matrix.
Rpp Rsp
Rr =
(2.17)
Rps Rss
The first index refers to incident polarization mode and the second to the emerging
polarization mode.
An ellipsometric characterization then requires measurement of at least three
values on ρ at three different χi and three pairs of Ψ and ∆ are defined. This is referred to as generalized ellipsometry. The complex-valued generalized ellipsometer
parameters are then defined according to
ρpp =
Rpp
= tan Ψpp ei∆pp
Rss
(2.18)
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2.3 Optical modeling
7
ρps =
Rps
= tan Ψps ei∆ps
Rpp
(2.19)
ρsp =
Rsp
= tan Ψsp ei∆sp
Rss
(2.20)
This gives us data in six different parameters Ψpp , Ψsp , Ψps , ∆pp , ∆sp and ∆ps . Depending on the sample properties and orientation, the off-diagonal normalized elements ρps and ρsp may be symmetric, antisymmetric, Hermitian, anti-Hermitian,
completely different or zero [10].
2.2.3
Infrared ellipsometry
Ellipsometry in the visual light region is very common for analysis of thin films.
It can be used to determine the thickness of the film with great accuracy. Spectroscopic infrared ellipsometry can be used for characterization of vibrations in
molecules. At first glance one might think that larger wavelengths of the infrared
light might reduce the sensitivity of the measurement but it is actually reliable
down to nanometer thick layers. Specific chemical information about the the sample can be determined in a spectrum that represents molecular vibrations.
This makes IR-ellipsometry a very useful tool for characterization of organic
materials such as proteins, polymers etc. When looking at proteins it is interesting
to note that the IR-spectra is divided into four different regions [11].
1. The region with vibrations between hydrogen and other elements such as
oxygen, nitrogen and carbon. This happens within the wavenumber range
4000 to 2500 cm−1 .
2. The region with vibrations from triple bindings, 2500 to 2000 cm−1 , where
C ≡ C and C ≡ N occur.
3. The region with double bindings, 2000-1500 cm−1 , were usually C=C, C=O
and C=N occur.
4. The region under 1500 cm−1 , also known as the fingerprint region where
similar molecules have different absorption patterns.
2.3
Optical modeling
Ellipsometry provides us with information about Ψ and ∆ for the sample. These
parameters depend also on wavelength and angle of incidence. To be able to
analyze the information we need an optical model. What kind of model depends
on the sample. With optical modeling we will develop a model for the sample and
then determine the unknown model parameters by non-linear regression analysis.
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8
Theory
The linear optical response of materials can be described with the refractive
index
N (λ) = n(λ) + ik(λ)
(2.21)
where n(λ) and k(λ) are the wavelength dependent real and imaginary part of
N (λ).
As an alternative one can also use the dielectric function
(λ) = 0 (λ) + i00 (λ) = N (λ)2
(2.22)
Often the objective of analyzis of ellipsometric data is to determine N (λ) of one
or several constituents of the sample.
In many cases a wavelength-by-wavelength approach is used. This means that
the N -value is extracted from the experimental data for each wavelength and
independent of all other spectral points. For this the thickness and N -spectra of
all the remaining sample constituents have to be known. However, in this project
dispersion models are used for the materials optical properties
Data
Modify
Model
Fit
Results
Figure 2.3: Illustration of the different steps in ellipsometric data analysis
Once an optical model is defined the data collected from the ellipsometry measurement can be fitted to the model. This is done by generating ellipsometry
values for the same condition as the measured data. In this work some parameters
are known since the sample is manufactured after specifications.
The Levenburg-Marquard algorithm is used to minimize the difference between
the generated values that the model gives us and the experimental values. This
algorithm is based on Mean Squared Error(MSE) and is carried out by altering
the unknown parameters in the optical model until a minimum in MSE is reached.
MSE is defined as
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2.3 Optical modeling
9
"
#
X Ψmod − Ψexp 2 ∆mod − ∆exp 2
1
M SE =
+
2N − M
σΨ,i
σ∆,i
(2.23)
where N is the number of measured Ψ and ∆ couples, M is the number of parameters that have been fitted and σΨ and σ∆ are the standard deviation of Ψ
and ∆, respectively. The general procedure is shown in Fig. (2.3). In practice
the user and/or program varies the fitting parameters until MSE has a minimum.
The fitting parameters values are then the result of the analysis.
2.3.1
Reflectance
In optical measurements the parameter usually obtained is the irradiance, which
will be denoted I. Now we can calculate the reflectance
2
Er Ir
2
<=
(2.24)
= = |R|
Ii
Ei
where Ir and Ii are incident and reflected irradiances and R the reflection coefficient.
Two-phase systems
The two-phase system, see Fig. 2.4, is the most simple case, i.e. a substrate with
refractive index N 1 in an ambient with index N 0 , and the Fresnel equations are
used in their original form . The Fresnel reflection coefficients will be denoted r
rp =
Erp
N1 cos θ0 − N0 cos θ1
tan (θ0 − θ1 )
=
=
Eip
N1 cos θ0 + N0 cos θ1
tan (θ1 + θ0 )
(2.25)
N0 cos θ0 − N1 cos θ1
sin (θ1 − θ0 )
Ers
(2.26)
=
=
Eis
N0 cos θ0 + N1 cos θ1
sin (θ1 + θ0 )
These equations relate the p- and s-components of the reflected fields, Erp och Ers
to components of the incident fields, Eip och Eis
From this we can with the help of Snell’s law derive an expression for the
complex refractive index
s
2
1−ρ
N1 = N0 sin θ0 1 +
tan2 θ0
(2.27)
1+ρ
rs =
and since
= N2
(2.28)
it follows that for the medium nr 1
1 = 0 sin θ0
2
1+
1−ρ
1+ρ
2
!
tan θ0
2
(2.29)
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10
Theory
θ0
d1
θ0
θ0
θ1
N0
N1
d1
N0
N1
θ1
θ2
N0
N1
θ1
dn-2
N2
Nn-2
θn-2
θn-1
Nn-1
Figure 2.4: Reflection and transmission in the 2-phase-, 3-phase- and nphasemodel.
Three-phase systems
First we restrict ourselves to discuss the case when the principal axes coincide with
an xyz coordinate system defined with the xz-plane equal to the plane of incidence,
with the x-axis along the surface and the z-axis pointing into the substrate. In the
three-phase-model a thin layer with the thickness d and refractive N1 = n1 + ik1
between two semi-infinite media are considered. The top layer is called the ambient
and has the real-valued refractive index N0 = n0 . The bottom layer is the substrate
and the refractive index for that layer is N2 = n2 + ik2 .
As usual we will assume a plane wave incident in the xz-plane at an angle of
incidence θ0 . When exploring the three-phase system we have to account for two
reflecting interfaces due to the interaction between the electromagnetic field of the
incidence wave and the thin film covered substrate.
What we are trying to do here is to find relations between the incident wave and
the reflected and transmitted waves. When trying to find these transmission and
reflection coefficients it is important to remember that the boundary conditions
apply differently for the s- and p-polarizations so we need to treat these separately.
