Department of Physics, Chemistry and Biology

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1 Department of Physics, Chemistry and Biology Master s Thesis Numerical calculations of optical structures using FEM Henrik Wiklund LITH-IFM-EX--06/1646--SE Department of Physics, Chemistry and Biology Linköpings universitet SE Linköping, Sweden

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3 Master s Thesis LITH-IFM-EX--06/1646--SE Numerical calculations of optical structures using FEM Henrik Wiklund Supervisor: Examiner: Hans Arwin ifm, Linköpings universitet Kenneth Järrendahl ifm, Linköpings universitet Linköping, 22 September, 2006

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5 Avdelning, Institution Division, Department Applied Optics Department of Physics, Chemistry and Biology Linköpings universitet SE Linköping, Sweden Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LITH-IFM-EX--06/1646--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Numeriska beräkningar av optiska strukturer med FEM Numerical calculations of optical structures using FEM Författare Author Henrik Wiklund Sammanfattning Abstract Complex surface structures in nature often have remarkable optical properties. By understanding the origin of these properties, such structures may be utilized in metamaterials, giving possibilities to create materials with new specific optical properties. To simplify the optical analysis of these naturally developed surface structures there is a need to assist data analysis and analytical calculations with numerical calculations. In this work an application tool for numerical calculations of optical properties of surface structures, such as reflectances and ellipsometric angles, has been developed based on finite element methods (FEM). The data obtained from the application tool has been verified by comparison to analytical expressions in a thorough way, starting with reflection from the simplest of interfaces stepwise increasing the complexity of the surfaces. The application tool were developed within the electromagnetic module of Comsol Multiphysics and used the script language to perform post-process calculations on the obtained electromagnetic fields. The data obtained from this application tool are given in such way that easily allows for comparison with data received from spectroscopic ellipsometry measurements. Nyckelord Keywords optics, optical structures, ellipsometry, FEM, Comsol Multiphysics

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7 Abstract Complex surface structures in nature often have remarkable optical properties. By understanding the origin of these properties, such structures may be utilized in metamaterials, giving possibilities to create materials with new specific optical properties. To simplify the optical analysis of these naturally developed surface structures there is a need to assist data analysis and analytical calculations with numerical calculations. In this work an application tool for numerical calculations of optical properties of surface structures, such as reflectances and ellipsometric angles, has been developed based on finite element methods (FEM). The data obtained from the application tool has been verified by comparison to analytical expressions in a thorough way, starting with reflection from the simplest of interfaces stepwise increasing the complexity of the surfaces. The application tool were developed within the electromagnetic module of Comsol Multiphysics and used the script language to perform post-process calculations on the obtained electromagnetic fields. The data obtained from this application tool are given in such way that easily allows for comparison with data received from spectroscopic ellipsometry measurements. v

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9 Acknowledgements This master s thesis has been done at the Laboratory of Applied Optics, Department of Physics, Chemistry and Biology, Linköping University, during the period March to September First of all I would like to thank the support team at Comsol AB, especially Magnus Olsson, who patiently answered my questions and helped me to use Multiphysics in an efficient way. I would like to thank everyone at the Laboratory of Applied Optics, especially my examiner Kenneth Järrendahl and supervisor Hans Arwin for their support and guidance during this work, above all for their constructive comments during the process of writing this report. I would also like to thank Sabyasachi Sarkar for the company during the late evenings spent at the department. Finally my thanks go to my family and friends whose support made this work possible. vii

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11 Contents 1 Introduction Background Task Thesis outline Summary Theory Electromagnetism Maxwell s equations Polarized light Describing materials Controlling electromagnetic properties Reflectance Far-field approximation Numerical calculations Finite Element Method (FEM) Electromagnetic boundary conditions Periodic boundary conditions Perfectly matched layers Total field formulation Scattered-field formulation Error estimation Comsol Multiphysics Comsol Script Optical Measurements by spectroscopic ellipsometry Summary Results and discussion Model development General configuration Two-phase system The model Calculation results Three-phase system ix

12 x Contents The model Calculation results Four-phase system The model Calculation results Lateral modulated model The model Calculation results Summary Conclusions and future work Conclusions Future work Enhanced techniques More advanced modeling Summary Bibliography 69 A Appendix 71 A.1 Left Handed Materials A.1.1 History A.1.2 Application areas List of Figures 74 List of Tables 75

13 Chapter 1 Introduction Complex surface structures in nature often have remarkable optical properties, many of these are not yet fully understood. By understanding the origin of these properties, such structures can be used in new materials, giving possibilities to obtain specific optical properties, chosen by design, in new materials. To enhance and simplify the process of analyzing these naturally occurring structures and the design of new materials, there is a need to assist data analysis and analytical calculations with advanced simulations in this field. 1.1 Background The intriguing optical properties found on butterflies (Lepidoptera) and beetles (Coleoptera) have for some time been at interest for the Laboratory of Applied Optics at Linköping University. During the discussing leading to this thesis, some different topics were proposed. For example an investigation of the optical properties of the wings of the butterflies Morpho rhetenor as well as studies on the rather new left handed materials, i.e. materials with negative index of refraction, were discussed. A common factor for these topics were the possibility to use numerical calculations as a complement to ellipsometry measurements. This led to the idea of developing an application tool for numerical calculations of the optical properties of surface structures. The program used in this work, Comsol Multiphysics, were chosen since knowledge about the product already existed within the group. To be able to understand which effects shown in simulations arising from limitations in the model and which ones representing the properties of the modeled structure it is important to explore and understand these limitations during the development process, so that the future result obtained from this application tool can be trusted. 1

14 2 Introduction 1.2 Task The objective of this thesis was to develop an application tool for two-dimensional finite-element calculations of optical properties of surface structures. The correctness of the application tool should be verified in a thorough way by comparison with available analytical expressions, starting with the simplest of interfaces and there after increasing the complexity stepwise. The final goal was to develop an application tool which later on shall be utilized to analyze naturally occurring complex surface structures, e.g. the structure on the wings of the butterfly Morpho rhetenor. 1.3 Thesis outline Chapter 2 This chapter covers the basic theory used in this thesis. It covers both the theory of electromagnetism and the basics of numerical calculations, especially the finite element method. A short introduction to ellipsometry is also given at the end. Chapter 3 Here the result from the calculations made during the process of development are presented together with a discussion about the accuracy for the different models. The accuracy is shown with both error-estimating calculations presented in tables and visual verification of the accuracy by the use of figures. Chapter 4 This chapter contains a conclusion of the overall performance of the developed application tool. The known limitations are described and guidelines for the use of the tool are presented. Future possibilities, both in terms of more advanced calculation techniques and different ways to describe the optical properties, as well as new structures to do calculations of, are also presented in this chapter. Appendix A A short description of the history of the concept metamaterials and some of its possible applications are presented in the appendix. 1.4 Summary In this chapter the purpose and goal of this thesis has been described. In the next chapter the basic theory of electromagnetism and numerical calculations will be presented and discussed. The models which have been used are described and the analytical expression for their optical properties are presented.

15 Chapter 2 Theory This chapter will give a brief overview of the theory used within this report. It will treat the classical theory of electromagnetism and the finite element method (FEM), which are the two basic parts of this thesis. Since the results given by the calculations mostly are to be compared to data measured with ellipsometry that topic will also be briefly discussed here. 2.1 Electromagnetism The basics for the application tool that have been developed in this work is the theory of electromagnetism. It is the equations unified by Maxwell that are used when the propagation and reflection of light is numerically calculated within the FEM based solver. This section will very briefly describe the basics of electromagnetism and try to give an overview of the physics behind the equations that are utilized in the models. The interaction between the electromagnetic wave and different media will be discussed in some detail, as well as the possibilities to control these properties by for example designing surface structures. Even though it is common to use CGS-units in electromagnetism, SI-units will be used all through the thesis Maxwell s equations In 1873 Maxwell [1] published a unified theory of electromagnetism, which today is recognized as classical electromagnetism. The four equations known as Maxwell s equations were developed by different physicists during the 19 th century, but Maxwell put them together and wrote them in a unified, modern, mathematical form. Maxwell s equations describe the behavior of the electromagnetic fields and their interaction with matter. In the general case, and written on differential form, 3

16 4 Theory the equations are, D = ρ (2.1) B = 0 (2.2) H = D t + J (2.3) E = B t (2.4) The four fields here 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 is the charge density. The equations describe, respectively, how electric fields are produced by electric charges, the absence of a similar magnetic charge, how magnetic fields are produced by currents and changing electric fields and how changing magnetic fields produce electric fields. The relation between the dielectric displacement field and the electric field, as well as the relation between the magnetic flux density and the magnetic field will be described in some detail later on. Plane waves It can rather easily be shown that one solution to Maxwell s equations is a harmonic oscillation, e.g. a plane wave. The electric field, the magnetic field and the direction of propagation given by the propagation vector q, are then orthogonal to each other and form a right-hand system. The amplitudes of the fields will also be proportional to each other according to, E 0 = cµh 0 (2.5) where c is the speed of light in the media and µ is the permeability. The electric field amplitude E 0 and the magnetic field amplitude H 0 are defined by, E = E 0 e i(ωt q r) (2.6) H = H 0 e i(ωt q r) (2.7) In the equation above ω is the angular frequency and r is the coordinate vector. Since the electric and magnetic fields in a plane wave are connected in such way, it is not necessary to describe all fields explicitly. This is used in the application tool, where the z-part of the electric and magnetic fields are used together with the wavelength and the direction of propagation to describe the fields. Electromagnetic spectrum The electromagnetic spectrum is divided into different classes of radiation. Radio waves are example of radiation in the longer wavelength regions, with wavelengths in the range of 10 3 to 10 8 m. On the other side of the spectrum X-rays with wavelengths in the range 10 8 to m and γ-rays with wavelengths below m are found. The wavelengths of interest for this thesis are in the visible and near infrared regions, i.e. between 400 nm and 1000 nm.

