Rigorous electromagnetic field calculations for advanced optical systems

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Propositions

accompanying the PhD thesis:

Rigorous Electromagnetic Field Calculations for Advanced Optical Systems

A.S. van de Nes, 13 December 2005, Delft.

I. A study of the fundamental conservation laws, derived from the Maxwell’s equations, is often non-trivial, and helps to increase basic understanding.

Paragraph 2.2 and 7.1 of this thesis II. The Debye-integral, with or without multilayer extension, has a much broader validity

domain than generally assumed.

Paragraph 4.7 and 7.1 of this thesis III. Angular momentum multiplexing is a very promising technique to increase the data

capacity for optical storage.

Paragraph 6.4 and 6.5 of this thesis IV. The optimum thickness of a metallic layer, embedded between a glass and air layer, for

the transmission of the near-field, is finite and non-zero.

Paragraph 7.2 of this thesis V. To increase the transmission of a small structure in a metallic layer, it is much easier to

adapt the field distribution, than to adapt the structure itself.

Paragraph 7.2 of this thesis VI. The distance for which the near-field still contributes to a spot focused behind a glass-air transition, is limited to a wavelength, this in contrast with the claim of D. Biss and T. Brown.

Comment in Optics Express 12, 967-969 (2004) VII. That the scientific value of theology is doubted by scientists from other fields

demonstrates their narrow perspective.

VIII. Delay leads in most cases to a more efficient solution of the problem.

IX. The claim that politicians should only listen to the opinion of the electorate is dangerous, since this puts a large amount of influence in the hands of the (non-elected) journalists.

X. The new bachelor/master system offers the universities an opportunity to concentrate their master education again purely on science.

XI. A sphere is almost always drawn incorrectly, using an oblique projection system for the axes, while an orthographic projection system is used for the sphere itself.

The sphere is drawn correctly for both systems on the cover of this thesis These propositions are regarded as defendable, and have been

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Stellingen

behorende bij het proefschrift:

Strenge Elektromagnetische Veldberekeningen voor Geavanceerde Optische Systemen

A.S. van de Nes, 13 december 2005, Delft.

I. Het bestuderen van de fundamentele behoudswetten, zoals afgeleid uit de Maxwell vergelijkingen, is vaak niet eenvoudig en helpt het begrip te verhogen.

Paragraaf 2.2 en 7.1 van dit proefschrift II. De Debye-integraal, al dan niet met multilagen-uitbreiding, is algemener geldig dan

wordt aangenomen.

Paragraaf 4.7 en 7.1 van dit proefschrift III. Multiplex gebruik van het hoekimpulsmoment is een veelbelovende techniek om de

capaciteit van optische data-opslag te vergroten.

Paragraaf 6.4 en 6.5 van dit proefschrift IV. De optimale dikte van een metaallaag, die zich tussen een glas- en een luchtlaag bevindt,

om de transmissie van het nabije-veld te verhogen, is eindig en niet nul.

Paragraaf 7.2 van dit proefschrift V. Teneinde de transmissie van een kleine structuur in een metaallaag te verhogen, is het

veel eenvoudiger om het veld aan te passen, dan om de structuur zelf aan te passen. Paragraaf 7.2 van dit proefschrift VI. De afstand waarvoor het nabije-veld nog een bijdrage levert aan een spot gefocusseerde net na een glas-lucht overgang, is gelimiteerd tot een golflengte, dit in tegenstelling tot de conclusie van D. Biss en T. Brown.

Reactie in Optics Express 12, 967-969 (2004) VII. Dat de wetenschappelijke waarde van theologie wordt betwijfeld door wetenschappers

uit andere velden geeft blijk van een beperkt inschattingsvermogen.

VIII. Uitstel leidt in de meeste gevallen tot een efficiëntere afhandeling van het probleem. IX. De claim dat politici alleen naar het volk moeten luisteren is gevaarlijk, en legt een grote

mate van invloed in de handen van de (niet-gekozen) journalisten.

X. Het nieuwe bachelor/master systeem biedt de universiteiten de mogelijkheid om de master opleiding weer volledig op de wetenschap te concentreren.

XI. Een bol wordt bijna altijd verkeerd getekend, waarbij gebruik gemaakt wordt van een scheef projectiesysteem voor de assen, maar van een orthografisch projectiesysteem voor de bol zelf.

De bol is correct getekend voor beide systemen op de omslag van dit proefschrift Deze stellingen worden verdedigbaar geacht en zijn als

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Rigorous Electromagnetic Field Calculations

for Advanced Optical Systems

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 13 december 2005 te 15:30 uur

door

Arthur Siewert VAN DE NES

doctorandus in de natuurkunde geboren te Amsterdam

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. J.J.M. Braat.

Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. ir. J.J.M. Braat, Technische Universiteit Delft, promotor Dr. S.F. Pereira, Technische Universiteit Delft

Prof. dr. H.P. Urbach, Technische Universiteit Delft Prof. dr. G.W. ’t Hooft, Universiteit Leiden

Dr. P. Török, Imperial College London, United Kingdom Prof. dr. G. Leuchs, University Erlangen-Nürnberg, Germany Prof. S.M. Barnett, University of Strathclyde, United Kingdom Prof. dr. ir. A. Gisolf, Technische Universiteit Delft, reservelid

Dr. S.F. Pereira heeft als begeleidster in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.

