Project work done at Linköping University

Studies of Nanostructured Layers with UV-VIS Spectroscopic Ellipsometry

by Tijs Mocking May 4th 2008

Supervisor: Hans Arwin Examiner: Kenneth Järrendahl

Laboratory of Applied Optics, Linköping University, Sweden

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Abstract In this report a model analysis is presented for three different nanostructured layers: silicon nanotips (SiNTs), gold nanosandwiches and Split Ring Resonators (SRR). The last two materials are metamaterials and both may show a negative refraction index. Experimental data are obtained for every sample using a variable angle spectroscopic ellipsometer. For the gold nanosandwiches, also an infrared ellipsometric measurement is done. For complex layers like these, advanced modeling is necessary. A recently developed analysis program including options for both anisotropic permittivity and permeability is used. A realistic model is presented in this report for the gold nanosandwiches, which also includes the magnetic activity in the layer itself. The results for the nanosandwiches are reasonable, and a magnetic oscillation is found in the horizontal plane at around 260 nm although it was not expected to have a resonance that far in the UV-range. For the SiNTs and the SRR it was not possible to create an acceptable model.

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Preface This report is written as a part of my internship for the Laboratory of Applied Optics on the University of Linköping, Sweden, which lasted three months. The internship is part of my study, Applied Physics on the University of Twente in The Netherlands. Doing this internship in a foreign country helped me to acquire some general knowledge in doing research and making a report and a presentation. Also it helped me to get some insights in scientific research and the people that are involved with the research. Furthermore during this internship I got in contact with many people from different countries and that was a very nice experience. The most important person who made this possible is Hans Arwin, from Linköping University who was my guide and supervisor during the internship. I want to thank him for making the internship possible and also for getting me into the floorball group. I also want to thank Kenneth Järrendahl who was my examiner and with whom I had some nice conversations. From the University of Twente I like to thank Herbert Wormeester. Without his relations in foreign countries it would not have been possible to do this internship. I also like to thank the Erasmus organization in The Netherlands who granted me with a Erasmus Internship Scholarship to help me finance this internship. I want to conclude with thanking the JA Woollam Co. Inc. for supplying me with a new version of their WVASE program to test and use during my internship.

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Table of contents ABSTRACT .............................................................................................................................. 2�

PREFACE ................................................................................................................................. 3�

TABLE OF CONTENTS ......................................................................................................... 4�

LIST OF SYMBOLS ................................................................................................................ 5�

1. INTRODUCTION ................................................................................................................ 6�

2. THEORETICAL ASPECTS ............................................................................................... 7�

2.1 ELLIPSOMETRY ................................................................................................................. 7�2.1.1 Standard Ellipsometry ........................................................................................... 7�2.1.2 Generalized Ellipsometry ..................................................................................... 8�

2.2 FIT PROCEDURE ............................................................................................................... 9�2.3 OPTICAL MODELS ............................................................................................................ 9�

2.3.1 Cauchy Model ........................................................................................................ 9�2.3.2 Lorentz Model ....................................................................................................... 10�2.3.3 Effective Medium Approximation (EMA) .......................................................... 10�2.3.4 Biaxial Layer ......................................................................................................... 12�

2.4 META-MATERIALS ........................................................................................................... 13�2.4.1 Negative Refractive Index .................................................................................. 14�2.4.2 The Split Ring Resonator (SRR) ....................................................................... 14�2.4.3 Gold Sandwiches ................................................................................................. 15�

2.5 THE HARMONIC OSCILLATOR ........................................................................................ 16�

3. EXPERIMENTAL ASPECTS .......................................................................................... 19�

3.1 VASE ELLIPSOMETER ................................................................................................... 19�3.2 IR VASE ........................................................................................................................ 20�3.3 THE DIFFERENT SAMPLES ............................................................................................. 20�

3.3.1 Silicon Nanotips ................................................................................................... 20�3.3.2 Gold Sandwiches ................................................................................................. 20�3.3.3 SRR ....................................................................................................................... 21�

4. EXPERIMENTAL RESULTS .......................................................................................... 22�

4.1 MEASUREMENTS ON SILICON NANOTIPS ...................................................................... 22�4.1.1 Substrate Characterization ................................................................................. 22�4.1.1 Silicon Nanotips ................................................................................................... 23�

4.2 MEASUREMENTS ON AU-SIO2-AU SANDWICHES .......................................................... 24�4.2.1 Substrate Characterization ................................................................................. 24�4.2.2 Au-SiO2-Au Sandwiches Structures................................................................. 25�4.2.3 IR VASE Measurement on Nanosandwiches ................................................. 28�

4.3 MEASUREMENTS ON COPPER NANOSTRUCTURES (SRR) ........................................... 30�

5. CONCLUSIONS AND DISCUSSION ............................................................................. 32�

6. REFERENCES ................................................................................................................... 33�

7. APPENDICES .................................................................................................................... 34�

APPENDIX A .......................................................................................................................... 34�APPENDIX B .......................................................................................................................... 36�

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List of Symbols b [kg/s2] Damping constant c [m/s] Speed of light (3*108) e [C] Elementary charge (1.6*10-19) f [ ] Fraction of a material k [ ] Extinction coefficient me [kg] Mass of an electron (9.1*10-31) n [ ] Refractive index r [m] Radius w [m] Width of a gold nano sandwich x0 [m] Initial position of an electron in x-direction z0 [m] Initial position of an electron in z-direction A, B, C [ ] Cauchy constants An [eV] Amplitude of resonance C [F] Capacitance E [V/m] Electric field En [eV] Position of resonance K [Kg/s] Spring constant L [H] Self inductance M [ ] Number of variable parameters in the model N [ ] Number of (Ψ, ∆) pairs Rp, Rs [ ] Reflection coefficient for p or s polarized light � [º] Phase of the reflected wave � [ ] Electric permittivity � [m] Wavelength � [ ] Magnetic permeability � [ ] Ratio of the reflection coefficients � [º] Standard deviations on the experimental data points � [s] Decay time � [º] Angle of incidence �i [ ] Ratio of the photon energies of the s-polarized light and the p-polarized light � [s-1] Angular frequency [º] Ellipsometric parameter (phase difference) [eV] Broadening of a resonance � [º] Ellipsometric parameter (amplitude ratio)

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1. Introduction Ellipsometry has become a very important tool for research nowadays. In the search for a negative refractive index in so called meta-materials, materials that have properties more due to their structure than their composition, ellipsometry is a suitable technique to use. Negative refractive index has been shown for micro wavelengths and 2D structures, but for optical frequencies and 3D structures it is still a challenge to find a negative refractive index. This is why a number of research groups is working on this subject using spectroscopic ellipsometry [1]. The properties of nanostructures are becoming more interesting for different usages. Trying to reduce the reflectivity of a surface E. Schubert [2] grows a SiO2 nanostructured layer on quartz while monitoring the reflectivity with ellipsometry. The same is done by S. Y. Bae et al. [3] using silicon nanotips, for use on a micro sun sensor for Mars rovers. The results presented by these researchers are very recent, but still there is a need for further methodological development in monitoring magnetic and electric activity in nanostructures. The objectives of this project are:

- Exploring the possibilities to extract nanostructural parameters from ellipsometric data, and - Exploring possibilities to perform ellipsometric modeling of metamaterials with potential negative refractive index.