17 2.1 Electromagnetism Polarized light It is necessary to describe the orientation of the fields in a propagating electromagnetic wave, i.e. the polarization, especially when studying reflection from surfaces. This is done by dividing the fields into two components, usually with the plane of incidence as reference. The plane of incidence is defined by the propagation vectors of the incident, reflected and refracted waves when studying reflection/- transmission at surfaces. When no oblique reflection or incidence occurs, a plane of propagation is chosen in such way that it simplifies the description of the polarization. The two components are the p-component, for which the electric field lies in the plane, and the s-component where the electric field is perpendicular to the plane. For light with angle of incidence θ 0 the p- and s-components are defined as shown in Fig E ip H ip N 0 θ 0 θ 0 E rp H rp E is H is N 0 θ 0 θ 0 E rs H rs N 1 θ 1 N 1 θ 1 y^ z^ x^ H tp E tp H ts E ts (a) p-components (b) s-components Figure 2.1: The p-polarized (a) and the s-polarized (b) parts of the reflected, transmitted and incident fields in reflection from a single surface with complex refractive index N 0 of the ambient medium and N 1 of the substrate. The angle of incidence θ 0 equals the angle of reflection, and θ 1 is the angle of refraction. For light propagating in an x-y-plane, as in the system above, the relation between the coordinate-based and the polarization-based description of the incident and reflected fields are, E ix = E ip cos θ 0 E rx = E rp cos θ 0 E iy = E ip sin θ 0 E ry = E rp sin θ 0 E iz = E is E rz = E rs. (2.8) Describing materials When considering an electromagnetic wave propagating through a medium, it is necessary to describe the way the electric and magnetic fields interacts with the

18 6 Theory material. Since the wavelengths of interest in this work are much larger than the size of the atoms, the atomic details in the interaction can be averaged into parameters more easily applied to general cases. The interaction with the electric field can be described with the electric permittivity ε, whereas the magnetic permeability µ describes the interaction with the magnetic field. These two electromagnetic parameters gives the complex refractive index N which for example easily can be connected to the speed of propagation of the wave or the way it refracts and reflects at interfaces. Permittivity ε The physical quantity permittivity, or electric permittivity, for a material describes how an electric field affects and is affected by a medium, i.e. it describes the materials ability to polarize in response to an applied electric field. Permittivity has the SI-units Farad per meter [F/m], and it gives the relationship between the electric displacement field D and the electric field E. It is usually complex valued and it varies with the angular frequency ω of the fields, D = εe = ε (ω) E = (ε (ω) iε (ω)) E (2.9) To give a rather simple description of how the permittivity may change with frequency one can think of the atoms and molecules as a set of harmonically bound electron oscillators with a resonance frequency ω 0. An electric field with a frequency far below ω 0 will displace the electrons from the atom core, inducing a polarization in the same direction as the field. As the frequency increases towards the resonant frequency, the electrons will be displaced further and further away from the nuclei cores, inducing a higher and higher polarization. Near the resonant frequency the polarization will be very large, and when the frequency of the field is larger than ω 0 the stored energy within the oscillations will be too large and the electrons response will be out of phase, i.e. there will be a negative response. With even higher frequencies the response will be of much lower amplitude, decreasing towards zero response, although it becomes positive again. Usually, in frequency-regions where the real part of the permittivity is negative, no propagating electromagnetic waves exist. In such regions the material is said to show metallic properties. For regions where the real part is positive, electromagnetic waves can propagate and the material is said to be dielectric. The imaginary part of the permittivity ε, represents the phase difference between the fields, and since such a delay of the response leads to losses the imaginary part also is a description of the attenuation. The permittivity of a material is normally presented by the relative permittivity ε r, given by, ε = ε r ε 0 or ε r = ε ε 0 (2.10) where ε 0 is the permittivity of free space, F/m.

19 2.1 Electromagnetism 7 Permeability µ Permeability, or magnetic permeability, is the degree of magnetization of a material that responds linearly to an applied magnetic field, i.e. it gives the relationship between the magnetic flux density B and the auxiliary field strength H. The SIunit of permeability is Henry per meter, [H/m]. Just like the permittivity it is frequency dependent and it may be complex valued, B = µh = (µ (ω) iµ (ω)) H (2.11) As in the case of permittivity, the imaginary part represents the phase delay in the response and may be related to the losses within the material. The frequency variation of magnetic response may be described with the harmonic oscillators as in the electric case, but now considering harmonically bound magnetic moments instead. It is usually said that µ r = 1 for large frequencies, e.g. in the visible range, since the magnetic dipoles cannot follow the rapid oscillations of such magnetic field. The permeability of a material is most often given as the relative permeability, in the same way as the relative permittivity, µ = µ r µ 0 or µ r = µ µ 0 (2.12) where µ 0 is the free space permeability, 4π 10 7 H/m. If the real part of the permeability is negative the medium will be opaque for electromagnetic waves, just as the case for permittivity. Refractive index N The complex refractive index N is the most commonly used parameter to describe optical properties of a medium, at least in elementary ray optics. It easily gives such electromagnetic properties as the phase velocity ν p, the group velocity ν g, and the refraction of a rays passing through an interface according to Snell s law. It is defined as, N = ± ε r µ r (2.13) The sign is here determined by requirements of causality (see appendix A.1). Since both ε and µ are frequency dependent and generally complex valued, the general form is, N = n (ω) ik (ω) (2.14) Here n is then the ordinary index of refraction and k is the extinction coefficient. Just like the name indicates, the extinction coefficient describes how the wave decays within the medium, according to, E (z) = E (0) e kz (2.15)

20 8 Theory where z is the direction of propagation. Considering real valued permittivity and permeability in eq on the preceding page, it is seen that if either ε r or µ r are negative, the refractive index will be complex valued, i.e. the medium will be opaque. But if both have the same sign, N will be real-valued and the material will be non absorbing Controlling electromagnetic properties In the same way as the individual atoms can be averaged out due to the large difference in size between the atoms and the wavelength of electromagnetic waves up to at least the infrared region, it is possible to replace the detailed interaction of more complex collections of structures with effective parameters of the same kind, as long as the size of the structures is smaller than the wavelength. The artificially created material, the metamaterial, can then be seen as a homogenous material with new effective parameters. For example it is common to talk about effective permittivity ε eff and effective permeability µ eff when discussing optical properties of metamaterials. In later years new ways to design materials have been invented, giving large possibilities for the physicist to, to some extent, control the permittivity and permeability after their own choice. Applications There are many different useful applications for materials with optical properties which has been controlled by design. It is for example possible to create a material with a specific refractive index and with matched optical impedance towards another material. The optical impedance is defined as, Z = µ ε, and Z 0 = µ0 ε 0 (2.16) where Z 0 is the optical impedance of free space. Surfaces with matched optical impedance, i.e. the optical impedance of the substrate and the optical impedance of the ambient media are equal, will not give rise to any reflection, and may thereby be used as antireflection coatings. Such coatings would be most useful in for example fiber optics. When constructing a metamaterial which has simultaneously negative permittivity and permeability the material will have a negative refractive index. These new so called left handed materials have a wide range of new and useful applications which currently are under exploration. A short description of the development of these materials and some possible applications will be given below, and some further information can be found in appendix A.1 on page 71. History In the science of condensed matter it is common to reduce the complexity of phenomenons by introducing composites. These composites consist of elementary building blocks of materials which behave according to some simplified dynamics. One of these composites is the plasmon, a collective oscillation of the electron density relative to the atom cores. These oscillations have a resonant frequency called

21 2.1 Electromagnetism 9 the plasma frequency, ω p. As discussed above it is close to the resonant frequency, or plasma frequency, that negative permittivity occurs. The plasma frequency is usually in the ultraviolet region of the spectrum. In 1996, a theoretical way [2] to bring the plasma frequency in a medium down to the GHz band was introduced. By construction of a metamaterial consisting of grids of thin metallic wires of the order of 1 µm in radius the plasma frequency could be depressed up to 6 orders of magnitude. This effect is achieved since the effective mass m eff of the electrons become as heavy as nitrogen atoms due to the narrowness of the wires, and due to that ω 2 is proportional to 1/m eff. In 1999, Pendry et al. [3] showed how to introduce a magnetic response into metamaterials by using special structures with the shape of a split ring of non-magnetic materials. In these materials an external applied magnetic field induces a current, which due to the structure produces an effective magnetic response. They predicted that this Split Ring Resonant (SRR), see Fig. 2.2, with a diameter of a few millimeters would get a magnetic resonance frequency a bit above 10GHz. This was later experimentally verified by Smith et al. [4]. Figure 2.2: A Split Ring Resonant (SRR), which has been introduced by Pendry et al. [3] and first fabricated by Smith et al. [4], in which currents may be induced when interacting with electromagnetic waves of specific wavelengths. The induced currents give rise to a magnetic response to the electromagnetic wave in frequency regions much higher than in naturally occurring materials. To be able to control the optical properties at higher frequencies, e.g. in the optical region, the size of these structures has to be decreased into the nanometer scale. This has, with some success, been done by Grigorenko et al. [5] by placing small gold nano-pillars in pairs upon a surface, which gave a similar result as the wires and SRRs. Another, more complex structure, was created by Shalaev et al. [6, 7] when they placed pairs of nano-rods upon each other in a specific pattern. One possible application area for the tool developed during this work would be to simplify the development of new surface structures with this kind of properties.