This work was supported by Philips Research, Eindhoven.

ISBN 90-9020229-3

Copyright c 2005 by A.S. van de Nes.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the author.

A free electronic version of this thesis can be downloaded from: http://www.library.tudelft.nl/dissertations

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Aan de nagedachtenis van mijn moeder Aan mijn vader

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Cover:

The frontside of the cover shows a schematic drawing of a sphere in an oblique coordinate system, the backside shows a schematic drawing of a sphere in an orthographic coordinate system.

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Introduction

In current and new optical design as well as in advanced research fields, a fundamental theo-retical background and reliable numerical calculation tools are required to obtain a thorough understanding of the optical system under study and to eventually improve its performance. As the optical systems get more complicated, often used approximations regarding the polarisation of the light and the interaction of the light with matter yield inaccurate results; therefore, more advanced tools are required. Even with a more advanced numerical simulation tool, improving an optical system often relies on trial and error methods, or a slightly more advanced optimisa-tion technique. In this thesis, we will develop the numerical calculaoptimisa-tion tools required to describe more advanced optical systems; we also try to gain a better fundamental understanding of the behaviour of the light in the optical system. The tools and obtained insights will prove applica-ble to optical data storage, microscopy, lithographic projection systems and many other fields of research where light is either focused to a tight spot with the dimension of the wavelength, or interacts with structures with the same typical dimensions. Application of the work presented in this thesis can also be of importance for other research fields, since tightly focused light beams are often used to obtain information about the studied object which interacts with the light, such as single molecules, quantum wells and dots, optical traps, and many other systems.

Illumination

Interaction with data

Detection

a

b

c

Figure 1.1: Typical optical system separated in (a) the illumination, (b) the interaction with the data,

and (c) the detection part.

Throughout this thesis we will separate the optical system in three fundamental parts, (a) the illumination, (b) the interaction with matter, and (c) the post-processing of the information, as shown in Fig.(1.1). For the illumination of a scattering structure containing information that has to be extracted from the system, a generalized imaging system is used. To be able to give

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2 Chapter 1 Introduction

an accurate description of the field in the focal region, it is not enough to know the intensity of the light, but amplitude, phase and polarisation information of the field entering the imaging system is required. An algorithm to accurately calculate the optical field distribution in the focal region is discussed in chapter 4. Due to the general validity of the presented technique, the field in the focal region can be obtained for various imaging systems, including amplitude- and phase-masks to modify the focal spot, immersion systems, the illumination of stratified media, and even near-field imaging systems. The interaction of the light in the focal region with a scatterer with the dimensions of the wavelength is treated in chapter 3. The optical system under study can be optimised by incorporating a-priori known information. The light used for illumination, the shape of the scatterer and the geometry in the focal region, as well as the collection of the light and the post-processing, have to be matched, yielding an inverse problem as discussed in chapter 5.

A recurring eﬀect in all possible optical systems discussed in this thesis, is the limitation of the minimum size of the spot, known as the diﬀraction limit, as well as the limitation of the amount of information which can be stored in, and retrieved from the system, as we discuss below.

1.1 Diﬀraction limit

For many optical systems, it is important to obtain an illumination volume as small as possible. The size to which the illumination volume can be reduced is limited in linear measure to λ/NA, a quantity which is known as the diﬀraction limit, where λ denotes the wavelength of the light and NA the numerical aperture of the imaging system. The numerical aperture of the optical system is related to the maximum illumination angle with respect to the optical axis, which is half of the opening angle of the illumination cone shown in Fig.(1.1). The diﬀraction limit is a fundamental limit and is related to the Heisenberg uncertainty relations.

Blu-ray Disc DVD CD λ=780 nm NA=0.45 1.2 mm substrate capacity 0.65 GBytes λ=650 nm NA=0.6 0.6 mm substrate capacity 4.7 GBytes λ=405 nm NA=0.85 0.1 mm cover layer capacity 25 GBytes

Figure 1.2: Pit and spot size for the three successive optical recording generations as determined by the

numerical aperture of the objective lens and the wavelength of the used light. This picture is taken from the Philips Research public relations web pages, [1].

In the case of optical data storage, the density of the stored information is determined by the size of the tiny depressions or bumps on the disk, commonly called pits, which are read and written optically. As shown in Fig.(1.2), the increase in information density for the successive optical recording generations, has been achieved by a stepwise increase of the numerical aperture and a decrease of the optical wavelength such that the resolution is limited by diﬀraction. The same resolution limit is found to determine the details which can be distinguished in far-ﬁeld microscopy, the minimum size of the lines, known as nodes, written on the integrated circuit in

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1.2 Improving the information density 3 optical lithography, and the smallest size of a cavity mode in an optical resonator. Since the limit only holds for far-field optics, for near-field microscopy the minimum feature size can be smaller than the diffraction limit at the cost of an exponential loss of detected light intensity.

1.2 Improving the information density

To overcome the fundamental diﬀraction limit, it is interesting to realise that the limit determines only the highest spatial frequency at which the intensity of the light is allowed to change. However, the electromagnetic ﬁeld, which in the optical domain is better known as light, contains more information aside from its amplitude, such as phase and polarisation. These additional quantities can be used to obtain extra information about a structure in the illumination volume interacting with the light.