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2. Theoretical aspects

2.1 Ellipsometry

2.1.1 Standard Ellipsometry Ellipsometry is an optical technique with which it is possible to measure the optical properties and also different other properties of thin films, for example their thickness. As an optical method, ellipsometry is nondestructive and contactless. Spectroscopic ellipsometry is a very sensitive technique and finds applications in many areas of research and industry. Ellipsometry works as follows: monochromatic light comes from a light source, and is polarized. A chopper can be placed just after the light source, which improves the signal-noise ratio by modulating the signal. After being polarized, the monochromatic light can pass an optional compensator (for example a quarter wave plate for giving a phase delay). Then the light hits the sample and the reflected light can pass an optional compensator again. Finally the light is measured and analyzed for changed polarization (see Fig.1) using an analyzer and a detector. There are many different compositions of an ellipsometric system, which each can have a specialized target. With ellipsometry it is possible to find the thicknesses of layers that are thinner than the wavelength of the probing light. It is sensitive to a single layer of atoms and even less. Also multilayer systems with different kind of layers can be examined with this technique. Optical inhomogeneous, isotropic and anisotropic layers can be measured. The polarization state of the light incident upon the sample can be described by its s and p-components. The s component is oscillating parallel to the sample surface and

Fig.1 - Schematic view of an ellipsometer where � being the angle of incidence. The incident light and the normal together forming the plane of incidence.

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is perpendicular to the plane of incidence, which is formed by the surface normal and the incident light. The p-component oscillates perpendicular to the s-component and the incoming beam. The complex-valued amplitudes of the s- and p-reflection coefficients are named Rs and Rp, respectively. 1) pi

p eR δ=pR and sis eR δ=sR

there |Rp| and |Rs| are the amplitudes and �p and �s are the phases of the reflection coefficients. The ratio of Rp and Rs is measured with standard ellipsometry.

2) ∆Ψ== i

s

p eR

R)tan(ρ

Here tan(�)=|Rp|/|Rs| and �=�p-�s are the amplitude ratio upon reflection and the phase shift, respectively. Optical parameters like layer thicknesses and refractive indices can normally not be calculated directly from measured � and � values. However the opposite, i.e. to calculate the � and � from an optical model with values on the thicknesses and refractive indices, is possible. This means that for gathering information about the sample it is necessary do some model analysis. With a layer model the results of the measurements are fitted, using a mean square minimization procedure, so parameters like for example thickness and optical constants can be determined. This model analysis has in this project been done in the program WVASE32tm from J.A. Woollam Co. Inc.

2.1.2 Generalized Ellipsometry Standard ellipsometry (or just short 'ellipsometry') is applied, when no s-polarized light is converted into p-polarized light or vice versa. When a sample is anisotropic, generalized ellipsometry can be applied. This includes measurements of at least three polarization changes at three different polarizations of the probe beam. Anisotropic samples are generally described by a non-diagonal Jones matrix

3) ��

���

�=

ssps

spppr

RR

RRR

At least three values of � at three different �i=Epi/Esi (ratio of the ingoing electric fields of the p- and s-polarized light) are required for an ellipsometric characterization. Now three pairs of � and � are defined. Three complex-valued ellipsometric parameters are measured: �pp, �ps and �sp which are defined as:

4a) ppipp

ss

pppp e

R

R ∆Ψ== )tan(ρ

4b) psips

pp

psps e

R

R ∆Ψ== )tan(ρ

4c) spisp

ss

spsp e

R

R ∆Ψ== )tan(ρ

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Now the Jones matrix can be rewritten as:

5) ��

���

�=

1ppps

spppss

r RRρρ

ρρ

Finding the values �pp, �ps and �sp is the objective in generalized ellipsometry. To improve the precision of the measurement one can measure at more then three �i’s.

2.2 Fit Procedure The program for analysis (WVASE32tm) has a procedure to determine the best fit that a model can give for measured data. This procedure works so that the mean square error (MSE) is calculated for a set of data and a model, and the fit for which the MSE is smallest is the best fit. WVASE32tm uses the following expression for calculating the MSE [5]:

6) �= ∆ �

�

�

�

��

�

�

��

��

�

∆−∆+

��

��

�

−−

=N

i i

ii

i

ii

MNMSE

1

2

exp,

expmod2

exp,

expmod

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σσψψ

ψ

where N is the number of (Ψ, ∆) pairs, M is the number of variable parameters in the model, exp

,iψσ is the measurement uncertainty for �, and exp,i∆σ is the measurement

uncertainty for , modiψ is the evaluated (fitted) point of � and exp

iψ is the measured

point of �, modi∆ is the evaluated (fitted) point of and exp

i∆ is the measured point of . The MSE is weighted by the uncertainties on each measured data point.

2.3 Optical Models For many materials the optical properties are known, but for many materials, like organic materials, or combinations of different materials, the optical properties are still unknown, or not available in the analysis program. There are models to describe some of the materials mentioned, and in this section a few different models will be described.

2.3.1 Cauchy Model The Cauchy model is a relation between the refractive index and the wavelength of the light for a material [5]. The Cauchy model can be used on transparent materials and the equation is given by:

7) 42)(λλ

λ CBAn ++=

Where n is the refractive index, � is the wavelength and A, B and C are coefficients which can be fitted using an analysis program. This model is very good for wavelengths in the visible region if a material is transparent, but quite poor in the

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infrared region. A different model is made by Sellmeier, who invented the Sellmeier equation which works very good in the UV, visible and infrared region. The Cauchy model will be used to model a glass substrate of which the material is unknown.