Reflection coefficients will be denoted Rp and Rs and the transmissions coefficients
will be denoted Tp and Ts .
The general form of the plane-wave solutions to the wave equation in the xzplane becomes
E(r) = E(z)ei(qx x−ωt)
(2.30)
where q is the propagation vector q = (qx , 0, qz )
If we assume an s-polarization we only have an y-component of E which gives
us
E(r) = (0, E0y , 0) ei(qx x−ωt)
(2.31)
In the ambient, the layer and in the substrate we will get different equations for
how E varies in the ambient
E0y (z) = Aeiq0z z + Be−iq0z z
z<0
(2.32)
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2.3 Optical modeling
In the layer
11
E1y (z) = Ceiq1z z + De−iq1z z
In the substrate
E2y (z) = F eiq2z (z−d)
0<z<d
z<d
(2.33)
(2.34)
where qz and −qz are the z-components of the wave vectors of the incident and
reflected waves and A, B, C, D and F are their complex amplitudes. Using this
we can define a reflection coefficient as the ratio B/A between the reflected and
incident fields. A transmission coefficient can be defined as F/A between the
transmitted and incident fields.
The boundary conditions on the tangential components of the E and H-field
require that Ex , Ey , Hx and Hy must be continuous at the interfaces z = 0 and
z = d. For the s-polarization there is no Hy -component so the only thing that
remains is to find the Hx -component. Using equation (2.4) and
B = µµ0 H
implies that
Hx =
i ∂E
(ωµ0 ) ∂z
and then using what we learned in Eqs. (2.32 - 2.34) we find that

q0z
iq0z z

− Be−iq0z z ), z < 0
 − ωµ0 (Ae
q1z
iq1z z
− De−iq1z z ), 0 < z < d
− ωµ0 (Ce
Hx (z) =

 − q2z F eiq2z (z−d) ,
z<d
ωµ0
(2.35)
(2.36)
(2.37)
Considering that the boundary conditions saying that Ey and Hx must be continuous at the interfaces z = 0 and z = d gives us
A+B =C +D
(2.38)
q0z (A − B) = q1z (C − D)
(2.39)
=F
Ceiq1z d + De−iq1z d = q2z F
iq1z d
Ce
q1z
+ De
−iq1z d
(2.40)
(2.41)
Now we eliminate C and D and if we solve the equations for the ratios B/A and
F/A we obtain
B
(q0z − q1z ) (q1z + q2z ) + (q0z + q1z ) (q1z − q2z ) ei2q1z d
=
A
(q0z + q1z ) (q1z + q2z ) + (q0z − q1z ) (q1z − q2z ) ei2q1z d
F
4q0Z q1Z eiq1Z d
=
A
(q0z + q1z ) (q1z + q2z ) + (q0z − q1z ) (qz − q2z ) ei2q1z d
(2.42)
(2.43)
For real wave vectors we have that
qiz =
2π
Ni cos θi
λ
(2.44)
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12
Theory
(i =0,1,2) and gives us the Fresnel coefficients for the s-polarizations as
r01 =
q0z − q1z
q0z + q1z
(2.45)
r12 =
q1z − q2z
q1z + q2z
(2.46)
t01 =
2q1z
q0z + q1z
(2.47)
t01 =
2q1z
q1z + q2z
(2.48)
and now the reflection and transmission coefficients can be rewritten as
R≡
r01 + r12 ei2β
B
=
A
1 + r01 r12 ei2β
(2.49)
T ≡
B
t01 t12 ei2β
=
A
1 + r01 r12 ei2β
(2.50)
where rlm is the Fresnel’s coefficient rp between the phases l and m and β is the
film phase thickness, which is given by
2πd
N1 cos θ1
(2.51)
λ
Here N1 is the complex index of refraction of the middle medium, d is the thickness
of the layer and θ1 is the angle of refraction. If the same derivation is performed for
the p-polarization this leads to the same expressions except that the coefficients
r01 , r12 , t01 and t12 will be associated with the p-versions of the Fresnel equations.
This is the final piece of the puzzle and we can now write
β=
Rp =
Erp
r01p + r12p ei2β
=
Eip
1 + r01p r12p ei2β
(2.52)
Rs =
Ers
r01s + r12s ei2β
=
Eis
1 + r01s r12s ei2β
(2.53)
n-phase systems
When adding more layers to the model the complexity of the analytical expression
for the reflectivity will increase. For solving these systems matrix-based methods
is used with the help of computers. The optical system involves m layers indexed
1, 2, 3..m of different materials between the semi-infinente ambient indexed 0
and the semi-infinite substrate indexed m + 1. We assume that all phases are
homogeneous and isotropic. All boundaries are parallel and abrupt. At any point
in the system we can mathematically resolve the electric field of the light into two
subfields, one corresponding to a wave traveling in the +z-direction, E+ and one
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2.3 Optical modeling
13
travelling in the -z-direction, E− . The total fields in any given z-plane is given by
E(z) = E + (z) + E − (z).
Consider the relation between the total fields in two different planes z1 and z2 .
Light propagation is described by linear equations and thus we geneally have
E + (z1 ) = S11 E + (z2 ) + S12 E − (z2 )
(2.54)
E − (z1 ) = S21 E + (z2 ) + S22 E − (z2 )
(2.55)
E(z1 ) = S(z1 , z2 )E(z2 )
(2.56)
or in matrix form
where
E(z1 ) =
E + (z1 )
E − (z1 )
E(z2 ) =
E + (z2 )
E − (z2 )
are called generalized field vectors. The matrix S(z1 , z2 ) is defined by
S11 S12
S(z1 , z2 ) =
S21 S22
(2.57)
(2.58)
The properties of the part of the system that lies between the planes z1 and z2
enter the matrix in the form of coefficients Sij In the case of two optical systems
the relation between the fields of the first and second optical system is related by
E1 = S1 E2
(2.59)
E2 = S2 E3
(2.60)
where S1 and S2 are the matrices containing the properties of the two optical
systems. If we eliminate the intermediate field vector E2 we can write
E1 = S1 S2 E3 = SE3
(2.61)
We call S = S1 S2 the scattering matrix of various optical systems. To derive the
reflection and transmission coefficients from the S-matrix we write equation (2.56)
as
+ +
E0
Em+1
S11 S12
=
(2.62)
−
S21 S22
E0−
Em+1
where E0 is the field vector in the ambient immediately adjacent to the interface,
Em+1 is the corresponding field vector in the substrate and S the total scattering
matrix for the whole system containing m layers on a substrate. E+
0 is the incident
−
+
wave, E−
is
the
reflected
wave,
E
the
transmitted
wave
and
E
0
m+1
1+m = 0 because
there is no backscattered wave in the substrate. From this follows
+ +
E0
S11 S12
Em+1
=
(2.63)
S21 S22
0
E0−
The overall complex transmission coefficient is given by
T =
+
Em+1
1
+ =
S
E0
11
(2.64)
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14
Theory
and the overall complex reflection coefficient by
R=
S21
E0−
=
S11
E0+
(2.65)
Scattering matrices are generally defined independently for the p- and s-polarizations,
Sp and Ss , respectively. The reflection and transmission coefficients for the p- and
s-directions become
S21p
1
Tp = S11p
Rp = S11p
(2.66)
Rs =
2.3.2
S21s
S11s
Ts =
(2.67)
1
S11s
Reflection at an anisotropic surface
Biaxially anisotropic substrate
We restrict ourselfs to discuss the case when the principal axes coincide with an xyz
coordinate system defined with the xz-plane equal to the plane of incidence with
the x-axis along the surface and the z-axis pointing into the substrate. There are
three different indices of refraction in an biaxial medium and they will be denoted
N1x , N1y and N1z . The reflection matrix is diagonal in this symmetry and the
diagonal elements are
rpp =
2
N1x N1z cos φ0 − N0 N1z
− N02 sin2 φ0
21
2 − N 2 sin2 φ
N1x N1z cos φ0 + N0 N1z
0
0
12
rss =
2.4
2
N0 cos φ0 − N1y
− N02 sin2 φ0
12
2 − N 2 sin2 φ
N0 cos φ0 + N1y
0
0
12
(2.68)
(2.69)
Effective medium approximation
A heterogeneous material can be modeled with a so called effective medium approximation generally known as EMA. The goal is to find an effective dielectric
function EM A expressed in terms of microstructure and complex-valued dielectric
functions of the components of a composite material. This EM A can account for
the essential features of the heterogeneous structure. The various optical quantities like reflection and transmission can be determined in a similar manner as for
homogeneous materials.