22 10 Theory Microstructures To change the effective permittivity and/or permeability of the material is not the only way to change the optical properties of a surface. For example the reflections of a microstructure can create optical properties of the surface that otherwise would be impossible to achieve by just designing the optical properties of the materials used. Such microstructures are common on the wings of butterflies, where complex microstructure creates ultra-high visibility and polarization effects induced by reflectivity. There are many microstructures created by nature that not yet are fully analyzed and understood. One of those are the structure on the wings of Morpho rhetenor, which is one of many microstructures that Vukusic et al. [8] are analyzing. The wings of the Morpho rhetenor have a complex multilayerlike structure that might be described as two-dimensional Christmas trees on the surface, which give rise to interesting optical effects, e.g. a strong reflection of blue light Reflectance The information extracted from the calculations in this work are the relationship between the incident and reflected fields. To verify the tool it is necessary to know what results to expect. From the simple cases, this can usually be expressed analytically. The Fresnel s equations are used when modeling layered structures. For models with a lateral modulated layer an effective media approximation (EMA) is used in addition. When doing measurements in optics, the parameter usually obtained is the intensity. Therefore the reflectance is the value extracted from the simulations. During this work the Fresnel s reflection coefficients will be denoted r whereas the total reflection coefficients will be denoted R. The reflectance will be denoted R, and the relation between these parameters are, R = I r I i = ( Er E i ) 2 = R 2 (2.17) where I r and I i are the reflected and incident intensities, respectively. Two-phase systems For the most simple case, a two-phase system (ambient-substrate), the Fresnel s equations are used in their original form. Fig. 2.1 show a two-phase system with all the reflected and transmitted parts of the fields marked out. r p = E rp = N 1 cos θ 0 N 0 cos θ 0 = tan (θ 1 θ 0 ) E ip N 1 cos θ 0 + N 0 cos θ 0 tan (θ 1 + θ 0 ) r s = E rs = N 0 cos θ 0 N 1 cos θ 0 = sin (θ 1 θ 0 ) E is N 0 cos θ 0 + N 1 cos θ 0 sin (θ 1 + θ 0 ) (2.18) (2.19) These equations relate the p- and s-components of the reflected fields, E rp and E rs, to the components of the incident fields, E ip and E is.

23 2.1 Electromagnetism 11 Three-phase systems When expanding to a three-phase system (ambient-layer-substrate) as shown in Fig. 2.3 on the next page, and thereby taking another reflecting interface into account, the relationship between the reflected and incident fields, i.e. the total reflection coefficients, will be described by, R p = E rp E ip = r 01p + r 12p e i2β 1 + r 01p r 12p e i2β (2.20) R s = E rs = r 01s + r 12s e i2β, (2.21) E is 1 + r 01s r 12s e i2β (2.22) where r lmp is the Fresnel s coefficient r p (eq. 2.18) between the phases l and m, and β is the film phase thickness, which is given by, β = 2πd λ N 1 cos θ 1 (2.23) Here N 1 is the complex index of refraction of the middle media, d is the thickness of the layer and θ 1 the angle of refraction. In Fig. 2.3 the components of the reflected and transmitted fields are shown individually. The total reflected field is the sum over all its components according to, E rp = E rs = E rp,j (2.24) j=1 E rs,j (2.25) j=1 n-phase systems When adding further layers in the model the complexity of the analytical expression for the reflectivity will increase rapidly. The expressions in eq and 2.27 are for the case of a four-phase system, shown in Fig When dealing with more layers a more flexible description, using matrices, are applied. Since this application tool only has been tested up to a four-phase system, that matrix description is left out. [ ] [ ] r01p + r 12p e i2β1 + r01p r 12p + e i2β1 r23p e i2β2 R p = E rp E ip = R s = E rs E is = (2.26) [1 + r 01p r 12p e i2β1 ] + [r 12p + r 01p e i2β1 r 23p e i2β2 [ ] [ ] r01s + r 12s e i2β1 + r01s r 12s + e i2β1 r23s e i2β2 (2.27) [1 + r 01s r 12s e i2β1 ] + [r 12s + r 01s e i2β1 ] r 23s e i2β2

24 12 Theory E ip E rp,1 E rp,2 E is E rs,1 E rs,2 N 0 N 1 d θ 0 θ 0 θ 1 N 2 θ 2 E ts,1 E tp,1 E ts,2 E tp,2 Figure 2.3: The incident, reflected and transmitted parts of the E-field in the three-phase model. The complex index of refraction of the ambient media, the layer and the substrate are N 0, N 1 and N 2 respectively. d is the thickness of the layer. The angle of incidence at the upper interface is θ 0 and θ 1 at the lower interface. The total reflected and transmitted p- and s-components of the fields are the sums of the components E rp,j, E rs,j, E tp,j and E ts,j. Lateral modulated models When modeling a laterally modulated surface layer according to Fig. 2.5(a), i.e. a microstructured surface layer consisting of the materials A and B with all internal boundaries orthogonal to the surface, the normal treatment used above will not be sufficient for an analytical expression of the reflectivity. An anisotropic description has to be used to describe the optical properties of the modulated layer, see Fig. 2.5(b). It can be shown that the macroscopic averages of the permittivity for the two extreme cases, shown in Fig. 2.6, when the applied electromagnetic field is parallel and normal respectively to the internal boundaries, are given by, ε = f A ε A + f B ε B (2.28) 1 ε = f A εa + f (2.29) B εb where f A and f B are the fractions of material A and B respectively, i.e. w A w B f A =, f B =, (2.30) w A + w B w A + w B

25 2.1 Electromagnetism 13 E ip E rp,1 E is N 0 θ 0 θ 0 E rs,1 N 1 d 1 θ 1 N 2 N 3 d 2 θ 2 θ 3 E tp,1 E ts,1 Figure 2.4: The incident, reflected and transmitted parts of the E-field in the four-phase model. The complex refractive indexes of the ambient media, the top layer, the bottom layer and the substrate are N 0, N 1, N 2 and N 3 respectively. The top layer has the thickness d 1 and the bottom layer d 2. The angles of incidence for the three interfaces are, mentioned from the top, θ 0, θ 1 and θ 2. The total reflected and transmitted p- and s-components of the fields are the sums of the components E rp,j, E rs,j, E tp,j and E ts,j. where w A and w B are defined according to Fig. 2.5(a). Thereby the anisotropic complex refractive indexes will be, N = ε r µ r (2.31) N = ε r µ r (2.32) The reflection coefficients R p and R s for the total system will have the same structure as in eq and 2.21 respectively, as before with polarization specific total reflection coefficients r 01p, r 01s, r 12p and r 12s, but also with polarization specific phase thicknesses β p and β s. R p = r 01p + r 12p e i2βp 1 + r 01p r 12p e i2βp (2.33) R s = r 01s + r 12s e i2βs 1 + r 01s r 12s e i2βs (2.34)

26 14 Theory E is E ip E rp E rs E is E ip E rp,1 E rp,2 E rs,1 E rs,2 N 0 θ 0 θ 0 N0 θ 0 θ 0 N 1A w A N 1B w B d N 1 N 1 d N 2 N 2 θ 2 Etp,1 Etp,2 θ 2 Etp E ts (a) Laterally modulated layer Ets,1 (b) Anisotropic layer Ets,2 Figure 2.5: a) A laterally modulated thin layer with the two composite materials A and B, with the complex refractive indexes N 1A and N 1B respectively, and the width of the composites are d A and d B. d is the thickness of the thin layer, θ 0 the angle of incidence and θ 2 is the angle of refraction for the lower interface. The refractive index of the ambient is N 0 and for the substrate N 2. b) The anisotropic model of the laterally modulated thin layer, where the composite layer has been replaced by an anisotropic medium with the complex refractive index N 1 in the direction parallel to the surface, and N 1 in the perpendicular direction. The total reflected and transmitted p- and s-components of the fields are as usual the sums of the components E rp,j, E rs,j, E tp,j and E ts,j respectively. E ε A ε B E ε A w A ε B w B w A w B (a) The parallel microstructure (b) The perpendicular microstructure Figure 2.6: The two extremes for a composite material. a) All boundaries in the composite are parallel to the applied electric field E. For this case the effective permittivity will be a volume average of the two composites. b) All boundaries are perpendicular to the applied electric field E. In this case the material with the lowest permittivity will dominate by screening effects.

27 2.1 Electromagnetism 15 where the single interface reflection coefficients and phase thicknesses are given by, r 01p = N 1 N 1 cos θ 0 N 0 N1 2 N 0 2 sin2 θ 0 (2.35) N 1 N 1 cos θ 0 + N 0 N1 2 N 0 2 sin2 θ 0 r 12p = N 1 N 1 cos θ 2 N 2 N1 2 N 2 2 sin2 θ 2 (2.36) N 1 N 1 cos θ 2 + N 2 N1 2 N 2 2 sin2 θ 2 r 01s = N 0 cos θ 0 N1 2 N 0 2 sin2 θ 0 N N 0 cos θ sin2 θ 0 (2.37) r 12s = N 2 cos θ 2 N1 2 N 2 2 sin2 θ 2 N N 2 cos θ sin2 θ 2 (2.38) β p = 2π d N 1 N1 2 λ N 0 2 sin2 θ 0 1 (2.39) β s = 2π d λ N 2 1 N 2 0 sin2 θ 0 (2.40) Far-field approximation When doing electromagnetic calculations of small structures, it is important to know if the resulting field is a near-field or a far-field. Near-fields are fields close to the structure and is dominated by evanescent waves whereas the far-field is further away and is dominated by propagating waves. A good example of the difference between the two fields can be found in the double-slit experiment. The normally observed field, the far-field, has a well known intensity peak in the middle of the interference pattern, whereas Chae K-M et al. [9] have shown with numerical calculations that the near-field has an intensity minimum at the center, i.e. there is a phase shift of π between the two interference patterns. It is also important to notice that no energy is transported by the evanescent waves in the near-field. It is thereby most important to be sure of what kind of fields that are obtained from the calculations. To ensure that it is the amplitude of the far-field that is retrieved from the calculations a far-field approximation can be performed. Such a transformation may be useful when dealing with fields similar to the one in Fig. 2.7, where the amplitude is rather complex close to the surface. In cases where reflections from a homogenous wave scattered at a flat surface which is infinite and homogenous in x-direction, there is no need for a far-field approximation, since there will be no difference in the structure of the field close to the surface and far away. The reason why the far-field and not the near-field is the field of interest is that the result will be compared to measurements performed a few decimeters distance