2.3.2 Lorentz Model The Lorentz oscillator model is very useful for metals. For analyzing a thin layer of metal on an opaque substrate the Lorentz model is the best choice to get reasonable results for the film thickness and optical constants. Also Lorentz oscillators are very useful for describing resonant absorption peaks. The Lorentz oscillator model is based on the assumption that electrons in a material react on an external electric field like a harmonically driven mass on a spring reacts on an external force [5]. In this model the mass is the electron and the spring is the force that holds the electron and the nucleus together. The Lorentz model is formulated as follows:

8) �= Γ−−

+∞=N

i ii

i

EiEEA

E1

221 )()(~ εε

Where )(~ Eε is the dimensionless dielectric function as a function of the photon energy E. N is the total number of oscillators and )(1 ∞ε is the real part of the dielectric function at very large photon energies. Each oscillator is described by three parameters; Ai, �i and Ei. Ai is the amplitude, �i is the broadening and Ei is the center energy of the ith oscillator.

2.3.3 Effective Medium Approximation (EMA) A heterogeneous material (multiple components) can be modeled with an Effective Medium Approximation (EMA). An EMA provides an expression for and average (effective) value on � which is called �EMA, This �EMA can be expressed in the dielectric functions and amounts of the different structures in the heterogeneous material. To give an example: let us consider two ideal two-component structures with composite materials A and B as shown in Fig.2.

Fig.2 - Two different two-component structures with in a) the material boundaries parallel to the applied electric field, and in b) the boundaries perpendicular to the field. Here �A and �B are the permittivity and fA and fB the volume fraction of material A and B, respectively [10].

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In Fig.2a) the boundaries of the two components in the heterogeneous material are parallel to the applied electric field. The dielectric function �EMA= �|| is then given by: 9) BBAA ff εεε +=|| where Af and Bf are the volume fractions of material A and B respectively and Aε and Bε are the dielectric functions. In Fig.2b) the boundaries are perpendicular to the applied electric field and then the dielectric function �EMA= ⊥ε is given by

10)

B

B

A

A ffεε

ε+

=⊥1

If the values for the volume fractions and the dielectric functions are set equal to each other for the two microstructures, it results in a different value for �EMA. These two examples show a phenomenon that is called screening. In the parallel situation the two components contribute according to their volume fraction. By definition there is no screening in this situation. In the other case where the boundaries are perpendicular to the electric field there is maximum screening. The material with the lowest � has the most influence on the ⊥ε . The material with the highest � will be “screened”. In this project some materials will be modeled using an EMA and an example of such a material is given in Fig.3. With an EMA or with a few different layers of EMA’s the optical properties of nanostructure are examined. The thickness of the EMA layer d and also the volume fraction of the material can be found. In the analysis of the experiments done, especially the Bruggeman EMA will be used. It is also called the Bruggeman Effective Medium Theory. The aggregate structure shown in Fig.4 can be approximated with a unit cell also shown in Fig.4. For the dielectric function �, Bruggeman derived the following equation:

Fig.3 - Schematic side view of a silicon substrate with a layer of silicon nanotips on top with height ’d’.

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11) 02

)1(2

=+−

−++−

εεεε

εεεε

B

BA

A

AA ff

This equation can be used when there are two different materials in the structure. However this equation can simply be altered to make it applicable on a structure with more then two materials. It then changes to:

12) 0...222

=++−

++−

++−

εεεε

εεεε

εεεε

C

CC

B

BB

A

AA fff

The Bruggeman theory is widely used in optical analysis, and it can be used for all values of fA.

2.3.4 Biaxial Layer All of the samples that are examined are anisotropic in at least two dimensions. When the optical axis is perpendicular to the surface the optical constants in the x- and y-directions (parallel to the surface) are isotropic, but for the z-direction they are different. Such a material is uniaxial, so �x = �y � �z . When the optical constants also differ for the x- and y-direction the material is biaxial, �x � �y � �z. This information is very important when one wants to make a model of the sample. A biaxial layer can be included in the model. There one can choose the optical constants per direction in the uni- or biaxial layer. When magnetic behavior of a material is included in the model the ‘meta6 layer’ is used. Here a similar thing happens. The permeability is not the same for different directions in the material. In the Meta6 layer the permittivity and the permeability will be calculated using the following equations: 13a) Eεε 0D = 13b) HB 0µµ= D is the electric displacement, and B is the magnetic flux density. In vacuum the permittivity and permeability are �0 and �0

Fig.4 - The random unit cell in the Bruggeman theory. The unit cell has probability fA to be material A and fB=1-fA to be material B [10].

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2.4 Meta-materials The term meta-material came to use quite recently. An artificial material that gains its optical properties from its structure more than from its composition is a meta-material. Also a meta-material is usually a material with “unusual” properties. Their three dimensional structure is periodic in two dimensions in most structures developed so far. The dimensions of the periodic structure are normally smaller than the wavelength of he electromagnetic waves it interacts with for these unusual properties to exist. See Fig.5 for an example of such a periodic structure. An example of these “unusual” properties is a possible negative refractive index. Such a meta-material will bend electromagnetic waves – such as microwaves, radio waves and visible light – the “wrong” way. The electromagnetic waves will not be refracted away from the source they are coming from, but they will refract back in the opposite direction towards the source as shown in Fig.6. This is consistent with Snell’s law if the refractive index is negative. These materials are also called “left handed meta-materials”. The existence of a negative refractive index has been proven already for the microwave frequency range [8]. The possibility of finding a negative refractive index is the reason why many research groups investigate meta-materials, since this property is not found in any existing natural material. There are already some applications of these materials introduced and suggested for medicine, lenses and

military application. A few examples are: the “super lens” and the “stealth suit” [14], [15]. There are some different kinds of meta-materials. The sizes of these materials depends on if the researchers want to find a negative refraction for a larger wavelength, or a shorter wavelength.

Fig.5 – An example of a metamaterial layer. These are small Split Ring Resonators which can exhibit a negative refractive index [13].

Fig.6 – The behavior of electromagnetic waves entering a material with a negative refractive index.