Another way to overcome the effect of the fundamental diffraction limit on the optical infor-mation density, is to use a-priori inforinfor-mation about the illuminated structure. If there is only a finite set of possibly illuminated structures, for example two orthogonally orientated lines of sub-wavelength dimensions, then the resolution does not have to be better than the diffraction limit to be able to identify which line is under study due to the difference in response for different polarisation directions.

Figure 1.3: Change of the k1-factor over the years to improve the information density in optical lithog-raphy. The use of resolution enhancing techniques (RET) is getting more and more important to improve the resolution even further. This picture is taken from Ref. [2].

In lithography, the minimum feature size on the photo-resist is proportional to k1λ/NA,

where k1 is determined by optical as well as material related eﬀects. A way to improve the

information density, is to change the type of illumination of the optical mask. In Fig.(1.3) [2], the trend of optimising the exposure pattern on the integrated circuit, as indicated by the factor

k1, is shown as a function of time. The factor is related to the minimum linewidth present in the

imaged pattern, and is changed by, for example, oﬀ axis illumination, the use of phase masks or the coherence of the used light. Even higher information densities can be obtained by using non-linear eﬀects, such as the non-linear response of the photo-resist on the intensity, allowing for even smaller imprinted features.

1.3 Scope of this thesis

In this thesis, we will discuss in great detail the electromagnetic ﬁeld, focused to and interacting with structures of wavelength dimensions. The previous two paragraphs have discussed the relevance of these fundamental issues, which are found in almost every advanced optical system.

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4 Chapter 1 Introduction

In chapter 2, we start with a short discussion of the theoretical background. The electromag-netic field has to obey Maxwell’s equations and the Lorentz law, from which additional relations can be derived such as the conservation laws, the boundary conditions and the reciprocity rela-tions. The fundamental physical quantities associated to the field satisfy a set of conservation laws, which can be used to obtain a better understanding of the behaviour of the light. Also presented in this chapter is the electromagnetic response of a single oscillating dipole, which will be used to obtain the electromagnetic field response of an arbitrarily shaped scatterer.

As discussed in chapter 3, we can introduce a set of Green’s tensors, which, if correctly defined, incorporate the response of any arbitrary shaped scatterer. These Green’s tensors can also be used for the propagation of the electromagnetic field known on a arbitrary shaped closed surface. The numerical calculation technique using the Green’s tensors has been verified with the results obtained from an analytical expression for one-, two- and three-dimensional test geometries. The extension of the Green’s tensor technique to apply to stratified media has been verified with the results of a modal decomposition technique. The numerical calculation load is mentioned for the algorithms presented in this chapter.

In chapter 4, we present several ways to obtain the electromagnetic field distribution in the focal region. One way to calculate this distribution is to decompose the field exiting the optical imaging system into Zernike polynomials, which yields a semi-analytic expression for the field in the focal region. Another fast and accurate method is to perform the Fourier decomposition in the azimuthal direction, but perform a numerical integral for the radial direction of the input field. With this technique it is relatively straightforward to obtain the field in the focal region even if it is embedded in a multi-layered environment. A set of examples for both techniques is used to show its application to a solid immersion system and to a pattern generator. Finally, we discuss some validity issues related to both techniques, and how a more general solution can be obtained.

To be able to optimise the optical systems under study, we show in chapter 5 that the ﬁeld can often be decomposed in a set of modes which are typical for the system. When the right set of modes is found, either analytically or numerically, it is much easier to improve the design or ﬁnd matching structures to maximise the information density. The technique to solve this inverse problem is discussed thoroughly for the scalar situation and consequently extrapolated to the vectorial situation. Next, an example is given where an optical mask is used to decrease the illumination volume as obtained with a low numerical aperture imaging system. For the high numerical aperture case, we show an optical mask which changes the polarisation direction of the light exiting the imaging system in order to yield an improved distribution in the focal region. As a last example, we use the modal decomposition to retrieve aberration information about the optical imaging system.

In chapter 6, experimental verification of the previously discussed calculation techniques is performed. The experimental setup consists of a stabilised optical interferometer allowing full control of the input of the light where all three basic quantities of the electromagnetic field that illuminates a set of advanced data structures can be measured. The different techniques required to control the state of the light and the main interferometer used in this setup are described, and the obtained results are presented. Next, the possibility of data multiplexing, using the polarisation and phase information as present in the angular momentum of the light, is explored both experimentally as well as numerically.

Finally, we have applied the acquired fundamental understanding as well as the calculation tools to a set of advanced optical systems, as presented in chapter 7. First, we discuss the validity of the conservation laws for high numerical aperture imaging systems. Next, we discuss the transmission through a sub-wavelength aperture and study the resonances of the electro-magnetic ﬁeld as a function of the design parameters. Finally, we discuss the near- and far-ﬁeld

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1.3 Scope of this thesis 5 distributions of the light as reﬂected from the scattering structures used in the next generation optical recording system.

In conclusion, in chapter 8, the most important observations are summarised and a general out-look for the work discussed in this thesis is given.

The mathematical deﬁnitions, some basic vector and tensor identities, important functions and a list of used symbols and their units can be found in appendix A. The appendices B-E contain important derivations and more detailed expressions to support the techniques discussed in chapters 3-5.