28 16 Theory Figure 2.7: The amplitude, shown as grayscale, of the reflected E-field at normal incidence on a dielectric surface with gold nano-pillars placed in pairs. away from the sample, which is in the far-field region. An approximation of the limit, d far, between the near-field and the far-field is, d far = 2L2 λ (2.41) where L is the period of the surface structure, or the dimension of an antenna. This is known as the antenna designer s formula [10]. An often used far field transformation is the Stratton-Chu formula [11], which main application area is radiation calculations of antennas. However, it is applicable on the type of field calculations in this work as well. In the Stratton-Chu approximation the resulting computed electric far-field E p, at point p, is given by, [ ] E p = ˆR 0 ˆn E a η ˆR 0 (ˆn H a ) e iqa ˆR 0 ds (2.42) S where E a and H a are the fields at point a on the surface S, just outside the scattering surface. ˆR 0 is the unit vector pointing from the origin to the field point p, ˆn is the outward normal from the surface at point a. η is an impedance constant of free space, q is the free space wave number (2π/λ or q ) and a is the vector from the origin to the surface point a. 2.2 Numerical calculations The numerical calculations, or simulations, are done with the Finite Element Method (FEM) using the commercial software Comsol Multiphysics. A two-

29 2.2 Numerical calculations 17 dimensional model of the surface is created and constraints and parameters are applied on each subdomain and boundary. To get a model close to reality some different methods are combined into an application tool that will have enough accuracy for our needs. In this section a rather brief introduction to FEM is given followed by more detailed introductions to the additional methods that have been applied Finite Element Method (FEM) The finite element method is a numerical procedure for finding approximate solutions of partial differential equations (PDE) over a model with specified boundary conditions. It is thereby a procedure that may be used to solve many different kind of problems in physics. The principle of the method is to replace an entire continuous domain by a number of subdomains, called elements. In these elements the unknown continuous functions are represented by simple interpolation functions with unknown coefficients. By doing so the number of degrees of freedom are reduced to a finite number, i.e. the solution of the entire system is approximated by a finite number of unknown coefficients. The solution is then obtained by linear or non-linear optimization. One major advantage of FEM is that the size of the elements can differ over the model, and thereby higher resolution and accuracy can be obtained at parts where so wanted, e.g. at interfaces or curves. It also makes it suitable for calculations in complex structures, which has lead to many applications in for example structural mechanics Electromagnetic boundary conditions The most important part of the model is the boundary conditions. These conditions describe how the interfaces affect and are affected by the electromagnetic fields, e.g. induced surface currents. They may be divided into subgroups, the internal and the external boundary conditions. Internal boundaries The internal boundaries are the ones inside the model, i.e. they have subdomains with additional conditions on both sides. Here the subindexes 1 and 2 refers to the subdomains on the different sides of the interface and ˆn is the surface normal. Continuity In the normal case the only condition to internal boundaries is the requirement of continuity of the tangential components of the fields, i.e. ˆn (E 1 E 2 ) = 0 ˆn (H 1 H 2 ) = 0 (2.43)

30 18 Theory Surface current and surface charge When introducing a surface current on the boundary J s or a surface charge ρ s only some small adjustments have to be done in the eq. 2.43, according to, ˆn (E 1 E 2 ) = ρ s ε ˆn (H 1 H 2 ) = J s (2.44) Impedance boundary condition The impedance boundary condition, also known as a mixed boundary condition, is used when the second medium is an imperfect conductor. where 1 ˆn ( E) iq 0 ˆn (ˆn E) = 0 µ r1 η (2.45) 1 ˆn ( H) iq 0 ηˆn (ˆn H) = 0 ε r1 (2.46) η = µ r2 /ε r2 and q 0 = ω ε 0 µ 0 (2.47) This condition may sometimes [12] be useful to give in a more generalized way, by exchanging the constants iq0 η with a constant γ e and introduce the term U on the right hand side according to, External boundaries 1 µ r1 ˆn ( E) γ eˆn (ˆn E) = U (2.48) The external boundaries are the ones that limit the model in size. represent either limiting materials or open space. They may Perfect electric/magnetic conductor The conditions for either perfect electric or magnetic conductor are given by eq and 2.50 respectively. ˆn E = 0 (2.49) ˆn H = 0 (2.50) Neutral For boundaries which are both perfect electric and magnetic conductors, the boundary conditions are, consequently, given by, (ˆn E) = 0, (ˆn H) = 0. (2.51)

31 2.2 Numerical calculations 19 Matched boundary condition The matched boundary is an open space boundary that may be used to introduce a plane electromagnetic wave, E 0 and H 0, with the propagation constant β. This type of boundary also allows for radiation out of the model for plane waves with the same direction of propagation as specified with the propagation constants. ˆn ( E) iβ (E (ˆn E) ˆn) = 2iβ (E 0 (ˆn E 0 ) ˆn) (2.52) ˆn ( H) iβ (H (ˆn H) ˆn) = 2iβ (H 0 (ˆn H 0 ) ˆn) (2.53) Weak formulation The weak formulation, or variational method, is another way to solve the original PDE problem. The Ritz method, also known as the Rayleigh-Ritz method [12], is one of the most commonly used variational methods within FEM. In this method the boundary-value problem is formulated in terms of a functional, i.e. a function of functions. The governing differential equations for the problem corresponds to the minimum of this functional, i.e. the solution can be obtained by minimizing the functional with respect to a test function. Denoting the functional F and the test function φ, which according to section is described with a finite number of coefficients corresponding to the interpolation functions within each element, the functional that is to be minimized can be written like, ( N ) F (φ) = F c i v i (2.54) where N is the total number of coefficients, c i is the coefficient to the corresponding interpolation function v i. The test function φ which minimizes the functional F is the solution to the problem. Since the variational method is difficult to use without the computational power provided with computers, it is rather unusual to describe physical problems in this way. For this reason it is most often necessary to obtain the functional F from the PDE description of the boundary-value problem. In the case of electrodynamics with complex values of the permittivity, permeability or boundary specific parameters like γ, the generalized variational principle have to be used for this calculation. The functional F is then given by, i=1 F (φ) = 1 2 Lφ, φ 1 2 Lφ, u + 1 φ, Lu φ, f (2.55) 2 where the inner product is defined as, φ, ψ = Ω φψdω (2.56) The operator L and the function f are taken from the PDE written on the form in eq An example, the Poisson s equation, is shown in eq u is the

32 20 Theory non-vector version of the inhomogeneous vector term in the boundary condition eq on page 18. Lφ = f (2.57) (ε E) = ρ { L = (ε ) f = ρ (2.58) In the case of vector functions the inner product is defined as, a, b = a bdv (2.59) V the function f is replaced by a vector function f and the inhomogeneous term u is replaced with U. For a electromagnetic boundary-value problem defined by the vector wave equation, ( ) 1 E q µ 0ε 2 r E = iq o Z 0 J (2.60) r on the volume V and a boundary condition on the boundary S according to eq the weak form can be derived as, F (E) = 1 [ ] 1 ( E) ( E) q 2 µ 0ε 2 r E E dv + r V [ γe ] 2 (ˆn E) (ˆn E) + E U ds Mesh S +iq 0 Z 0 V E JdV (2.61) The mesh is the net which divides the model into a finite number of elements. It can be created in some different ways, but there are two methods which mainly are used. One is unstructured with triangular elements and the other is structured with squared elements. Both methods have their advantages and disadvantages. For example, it is preferable to have equal meshes at boundaries linked by periodic boundary conditions (see Sec ), which may be accomplished by a structured mesh. On the other hand such method of meshing will require that the subdomains of the model all are fairly rectangular in shape. When using triangular elements, it is easier to have freely shaped geometries in the model. The unstructured way have another advantage in that the random orientation of the elements actually helps minimizing phase-errors occurring due to for example numerical anisotropic

33 2.2 Numerical calculations 21 phase-velocities. The number of finite elements used, how fine the mesh is, is most important. With too few elements the solution may not converge. On the other hand, if the mesh is too fine, the simulation will take much more computational power than actually needed. Since the meshing is very flexible in FEM, it is easy to use a finer mesh at the parts of the model where this is needed, for example areas close to interfaces. A rule of thumb is to have at least five to ten elements per wavelength on open subdomains, and even denser close to surfaces and corners Periodic boundary conditions When studying the optical properties of a surface structure, it would be convenient to work with an unlimited sample, so that no effects of the physical limitations of the sample would interfere with the result. But when dealing with nano-sized surface structures it is not even reasonable to perform large scale surface simulations with ordinary computer power. This problem is possible to overcome with Periodic Boundary Conditions (PBC), in the cases where the surface structure has some kind of periodicity. The PBC links the sides opposite to each other and will thereby, in a way, make the model infinitely long in that direction, repeated with the periodicity introduced in the condition linking the boundaries together. The PBC can be used to make the linked boundaries identical to each other, but it is often more convenient to introduce a different relationships between the two boundaries. Those, more generalized, conditions are sometimes called linked boundary conditions. When using models where the E-field only has one of the p- and s-components, the boundary conditions corresponding to perfect electric/magnetic conductors, eq and 2.50 are used for the boundaries where periodic boundary conditions are applied. For models which have both components of the electric field, the neutral boundary condition, eq is used instead Perfectly matched layers In some directions there might be an infinite continuation of the media. It is then necessary to use a boundary condition which can represent this infinite continuation, also called open boundary. The best way to do this is to introduce a Perfectly Matched Layer (PML). The PML [13, 12] is used to limit the reflections from this kind of open, free-space, boundaries. They are expansions of the model in the directions where infinity is to be simulated. By changing the way that the permeability and permittivity are defined for the subdomain, a gradually increasing absorption is achieved. This absorption may also be seen as a coordinate transformation which makes the optical path length infinite. The typical length of the PML is about one wavelength, but this is of course a question of the quality needed in the simulation versus the