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Some examples of different kinds of meta-materials:

- Single or double Split Ring Resonators - Area’s with ellipsoidal voids in a metal sheet - Wire pairs meta-material - Plate pairs - Metal sandwiches

2.4.1 Negative Refractive Index For describing the behavior of light in a medium the refractive index N=n+ik is very important. In this definition n is the real part of the refractive index, and k is the imaginary part, also the extinction coefficient. The real part of the refractive index n is positive for all naturally existing materials, but the existence of n < 0 is possible. The refractive index can also be calculated with N εµ±= where "' εεε i+= is the permittivity and "' µµµ i+= is the permeability of the material with a real and an imaginary part for both � and �. Some metals (for example silver and gold) already have a negative permittivity for optical frequencies, however, to have a negative refractive index, also the permeability should be smaller then zero: � < 0 and � < 0 [6]. Of course the multiplication of � and � will be positive if both values are smaller than zero, and in this situation the negative value of the root in N εµ±= should be taken. The behavior of electromagnetic waves in a medium with a negative refractive index is shown in Fig.6. That both the permittivity and the permeability are smaller than zero is a sufficient but not a necessary condition for negative refraction. The necessary condition to have a negative refractive index is 0'"'" <+ εµµε , however the materials with just one of the �- and �- values negative will not by useful for application, because such a material will have a large loss [6].

2.4.2 The Split Ring Resonator (SRR) Two concentric rings with a gap in both rings as shown in Fig.7 are known as a Split Ring Resonator (SRR). These SRR’s are the basis of most of the meta-materials with negative magnetic permeability nowadays. It works as follows. Because of the magnetic field in the electromagnetic radiation charges are moved in the rings and will accumulate near the gaps. The gaps in the rings prevent the charges to follow the ring, which means that some charges cross the distance w to the other ring and in this way completing the L-C circuit, which results in a negative magnetic permeability. The capacitance across the rings causes this SRR to be

Fig.7 – Schematic top view of a SRR with definitions of the sizes.

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resonant [9]. Resonance occurs at the resonance frequency: LC

m

1=ω and at this

frequency there is a large absorbance of the external magnetic field. In Fig.8 the resulting permeability is shown. The real part of � is shown in the left graph and the imaginary part is shown in the graph in Fig.8.

If the layer of SRR’s now also has a negative permittivity for these frequencies, the material can exhibit a negative refractive index. As said before this has already been demonstrated in the microwave range, but for optical frequencies it is still a challenge to get a negative refractive index.

2.4.3 Gold Sandwiches Another meta-material that is expected to show some negative refractive index is shown in Fig.9. On a glass surface there are a number of gold/SiO2/gold sandwiches. The figure is just a 2D version with cylindrical structures which shows one cylindrical gold sandwich, but the structures are in fact three dimensional. The thicknesses of the Au-layers are 20 nm and the thickness of the SiO2 is 20 nm for one type of samples and 40 nm in another type.

Fig.8 – In the left graph the real part of the permeability is shown for different resistances of the material. To the right the imaginary part of the permeability is shown for r2=1.5 mm and w=0.2 mm [9].

Fig.9 – Schematic side view of a gold sandwich with a layer of SiO2 between gold layers on a glass substrate [6].

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In this structure an effective permeability was predicted because of localized plasmonic resonances [4], [5]. In these structures there are two different resonances possible: a symmetric and an asymmetric resonance, which result in an effective permittivity and an effective permeability respectively. These resonances correspond to an electric and a magnetic resonance. In Fig.10 the magnetic field and the electric field are shown for a different kind of nano sandwiches, with silver instead of gold. The 2D structure of Fig.9 is elongated for infinite length in the y-direction for this example (directions as in Fig.9).

The arrows are the electric displacement and the colors represent the value of the magnetic field. In the figure it is clearly visible where the permittivity and the permeability come from. For a measurement with a wavelength of 1720 nm the electric displacement vectors are opposite to each other in the different nanostrips. This gives a strong magnetic field in between the nanostrips and that gives rise to a magnetic permeability. For 910 nm the electric displacement vectors are parallel, which give a symmetric resonance and thus an effective permittivity. The wavelengths for which this phenomenon should appear depend largely on the width of the nanostrips and also on the aspect ratio. The resonances of the permittivity and the permeability ideally should overlap, because then there is a possibility that both are negative, and then it might be possible to find a negative refractive index. As said before the measurements for this project are done on gold sandwiches, not on nanostrips. However, it might be possible to find this negative refractive index in these gold nanosandwiches.

2.5 The Harmonic Oscillator Due to an applied electromagnetic field an electron in a symmetric structure moves. Magnetic resonances are expected, but it will be interesting to find more information about the fundamental origin of these resonances. Do these resonances origin from the ordinary movement of an electron in an atomic structure, or do they come from the microscopic structure? The movement of an electron in an atomic structure can be approximated with the Harmonic Oscillator model. Due to the electric field the charge starts moving, and

Fig.10 – Field maps in the sandwich structure during a spectroscopic scan for 1720 nm (left) and 910 nm (right). The arrows represent the electric displacement and the colormap represents the magnetic field [6].

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because of the magnetic field the electron starts moving in a perpendicular direction as shown in Fig.11. The following equations of motions describe the situation shown in Fig.11

14a) 2

2

0 dtxd

mdtdx

bKxeEKx e−−+−=

14b) 2

2

00 dt

zdm

dtdz

bKzecE

dtdx

Kz e−−+−=

Where b=me/� is the damping constant with the mass of an electron me and a decay time . The spring constant K is given by K=2*me with angular frequency . The initial position of the electron is given by x0 and z0. e is the constant elementary charge and c0 is the speed of light in vacuum. E is the applied electric field. With this information an approximation of the movement of an electron in an electromagnetic field can be made using the Simulink function in MATLAB. In this way it can be shown if there is an electric and/or a magnetic resonance. For the Simulink model, see Appendix A. According to this model the electron would move in its structure as shown in Fig.12 The displacement in x-direction is much larger than the displacement in the z-direction. This means that the electric field has a much larger influence on the electrons in a structure than the magnetic field. So if a resonance in the magnetic permeability is found during any measurement, one can conclude that this is not due to the movement of the electrons, but it is more due to the microscopic structure of the layer that is examined. This is the property of a metamaterial.

Fig.11 – An electron with mass me, oscillating in a symmetric atomic structure. The E-field is pointing in the x-direction, while the B-field points perpendicular to the E-field. The electron is moving in the x- and z-direction.

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Fig.12 – The movement of an electron in an xz-plane due to the electromagnetic field from a plane wave propagating in the z-direction. Notice that there is a 20 order difference in the scales.