Initially we assume that heterogeneous materials are macroscopically uniform
and we also assume that components of the heterogeneous material are isotropic
so they can be characterized by a scalar dielectric function.
Consider a simple hypothetical two-component composite of materials A and
B, see fig. 2.5. The macroscopic average dielectric function k in the case the
boundaries are parallel to the applied field is given by
k = fA A + fB B
(2.70)
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2.4 Effective medium approximation
15
where fA and fB are the materials volume fractions. When the field and the sample
boundaries are perpendicular, the macroscopic average is given by
⊥ =
fA
A
1
+
(2.71)
fB
B
Due to the limitations on the validity of the effective medium approximations there
are many different theories which are applicable to different microstructures.
E
A f
A
εA
B fB
εB
E
A
fA
εA
B
fB
εB
Figure 2.5: Microstructures for a two-component composite. To the left all the
boundaries are perpendicular to E and to the right the boundaries are parallel to
the applied field E
2.4.1
The Bruggeman effective medium
One theory that is often used in optical analysis is the Bruggeman effective medium
[12]. This theory describes an aggregate microstructure where materials A and
B are randomly mixed. The approximation assumes that the microstructural
dimensions are much smaller than the wavelength λ.
In this theory we approximate that the unit cell is a sphere whose dielectric
function is A with the probability fA and B with the probability fB = 1 − fA .
Bruggeman gives us this equation for the composite dielectric function A − B − + (1 − fA )
=0
(2.72)
A + 2
B + 2
One of the great advantages of the Bruggeman model is that it is symmetric and
can be used for all values of fA .
So far all particles discussed have been assumed to be spherical. However,
the shape of the particle will affect the local field. To coupe with this problem
there are some correction terms called depolarization factors, table 2.1. When a
dielectric object is placed in an electric field E0 the surface bound charge creates
a depolarization field E1 which reduces the total field in the object. The electric
field can be described as
fA
E1 = E0 −
qP
0
(2.73)
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16
Theory
where q is the depolarization factor and P is the polarization inside the object.
The factor q is dependent on the geometric shape of the object and 0 is the electric
constant.
Depolarization Factors
Shape
Axis
Sphere
any
Thin slab
normal
Thin slab
in plane
Long circular cylinder longitudinal
Long circular cylinder transverse
q
1/3
1
0
0
1/2
Table 2.1: Depolarization factors
2.5
Euler angles
In this project measurements are done at many different sample angles (rotational
azimuths) and it will be necessary to figure out what movement respond to which
Euler angle. Any rotation can be described using three angles fig. 2.6. The angles
that give the three rotation matrices are called Euler angles.
The intersection of the xy and the x´y´ coordinate planes is called the line of
nodes. Φ is the angle between the x-axis and the line of nodes. θ is the angle
between the z-axis and the z´-axis. Ψ is the angle between the line of nodes and
the x´-axis.
z
y’
z’
θ
x’
ψ
x
Φ
y
Line of nodes
Figure 2.6: Euler angles
In a matrices representation one can write the rotation as
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2.6 Capillary action
cos φ
M =  − sin φ
0
17

sin φ 0
(1
cos φ 0   0
0
1
0
0
cos θ
− sin θ

0
cos ψ
sin θ   − sin ψ
cos θ
0

0
0 
1
(2.74)
The problem is that it has never been agreed upon the order in which the rotations
are applied and even the axes about which they are applied. This means that I
must somehow define which rotation affects the results in my measurements.

2.6
sin ψ
cos ψ
0
Capillary action
Capillary action is the force that drives liquids through capillaries. It is caused by
the relationship between adhesion which is the liquids attractive force against the
capillary wall and the cohesive force between the liquids molecules. This is the
same phenomenon that helps trees to suck up water from the ground.
In the case of the silicon trenches the capillary force will keep the water out of
the trenches since the height of the liquid column is given by
h=
2γ cos θ
ρgr
(2.75)
where γ is the surface tension (J/m2 ), θ is the contact angle, ρ is the liquid density,
g is the acceleration due to gravity and r is the with of the trench.
2.7
Proteins
The word protein comes from the greek word "prota" which means "of primary
importance" and was first described and named by the Swedish chemist Jakob
Berzelius in 1838. Proteins are large organic compounds made of amino acids that
are arranged in a linear chain and joined together by peptide bonds between the
carboxyl and amino acid residues.
Proteins are linear polymers built from 20 different L-α-amino acids. All amino
acids share common structural features including an α-carbon to which an amino
group, a carboxyl group and a variable side chain are bonded.
The amino acids in a polypeptide chain are linked by peptide bonds formed in a
dehydration reaction. Once linked in the protein chain, an individual amino acid
is called a residue and the linked series of carbon, nitrogen, and oxygen atoms
are known as the main chain or protein backbone. The peptide bond has two
resonance forms that contribute some double bond character and inhibit rotation
around its axis, so that the α carbons are roughly coplanar.
The isoelectric point is the pH at which a protein carries no net electrical
charge. It is important to know the isoelectric point to determine what kind of
buffer should be used for protein adsorption. This is to minimize the electrostatic
forces that counteract the adsorption [13].
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18
2.7.1
Theory
Albumin
Albumin is the most abundant protein in the human blood plasma. It is produced
in the liver. Albumin comprises about half of the blood serum protein. The
protein has many functions such as maintaining osmotic pressure, transporting
thyroid hormones that control how fast the body burns energy, transporting fat
soluble proteins, transporting many drugs, competitively bind calcium ions and
buffering pH. Figure 2.7 shows the structure of human albumin.