34 22 Theory computational power that are available. For a PML that are to absorb radiation in the y-direction the permeability and permittivity matrices will be multiplied with the following operator, where, L = L xx = s ys z s x L xx L yy L zz L yy = s zs x s y (2.62) L zz = s xs y s z (2.63) s x = 1 s y = a ib s z = 1. (2.64) For example the relative permittivity in such case would be given by, ε r = ε r L = ε rl xx ε r L yy 0 (2.65) 0 0 ε r L zz The constants a and b are to be set depending on the size of the PML and how fine the mesh in that subdomain is. The smaller the subdomains are, the higher value of the coefficients will be needed. The attenuation of the field in the example above is described by, E = E 0 e bqy y (2.66) when propagating over a distance y with the y-component q y of the wave vector q. The real part of s y will affect the additional absorption of the already decaying evanescent waves in the PML. These PMLs would in an analytical solution not give any reflections at all. This is unfortunately not the case in numerical calculations, and some reflections will always occur when using FEM. However, they are minimized to a very acceptable level with this implementation. Within this report, no simulation where the substrate is a left hand material is performed, but since it might be a natural development in the future, it is worth mentioning that the ordinary PML does not work if applied to such materials [14]. Instead a modified version, especially made for left handed materials will have to be applied Total field formulation The most straight forward way to introduce an incident field is to introduce it on the outer (upper) boundary. The wave will then travel from this boundary down towards the scattering surface, and the reflected wave will propagate up towards the upper boundary again. The solution provided by FEM is the total field

35 2.2 Numerical calculations 23 and the reflected wave may be distinguished by subtraction of the incident wave from the total field. This is not all too difficult, since the analytical expression for the incident wave is well defined. However, the propagation of the wave is calculated numerically, and some difference between the analytical expression and the simulated field will occur. This is usually a rather small difference, but since the amplitude of the incident field in many cases are considerably larger than the amplitude of the reflected field, such differences may have a significant influence on the extracted information of the reflected field. Another problem that might occur when using a total field formulation is when the method is combined with a PML. The incident wave will then be introduced above the PML, and the amplitude of the wave will be drastically decreased on the way down through the PML. This introduces a numerical uncertainty in the amplitude of the effective incident field, below the PML. Thereby the accuracy of the calculations is further decreased Scattered-field formulation An alternative to the total field formulation is the scattered-field formulation [12, 15], where the incident field is introduced directly on the scatterer. The implementation of the scattered-field formulation uses the weak formulation (variational calculation). When considering a volume V bounded by the surface S, the electromagnetic fields generated by an internal current density J i can be described with the curlcurl equation obtained from Maxwell s equations, [ 1 µ E ] + ε 2 E t 2 + σ E t = J i t r V (2.67) where the magnetic field has been eliminated with aid of the constitutive relations and σ is the conductivity of the media. By substituting E with E inc + E sc into eq with J i = 0, the incident field E inc may be separated from E sc according to, [ 1 ] µ E sc + ε 2 E sc t 2 [ ] 1 = µ E inc ε 2 E inc t 2 + σ E sc t = σ E inc t (2.68) The right hand side may now be replaced by an equivalent current source, J eq according to eq. 2.67, i.e. [ ] J eq 1 = t µ E inc + ε 2 E inc t 2 + σ E inc (2.69) t When applying eq in eq. 2.68, [ ] 1 µ E sc + ε 2 E sc t 2 + σ E se t = J eq t (2.70)

36 24 Theory J eq is only nonzero in the region with µ, ε or σ are different from the ambient media, according to eq where J i is zero due to the absence of currents. Together with an impedance boundary condition according to, [ ] 1 ˆn µ E sc + Y c t [ˆn ˆn E sc] = 0 (2.71) the weak form of eq will be, { 1 µ [ N i] [ E sc ] + εn i 2 E sc t 2 + σn i E sc t + V + 1 µ [ N i] [ E inc ] + εn i 2 E inc t 2 + σn i E } inc dv + t { + Y c [ˆn N i ] t [ˆn E sc] 1 } µ [ˆn N i] [ E inc ] ds = 0 (2.72) S where the intrinsic admittance of the infinite medium Y c is ε/µ, N i are the interpolation vector functions and ˆn is the outward normal to the surface S. When written on this form, the incident field is involved in the volume integral over the entire computational domain V as well as in the surface integral over S. To increase the efficiency of the calculations, the knowledge of the incident field outside the scatterer, in terms of fulfilled equations, may be utilized to reduce the weak form to, { 1 µ [ N i] [ E sc ] + εn i 2 E sc t 2 + σn i E } sc t + dv + V + V s { 1 µ [ N i] [ E inc ] + εn i 2 E inc t 2 + σn i E } inc dv + t + S S s Y c [ˆn N i ] t [ˆn E sc] ds+ 1 µ [ˆn s N i ] [ E inc ] ds = 0 (2.73) where S s is the surface of the scatterer, V s its volume and ˆn s its outward normal. From eq it is seen that the incident wave only will be introduced in the weak terms inside the scatterer and on its outer boundaries. By introducing the incident wave into the weak formulation directly there will be an increase in accuracy when extracting the amplitude of the reflected wave, since no subtraction will have to be made. On the other hand, when looking at the transmitted field the incident field will have to be added to the result, since the field obtained from the finite element calculation will be the difference between the total field and the incident field.

37 2.2 Numerical calculations Error estimation It is important for the verification process to have some quantities for determination of the accuracy of the simulated data. The parameters which are to be compared for each model setup are the reflectances R s and R p as well as the parameters used in ellipsometry measurements, Ψ and. These parameters are calculated for each angle of incidence θ and wavelength λ. Mean squared error The method for estimating the accuracy throughout this thesis were chosen to be the Mean Squared Error (MSE). MSE = 1 N + 1 N (X ri X ci ) 2 (2.74) i=1 where X r is the reference (analytic) value, X c the calculated value and N the number of points that are compared. The mean squared error is exactly what it seems to be, the mean value of the squared error in each point of measurement. To be able to get a single value for each model the sum is taken over both angle of incidence θ and wavelength λ. For an easier implementation the normalization is changed from N + 1 to N, which is due to that the summation will have to be divided into different steps. Since N usually is large, e.g. 13 wavelengths with 40 angles of incidence each gives 520 evaluated points, this will have a very limited effect on the result. ( ) 1 MSE = (X rλθ X cλθ ) 2 (2.75) 1 N λθ + 1 λ,θ (X rλθ X cλθ ) 2 1 N λ λ N θ One advantage by using this error estimation is that it is also used in some ellipsometry measurements, where parameters are adjusted to make a model fit the measured data. As a complement to the single MSE value calculated for each model, the extreme values of the MSE where only the angle is varied can also be presented where so is needed. This data will help to discover if there are some wavelengths for which the model works less well Comsol Multiphysics The program chosen as environment for the application tool is Comsol Multiphysics ver. 3.2, from here on only referred to as Multiphysics. Multiphysics were in the beginning a PDE tool box for Mathworks MATLAB which later became an add-on application with the name FEMLAB. The program is now independent but still compatible with MATLAB. It is a multipurpose FEM program which can be used for many different types of simulations within a wide range of areas in physics. In addition to the FEM solver, the program contains modules for the different areas of physics. These modules have predefined sets of equations and variables used for solving problems of the specified type. Multiphysics has also θ

38 26 Theory a very strong advantage when it comes to linking problems from different areas together Comsol Script The new version of Multiphysics contains the possibility to easily operate the program from a consol, just like FEMLAB could be controlled from MATLAB. This script approach gives more flexible ways for post processing the data obtained from the solution, as well as doing sets of multiple simulations while varying a larger amount of parameters. It can be seen as a small version of the MATLAB interface, which is more than enough for the average user. It is also possible to create a graphical user interface of your own with the script language combined with JAVA, which opens up for large possibilities in creating your own specialized application tool with Multiphysics as backbone. 2.3 Optical Measurements by spectroscopic ellipsometry Ellipsometry is a commonly used optical technique for determining properties of surfaces and thin films. By studying the change of polarization when a monochromatic plane wave is reflected at oblique incidence it is possible to determine the complex reflectance ratio, which is defined as ρ = R p = R p R s ei(δp δs) (2.76) R s where R p and R s are the complex reflection coefficients for the p- and s- parts, respectively. ρ is usually expressed using the phase change,, and the amplitude ratio, tan Ψ, where tan Ψ = R p and = δ p δ s (2.77) i.e. R s ρ = tan Ψe i. (2.78) By varying the angle of incidence, frequency and the polarization of the incident wave enough information can be gathered to retrieve the electromagnetic properties of the material, as well as the thickness of thin layers. Since the relative amplitude change is a ratio between the p- and s-components the result will be independent of the intensity of the source.

39 2.4 Summary Summary In this chapter the basic theory of FEM has been discussed, including the weak formulation and the scattered-field formulation. The notations for the parameters that are obtained from the calculations are defined, and the analytical expressions for the reflectance in the different models are displayed. In the next chapter this will be used to describe the development and verification process of the application tool. The obtained accuracy in the different models will also be shown.