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Chapter 2

Theoretical background

In this chapter, we introduce the set of Maxwell equations required to describe the behaviour of the electromagnetic field in the optical domain. From Maxwell’s equations we can obtain other observable quantities such as the energy density, the linear, and the angular momentum density associated with the field and find conservation laws applying to these quantities. Furthermore, we derive the electromagnetic field radiated by a single oscillating electric or magnetic dipole. Next, we introduce the radiation and boundary conditions necessary to have a physical inter-pretation of the electromagnetic field. Finally, we discuss the reciprocity theorem describing the relation between the electromagnetic field at the source and observation point. Because of the mathematical nature of this chapter, we refer to appendix A for the fundamental algebra, adopted notation and a list of the symbols.

2.1 Maxwell’s equations

The behaviour of the electromagnetic ﬁeld is described by the well-known set of Maxwell equa-tions, which have been experimentally obtained by Gauss, Faraday and Ampere and were later corrected to its current form by Maxwell. In most introductory textbooks about electromag-netism [3, 4] and optics [5, 6] they are given in their diﬀerential form,

∇ · D = f , (2.1a) ∇ × E = −∂B ∂t , (2.1b) ∇ · B = 0 , (2.1c) ∇ × H = Jf + ∂D ∂t . (2.1d)

WhereD and E describe the electric displacement and the electric ﬁeld, respectively, and B and

H describe the magnetic field and the magnetic induction, respectively. The free current density Jf describes the flow of the free charge density f through the medium. The subtle difference

between both terms for the electric ﬁeld becomes clear when an interaction with a medium takes place. For linear media, the electric displacement is described by

D = 0E + Pe

= 0(I + χe)E

= E , (2.2)

with 0 the electric permittivity of vacuum,Pe the electric polarisation, χe the electric

suscepti-bility tensor and the electric permittivity tensor of the medium. Therefore, we can replace the 7

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8 Chapter 2 Theoretical background

electric displacement by the medium-dependent electric permittivity tensor times the electric ﬁeld. For linear media, the magnetic induction is described by

H = 1 μ0B − Pm = 1 μ0B − χmH , B = μ0(I + χm)H = μH , (2.3)

with μ0 the magnetic permeability of vacuum,Pm the magnetic polarisation, χm the magnetic

susceptibility tensor and μ the magnetic permeability tensor of the medium. Therefore, we can replace the magnetic ﬁeld by the medium-dependent magnetic permeability tensor times the magnetic induction. Now we can establish the expressions for the most fundamental properties associated with the electromagnetic ﬁeld. The expression for the electromagnetic force density, which is an experimentally obtained expression known as the Lorentz force law, is given by

F = E + J × B , (2.4)

where the force is experienced by an external charge density interacting with an electric field, or, when moving with speed v, an effective external current density J = v interacting with a magnetic field. Since, in general, there is a lot of confusion about the nomenclature of the magnetic field, in the remaining part of this thesis we will express the electromagnetic field in terms of E and H, where we restrict the term magnetic field to the magnetic induction H. To describe the interaction of the electromagnetic field with different types of media, we need one more empirical relation, known as Ohm’s law, which defines a linear relationship between the free current in a medium and the electromagnetic field

Jf = σFn

= σ (E + μv × H)

≈ σE , (2.5)

with Fn =F/ the normalised electromagnetic force, σ the conductivity of the medium and v

the average velocity and direction of the charged particles. In general this velocity is low and the magnetic contribution can be neglected.

Since we are going to describe optical wave phenomena, we explicitly expect a periodic time dependence with oscillation frequency ω for the electromagnetic ﬁeld.

E (r, t) = ReE (r) e−ıωt = 1 2 E (r) e−ıωt_{+ E}∗_{(r) e}ıωt _, _(2.6) H (r, t) = ReH (r) e−ıωt = 1 2 H (r) e−ıωt_{+ H}∗_{(r) e}ıωt _. _(2.7)

The electric and magnetic fields are physical quantities, therefore we only have to consider the real part. In our description we will not include dispersion effects in the electric permittivity or magnetic permeability, since we assume a single harmonic oscillation frequency. This as-sumption, however, is not a restriction to the theoretical description, since frequency-dependent effects can be included in the calculations for each different frequency, or similarly, for each dif-ferent wavelength. Since the propagation speed of the electromagnetic field in vacuum is given

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2.2 The conservation laws 9 by c = (μ00)−1/2, we can associate a wavelength in vacuum to the path traveled by the light in a single oscillation λ = 2πc/ω. Next, we split the free charge and current density in a material-dependent part and an externally applied part, ρ and J , transforming Maxwell’s equations to

∇ · E = ρ , (2.8a)

∇ × E = ıωμH , (2.8b)

∇ · μH = 0 , (2.8c)

∇ × H = J − ıωE , (2.8d)

where we included the contribution of any free electrons of the medium in the electric permitti-vity, and assume isotropic media, yielding for the electric permittivity and magnetic permeability a diagonal matrix with identical elements which can be replaced by two scalar values,

 = αα+

ıσ

ω , (2.9)

μ = μαα. (2.10)

For stratiﬁed media, it will prove useful to deﬁne the normalised electric permittivity i = /0

and magnetic permeability μi = μ/μ0, with i∈ N a positive integer denoting the layer number.