Albumin is synthesized in the liver as preproalbumin which has an N-terminal
peptide that is removed before the protein is released from the rough endoplasmic
reticulum. The product, proalbumin, is in turn cleaved in the Golgi vesicles to
produce the secreted albumin.
The reference range for albumin concentrations in blood is 30 to 50 g/L.
Low blood albumin levels (hypoalbuminemia) can be caused by the liver disease cirrhosis of the liver (most commonly), decreased production (as in starvation), excess excretion by the kidneys (as in nephrotic syndrome), excess loss
in bowel (protein losing enteropathy) malnutrition, malabsorption, neoplasia and
pregnancy.
Molecular dimensions of human albumin is 3 × 3 × 8 nm. This means that
it should not have a problem fittning in the treanches. It has a serum half-life of
approximately 20 days and a molecular mass of 67 kDa. The isoelectric point is
located at pH 5.8.
Figure 2.7: Sturcture of human albumin
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2.7 Proteins
2.7.2
—
19(31)
19
Protein adsorption
Adsorption is a process that occurs when a gas or liquid solute accumulates on
the surface of a solid or, more rarely, a liquid (adsorbent), forming a molecular or
atomic film (the adsorbate). It is different from absorption, in which a substance
diffuses into a liquid or solid to form a solution. The term sorption encompasses
both processes, while desorption is the reverse process.
Adsorption is operative in most natural physical, biological, and chemical systems, and is widely used in industrial applications such as activated charcoal,
synthetic resins and water purification. Adsorption, ion exchange and chromatography are sorption processes in which certain adsorptive are selectively transferred
from the fluid phase to the surface of insoluble, rigid particles suspended in a vessel
or packed in a column.
Protein adsorption is controlled by the characteristics of the surface, the protein
and by the buffer. Properties such as electrical charge, molecular orientation and
pH of the buffer all influence the adsorption process [14].
Two main factors that control protein adsorption are
• Hydrophobic interaction in the proteins
• Electrostatic forces
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—
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Theory
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Chapter 3
Experimental and
instrumentation
3.1
Ellipsometer
The ellipsometer used for the measurements is an IR-VASE from J.A. Woollam
Co. Inc., USA. IR-VASE is a spectroscopic ellipsometer that covers a spectral
range from 300 to 5000 cm−1 The IR-ellipsometer is based on the concept of a
rotating compensator [9].
The input unit of the IR-VASE contains mirrors that focus the polarized radiation on the sample that is being measured. Polarization is achieved by passing the
radiation through a wire grid polarizer. This polarizer is mounted to a goniometer
which allows the polarization of the radiation to be adjusted relative to the plane
of incidence.
The input unit also contains an alignment laser, a tool that helps to set the
sample surface perpendicular to the plane of incidence. For the analysis of the
experimental data the WVASE program from J.A. Woollam Co. Inc., USA was
used.
3.2
Silicon samples
The samples used for the measurements were made at KTH by Xavier Badel. The
samples were made using the lithography and electrochemical etching method [15].
This is a technique that makes it easy to make 3-D etchings in silicon. The idea
is to have trenches etched in the sample. The walls are about 300 nm wide, 9 µm
high and spaced with a distance center-to-center of 1500 nm. The sample is shown
in Fig. 3.1. The etched area of the sample is circular with a diameter of about 1.6
cm. As it turns out the periphere of the sample can not be used for measurements.
21
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22
—
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Experimental and instrumentation
Figure 3.1: The trenches in the silicon sample
3.2.1
Euler angles
In the measurements the Euler angles were defined in according to fig. 3.2. This is
how the angles are represented in the WVASE-program that was used to process
the experimental data and also used to build the model. The Φ-direction is the
in the a wall of the sample. θ is 90◦ and is the direction along the normal of the
wall. Ψ is the sample rotation. There is a build in idiosyncrasy in the WVASE
system. The legends on WVASE experimental data graphsis the sample rotation
angle is shown as θ = xx◦ , but is more closely related to Ψ in the Euler angles.
Therefor θ will be the sample rotation from here on.
3.3
Protein adsorption
Before applying the protein the trenches must be filled with water. Because of the
capillary forces we first need to put the sample in alcohol, which have different
characteristics than water and therefore has no problem making its way down into
the trenches. The sample was put in 30 ml of methanol that was diluted with 15
ml of water every 24:th hour until the alcohol percentage was down under 0.1%.
10 mg of human serum albumin was dissolved in 10 ml PBS ( PhosphateBuffered Saline). The PBS has a pH of 7.4 which is close to the pH in the human
body. The silicon sample was put in 26 ml of PBS since that is the amount it took
to cover it completly. Then 3 ml of the protein solution was added. The sample
was left in the solution för 60 minutes whereafter it was dried with nitrogen gas.
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3.4 Measurements
23
Wall in the sample
z
x
y
θ
Φ
ψ
Substrate
Figure 3.2: Definition of euler angles during measurements in the WVASEprogram.
3.4
Measurements
Three different silicon samples were measured with the IR-VASE. The angles of
incidence were 25◦ , 45◦ and 65◦ in the range of 300 - 5000 cm−1 and the resolution
was 8 cm−1 . The sample was rotated 360◦ and the measurements for all three
angles of incidence was carried out every 45◦ . One silicon sample was prepared
with proteins and measured in the same way.
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—
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Experimental and instrumentation
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Chapter 4
Results and discussion
4.1
Experimental results
As we can see in Fig. 4.1 and 4.2 there are three different Ψ- and ∆-curves for each
angle af incidence and θ = 0◦ . In Fig. 4.3 and Fig. 4.4 we can see the results for
the three different incidence angles. In the appendix Fig. A.1 to Fig. A.28 show
the curves for different θ. AnE is the Rpp /Rss -ratio, Aps is the Rps /Rpp -ratio and
Asp is the Rsp /Rss -ratio. Information is located between 6 µm and 18 µm. Above
and below these limits the noise was to large for any usable information to be seen.
Experimental Data
100
Exp AnE 25° θ=0°
Exp Aps 25° θ=0°
Exp Asp 25° θ=0°
Ψ in degrees
80
60
40
20
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.1: Experimental values of Ψ for the incident angle 25◦ and θ = 0◦ .
25
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26
Results and discussion
Experimental Data
300
Exp AnE 25° θ=0°
Exp Aps 25° θ=0°
Exp Asp 25° θ=0°
∆ in degrees
200
100
0
-100
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.2: Experimental values of ∆ for the incident angle 25◦ and θ = 0◦ .
Experimental Data
100
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
Ψ in degrees
80
60
40
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
20
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.3: Experimental values of Ψ for all three incident angles and θ = 0◦ .
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4.2 Optical model devolepment
27
Experimental Data
300
∆ in degrees
200
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
100
0
-100
6
8
10
12
14
Wavelength (µm)
16
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
18
Figure 4.4: Experimental values of ∆ for all three incident angles and θ = 0◦ .