40 28 Theory

41 Chapter 3 Results and discussion This chapter will describe the steps performed during the process of development and verification of the application tool, i.e. the implementation of the theory described in Sec. 2. It begins with a section about the basic settings utilized in the tool and continues with separate sections for the different types of structures that have been simulated. The main purpose of the models created in this work is to verify the accuracy of the obtained result. Due to this, the focus will be to find out where the calculations start to differ from the analytical expressions, and if possible, ways to modify the models so that the differences may be minimized. If the accuracy for some cases can not be verified, it is important to map out the limits so that the results obtained from unknown structures can be trusted. Unfortunately, it is not possible to do simulations with all different combinations of parameters, but the simulations presented in this chapter are selected so that they will cover as large range of combinations as possible. 3.1 Model development To obtain a high efficiency the models are initially created within Multiphysics where it is easy to create a geometry, apply boundary conditions, define the necessary expressions, e.g. for the incident wave, and set the optical properties of the different domains. The first step of the verification process, comparison between the calculated and analytical values for the reflectances R s and R s, and the ellipsometric angles Ψ and, for a single frequency may also easily be performed in Multiphysics. Smaller simulation sets can be performed for which the resulting fields easily can be graphically examined, i.e. the plane wave approximation may be verified and abnormalities may be noticed. The finished model is then exported to a script file, where possibilities to easily modify the size of the model, the parameters of the materials etc are introduced as well as more advanced post-processing calculations on the obtained fields. This section will describe the configuration of the models which are used throughout this work. 29

42 30 Results and discussion 1000 nm 1250 nm PML 1000 nm Ambient 500 nm Substrate 1250 nm PML Figure 3.1: The four standard domains in the basic geometry. Geometry The geometry of the multiple-phase models are more or less identical, except for the layers introduced between the ambient media and the substrate (Fig. 3.1). The width is chosen so that it allows for at least one period of the incident wave always to fit within the model, i.e nm. The heights of the two PMLs are rather large, 1250 nm, which in this model represents from about one to three wavelengths. Since no measurements of the transmission is made on these models, the substrate domain is only 500 nm thick whereas the ambient domain is 1000 nm General configuration The major parts of the configuration of the models are the same. The ambient area is for example always air, i.e. the relative permittivity as well as the relative permeability of the ambient domains are both set to unity. Equations for the incident field are added, so that the only parameters connected to the incident field needed to be changed during the set of simulations are the polarization specific amplitudes E 0p and E 0s as well as the wavelength and the angle of incidence. The polarization specific amplitudes may be given in a time dependent way, i.e. the polarization may be changed over time.

43 3.1 Model development 31 Structure Ψ R p R s Triangles 3.08e e e e-7 Squares 2.05e e e e-9 Table 3.1: The mean squared error for two-phase systems meshed with an unstructured mesh (triangles) and a structured mesh (squares). The relative permittivity of the substrate is 4.00 and both meshes contain elements, which represents degrees of freedom. The wavelength in these simulations was 400 nm, i.e. the wavelength where the density of the mesh is most important for the simulation result. Perfectly matched layers As mentioned above, the geometry for the PML is rather large in this application tool. Thereby the need for strong absorption is reduced and the absorption coefficients defined in Sec , the constants a and b, may be set to unity without getting unwanted reflections of noticeable amplitude from these layers. Scattered-field formulation The scattered-field formulation is used to introduce the incident field in the model. As described in Sec the incident wave is to be introduced on the boundaries of the scattering object as well as on the areas (domains) within. Since the scattering object in this application tool is a semi-infinite medium, the only boundary where the modification will have to be made is the boundary between the ambient media and the top of the layer/substrate. The additional term in the area part of the weak formulation is to be added in all areas below that boundary. Since there currently does not exist any way to introduce new terms in Multiphysics, the boundary term is introduced as a surface current. For the domain the changes are added to the ordinary weak formulation term. Mesh To obtain a good result the maximum size of the elements are set to 25 nm. This value is only a guideline for the program, and the program adapts the mesh so that the density is increased in areas where so is needed. The two different structures of the mesh described in Sec have been evaluated. As shown in Table 3.1 the obtained result for the mean squared error of the reflectance is better with the structured method, using squares. For the parameters used within ellipsometry, Ψ and, there are a slight advantage for the unstructured method, using triangles. In addition, the purpose of this application tool is to be used on advanced optical structures, which are to be rather freely shaped, making only the unstructured mesh applicable. That is, the unstructured mesh was chosen as the default mesh.

44 32 Results and discussion Periodic boundary conditions The left and right sides of the model are connected with periodic boundary conditions of the type, E target = E source e iqx x (3.1) where q x is the x-component of the wave vector, x is the width of the model, E source the field on the source boundary and E target the field on the target boundary. This means that the periodicity of the model is determined by the incident wave, i.e. by the wavelength and angle of incidence. The periodicity of the surface structure is only taken into account when creating the geometry of the model, and it might thereby be a mismatch between the periodicity of the incident light and the surface structure. This mismatch might influence the result in a negative way, but there is no way to eliminate it. Preferable the size of the model should be adapted to both these periodicities, i.e. the size should be a multiple of both the periodicities. This is unfortunately not possible since the model used already is close to the limit of what can be calculated on an ordinary equipped desktop computer. Electromagnetic boundary conditions For the boundaries which are connected with a periodic boundary condition, i.e. the sides of the model, the electromagnetic condition on the boundary will be of a type closely related to the neutral type (eq. 2.51) according to, (ˆn E) z = 0, (ˆn H) z = 0. (3.2) This pair of conditions restricts the possibilities for currents in the z-direction of the boundary, i.e. out of the two-dimensional model. The internal boundaries will be set to be continuous (eq. 2.43), even though the boundary between the layer/substrate and the ambient area will be altered to include a surface current. The two boundaries on the outside of the perfectly matched layers will be of the matched type according to eq to minimize the reflections from these boundaries. Data extraction The result obtained from FEM calculations are the complex description of the fields in terms of E z and H z. From this data the reflectances R p and R s, and the ellipsometric angles Ψ and are to be extracted. The reflected field is assumed to be a plane wave, and the amplitude (absolute value) of the p- and s-part of the wave is approximated by averaging through integration over the upper part of the model, just below the PML. The p-component is calculated with the use of H z, by which the corresponding E p easily may be obtained according to eq. 2.5 on page 4 while the s-component equals E z. It is a bit more difficult to extract the phase from the resulting data. The phase

45 3.1 Model development 33 may in some cases be close to 180 degrees, and due to the numerical calculation, it may point wise change from 180 to -180 degrees, which in the case of an averaged value retrieved by integration could give an approximated phase of 0 degrees. Since the phases from the p- and s-components are to be compared to give the phase change, it would be possible to integrate this difference instead. But the problem would still not be avoided since this difference also might be in the region of ±180 degrees. For this reason integration is avoided when extracting the phase information in this work. It would be possible to use logical expressions within the integration to obtain the phase. These logical expressions could for example compare the result with the phase for the previously calculated angle of incidence or with the result which is expected by theoretical calculations. This is though avoided, to keep the computational time needed for the post-processing low. The phase information is thus extracted from a limited number of points and these values are analyzed to give such correct description of the phase change as possible, i.e. the phase shifts at ±180 degrees are compensated for with logical expressions. It is a bit more difficult to extract the phase from the resulting data. The phase may in some cases be close to 180 degrees, and due to the numerical calculation, it may point wise change from 180 to -180 degrees, which in the case of an averaged value retrieved by integration could give an approximated phase of 0 degrees. Since the phases from the p- and s-components are to be compared to give the phase change, it would be possible to integrate this difference instead. But the problem would still not be avoided since this difference also might be in the region of ±180 degrees. For this reason integration is avoided when extracting the phase information in this work. It would be possible to use logical expressions within the integration to obtain the phase. These logical expressions could for example compare the result with the phase for the previously calculated angle of incidence or with the result which is expected by theoretical calculations. This is though avoided, to keep the computational time needed for the post-processing low. The phase information is thus extracted from a limited number of points and these values are analyzed to give such correct description of the phase change as possible, i.e. the phase shifts at ±180 degrees are compensated for with logical expressions. Far-field approximation Multiphysics has built-in support for far-field calculations of the electromagnetic field with the Stratton-Chu formula (eq. 2.42) described in Sec One limitation with this built-in function is that it only takes the actual model into account when integrating, i.e. the infinite extension of the model which is accomplished by use of the periodic boundary conditions are not introduced in these calculations. Thereby this built-in function will not work for this application tool and far-field calculations would have to be calculated separately by translation of the field obtained by FEM and integrating over a large number of these fields placed next to each other, thereby approximating an infinite extension of the sample. This type

46 34 Results and discussion of far-field calculation has not yet been implemented in this application tool. Simulation series As mentioned in Sec the wavelength regions of interests are the visible and near-infrared regions, i.e. 400 nm to 1000 nm. The angles of incidence, usually of interest in ellipsometry, are between 30 and 75 degrees. Since the main goal for this work is to verify the obtained fields, the calculations will be made with a focus on covering a large range of wavelengths and angles of incidence, instead of obtaining high resolution of these parameters. When using the finished application tool the priorities will most likely be the opposite. The series of calculation will per default cover wavelengths from 400 nm to 1000 nm with a step of 50 nm whereas the angles of incidence will go from 30 to 75 degrees with a step of one degree.