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3. Experimental aspects

3.1 VASE Ellipsometer VASE stands for Variable Angle Spectroscopic Ellipsometer. For the experiments a VASE from J. A. Woollam Co., Inc. was used. A schematic picture of the VASE is given in Fig.13. The light source is a xenon arc lamp with a monochromator.

The instrument can measure with wavelengths between 188 nm and 1700 nm. The angle of incidence can be varied between 20 and 90 degrees. One should perform the following steps to perform a measurement. The first action is to mount a sample on the sample holder. This sample stays on the holder because of a vacuum pump. Then the alignment process is performed. The alignment detector can be removed after aligning, but this is not necessary if the measurements are done below 5 eV. These steps should be done on a silicon calibration sample first. After a calibration, real measurements on samples under investigation can be done. Typical experimental parameter settings are:

- Wavelength range: 200 – 1700 nm - Angle of incidence range: 40 – 80 degrees - Time of measurements: 30 to 200 minutes.

Some pictures of the experimental setup are shown in Appendix B.

Fig.13 – The VASE ellipsometer with angle of incidence � with the normal. Here a rotating analyzer VASE is shown.

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3.2 IR VASE Measurements in the near and far infrared range were performed using an InfraRed Variable Angle Spectroscopic Ellipsometer (IR VASE) from J.A. Woollam, Co., Inc. The principle of the IR VASE measurement is similar to the VASE experiment for the visible range. This IR VASE has a spectral range from 2 to 30 �m, and the angle of incidence can be varied from 30° to 90° with a precision better than 0,005°. A special feature of the IR VASE is the Rotating Compensator Ellipsometer (RCE) configuration, which provides very accurate “delta” data from 0� to 360� as well as advanced measurements like anisotropy, Mueller matrix and depolarization measurements.

3.3 The Different Samples The following samples have been used for measurements:

• Silicon nanotips (SiNT) on silicon • Sandwiched gold/SiO2/gold nanodots on glass • Split Ring Resonators (SRR)

3.3.1 Silicon Nanotips Two different samples with SiNTs were available: ECR 1006 and ECR 1008. An example of a layer with nanotips is shown in Fig.14. The properties of the different structures are: ECR 1006 - SiNTs Length = 950 ~ 900 nm Diameter = 100 ~ 140 nm Spacing = ~ 200 nm ECR 1008 - SiNTs Length = ~1350 nm Diameter = 100 ~ 130 nm Spacing = ~ 200 nm The percentage of void in the SiNT (Silicon NanoTips) is about 85%. This percentage changes for different depths in this SiNT-layer.

3.3.2 Gold Sandwiches A schematic example of the layer layout is shown in Fig.15. The sandwiches have three layers and they are placed on a glass substrate. The gold layers are 20 nm each, but the SiO2-layer varies between 20 nm and 40 nm for two different samples.

Fig.14 – SEM image of the ECR 1008 SiNT. The tips have a height of about 1350 nm.

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The diameter of such a sandwich is approximately 170 nm. It is predicted that in these sandwiches there will be electric resonances but also magnetic resonances, which should make a negative refractive index at optical frequencies possible.

3.3.3 SRR The third substrate that is examined is the SRR. An SRR is shown in Fig.16. All the different sizes of the structures are given in the figure. These structures are copper structures embedded in a silicon surface. The height of the copper structure is about 800 nm.

Fig.15 – Schematic side view of cylindrical gold sandwiches on a glass substrate. The thickness of the SiO2 layer d is 20 nm or 40 nm.

Fig.16 – SEM micrograph of the SRR pattern on the silicon substrate. [9]

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4. Experimental results

4.1 Measurements on Silicon NanoTips The substrates for the samples are measured first to make sure that a good approximation of the silicon substrate including the oxide layer can be made. After that the real SiNTs structures are examined.

4.1.1 Substrate Characterization The substrates consist of silicon with a top layer of silicon dioxide, because the oxygen in the air reacts with the silicon. The model used is shown in Fig.17. An intermix layer is included to model the interface and this gives a better result for the fit. The mean square error is: MSE=0.438. The experimental and fit data are shown in Fig.18.

0 si 1 mm1 Intermix 1.021 nm2 sio2 2.361 nm

Fig.17 – Layer structure of the substrate on which the silicon nanotips had been grown.

Fig.18 – Experimental data and fit data of the substrate on which the SiNTs had been grown.

23

4.1.1 Silicon Nanotips

Because it is expected that the widths of the nanotips changes for different heights above the surface, a model as shown in Fig.19 has been made for a measurement on a ECR 1006 sample. The nanotips are expected to have a void fraction of about 85%, but the higher one gets above the surface, the more narrow the SiNTs will become, so there is more void at that height. The different layers of the nanotips are modeled as an effective media layer. It should be possible to approximate the optical constants with

a mixture of silicon and void. The result of this modeling is shown in Fig.20. It is clear that the result is not as good as would be expected. The multilayer EMA model is not sufficient for modeling this complicated nanostructure and another way of modeling this structure should be found. The optical constants that describe the optical behavior of the SiNT layer are shown in Fig.21. There are big differences in these values when changing the wavelength of the light or the angle of incidence.

Fig.19 – Optical model of the SiNTs sample. The nanotip layer is assumed to be anisotropic. In the horizontal plane the layer is symmetric, but not in the vertical plane. The EMA model is used to approximate the SiNTs layer.

0 si 1.004 mm1 sio2 2.361 nm2 ema (si)/19% sio2 1.870 nm3 sio2 2.200 nm4 ema2 (si)/85% void 0.000 nm5 ema3 (si)/85% void 0.000 nm6 biaxial 124.776 nm7 ema4 (si)/90% void 0.000 nm8 ema5 (si)/90% void 0.000 nm9 biaxial2 382.172 nm10 ema6 (si)/95% void 0.000 nm11 ema7 (si)/95% void 0.000 nm12 biaxial3 51.787 nm13 ema8 (si)/97% void 0.000 nm14 ema9 (si)/97% void 0.000 nm15 biaxial4 10.239 nm16 ema10 (si)/98% void 0.000 nm17 ema11 (si)/98% void 0.000 nm18 biaxial5 4.914 nm

Fig.21 – Measured graphs of the refractive index and the extinction coefficient for two different angles of incidence.