4.2
4.2.1
Optical model devolepment
Silicon trenches
For the optical model of the silicon trenches one could easily make a variety of
different models that may vary in complexity. While using the Bruggeman model
we assumed that trench walls are like thin slabs which gives us the depolarization
factors 1 and 0 for the optical properties normal to and in plane eith the walls
respectively, see table 2.1.
The general idea is to build the model from silicon substrate that has the
thickness 1 mm and then try to simulate the voids in the sample with the help
of EMA-materials. The model is shown in Fig. 4.5 with the silicon substrate as
layer 0. Since the sample is etched in a way that could be described like walls
the model must have different characteristics in different directions and therefore
a biaxial model was to be used. Early on it was realized that just one biaxial
was not enough so the model was expanded into a three layer biaxial model. This
is because the walls do not have the same distance between them at the bottom
compared to in the center and the top of the sample. These three biaxial layers
are illustrated in the model in Fig. 4.5 and are numbered 13, 14 and 15.
Each of the three biaxial-layers is connected to two different EMA-materials
that have the same percentage of void. These two EMA-materials are coupled to
each other so that the void always is the same for both of them when they are
fitted to the experimental data. Since there are three biaxial-layers the model gives
us three pairs of EMA-materials. Each of the six EMA-materials is connected to
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—
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Results and discussion
different p-doped silicon materials. The p-doped silicon is connected to silicon
with free carriers-materials. For these materials different resistances have to be
fitted on account of the direction. The layers marked 1 to 6 and -1 to -8 are
help layers. Some of the offsets in these layers are to high to have a physicall
explanation. To get a better model it is needed to develop specific layers adapted
for silicon trenches. In Figs. 4.6 to 4.7 the experimental values and the model are
shown at an angle of incidence of 25◦ when θ is 0. In Figs. B.1 to B.2 the results
are shown for 45◦ and 65◦ when θ is 0◦ . In figs. 4.10 and 4.11 we can observe that
the model is still accurate when θ is changed to 45◦ .In appendix B Figs. 4.10 and
B.24 the model and the experimental values for all three angles of incidence are
shown when θ vary from 45◦ to 135◦ in intervals of 45◦ . We only need to follow
the rotation of the sample to 135◦ because of the symmetry of the structure the
data will just be a repetition of the first 180◦ .
The thickness of the three biaxial layers seen in Fig 4.5 is what we can expect
from the specifications of the sample. Since we know the specifications for the
sample we know that the height should be 9 µm. Knowing that the walls are 300
nm thick and the distance center to center of the walls is 1500 nm this gives us a
void that is 80 %. From the model Fig. 4.5 we see that adding the thickness of the
three biaxial layers gives us the total height of 8.67 ± 0.08 µm. When calculating
the void percentage average we get that in the model it is 77.9 %. When the
model that was constructed for the experimental data from the measurements on
the silicon sample was fitted the Mean square error was 3.15 which is considered
a good result.
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4.2 Optical model devolepment
15 biaxial
14 biaxial
13 biaxial
12 ema6 (si_p_doped6)/59.6% void
11 ema5 (si_p_doped5)/59.6% void
10 ema4 (si_p_doped4)/83% void
9 ema3 (si_p_doped3)/83% void
8 ema2 (si_p_doped2)/77.5% void
7 ema (si_p_doped1)/77.5% void
6 si_p_doped6
5 si_p_doped5
4 si_p_doped4
3 si_p_doped3
2 si_p_doped2
1 si_p_doped1
0 si_p_doped
-1 si_p
-2 silicon with free carriers
-3 silicon with free carriers1
-4 silicon with free carriers2
-5 silicon with free carriers5
-6 silicon with free carriers6
-7 silicon with free carriers7
-8 silicon with free carriers8
29
0.484542 µm
7.356988 µm
0.826717 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
1 mm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
0.000000 µm
Figure 4.5: Three layer biaxial model for silicon trenches
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30
Results and discussion
Generated and Experimental
100
Model Fit
Exp AnE 25° θ=0°
Exp Aps 25° θ=0°
Exp Asp 25° θ=0°
Ψ in degrees
80
60
40
20
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.6: Experimental and model values for Ψ for the incident angle 25◦ and θ
is 0◦ .
Generated and Experimental
400
Model Fit
Exp AnE 25° θ=0°
Exp Aps 25° θ=0°
Exp Asp 25° θ=0°
∆ in degrees
300
200
100
0
-100
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.7: Experimental and model values for ∆ for the incident angle 25◦ and
θis 0◦ .
main: 2007-12-5 15:34
—
31(43)
4.2 Optical model devolepment
Generated and Experimental
100
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
80
Ψ in degrees
31
60
40
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
20
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.8: Experimental and model values for Ψ for all the incident angles at θ
= 0◦ .
Generated and Experimental
400
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
∆ in degrees
200
0
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
θ=0°
-200
-400
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.9: Experimental and model values for ∆ for all the incident angles at θ
= 0◦ .
main: 2007-12-5 15:34
—
32(44)
32
Results and discussion
Generated and Experimental
80
Model Fit
Exp AnE 25° θ=45°
Exp Aps 25° θ=45°
Exp Asp 25° θ=45°
Ψ in degrees
60
40
20
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.10: Experimental and model values for Ψ for the incident angle 25◦ and
θ = 45◦ .
Generated and Experimental
300
∆ in degrees
200
100
0
Model Fit
Exp AnE 25° θ=45°
Exp Aps 25° θ=45°
Exp Asp 25° θ=45°
-100
-200
6
8
10
12
14
Wavelength (µm)
16
18
Figure 4.11: Experimental and model values ∆ for the incident angle 25◦ and θ =
45◦ .
main: 2007-12-5 15:34
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33(45)
4.3 Sample with protein
4.3
33
Sample with protein
After exposing the sample to proteins the same measurements ware carried out
again. According to previous studies [1] we know that the proteins should show
optical signatures visible at the frequency 1537 ± 3 cm−1 . In Fig. 4.12 we show
that there is a small peak at 1540 cm−1 when θ = 0◦ .
Experimental Data
55
Exp AnE 45° θ=0°
Exp AnE 45° θ=0°
Ψ in degrees
50
45
40
35
30
25
1400
1500
1600
Wave Number (cm
-1)
1700
1800
Figure 4.12: Ψ for clean silicon sample (dashed curve) and for same sample after
expsure to protein (solid curve). A small peak is visible at 1540 cm−1 .
4.4
Conclusions and suggestion of further work
The main object for this project was to develop an optical model for the silicon
trenches with the help of experimental data from infrared ellipsometry and try
to detect proteins in the silicon structure. For the optical model a three layered
biaxial model was developed. This gave us a model which was accurate but not
perfect. The MSE was calculated to 3.54 and could not be improved by adding
more layers. The thicknesses of the three biaxial layers are what we expected from
the specifications of the sample and therefore give us an indication that the model
is correct. Perhaps a completely different type of model could be developed. It
has been suggested that a model based on photonic crystals could be used instead
of the biaxial model used in this project [3]. FEM-simulations could be used to
develop a theoretical model for the structures .