If we assume no local charges or externally applied currents, ρ = 0 and J = 0, Maxwell’s equations can be uncoupled yielding the homogeneous wave equations

μ∇ × 1 μ∇ × E − κ 2_{E = 0 ,} _(2.11a) ∇ × 1 ∇ × H − κ 2_{H = 0 ,} _(2.11b)

with the wave number κ2 = |κ|2 = μω2. Following from Maxwell’s equations it is clear that the electric and magnetic ﬁeld components are both perpendicular to each other and to the propagation direction. Finally, we deﬁne a set of useful material-dependent quantities,

ni =√iμi , (2.12) Zi = μi i , (2.13) κi = 2π λ ni , (2.14) Di = Im 1 κi , (2.15)

which are the complex refractive index, the impedance, the wave number and the skin depth in a medium i, respectively.

2.2 The conservation laws

Probably the most fundamental understanding of various eﬀects in physics is acquired by study-ing conservation laws. In this section we obtain conservation laws for the charge density, the energy density, the linear momentum density and the angular momentum density in a similar way as in Ref. [4]. In section 7.1 we will use the conservation laws to study the transfer of momentum and angular momentum from the illumination to a set of structures, and vice versa. In this derivation we restrict ourselves to piece-wise constant and dispersion-free media, for a

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more complete derivation and discussion on the validity for general materials and to incorporate material transitions, we refer to Ref. [7].

The relation between the charge and current density is found by taking the divergence of Eq.(2.1d), using the vector identity Eq.(A.11c), interchanging the order of the time derivative and the divergence, and ﬁnally substituting the result in Eq.(2.1a) to obtain

∂

∂t =−∇ · J , (2.16)

which is the conservation law of charge, describing, for a certain integration volume, the change of the charge density in time as a function of the flow of the current density. Since the charge density is proportional to the electric field, this has the same time dependence, and therefore, the time-independent analogue of this law is defined as

ıωρ =∇ · J . (2.17)

It is also possible to obtain a conservation law for the energy density. Since interaction with matter can change the motion of the matter, we have to introduce the energy density associated to the mechanical part Um. The change in energy stored in the electromagnetic ﬁeld

as a function of time is equal to the work done by the force acting on a charge density ρ moving with speed v through the ﬁeld. Since the force due to the magnetic ﬁeld is perpendicular to the moving charge, it does no work. With the use of Eqs.(2.1), Eq.(2.4) and the vector identity Eq.(A.11a), we obtain Poynting’s theorem

∂Um ∂t = v· F = v· (E + μJ × H) =J · E = ∇ × H − ∂E ∂t · E =−∇ · (E × H) + H · (∇ × E) −∂E ∂t · E =−∇ · (E × H) − H · ∂μH ∂t − ∂E ∂t · E =−∇ · S −1 2 ∂ ∂t(μH · H + E · E) , ∂ ∂t(Um+ Ueh) =−∇ · S , (2.18)

where we have deﬁned the Poynting vector and the electric, magnetic and total energy density, respectively, as S = E × H , (2.19) Ue= 2E 2_, _(2.20) Uh= μ 2H 2 _, _(2.21) Ueh= Ue+ Uh. (2.22)

The ﬁnal line of Eq.(2.18) denotes the conservation law of energy. From the time-independent conservation law we can obtain additional relations; therefore, we use Eq.(2.6) and a similar

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2.2 The conservation laws 11 equation for the current density to rewrite the third line of Eq.(2.18) as

∂Um ∂t = 1 4 J · Ee−2ıωt_{+ J}_{· E}∗_{+ J}∗_{· E + J}∗_{· E}∗_e2ıωt = 1 2Re J · Ee−2ıωt_{+ J}∗_{· E} _, _(2.23)

which yields for the time averaged quantity ∂Um ∂t = 1 2Re [J ∗_{· E] .} _(2.24)

Note that this is equivalent with taking the cycle average. The right hand side of this conserva-tion law can be written as

J∗_{· E = [∇ × H}∗_{− ıω}∗_E∗_]_{· E}

=− [∇ · (E × H∗)− (∇ × E) · H∗+ ıω∗E∗· E] =− [∇ · (E × H∗)− ıωμH · H∗+ ıω∗E∗· E]

=−2∇ · S − 4ıω(U∗_e− Uh) , (2.25)

where we have identiﬁed

S = 1 2(E× H ∗_{) ,} _(2.26) Ue= 4|E| 2 _, _(2.27) Uh = μ 4|H| 2 , (2.28)

which can be recognized as the complex analogue of the Poynting vector and the electric and magnetic energy density, respectively. Note that the time averaged quantities Eqs.(2.19-2.21) relate to these time-independent quantities as

S = 1 2Re [E× H ∗_{] = Re [S] ,} _(2.29) Ue = 1 2Re 2E · E ∗_{= Re [U}_e_{] ,} _(2.30) Uh = 1 2Re _μ 2H · H ∗_{= Re [U}_h_{] .} _(2.31)

Finally, substituting Eq.(2.25) in Eq.(2.24) yields the complex analogue of the energy conserva-tion law ∂Um ∂t =−Re [∇ · S + 2ıω(U∗_e− Uh)] , (2.32)

which is known as the complex analogue of Poynting’s theorem. The real part of the Poynting vector denotes the power density flowing through the volume, and the imaginary part denotes the average reactive power available, given by the difference in energy stored in the electric and magnetic field.