Optical constants Si Nanotips

Wavelength (nm)400 600 800 1000 1200 1400

<n>

<k>

0.6

0.8

1.0

1.2

1.4

1.6

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Exp < n>-E 50°Exp < n>-E 70°Exp < k>-E 50°Exp < k>-E 70°

Si Nanotips

Wavelength (nm)400 620 840 1060 1280 1500

Ψ in

deg

rees

0

10

20

30

40

50

60Model Fit Exp E 50°Exp E 60°Exp E 70°Exp E 80°

Fig.20 – Experimental and fit data of the measurement on the silicon nanotips.

24

4.2 Measurements on Au-SiO2-Au sandwiches First a measurement on a bare glass surface without the gold sandwiches was done. This is the characterization of the substrate. If this is done, better results for the real measurements can be obtained. Subsequently these experimental results will be used for fitting, using a model that does not include the magnetic resonances in the material. Also a model will be made that does include these resonances.

4.2.1 Substrate Characterization The experimental results are fitted using a Cauchy model. The Cauchy parameters are as follows: An=1.494, Bn=6.412*10-3 and Cn=-8.604*10-5. The experimental results and the fit are shown in Fig.22. The error of the fit is: MSE=0.536. The optical constants of the Cauchy layer are shown in Fig.23 and it is clear that this is the result for a glass substrate, because the refractive index n is around 1.50 which is the n-value for glass. The refractive index gets higher in the UV-range. The extinction coefficient stays zero for all wavelengths.

Substrate Characterization

Wavelength (nm)400 600 800 1000 1200 1400

Ψ in

deg

rees

0

5

10

15

20

25

30

35

Model Fit Exp E 40°Exp E 50°Exp E 60°Exp E 70°Exp E 80°

Fig.22 – Experimental and fit data of the substrate on which the gold sandwiches are grown.

Fig.23– Experimental and fit data of the real part and the imaginary part of the refractive index of the substrate.

cauchy Optical Constants

Wavelength (nm)200 400 600 800 1000 1200 1400

Inde

x of

refra

ctio

n '

n'

Extinction C

oefficient 'k'

1.490

1.500

1.510

1.520

1.530

1.540

1.550

0.00

0.02

0.04

0.06

0.08

0.10

nk

25

4.2.2 Au-SiO2-Au Sandwiches Structures

For modeling the actual Au-SiO2-Au sample the Cauchy parameters in section 4.2.1 are used in the model as the substrate of the sample. In Fig.24 the model is shown that is used for fitting the experimental results without including magnetic effects. Because the material is asymmetric (the x and y directions are similar and form the horizontal plane, but the vertical z-direction is different), a biaxial layer is used. Two different Lorentzian oscillators are used to make an accurate approximation of the gold sandwich layer.

0 substrate nanosandwich 1 mm1 lorentz 0.000 nm2 lorenz2 0.000 nm3 biaxial 59.072 nm

In Fig.25 the experimental data and the fit resulting from the model shown in Fig.24 are shown. The model gives an MSE of 1.482, so this fit is quite accurate. In Fig.26 the model is shown for modeling the ex

perimental results when also magnetic behavior in the material is included. The meta6 layer is the layer where one can decide if the magnetic resonances in the material should be included or not. For approximating the permeability in the layer a Lorentzian-oscillator is used. In references [11] and [12] it is explained why the Lorentzian-oscillator is a good approximation for the behavior of the permeability.

Fig.24 –optical model of the gold nano sandwiches sample, without including magnetic effects. The Lorentz layer simulates the electric resonances in the x- and y- directions and the Lorentz2 layer simulates the response in z-direction.

Fig.25 – Experimental and fit data for gold nanosandwiches on a surface. This measurement was done on a sample with gold nanosandwiches that had layers of 20 nm Au, 20 nm SiO2 and 20 nm Au.

Gold sandwiches 20-20-20

Wavelength (nm)200 400 600 800 1000 1200 1400

Ψ in

deg

rees

0

5

10

15

20

25

Model Fit Exp E 56°Exp E 58°Exp E 60°Exp E 62°Exp E 64°Exp E 66°Exp E 68°Exp E 70°

26

The experimental data and the resulting fit are shown in Fig.27. The mean square error is MSE=1.151. The fit is quite good, and using magnetic resonances in the model for the gold sandwiches gives a better result than when the magnetic behavior is neglected. There is one magnetic resonance, and it is observed in the Lorentz3 sub layer, which means that there is a magnetic resonance in the horizontal plane. The permeability is shown in Fig.28. The magnetic resonance is situated around 260 nm, in the UV-range. The permeability in Lorentz4 has a constant value of �=1.251.

Gold sandwiches 20-20-20

Wavelength (nm)0 300 600 900 1200 1500 1800

Ψ in

deg

rees

0

5

10

15

20

25

Model Fit Exp E 56°Exp E 58°Exp E 60°Exp E 62°Exp E 64°Exp E 66°Exp E 68°Exp E 70°

0 substrate nanosandwich 1 mm1 lorentz 0.000 nm2 lorenz2 0.000 nm3 biaxial 0.000 nm4 lorentz3 0.000 nm5 lorentz4 0.000 nm6 biaxial2 0.000 nm7 meta6 61.423 nm Fig.26 – Layer model of the gold nano

sandwiches sample, when concerning the magnetic response of the material. The biaxial layer simulates the electric response, and the biaxial2 layer the magnetic response, which is also uniaxial. These two come together in the Meta6 layer.

Fig.27 – Experimental and fit data for gold nanosandwiches on a surface. Again all three layers of the sandwiches are 20 nm thick. While fitting, also the magnetic response of the material was taken into account.

Fig.28 – The magnetic response of the gold nanosandwiches. The magnetic resonance is situated around 260 nm and it is observed in the horizontal plane.

lorentz3 Optical Constants

Wavelength (nm)0 300 600 900 1200 1500 1800

Rea

l per

mea

bilit

yIm

ag permeability

1.110

1.115

1.120

1.125

1.130

1.135

1.140

0.005

0.010

0.015

0.020

0.025

0.030

ε1ε2

27

Table.1 Oscillator parameters per layer Layer Number of oscillator �� Am � (eV) En (eV) Lorentz 1 1.237 0.477 0.140 1.475 Lorentz 2 1.237 1.120 0.322 1.861 Lorentz2 1 0.981 0.188 0.172 1.540 Lorentz2 2 0.981 0.042 0.300 2.451 �� Lorentz3 1 1.091 0.904 7.648 5.434 Lorentz4 1 1.251 0 0 0

There are multiple oscillators in the dielectric functions of different layers. In Table.1 the properties of the oscillators in the different layers are shown. Here �� is the permittivity of the layer and �� the permeability for low or values of the wavelength. Am is the amplitude and � is the broadening of the resonance. En is then the position of the resonance in the energy spectrum. Lorentz2 and Lorentz4 are the vertical directions, and Lorentz and Lorentz3 are the horizontal directions. So there is a resonance in the permeability for the horizontal direction, and the permeability is constant for the vertical direction. There are two dielectric resonances for both the x- and y-direction and the z-direction. The dielectric functions of the Lorentz and Lorentz2 layer are shown in Fig.29, and the optical constants of the Meta6 layer are shown in Fig.30. Most optical variation is seen between 500 nm and 1000 nm.