In the case with the protein we found a peak where is expected to be which is a
main: 2007-12-5 15:34
34
—
34(46)
Results and discussion
good indication that it is possible to study absorption in more complex structures.
The peak was not very strong and it can probably be made more visible if exposing
the samples to the protein for a longer time period. An optical model for the
proteins could be developed with the help of Gauss oscillators.
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—
35(47)
Bibliography
[1] H. Arwin, A. Askendahl, P. Tengvall, D. Thompson, and J. A. Woollam,
“Infrared ellipsometry studies of the thermal stability of protein monolayers
and multilayers.” submitted, 2007.
[2] H. Arwin, L. M. Karlsson, A. Kozarcanin, D. Thompson, T. Tiwald, and
J. Woollam, “Carbonic anhydrase adsorption in porous silicon studied with
infrared ellipsometry,” phys stat soli, no. 8, 2005.
[3] D. W. Thompson, 2007. Private communication.
[4] X. Badel, Electrochemically ethed pore arrays in silicon for X-ray imaging.
PhD thesis, KTH Microelectronics and Information Technology, 2005.
[5] J. C. Maxwell, A treatise on electricity and magnetism. Oxford: Claredon
Press, 1873.
[6] P. Drude Ann. Phys. Chemie, vol. 32, p. 584, 1887.
[7] P. Drude Ann. Phys. Chemie, vol. 38, p. 481, 1890.
[8] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light. NorthHolland, 1986.
[9] R. W. Collins, D. E. Aspnes, and E. A. I. (Eds.), “Proceedings of the 2nd international conference on spectroscopic ellipsometry,” Thin Solid Films, vol. 313314, 1998.
[10] H. Arwin, “Ellipsometry based sensor systems,” in Encyclopededia of sensors
(C. A. Grimes, E. C. Dickey, and M. V. Pishko, eds.), pp. 329–357, American
Scientific Publishers, 1999.
[11] B. Stuart, Biological Applications of Infrared Spectroscopy. John Wiley and
sons, Ltd, University of Greenwich, 1997.
[12] D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten
von heterogenen substanzen,” Ann. Phys. (Leipzig), vol. 24, pp. 636–679,
1935.
[13] J. Tooze, Introduction to Protein Structure 2nd ed. Garland Publishing: New
York, NY, 1999.
35
main: 2007-12-5 15:34
36
—
36(48)
Bibliography
[14] J. Gustavsson, “Protein adsorption on porous silicon,” Master’s thesis, Department of Physics and Measurement Technology, Linköping, 1999.
[15] P. Kleimann, X. Badel, and J. Linnros, “Towards the formation of 3d nanostructures by electochemical etching of silicon,” Electrochemical etched pore
arrays in silicon for X-ray imaging detectors, 2005.
main: 2007-12-5 15:34
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Appendix A
Experimental results
37
main: 2007-12-5 15:34
—
38(50)
38
Experimental results
Experimental Data
100
Exp AnE 45° θ=0°
Exp Aps 45° θ=0°
Exp Asp 45° θ=0°
60
40
20
0
6
8
10
12
Wavelength (µm)
14
Figure A.1: Experimental values Ψ
for the incident angle 45◦ and θ =
0◦ .
14
16
10
Exp AnE 65° θ=0°
Exp Aps 65° θ=0°
Exp Asp 65° θ=0°
12
14
Wavelength (µm)
0
6
8
16
Figure A.5: Experimental values Ψ
for the incident angle 25◦ and θ =
45◦ .
10
12
Wavelength (µm)
14
16
Experimental Data
200
∆ in degrees
Ψ in degrees
8
16
Figure A.4: Experimental values ∆
for the incident angle 65◦ and θ =
0◦ .
Exp AnE 25° θ=45°
Exp Aps 25° θ=45°
Exp Asp 25° θ=45°
20
14
Experimental Data
300
40
10
12
Wavelength (µm)
100
-100
Experimental Data
60
8
Figure A.2: Experimental values ∆
for the incident angle 45◦ and θ =
0◦ .
∆ in degrees
Ψ in degrees
10
12
Wavelength (µm)
Figure A.3: Experimental values Ψ
for the incident angle 65◦ and θ =
0◦ .
80
6
200
20
8
0
300
40
0
6
100
-100
Exp AnE 65° θ=0°
Exp Aps 65° θ=0°
Exp Asp 65° θ=0°
60
0
6
16
Experimental Data
80
Exp AnE 45° θ=0°
Exp Aps 45° θ=0°
Exp Asp 45° θ=0°
200
∆ in degrees
Ψ in degrees
80
Experimental Data
300
18
Exp AnE 25° θ=45°
Exp Aps 25° θ=45°
Exp Asp 25° θ=45°
100
0
-100
6
8
10
12
14
Wavelength (µm)
16
Figure A.6: Experimental values ∆
for the incident angle 25◦ and θ =
45◦ .
18
main: 2007-12-5 15:34
—
39(51)
39
Experimental Data
70
Exp AnE 45° θ=45°
Exp Aps 45° θ=45°
Exp Asp 45° θ=45°
60
200
∆ in degrees
Ψ in degrees
50
Experimental Data
300
40
30
20
Exp AnE 45° θ=45°
Exp Aps 45° θ=45°
Exp Asp 45° θ=45°
100
0
10
0
6
8
10
12
Wavelength (µm)
14
16
Figure A.7: Experimental values Ψ
for the incident angle 45◦ and θ =
45◦ .
-100
Ψ in degrees
20
8
10
Wavelength (µm)
12
14
Figure A.9: Experimental values Ψ
for the incident angle 65◦ and θ =
45◦ .
0
6
8
10
Wavelength (µm)
200
∆ in degrees
Ψ in degrees
40
20
8
10
12
14
Wavelength (µm)
16
Figure A.11: Experimental values of
Ψ for all three incident angles and θ
= 45◦ .
12
14
Experimental Data
300
60
0
6
-100
Figure A.10: Experimental values ∆
for the incident angle 65◦ and θ =
45◦ .
Experimental Data
80
16
Exp AnE 65° θ=45°
Exp Aps 65° θ=45°
Exp Asp 65° θ=45°
100
10
0
6
14
Experimental Data
200
∆ in degrees
Exp AnE 65° θ=45°
Exp Aps 65° θ=45°
Exp Asp 65° θ=45°
30
10
12
Wavelength (µm)
300
50
40
8
Figure A.8: Experimental values ∆
for the incident angle 45◦ and θ =
45◦ .
Experimental Data
60
6
100
0
-100
6
18
8
10
12
14
Wavelength (µm)
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
16
18
Figure A.12: Experimental values of
∆ for all three incident angles and θ
= 45◦ .
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—
40(52)
40
Experimental results
Experimental Data
80
Exp AnE 25° θ=90°
Exp Aps 25° θ=90°
Exp Asp 25° θ=90°
40
20
0
6
8
10
12
14
Wavelength (µm)
16
Figure A.13: Experimental values Ψ
for the incident angle 25◦ and θ =
90◦ .