47 3.2 Two-phase system Two-phase system The model Figure 3.2: The geometry for the two-phase system. The upper and lower domains are perfectly matched layers, which approximates infinite extension of the two phases in vertical direction. The left and right boundaries are connected with periodic boundary conditions which results in an infinite extension of the model in the horizontal direction. The incident wave is introduced on the middle boundary with the scattered-field formulation. The color (grayscale in printed version) represents the instantaneous electric field, where the illustrated field in the two upper domains is the reflected field and in the two lower domains the transmitted. The wave fronts of the incident field are indicated with contour lines. The width of the model used in this application tool is illustrated with the two vertical lines in the middle of the figure. In this simulation the angle of incidence is 75 degrees and the relative permittivity of the substrate is The first case is a two-phase system, i.e. reflection from a single surface. It is modeled with four subdomains according to Fig. 3.2, where the two upper domains represents the ambient media and the two lower domains represents the substrate. The middle domains are normal media whereas the most upper and lower domains are perfectly matched layers representing semi-infinite extension in vertical direction of the two phases. An infinite extension of the sample in horizontal direction is accomplished by periodic boundary conditions linking the right and left sides together. The incident field is introduced on the middle boundary by the use of

48 36 Results and discussion weak terms, according to the scattered-field formulation in Sec The width of the used model, shown by the two vertical lines in Fig. 3.2, is 1 µm Calculation results At first, the simulations were done with the different polarization parts separately, i.e. individual models were created for calculation of the p- and the s-parts. When the correctness of these both models were verified an integrated model was created. From this model the result was compared with analytical expressions for a large number of different substrate permittivities. The result from a selection of these calculations is presented in Table 3.2. From this data it is seen that there is a good agreement between the calculated values and the analytical expressions for the reflectances R p and R s. The mean squared error for the p-component is of the magnitude 10 8 and for the s-component the result is only a little less accurate, about For the ellipsometric angles Ψ and the MSE values are somewhat higher. However, the reflectance data are ratios in the region [0, 1] and the ellipsometric angles are measured in degrees, i.e. in the region ] 180, 180], which automatically gives rise to higher mean squared errors since no normalization is done to simplify the comparison. The obtained MSE-values for these models are overall indications of good accuracy for the two-phase model. The result from two of the series, i.e. calculations with identical material properties and where only the frequency and angle of incidence are varied, are presented in Fig. 3.3 and 3.4. The first figure illustrates the model with a relative permittivity of the substrate set to 4.84, i.e. the index of refraction of the substrate is Figure 3.3(a) shows how the MSE of the reflectances R p and R s vary with the wavelength. Shown on equal axis only the s-part (dashed) is visible since the p-part is of much smaller magnitude. The MSE value for R s increases towards longer wavelengths, but it remains on a reasonable low level throughout the wavelengths of interest. Figure 3.3(b) shows the mean squared error for the ellipsometric angles Ψ (dashed) and. The mean squared error for Ψ is as seen much lower than for the phase change, which is around 0.01 for the extreme wavelengths. The angular variation of the reflectances R p and R s are shown in Fig. 3.3(c), the upper line consists of both the analytical and theoretical data of the s-component whereas the lower curve consists of the data for the p-component. The data is taken from a simulation with a wavelength of 650 nm, i.e. red light. In Fig. 3.3(d) the ellipsometric angles Ψ and are shown as functions of angle of incidence for the wavelength 650 nm.

49 3.2 Two-phase system 37 (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.3: Results from calculations on a two-phase system with the relative permittivity of the substrate set to a) The mean squared error of the reflectances R p and R s (dashed) as function of wavelength. b) The mean squared error of the ellipsometric angles Ψ (dashed) and as function of wavelength. c) The reflectances R p and R s as functions of angle of incidence with the wavelength set to 650 nm. d) The ellipsometric angles Ψ and as functions of angle of incidence with the wavelength set to 650 nm.

50 38 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.4: Results from calculations on a two-phase system with the relative permittivity of the substrate set to a) The mean squared error of the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error of the ellipsometric angles Ψ (dashed) and as functions of wavelength. c) The reflectances R p and R s as functions of angle of incidence with the wavelength set to 400 nm. d) The ellipsometric angles Ψ and as functions of angle of incidence with the wavelength set to 400 nm.

51 3.2 Two-phase system 39 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e0 8.34e e e e e e-6 avg. 8.47e e e e-6 Table 3.2: Mean squared error for two-phase systems with different relative permittivity ε r1 in the scattering media. With a few exceptions for the phase change the result is very acceptable. The details of the marked simulation series are shown in separate figures. In Fig. 3.4 the result from a system with a substrate relative permittivity of 8.41, i.e. with the complex index of refraction N = 1.9, is displayed. As seen in Table 3.2 the result from this calculation is one of the less accurate, especially concerning the ellipsometric angle. From Fig. 3.4(b) it is seen that the wavelength 400 nm is the main reason for this large inaccuracy. This can be explained by Fig. 3.4(d) which shows how vary with the angle of incidence for this wavelength. It is seen that the calculated value for at 71 degrees differs from the analytical value by about 20 degrees. For the other parts of this calculation the result is rather good. Table 3.3 shows the result from simulation series done with increasing absorption in the substrate. The accuracy is not significantly decreased with the increase of absorption. The result from a simulation with the relative permittivity i, i.e. the complex index of refraction N i, is shown in Fig The variation of the MSE shows that the errors are larger for longer wavelengths, especially at the end-value of 1000 nm.

52 40 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.5: Results from calculations of an absorbing two-phase system with the relative permittivity of the substrate set to i, i.e. with a complex index of refraction of N i. a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 1000 nm. d) The ellipsometric data Ψ and as function of angle of incidence for the wavelength 1000 nm.

53 3.2 Two-phase system 41 Ψ R p R s i 6.67e e e e i 1.44e e e e i 3.12e e e e i 5.02e e e e-6 avg. 2.56e e e e-6 Table 3.3: Mean squared error for two-phase systems with varying complex permittivity ε r1 in the scattering media. The details of the marked simulation serie is shown in a separate figure.

54 42 Results and discussion 3.3 Three-phase system The model Figure 3.6: The geometry for the three-phase system. The upper and lower domains are perfectly matched layers, which approximates infinite extension of the two outer phases in vertical direction. The middle domain is a thin layer of a third media. The left and right boundaries are connected with periodic boundary conditions which results in an infinite extension of the model in the horizontal direction. The incident wave is introduced on the upper boundary of the thin layer with the scatteredfield formulation. The color (grayscale in printed version) represents the instantaneous electric field, where the illustrated field in the two upper domains is the reflected field and in the three lower domains the transmitted. The wave fronts of the incident field are indicated with contour lines. The width of the model used in this application tool is illustrated with the two vertical lines in the middle of the figure. In this simulation the angle of incidence is 75 degrees with the thickness 500 nm of the thin layer and the relative permittivities for the thin layer and the substrate are 4.00 and 9.00, respectively. The next step is the three-phase system shown in Fig. 3.6, i.e. reflections from a thin layer on substrate. This is modeled with an extra domain between the ambient air and the substrate. The infinite extension of the model is obtained by a similar use of PML and PBC as in the case of a two-phase system. The thin layer adds two (three) extra parameters which defines the model, the relative

55 3.3 Three-phase system 43 ε r1 ε r2 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 1.05e e e e-6 Table 3.4: The mean squared error for three-phase systems with the thickness of the middle media set to 10 nm and with varying relative permittivities. The relative permittivity of the thin layer and for the substrate are denoted ε r1 and ε r2, respectively. The details of the marked simulation serie is shown in a separate figure. permittivity (and permeability) of the thin layer and the thickness of the layer. Since it is very time consuming to run large series of simulations, only a few values of each parameter will be tested in this verification process. The chosen thicknesses are 10, 100 and 1000 nm and the relative permittivities used, ε r1 for the thin layer and ε r2 for the substrate, will be 2.25, 4, 6.25 or 9, i.e. the index of refraction will be 1.50, 2.00, 2.50 and There will also be a smaller set of simulation series with complex permittivities, i.e. absorbing materials Calculation results To structure the data obtained from the simulations the result are presented with respect to the thickness of the layer in Tables 3.4, 3.5 and 3.6 where the thickness is 10, 100 and 1000 nm respectively. For each of these tables the result from the simulation series with the relative permittivities set to 6.25 for the thin layer and 4.00 for the substrate are shown in Fig. 3.7, 3.8 and 3.9. For each serie the variation with respect to the angle of incidence of the reflectances and the ellipsometric angles are shown for the shortest wavelength, i.e. 400 nm.

56 44 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.7: Results from calculations of a three-phase system with the thickness 10 nm of the middle media. The relative permittivities for the thin layer, ε r1, and the substrate, ε r2 are set to 6.25 and 4.00 respectively, i.e. the index of refraction for the thin layer and the substrate are 2.25 and 2.00 respectively. a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

57 3.3 Three-phase system 45 ε r1 ε r2 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 2.45e e e e-6 Table 3.5: The mean squared error for three-phase systems with the thickness of the middle media set to 100 nm and with varying relative permittivities. The relative permittivity of the thin layer and for the substrate are denoted ε r1 and ε r2, respectively. The details of the marked simulation serie is shown in a separate figure.

58 46 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.8: Results from calculations of a three-phase system with the thickness 100 nm of the middle media. The relative permittivities for the thin layer, ε r1, and the substrate, ε r2 are set to 6.25 and 4.00 respectively, i.e. the index of refraction for the thin layer and the substrate are 2.25 and 2.00 respectively. a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

59 3.3 Three-phase system 47 ε r1 ε r2 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 3.67e e e e-6 Table 3.6: The mean squared error for three-phase systems with the thickness of the middle media set to 1000 nm and with varying relative permittivities. The relative permittivity of the thin layer and for the substrate are denoted ε r1 and ε r2, respectively. The details of the marked simulation serie is shown in a separate figure.