Lorentz and Lorentz2 Optical Constants

Wavelength (nm)0 300 600 900 1200 1500 1800

Rea

l Die

lect

ric C

onst

ant,

ε 1

Imag D

ielectric Constant,

ε2

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

ε1xε1zε2xε2z

Fig.29 – Dielectric functions (real and imaginary parts) of the gold nanosandwich layer.

Optical constants Meta6 layer

Wavelength (nm)300 600 900 1200 1500 1800

<n> <

k>

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Model Fit Exp < n>-E 56°Model Fit Exp < k>-E 56°

Fig.30 – Measured graph of the refractive index and the extinction coefficient for two different angles of incidence.

28

4.2.3 IR VASE Measurement on Nanosandwiches T. Driscoll et al. [16] have found that in a structure like Split Ring Resonator a resonance in the permeability could be found at long wavelengths in the infrared range. This could also count for the gold nanosandwiches. However, a resonance in the IR region has not been found yet using the VASE for the visible range, so by using the IR VASE maybe a resonance can be found. A measurement on only the glass substrate can be seen in Fig.31. The optical constants of the Cauchy layer of the substrate in chapter 4.2.1. are used as the initial values and then the optical constants n and k of the layer are fitted to the experimental data. The resulting fit and the experimental data are shown in Fig.31. The resulting value for the MSE is MSE = 1,16. The optical constants n and k for the glass substrate are shown in Fig.32.

Optical constants IR

Wavelength (nm)0 3000 6000 9000 12000 15000

<n> <

k>

0.6

0.9

1.2

1.5

1.8

2.1

2.4

-0.3

-0.0

0.3

0.6

0.9

1.2

1.5Model Fit Exp < n>-E 50°Exp < n>-E 70°Model Fit Exp < k>-E 50°Exp < k>-E 70°

Substrate Characterization

Wavelength (nm)0 2000 4000 6000 8000 10000 12000

Ψ in

deg

rees

0

10

20

30

40

50Model Fit Exp E 50° Visible rangeExp E 60° Visible rangeExp E 70° Visible rangeExp E 50°Exp E 60°Exp E 70°

Fig.31 – Experimental and fit data of the glass substrate in the IR range. Also the data measured in the visible range are plotted in the graph. The gap at 1400-1800 nm arises from the fact that both the VIS and IR VASE are not accurate enough for these wavelengths.

Fig.32 – Optical constants of the substrate. Both the experimental and fit data are plotted.

29

The experimental data of the actual measurement on the nanosandwiches is shown in Fig.33. Making a fit that was accurate for both the visible range and the IR range did not work, but these experimental data give some important information. The result is almost the same for the measurement on the glass substrate. This means that the layer with sandwiches has no major effect on the light reflecting on the surface. If there would be a resonance in the permeability in the IR range, some activity in the � would be expected. However, the fact that this is not observed in the results must mean that there is no resonance in the permeability in the IR range, also according to the experimental data for � that are shown in Fig.34. The action starts around 7000 nm, which is the same for the data and this is typical for a glass sample and it looks like there is no magnetic action in the IR region.

Fig.33 – Experimental data of the nanosandwich sample. Both the visible and IR range measurements are plotted. The result is very similar to the result of only the glass substrate.

Gold nanosandwiches IR measurement

Wavelength (nm)0 2000 4000 6000 8000 10000 12000

Ψ in

deg

rees

0

10

20

30

40

50Exp E 60° Visible rangeExp E 70° Visible rangeExp E 50°Exp E 60°Exp E 70°

Gold nanosandwich IR measurement

Wavelength (nm)0 2000 4000 6000 8000 10000 12000

∆ in

deg

rees

-100

0

100

200

300Exp E 60°Exp E 70°Exp E 50°Exp E 60°Exp E 70°

Fig.34 – Experimental � data of the nanosandwich sample. Optical activity starts at 7000 nm and higher.

30

4.3 Measurements on Copper Nanostructures (SRR) Because it was not possible to measure on the substrate on which the SRRs were grown, the approximation is made that the substrate consists of silicon with a layer of 2 nm silicon dioxide on it. The fabrication process of the SRR’s is explained in reference [9]. The layer model is shown in Fig.35. Also a generalized ellipsometry measurement is done to find out if there is some anisotropy in the structure.

The experimental and fitting data for this metamaterial are shown in Fig.36. The mean square error is: MSE=47.76. The resulting fit is not very good. More research should be done to find the right model for this sample. Then it will be possible to find information about the optical constants of this metamaterial.

Fig.35 – Optical model simulating the Cu nanostructure on the substrate.

Fig.36 – Experimental and fit data for the copper Split Ring Resonators. The fit is not accurate, and there is some anisotropy measured for wavelengths higher than 900 nm. The Aps and Asp data hold the information about the anisotropy. The AnE data are used to find the optical parameters of the sample.

Split Ring Resonator

Wavelength (nm)0 400 800 1200 1600 2000 2400

Ψ in

deg

rees

0

20

40

60

80Model Fit Exp AnE -50°Exp AnE -40°Exp AnE -30°Exp Aps -50°Exp Aps -40°Exp Aps -30°Exp Asp -50°Exp Asp -40°Exp Asp -30°

0 si 1 mm1 sio2 2.000 nm2 lorentz 0.000 nm3 lorenz2 0.000 nm4 biaxial 819.890 nm

31

The optical constants of the biaxial layer are obtained from the experimental data and they are shown in Fig.37. The graphs of these optical constants are quite special, because the refractive index is smaller than one for wavelengths smaller than 1500 nm. But the refractive index of a vacuum is only one. This could mean that the complexity of the SRR-layer and -structure have influenced the refractive index to behave like this. The optical model should be optimized to be able to make some reasonable conclusion on this particular behavior. This near zero refractive index happens for both the angles of incidence, so with a larger sample this also could be tested for larger (more standard) angles of incidence.