18
30
20
8
10
12
14
Wavelength (µm)
16
Figure A.15: Experimental values Ψ
for the incident angle 45◦ and θ =
90◦ .
18
20
10
0
6
8
10
12
Wavelength (µm)
14
Figure A.17: Experimental values Ψ
for the incident angle 65◦ and θ =
90◦ .
18
Exp AnE 45° θ=90°
Exp Aps 45° θ=90°
Exp Asp 45° θ=90°
0
6
8
10
12
14
Wavelength (µm)
16
18
Experimental Data
Exp AnE 65° θ=90°
Exp Aps 65° θ=90°
Exp Asp 65° θ=90°
200
∆ in degrees
30
16
Experimental Data
300
Ψ in degrees
40
12
14
Wavelength (µm)
Figure A.16: Experimental values ∆
for the incident angle 45◦ and θ =
90◦ .
Exp AnE 65° θ=90°
Exp Aps 65° θ=90°
Exp Asp 65° θ=90°
50
10
100
-100
Experimental Data
60
8
200
∆ in degrees
40
10
6
6
300
Ψ in degrees
50
0
Figure A.14: Experimental values ∆
for the incident angle 25◦ and θ =
90◦ .
Exp AnE 45° θ=90°
Exp Aps 45° θ=90°
Exp Asp 45° θ=90°
60
100
-100
Experimental Data
70
Exp AnE 25° θ=90°
Exp Aps 25° θ=90°
Exp Asp 25° θ=90°
200
∆ in degrees
Ψ in degrees
60
Experimental Data
300
16
100
0
-100
6
8
10
12
Wavelength (µm)
14
Figure A.18: Experimental values ∆
for the incident angle 65◦ and θ =
90◦ .
16
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—
41(53)
41
Ψ in degrees
60
40
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
20
0
6
8
10
12
14
Wavelength (µm)
16
Figure A.19: Experimental values of
Ψ for all three incident angles and θ
= 90◦ .
30
20
8
10
12
14
Wavelength (µm)
16
Figure A.21: Experimental values Ψ
for the incident angle 25◦ and θ =
135◦ .
18
20
8
10
12
14
Wavelength (µm)
16
Figure A.23: Experimental values Ψ
for the incident angle 45◦ and θ =
135◦ .
10
12
14
Wavelength (µm)
16
18
Experimental Data
100
Exp AnE 25° θ=135°
Exp Aps 25° θ=135°
Exp Asp 25° θ=135°
0
-100
6
8
10
12
14
Wavelength (µm)
16
18
Experimental Data
200
∆ in degrees
Ψ in degrees
Exp AnE 45° θ=135°
Exp Aps 45° θ=135°
Exp Asp 45° θ=135°
40
0
6
8
300
80
60
6
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
Figure A.22: Experimental values ∆
for the incident angle 25◦ and θ =
135◦ .
Experimental Data
100
-100
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
200
10
0
6
0
300
∆ in degrees
40
100
Figure A.20: Experimental values of
∆ for all three incident angles and θ
= 90◦ .
Exp AnE 25° θ=135°
Exp Aps 25° θ=135°
Exp Asp 25° θ=135°
50
Ψ in degrees
18
Experimental Data
60
Experimental Data
θ=90° 300
θ=90°
θ=90°
θ=90°
θ=90°
θ=90° 200
θ=90°
θ=90°
θ=90°
∆ in degrees
Experimental Data
80
18
100
Exp AnE 45° θ=135°
Exp Aps 45° θ=135°
Exp Asp 45° θ=135°
0
-100
6
8
10
12
14
Wavelength (µm)
16
Figure A.24: Experimental values ∆
for the incident angle 45◦ and θ =
135◦ .
18
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—
42(54)
42
Experimental results
Experimental Data
100
200
Exp AnE 65° θ=135°
Exp Aps 65° θ=135°
Exp Asp 65° θ=135°
∆ in degrees
Ψ in degrees
80
60
40
20
8
10
12
14
Wavelength (µm)
16
Figure A.25: Experimental values Ψ
for the incident angle 65◦ and θ =
135◦ .
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
80
60
40
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
8
10
12
14
Wavelength (µm)
16
Figure A.27: Experimental values of
Ψ for all three incident angles and θ
= 135◦ .
Exp AnE 65° θ=135°
Exp Aps 65° θ=135°
Exp Asp 65° θ=135°
0
-100
6
8
10
12
14
Wavelength (µm)
16
18
Experimental Data
300
200
20
0
6
100
Figure A.26: Experimental values ∆
for the incident angle 65◦ and θ =
135◦ .
Experimental Data
100
Ψ in degrees
18
∆ in degrees
0
6
Experimental Data
300
18
100
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
0
-100
6
8
10
12
14
Wavelength (µm)
16
Figure A.28: Experimental values of
∆ for all three incident angles and θ
= 135◦ .
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
θ=135°
18
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—
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Appendix B
Experimental and model
results
43
main: 2007-12-5 15:34
44
—
44(56)
Experimental and model results
Generated and Experimental
Ψ in degrees
80
Model Fit
Exp AnE 45° θ=0°
Exp Aps 45° θ=0°
Exp Asp 45° θ=0°
60
40
200
8
10
12
Wavelength (µm)
14
Figure B.1: Experimental and model
values for Ψ for the incident angle 45◦
and θ = 0◦ .
16
-400
8
40
20
10
12
Wavelength (µm)
14
16
Generated and Experimental
200
∆ in degrees
Ψ in degrees
6
300
Model Fit
Exp AnE 65° θ=0°
Exp Aps 65° θ=0°
Exp Asp 65° θ=0°
60
Model Fit
Exp AnE 45° θ=0°
Exp Aps 45° θ=0°
Exp Asp 45° θ=0°
Figure B.2: Experimental and model
values for ∆ for the incident angle
45◦ and θ = 0◦ .
Generated and Experimental
80
0
6
0
-200
20
0
6
Generated and Experimental
400
∆ in degrees
100
Model Fit
Exp AnE 65° θ=0°
Exp Aps 65° θ=0°
Exp Asp 65° θ=0°
100
0
-100
8
10
12
Wavelength (µm)
14
Figure B.3: Experimental and model
values for Ψ for the incident angle 65◦
and θ = 0◦ .
16
-200
6
8
10
12
Wavelength (µm)
14
Figure B.4: Experimental and model
values for ∆ for the incident angle
65◦ and θ = 0◦ .
16
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—
45(57)
45
Generated and Experimental
60
50
30
20
0
6
8
10
Wavelength (µm)
12
14
Figure B.5: Experimental and model
values for Ψ for the incident angle 65◦
and θ = 45◦ .
Generated and Experimental
80
0
-200
Model Fit
Exp AnE 65° θ=45°
Exp Aps 65° θ=45°
Exp Asp 65° θ=45°
40
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
20
6
8
10
Wavelength (µm)
12
14
Figure B.6: Experimental and model
values for ∆ for the incident angle
65◦ and θ = 45◦ .