Furthermore, it is possible to obtain a conservation law for the linear momentum density, again acknowledging a mechanical contribution Pm. The change in momentum of the

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density ρ moving with a speed v.

∂Pm ∂t =F = E + μJ × H = (∇ · E) E + μ ∇ × H −∂E ∂t × H = (∇ · E) E + (∇ · μH) H − μH × ∇ × H − μ∂ ∂t(E × H) + μE × ∂H ∂t = (∇ · E) E + (∇ · μH) H − μH × ∇ × H − E × ∇ × E − ∂ ∂t(μE × H) , ∂ ∂t(Pm+Peh) = (∇ · E) E + (∇ · μH) H − 1 2∇μH 2_{+ μ}_{H · ∇H −}1 2∇E 2₊_{E · ∇E} =−∇ · T , (2.33)

where we used Eqs.(2.1), vector identity Eq.(A.11d), added for sake of symmetry (∇·μH)H = 0 and deﬁned the momentum associated to the electromagnetic ﬁeld and the Maxwell stress tensor as

Peh = μS , (2.34)

T = UehI− EE − μHH . (2.35)

Note the introduction of the tensors ∇E, ∇H, EE and HH, in accordance with the deﬁnition in Eq.(A.1). The last line of Eq.(2.33) denotes the conservation law of linear momentum. Now, we express the second line of Eq.(2.33) in its time-independent components

(∇ · E) E + μJ × H = 1 2Re

(∇ · E) Ee−2ıωt_{+ (∇ · E) E}∗_{+ μJ}_{× He}−2ıωt_{+ μ}∗_{J × H}∗ _,

(2.36) which yields for the time averaged expression

∂Pm ∂t = 1 2Re [(∇ · E) E ∗_{+ μ}∗_{J × H}∗_{] .} _(2.37) We ﬁnd for

(∇ · E) E∗+ μ∗J × H∗= (∇ · E) E∗+ (∇ · μ∗H∗) H + μ∗(∇ × H + ıωE) × H∗ = (∇ · E) E∗+ (∇ · μ∗H∗) H− μ∗H∗× ∇ × H + ıωμ∗E × H∗ = (∇ · E) E∗+ (∇ · μ∗H∗) H− μ∗H∗× ∇ × H − E × ∇ × E∗ = (∇ · E) E∗−1 2∇E 2_{+ E}_{· ∇E}∗ + (∇ · μ∗_H∗_{) H} ₋1 2∇μ ∗_H2_{+ μ}∗_H∗_{· ∇H} =−2∇ · T , (2.38) with T = Re [T] = Re UeI− 2EE ∗₊_U hI− μ 2HH ∗∗ _, _(2.39) resulting in ∂Pm ∂t =−Re [∇ · T] . (2.40)

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2.3 The electric and magnetic dipole 13 Finally, it is possible to obtain a conservation law for the angular momentum density. The change in angular momentum of the electromagnetic ﬁeld as a function of time is equal to the electromagnetic torque working on a charge density ρ moving with a speed v.

∂Lm

∂t =N

= r× F ,

∂

∂t(Lm+Leh) =−∇ · M , (2.41)

where we deﬁned the angular momentum and the angular momentum ﬂux as

Leh= r× Peh= μr× S , (2.42)

M = r × T . (2.43)

The last step in Eq.(2.41) can only be fully understood by a relativistic derivation as shown in Refs. [4, 8, 9]. Due to the symmetry of the problem with the solution for the linear momentum, we follow the same path and only give the result

∂Lm

∂t

=−Re [∇ · M] , (2.44)

with the time averaged angular momentum ﬂux

M = Re [M] = Re [r × T] . (2.45)

2.3 The electric and magnetic dipole

In this section we derive the electromagnetic ﬁeld for both an externally applied electric and magnetic source. We will need the descriptions presented here for the electric and magnetic dipole, to be used in chapter 3 as a generic point-source to obtain the electromagnetic ﬁeld distribution of an arbitrarily shaped scatterer. The set of Maxwell equations describing the problem is

∇ · E = ρe, (2.46a)

∇ × E = ıωμH − Jm, (2.46b)

∇ · μH = ρm, (2.46c)

∇ × H = −ıωE + Je, (2.46d)

where we included the possibility of an externally applied electric, Je, as well as magnetic, Jm,

current density describing the ﬂow of the electric, ρe, and magnetic, ρm, charge densities,

respec-tively. Although including the magnetic charge and current density in Maxwell’s equations is in general not necessary, we will use it later on to describe the eﬀect of an externally applied virtual magnetic current density. From the set of Maxwell equations with a source term, Eqs.(2.46), we obtain for the inhomogeneous set of wave equations

μ∇ × 1 μ∇ × E − κ 2_{E = ıωμJ} e− μ∇ × 1 μJm , (2.47a) ∇ ×1 ∇ × H − κ 2_{H = ıωJ} m+ ∇ × 1 Je, (2.47b)

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with, using the conservation law of charge Eq.(2.17), ∇ · E = ρe= 1 ıω∇ · Je, (2.48a) ∇ · μH = ρm= 1 ıω∇ · Jm, (2.48b)

Since the electric current density originates from an electric charge distribution and the magnetic current density originates from a magnetic charge distribution, we know that by integration by parts, Je(r)dr =− r [∇ · Je(r)] dr =−ıω rρe(r)dr =−ıωp , (2.49a) Jm(r)dr =− r [∇ · Jm(r)] dr =−ıω rρm(r)dr =−ıωm , (2.49b)

where the electric p and magnetic m dipole moment are given by the integral over the electric and magnetic charge density, respectively.