Fig.37 – Refractive index and extinction coefficient for two different angles of incidence. The refractive index reaches very high values, while the imaginary part of the refractive index gets below zero for some values of the wavelength.

biaxial Optical Constants

Wavelength (nm)0 300 600 900 1200 1500 1800

Inde

x of

refra

ctio

n 'n

'

Extinction C

oefficient 'k'

0.87

0.90

0.93

0.96

0.99

1.02

0.00

0.03

0.06

0.09

0.12

0.15

nxnzkxkz

32

5. Conclusions and Discussion There were three different kinds of samples on which measurements were performed. For two of these samples, SiNTs and SRRs, there was not enough time to get acceptable models, but still the experimental results gave some interesting information. One sample however, the gold nanosandwiches, gave less problems in modeling, and a model including the magnetic behavior of the nanosandwich layer has been made. SiNTs A model of the substrate was easily made with a very small error. The model for the SiNTs layer however was not very good. Mathematically the model seemed appropriate, but is physically not correct. The model was not sufficient for a complicated structure like the SiNTs. SRR Measurements on just the substrate were not possible for this sample. An assumption has been made to model the substrate. A model has been made for this sample, but also for this sample the model failed to give an accurate fit to the experimental data. It gives some global approximation, but it is not good enough to make conclusions from. The experimental data however give some special information about the refractive index that which is smaller than one for wavelengths up to 1500 nm. Gold Nanosandwiches The glass substrate is modeled using a Cauchy layer, which gave a quite accurate fit. The gold nanosandwich layer is modeled in the spectral range up to 1700 nm using Lorentzian oscillators for a uniaxial layer. In the infrared a point to point fit was done. The fit of the gold nanosandwiches was acceptable, but when the magnetic behavior of the complex structure was included in the model using a new version of the WVASE32tm program the fit was even more accurate, and a magnetic oscillation is found in the horizontal plane in the UV-range. Still one cannot say that it is a fact that this magnetic oscillation is exactly there. The fit should still be better if one wants to conclude that with more certainty. Oscillations were expected in the IR-range, that is why an IR measurement has been done, but the result was different than was expected. There was no magnetic activity in the IR-range, the measurement on only the substrate looked almost identical to the measurement on the sample including the gold nanosandwiches. In future experiments different approaches or improvements for modeling the SiNTs and the SRR structures should be developed. The SiNTs and the SRR hold very interesting information on how the microstructure of a layer has more influence on the optical behavior of the layer than the composition of this layer. The version of the WVASE32tm program in which magnetic behavior could be included was only a tryout that seemed to work acceptable. This program should be tested more so the value of the program will be known when one wants to model different complex metamaterials.

33

6. References [1] N. Liu and H. Guo, “Three-dimensional photonic metamaterials at optical

frequencies,” Nature Mat., 7, pp. 31-37, 2008 [2] E. Schubert, “Sub-Wavelength Antireflection Coatings From Nanostructure

Sculptured Thin Films,” Phys., 47, pp. 545–550, 2007 [3] S. Y. Bae, C. Lee, S. Mobasser and H. Manohara, “Silicon Nanotips

Antireflection Surface for Micro Sun Sensor,” Nanotechn., 2, pp. 527–530, 2006

[4] N. Feth, C. Enkrich and M. Wegener, “Large-area magnetic metamaterials via compact interference lithography,” Opt. Express, 15, pp. 501-507, 2007

[5] J. A. Woollam Co., Inc., “Guide to using WVASE32TM,” WexTech Systems, Inc., New York, 1995

[6] U. K. Chettiar, A. V. Kildishev, T. A. Klar† and V. M. Shalaev, “Negative index meta-material combining magnetic resonators with metal films,” Opt. Express, 14, pp. 7872-7877, 2006

[7] M. W. McCall, A. Lakhtakia and W. S. Weiglhofer, “The negative index of refraction demystified,” Eur. J. Phys., 23, pp. 353–359, 2002

[8] R. A. Shelby, D. R. Smith and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science, 292, pp. 77-79, 2001

[9] P. Han and W. Ni, “Optical Studies of Cu-based Meta-materials,” Thesis report, 2007

[10] H. Arwin, “Thin Films Optics and Polarized Light,” Lecture notes Linköping University, 2007

[11] T. Driscoll, D. N. Basov, W. J. Padilla, J. J. Mock and D. R. Smith “Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements,” Phys. Rev. B75, 115114, 2007

[12] A. Ishimaru, S. Lee, Y. Kuga and V. Jandhyala, “Generalized Constitutive Relations for Metamaterials Based on the Quasi-Static Lorentz Theory,” IEEE trans. antennas propagate. 51, pp. 2550-2557, 2003

[13] The Spokesman-Review, “Building for the future,” 2006 http://www.spokesmanreview.com/cover.htm

[14] A. Cho, “High-tech metamaterials could render objects invisible,” Science, 312, pp.1120, 2006

[15] F. Mesa, M. J. Freire, R. Marqués, and J. D. Baena, “Three-dimensional superresolution in metamaterial slab lenses: Experiment and theory,” Phys. Rev. B72, 235117, 2005

[16] T. Driscoll, G. O. Andreev and D. N. Basov, “Quantitative investigation of a terahertz artificial magnetic resonance using oblique angle spectroscopy,” Appl. Phys. Lett., 90, 092508, 2007

34

7. Appendices

Appendix A

Fig.A1 – Simulink model for finding a solution for the coupled differential equations which give the movement of an electron in an electromagnetic field. The upper half of the model calculates the x-movement (in reaction to an electric field in the x-direction), while the lower half calculates the z-movement (in reaction to the magnetic field in the y-direction). The differential equations are coupled by the first derivate of the x-displacement that is multiplied with the electric field (given as a sine wave) and the e/c0 constant. The result is plotted separately and also in a XY graph.

35

Fig.A2 – Displacement of the electron in the horizontal direction. On the vertical axis the position in meters is plotted against the time on the horizontal axis.

Fig.A3 – Displacement of the electron in the vertical direction. Very interesting is the amount of displacement in this direction. It is a negligible value.

Fig.A4 – The driving force: the electrical field. On the vertical axis the electric field in V/m is plotted.

36

Appendix B

Fig.B1 – The VASE Ellipsometer and the lamp plus the monochromator.

Fig.B2 – Front view of the VASE Ellipsometer.