Generated and Experimental
θ=45°
300
θ=45°
θ=45°
θ=45°
200
θ=45°
θ=45°
θ=45°
100
θ=45°
θ=45°
∆ in degrees
Ψ in degrees
60
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
0
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
θ=45°
-100
8
10
12
14
Wavelength (µm)
16
Figure B.7: Experimental and model
values of Ψ for all three incident angles and θ = 45◦ .
Generated and Experimental
80
18
-200
6
8
10
12
14
Wavelength (µm)
Model Fit
Exp AnE 25° θ=90°
Exp Aps 25° θ=90°
Exp Asp 25° θ=90°
∆ in degrees
Ψ in degrees
200
20
8
10
12
14
Wavelength (µm)
16
Figure B.9: Experimental and model
values for Ψ for the incident angle 25◦
and θ = 90◦ .
18
Generated and Experimental
Model Fit
Exp AnE 25° θ=90°300
Exp Aps 25° θ=90°
Exp Asp 25° θ=90°
40
16
Figure B.8: Experimental and model
values of ∆ for all three incident angles and θ = 45◦ .
60
0
6
100
-100
10
0
6
200
∆ in degrees
Ψ in degrees
40
Generated and Experimental
300
Model Fit
Exp AnE 65° θ=45°
Exp Aps 65° θ=45°
Exp Asp 65° θ=45°
18
100
0
-100
6
8
10
12
14
Wavelength (µm)
16
Figure B.10:
Experimental and
model values ∆ for the incident angle
25◦ and θ = 90◦ .
18
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—
46
46(58)
Experimental and model results
Generated and Experimental
80
Model Fit
Exp AnE 45° θ=90°
Exp Aps 45° θ=90°
Exp Asp 45° θ=90°
200
∆ in degrees
Ψ in degrees
60
40
20
100
0
Model Fit
Exp AnE 45° θ=90°
Exp Aps 45° θ=90°
Exp Asp 45° θ=90°
-100
0
6
8
10
12
14
Wavelength (µm)
16
18
Figure B.11:
Experimental and
model values for Ψ for the incident
angle 45◦ and θ = 90◦ .
-200
8
10
12
14
Wavelength (µm)
∆ in degrees
Ψ in degrees
20
100
0
Model Fit
Exp AnE 65° θ=90°
Exp Aps 65° θ=90°
Exp Asp 65° θ=90°
-100
10
12
Wavelength (µm)
14
16
Figure B.13:
Experimental and
model values for Ψ for the incident
angle 65◦ and θ = 90◦ .
Generated and Experimental
80
Ψ in degrees
60
40
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
20
-200
6
8
10
12
Wavelength (µm)
10
12
14
Wavelength (µm)
16
Figure B.15:
Experimental and
model values of Ψ for all three incident angles and θ = 90◦ .
16
Generated and Experimental
θ=90°
300
θ=90°
θ=90°
θ=90°
200
θ=90°
θ=90°
θ=90°
θ=90°
100
θ=90°
0
-100
8
14
Figure B.14:
Experimental and
model values for ∆ for the incident
angle 65◦ and θ = 90◦ .
∆ in degrees
8
18
200
40
0
6
16
Generated and Experimental
300
Model Fit
Exp AnE 65° θ=90°
Exp Aps 65° θ=90°
Exp Asp 65° θ=90°
60
6
Figure B.12:
Experimental and
model values for ∆ for the incident
angle 45◦ and θ = 90◦ .
Generated and Experimental
80
0
6
Generated and Experimental
300
18
-200
6
8
10
12
14
Wavelength (µm)
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
16
Figure B.16:
Experimental and
model values of ∆ for all three incident angles and θ = 90◦ .
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
θ=90°
18
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47(59)
47
Generated and Experimental
Model Fit
Exp AnE 25° θ=135°
Exp Aps 25° θ=135°
Exp Asp 25° θ=135°
Ψ in degrees
60
40
20
200
8
10
12
14
Wavelength (µm)
16
Figure B.17:
Experimental and
model values for Ψ for the incident
angle 25◦ and θ = 135◦ .
18
0
-200
Model Fit
Exp AnE 25° θ=135°
Exp Aps 25° θ=135°
Exp Asp 25° θ=135°
20
8
10
12
14
Wavelength (µm)
16
Figure B.19:
Experimental and
model values for Ψ for the incident
angle 45◦ and θ = 135◦ .
18
60
40
0
6
Model Fit
Exp AnE 45° θ=135°
Exp Aps 45° θ=135°
Exp Asp 45° θ=135°
8
10
12
14
Wavelength (µm)
10
12
14
Wavelength (µm)
16
Figure B.21:
Experimental and
model values for Ψ for the incident
angle 65◦ and θ = 135◦ .
18
Generated and Experimental
200
100
Model Fit
Exp AnE 65° θ=135°
Exp Aps 65° θ=135°
Exp Asp 65° θ=135°
0
8
16
300
20
0
6
18
Generated and Experimental
400
Model Fit
Exp AnE 65° θ=135°
Exp Aps 65° θ=135°
Exp Asp 65° θ=135°
16
Figure B.20:
Experimental and
model values for ∆ for the incident
angle 45◦ and θ = 135◦ .
∆ in degrees
80
12
14
Wavelength (µm)
100
-100
Generated and Experimental
100
10
200
∆ in degrees
Model Fit
Exp AnE 45° θ=135°
Exp Aps 45° θ=135°
Exp Asp 45° θ=135°
40
0
6
8
300
80
60
6
Figure B.18:
Experimental and
model values ∆ for the incident angle
25◦ and θ = 135◦ .
Generated and Experimental
100
Ψ in degrees
100
-100
0
6
Ψ in degrees
Generated and Experimental
300
∆ in degrees
80
18
-100
6
8
10
12
14
Wavelength (µm)
16
Figure B.22:
Experimental and
model values for ∆ for the incident
angle 65◦ and θ = 135◦ .
18
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48
—
48(60)
Experimental and model results
Generated and Experimental
Model Fit
Exp AnE 25°
Exp AnE 45°
Exp AnE 65°
Exp Aps 25°
Exp Aps 45°
Exp Aps 65°
Exp Asp 25°
Exp Asp 45°
Exp Asp 65°
Ψ in degrees
80
60
40
20
0
6
Generated and Experimental
400
θ=135°
θ=135° 300
θ=135°
θ=135°
θ=135° 200
θ=135°
θ=135°
θ=135° 100
θ=135°
∆ in degrees
100
0
-100
8
10
12
14
Wavelength (µm)
16
Figure B.23:
Experimental and
model values of Ψ for all three incident angles and θ = 135◦ .
-200
18
6
8
10
12
14
Wavelength (µm)
Model Fit
Exp AnE 25° θ=135°
Exp AnE 45° θ=135°
Exp AnE 65° θ=135°
Exp Aps 25° θ=135°
Exp Aps 45° θ=135°
Exp Aps 65° θ=135°
Exp Asp 25° θ=135°
16
Exp Asp 45° 18
θ=135°
Exp Asp 65° θ=135°
Figure B.24:
Experimental and
model values of ∆ for all three incident angles and θ = 135◦ .