60 48 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.9: Results from calculations of a three-phase system with the thickness 1000 nm of the middle media. The relative permittivities for the thin layer ε r1 and the substrate ε r2 are set to 6.25 and 4.00 respectively, i.e. the index of refraction for the thin layer and the substrate are 2.25 and 2.00 respectively. a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

61 3.3 Three-phase system 49 Max size. Ψ R p R s 25 nm 1.61e e e e-6 20 nm 5.97e e e e-6 Table 3.7: The mean squared error for three-phase systems with the thickness of the middle media set to 1000 nm and with varying relative permittivities. The relative permittivities were 6.25 for the thin layer, ε r1, and 4.00 for the substrate, ε r2. The details from both these simulation series are shown in separate figures. From these Tables (3.4, 3.5 and 3.6) and figures ( and 3.9) it is seen that the accuracy for these three-phase simulations are rather good. The MSE tends to increase towards longer wavelengths but remains on a reasonable level for this application tool. In Fig. 3.9(b) there are indications of a rather strong increase in the difference between the calculated and the analytical values of the ellipsometric angle for the shorter wavelength, especially the end-value 400 nm. This could be an effect of that the density of the mesh is too small, i.e. to large elements are used. A reduction of the maximum allowed size of the elements from 25 to 20 nm gave the results shown in Table 3.7 and Fig When comparing Fig. 3.9(b) and 3.10(b) it is seen that the MSE for at the end-value wavelength 400 nm has decreased from above 0.1 to below 0.02 whereas the value for the other end-value of the wavelength, 1000 nm, is unchanged at a value around A decrease of the maximum allowed size of the elements from 25 to 20 nm increases the number of elements from almost for this model to about elements. This represents an increase in degrees of freedom from almost to about , which makes the requirements of available memory and computational time significantly larger. Table 3.8 and Fig presents data from simulation series of a model with an absorbing substrate. To reduce the number of simulations the thickness 100 nm were chosen for all series. As seen the accuracy is not significantly lower than for the non-absorbing series, and the result may be seen as acceptable. In Table 3.9 the result from similar simulation series done with an absorbing layer and a non-absorbing substrate are presented. The results here are overall good and with a strong absorption in the layer it can be seen that the change of substrate has a limited effect on the accuracy.

62 50 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.10: Results from calculations of a three-phase system with the maximum allowed size of the elements in the mesh reduced from 25 nm to 20 nm. The thickness is 1000 nm of the middle media and the relative permittivities for the thin layer ε r1 and the substrate ε r2 are, as before, set to 6.25 and 4.00 respectively. a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

63 3.3 Three-phase system 51 ε r1 ε r2 Ψ R p R s i 1.13e e e e i 6.44e e e e i 4.27e e e e i 1.31e e e e i 5.54e e e e i 3.40e e e e i 3.41e e e e i 5.83e e e e-6 avg. 2.33e e e e-6 Table 3.8: The mean squared error for three-phase systems with the thickness of the middle media set to 100 nm and with varying real valued relative permittivity of the thin layer ε r1 and with a complex valued relative permittivity ε r2 of the substrate. The details of the marked simulation serie is shown in a separate figure. ε r1 ε r2 Ψ R p R s i e e e e i e e e e i e e e e i e e e e i e e e e i e e e e i e e e e i e e e e-6 avg. 2.39e e e e-6 Table 3.9: The mean squared error for three-phase systems with the thickness of the middle media set to 100 nm and with two different complex valued relative permittivities of the thin layer ε r1 and with varying values of the relative permittivity of the substrate ε r2.

64 52 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.11: Results from calculations of a three-phase system with complex relative permittivity of the substrate. The thickness of the thin layer is 100 nm and the relative permittivity ε r1 is set to The substrate complex relative permittivity ε r2 is set to i, i.e. the complex index of refraction for the substrate is N i whereas for the thin layer N 1 = a) The mean squared error for the reflectances R p and R s (dashed) as functions of wavelength. b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 900 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 900 nm.

65 3.4 Four-phase system Four-phase system The model Figure 3.12: The geometry for the four-phase system. The upper and lower domains are perfectly matched layers, which approximates infinite extension of the two outer phases in vertical direction. The two middle domains are thin layers of different medias. The left and right boundaries are connected with periodic boundary conditions which results in an infinite extension of the model in the horizontal direction. The incident wave is introduced on the upper boundary of the upper thin layer with the scattered-field formulation. The color (grayscale in printed version) represents the instantaneous electric field, where the illustrated field in the two upper domains is the reflected field and in the four lower domains the transmitted. The wave fronts of the incident field are indicated with contour lines. The width of the model used in this application tool is illustrated with the two vertical lines in the middle of the figure. In this simulation the angle of incidence is 75 degrees with the thicknesses 500 nm of the two thin layers and the relative permittivities for the layers and the substrate set to 9.00, 4.00 and 9.00, respectively. The last model only using homogenous thin layers is the four-phase model, shown in Fig. 3.12, i.e. reflections from a pair of thin layers on a substrate. As in the previous models, PML and PBC are used to obtain infinite extensions and the layers are introduced between the ambient media and the substrate. The boundary which introduces the incident wave is the upper boundary of the upper thin layer,

66 54 Results and discussion ε r1 ε r2 ε r3 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 1.22e e e e-6 Table 3.10: Mean squared error for the four-phase system with the thickness 10 nm of both layers. The upper layer has the relative permittivity ε r1, the bottom layer ε r2 and the substrate ε r3. i.e. the boundary of the scattering media. These two thin layers introduces four (six) new parameters which defines each model, the relative permittivities (and relative permeabilities) and the thicknesses of the two layers. The permittivities chosen for this verification process are 4.00, 6.25 and 9.00, i.e. the indexes of refraction 2.00, 2.50 and 3.00, and the thicknesses are 10, 100 and 1000 nm. In the case of absorbing medias is the imaginary part of the complex relative permittivity set to 10i Calculation results The data from simulation series done with the layers of equal thickness, 10, 100 and 1000 nm, are shown in Table 3.10, 3.11 and 3.12 respectively. The average result shows that the accuracy decreases slightly with the increasing size, especially for the ellipsometric angle. Figure 3.13 presents the detailed result from the simulation on a model with the relative permittivities ε r1, ε r2 and ε r3 set to 4.00, 6.25 and 9.00, respectively. From Fig. 3.13(b) it is seen that the major part of the inaccuracy originates from simulations with the wavelength 450 nm. The variation of as function of angle of incidence for this wavelength is shown in Fig From this figure it is seen that the deviation between the calculated values and the analytical expressions are increased around the angles of incidence 65 to 70 degrees, i.e. close to the steep slope, otherwise the fit is rather good. The accuracy for the model with a thin layer, 10 nm, on top of a thicker layer, 100 nm, is presented in Table The result here does not differ much from the previously presented models and as can bee seen in Fig the major part of inaccuracy is due to a too sparse mesh for the shortest wavelength, 400 nm, according to previous discussions. In Table 3.14, where the result from models with increasing absorption is presented, the accuracy here is on a similar level. As usual the inaccuracy is highest for the ellipsometric angle, but it is still acceptable.

67 3.4 Four-phase system 55 ε r1 ε r2 ε r3 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 1.82e e e e-6 Table 3.11: Mean squared error for the four-phase system with the thickness 100 nm of both layers. The upper layer has the relative permittivity ε r1, the bottom layer ε r2 and the substrate ε r3. ε r1 ε r2 ε r3 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 2.64e e e e-6 Table 3.12: Mean squared error for the four-phase system with the thickness 1000 nm of both layers. The upper layer has the relative permittivity ε r1, the bottom layer ε r2 and the substrate ε r3. The details of the marked simulation serie is shown in a separate figure. ε r1 ε r2 ε r3 Ψ R p R s e e e e e e e e e e e e e e e e e e e e e e e e-6 avg. 5.83e e e e-6 Table 3.13: Mean squared error for the four-phase system with the thickness 10 nm and 1000 nm of the upper and the lower layer, respectively. The upper layer has the relative permittivity ε r1, the bottom layer ε r2 and the substrate ε r3. The details of the marked simulation serie is shown in a separate figure.

68 56 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.13: The results from calculations on a four-phase system with both thicknesses set to 1000 nm, the relative permittivities in the upper and the lower layer are set to 4.00 and 6.25 respectively. The relative permittivity in the substrate is a) The mean squared error of the reflectances R p and R s (dashed). b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

69 3.4 Four-phase system 57 Figure 3.14: The variations of the ellipsometric angle for a four-phase system with both thicknesses of the thin layers set to 1000 nm, and the relative permittivities for the upper layer, lower layer and the substrate set to 4.00, 6.25 and 9.00 respectively. The wavelength for this calculation was 450 nm. ε r1 ε r2 ε r3 Ψ R p R s e e e e i 1.08e e e e i i 3.16e e e e i i i 3.17e e e e-6 avg. 2.15e e e e-6 Table 3.14: Mean squared error for the absorbing four-phase system with the thickness 100 nm of both the upper and the lower layer. The complex relative permittivity for the upper and lower layer, ε r1 and ε r2 as well as for the substrate ε r3 are gradually increased.

70 58 Results and discussion (a) MSE for R p and R s (b) MSE for Ψ and (c) R p and R s (d) Ψ and Figure 3.15: The results from calculations on a four-phase system with the thickness of the upper layer set to 10 nm and the thickness of the lower layer set to 1000 nm. The relative permittivities in the upper and the lower layer are set to 9.00 and 6.25 respectively. The relative permittivity in the substrate is a) The mean squared error of the reflectances R p and R s (dashed). b) The mean squared error as function of wavelength for the ellipsometric angles Ψ (dashed) and. c) The reflectances R p and R s as function of angle of incidence for the wavelength 400 nm. d) The ellipsometric angles Ψ and as function of angle of incidence for the wavelength 400 nm.

71 3.5 Lateral modulated model Lateral modulated model The model Figure 3.16: The geometry for the lateral modulated system. The upper and lower domains are perfectly matched layers, which approximates infinite extension of the two outer phases in vertical direction. The thin layer in the middle is a mixture of two different medias with different optical properties. The left and right boundaries are connected with periodic boundary conditions which results in an infinite extension of the model in the horizontal direction. The incident wave is introduced on the upper boundary of the upper thin layer with the scattered-field formulation. The color (grayscale in printed version) represents the instantaneous electric field, where the illustrated field in the two upper domains is the reflected field and in the lower domains the transmitted. The wave fronts of the incident field are indicated with contour lines. The width of the model used in this application tool is illustrated with the two vertical lines in the middle of the figure. In this simulation the angle of incidence is 75 degrees with the thickness 500 nm of the thin layer and the relative permittivities for modulated thin layer and the substrate are 9.00, 4.00 and 9.00, respectively. The lateral modulated system is the most advanced optical structure for which the application tool has been tested within this thesis. The thin layer between the ambient media and the substrate here consists of two different materials, with all internal boundaries between the two materials perpendicular to the surface.