Now, if we assume an externally applied electric current density, originating from a single oscillating electric dipole source with coordinates r, therefore setting Jm = 0, we obtain for the

wave equations μ∇ × 1 μ∇ × E − κ 2_{E = ω}2_μδ(r_{− r}_{)p ,} _(2.50a) ∇ ×1 ∇ × H − κ 2_{H = −ıω∇ ×}1 δ(r− r _{)p .} _(2.50b)

The electromagnetic ﬁeld which satisﬁes these equations will be shown in section 3.1, where we use the electric dipole source to construct a set of Green’s tensors required to treat scattering and propagation problems. If we assume an externally applied magnetic current density, originating from an oscillating magnetic dipole source with coordinates r, therefore setting Je = 0, we

obtain for the wave equations

μ∇ × 1 μ∇ × E − κ 2_{E = ıωμ∇ ×} 1 μδ(r− r _{)m ,} _(2.51a) ∇ × 1 ∇ × H − κ 2_{H = ω}2_δ(r_{− r}_{)m .} _(2.51b)

These equations are, as is well-known, mathematically similar to Eqs.(2.50) when the electric and magnetic ﬁeld, electric and magnetic source, and electric permittivity and magnetic permeability change roles. For the magnetic dipole source we can construct another set of Green’s tensors required to treat scattering and propagation problems.

2.4 Radiation and boundary conditions

For the transition from one material to another, as depicted in Fig.(2.1), the electromagnetic ﬁeld has to satisfy a set of boundary conditions. These boundary conditions can be obtained by integration of the electromagnetic ﬁeld over the surface where the material transition takes place, as discussed in more detail in Refs. [3,4,6]. Such an integration of the source-less Maxwell equations, Eqs.(2.8) with ρ = 0 and J = 0, yields

1E1⊥= 2E2⊥ , (2.52a)

E₁= E₂, (2.52b)

μ1H1⊥= μ2H2⊥ , (2.52c)

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2.5 Reciprocity theorem 15

x

y

z

ε

1

,

μ

1

ε

2

,

μ

2

E

1

H

1

E

2

H

2 ^ ^ ^

Figure 2.1: Material transition at a plane

per-pendicular to the z-axis. The electromagnetic ﬁeld vectors E and H are decomposed in components parallel and perpendicular to the surface.

|

r|→∞

E,H

^

x

z

y

r

^ ^

Figure 2.2: Radiation condition for|r| → ∞, the

electromagnetic ﬁeld vectors E and H have to be ﬁnite and tangential.

The obtained boundary conditions determine a relation between the ﬁeld components on both sides of the medium transition.

To be able to obtain a physically meaningful result for the homogeneous, Eqs.(2.11), and the inhomogeneous, Eqs.(2.47), wave equations, the field has to fulfil the radiation condition, as discussed in Refs. [7, 10]. For any source of finite extent, we know from the energy conservation law that the total emitted energy of the electromagnetic radiation should be finite, even when approaching infinity. Also, we know that we have to fulfil the boundary conditions everywhere, which means that, when approaching infinity, the field should be purely tangential, as depicted in Fig.(2.2). The Silver-Müller radiation conditions for the electric and magnetic field is given by

lim

r→∞(r× ∇ × E + ıκrE) = 0 , (2.53a)

lim

r→∞(r× ∇ × H + ıκrH) = 0 , (2.53b)

with r the vector pointing toward the observation point, and its origin chosen within a finite distance of all the source elements. Due to the constraint on the location of the sources, the relation between electric and magnetic field is determined by the source-less Maxwell equations, and we obtain ˆr × E = (μ/)H and ˆr× H = −(/μ)E with the distance r approaching infinity. This radiation condition has to be met for both the electric and the magnetic field emitted by any source with finite extent.

2.5 Reciprocity theorem

In general, it is possible to establish a relation between the electric ﬁeld at location r2 emitted by an electric source Je1 and the electromagnetic ﬁeld at location r1 emitted by an electric

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16 Chapter 2 Theoretical background r1 p1 r2 p2 E2(r1),H2(r1) E1(r2),H1(r2) ε(r) μ(r)

Figure 2.3: Relation between the electric and magnetic ﬁeld at location r₁and r₂ as obtained from an electric oscillating dipole at location r₂ and r₁, respectively.

be derived as follows ∇ · [E1× H2− E2× H1] = [(∇ × E1)· H2− E1· (∇ × H2)] − [(∇ × E2)· H1− E2· (∇ × H1)] = [ıωμH1· H2− E1· (−ıωE2+ Je2)] − [ıωμH2· H1− E2· (−ıωE1+ Je1)] = [E2· Je1− E1· Je2] , (2.54)

since μ and are independent of the emitting source, the electromagnetic ﬁeld as obtained from Je1 is denoted with the subscript 1, and the electromagnetic ﬁeld as obtained from Je2

is denoted with a subscript 2. When we use an electric dipole as the source, as discussed in section 2.3, Jei= δ(r− ri)pi, the volume integral for both sides of Eq.(2.54) transforms in

ˆ n · [E1(r)× H2(r)− E2(r)× H1(r)] dAr= [E2(r)· δ(r − r1)p1 −E1(r)· δ(r − r2)p₂] dr , (2.55) where we used the divergence theorem, Eq.(A.13a). Applying the radiation condition from sec-tion 2.4, we can write for r approaching inﬁnity ˆr ×Ei = (μ/)1/2Hi. Since the electromagnetic

ﬁeld is required to be transversal at inﬁnity, we see, after some algebra using the vector identities from section A.3, that the terms at the left hand side of the equation cancel each other, yielding for the right hand side

E2(r1)· p₁= E1(r2)· p₂. (2.56)

We can establish another relation for the magnetic ﬁeld as obtained from either of the electric sources, ∇ · [H1× H2]− μ∇ · [E1× E2] = [(∇ × H1)· H2− H1· (∇ × H2)] − μ[(∇ × E1)· E2− E1· (∇ × E2)] = [(−ıωE1+ Je1)· H2− H1· (−ıωE2+ Je2)] − μ[ıωμH1· E2− ıωμE1· H2] = [H2· Je1− Je2· H1] . (2.57)

Again, taking the integral on both sides, and using the radiation condition to show that the terms on the left hand side cancel each other at inﬁnity, we obtain

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2.5 Reciprocity theorem 17 Finally, in a similar way, it is possible to obtain the relations for the electric and magnetic ﬁeld as obtained from a magnetic source, in which case we obtain

E2(r1)· m1 = E1(r2)· m2 (2.59)

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Chapter 3

Scattering structures

Only recently, current computing power has become advanced enough to be able to rigorously calculate the electromagnetic ﬁeld produced by scattering structures. Before such a tool was available, only semi-analytical analysis was possible, restricting the studied objects to highly symmetric conﬁgurations. For structures much larger than the wavelength, crude approxima-tions have sometimes been made to get around this limitation. However, for structures of the order of the wavelength or smaller, a rigorous calculation tool is required.

In this chapter, we calculate the electromagnetic field distribution of an arbitrarily shaped scatterer illuminated by a monochromatic light source. Therefore, we introduce a set of Green’s tensors, which contain the electromagnetic response of an electric or magnetic dipole oscillating along one of the three coordinate axes, section 3.1. Next, we can subdivide the scatterer in cells, small enough that the electromagnetic field can be considered constant. Finally, it is possible to obtain the total effect of all scattering cells on the electromagnetic field distribution of the initial situation by applying an iterative algorithm, as discussed in section 3.2. A thorough discussion on the required Green’s tensor, in a single homogeneous medium as well as in stratified media, is given in appendix B and can be found to a certain extent in Refs. [7,11–13]. Aside from providing a solution to the scattering problem in the form of a volume integral, in section 3.2, the Green’s tensors can be used to propagate the field to an arbitrary position in space if the electromagnetic field is known on a closed surface, as discussed in section 3.3. In section 3.4, we compare three analytically known results: the sphere, the cylindrical rod extending to infinity, and the multilayer stack, with the numerically calculated results using the Green’s tensor technique. An explanation on how to obtain the electromagnetic field for the three analytical solutions can be found in appendix D. Furthermore, we compare the results obtained with the Green’s tensor technique to the results obtained with a Fourier modal method technique for a trench in a multi-layered medium. Finally, the computational load of the Green’s tensor technique is discussed in section 3.5.

3.1 Green’s tensors

In this section we present a theoretical basis for obtaining a solution of Maxwell’s equations with an external virtual source. For this purpose we introduce a set of complex valued tensors which provide a solution for a generic situation, the electric or magnetic dipole, denoted as Green’s tensors. The complete derivation of Green’s tensors can be found in appendix B.

As a generic situation, we consider the electromagnetic ﬁeld distribution for a single electric dipole placed at position rwith an arbitrary oscillation direction p satisfying the inhomogeneous wave equations, as schematically depicted in Fig.(3.1). From Eqs.(2.50) it is clear that the electric ﬁeld should be proportional to the vector p. As a solution for this generic situation, we

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20 Chapter 3 Scattering structures r Gee,Geh Ghe,Ghh ε(r) μ(r) x y z I ^ r’ ^ ^

Figure 3.1: Three orthogonal oscillating dipoles located at r cause an electromagnetic ﬁeld response at r as stored in the Green’s tensors. The discussed conﬁguration allows for material transitions in the plane perpendicular to the z-axis.

introduce a set of Green’s tensors Geeand Geh to store, respectively, the electric and magnetic

ﬁeld response for three electric dipoles oscillating along each of the orthogonal axes in the Cartesian basis. Equivalently, for a single magnetic dipole placed at position r with an arbitrary oscillation direction m, the set of inhomogeneous wave equations, Eqs.(2.51), have to be satisﬁed. Therefore, we introduce a second set of Green’s tensors Ghe and Ghh to store, respectively, the

electric and magnetic ﬁeld response for three magnetic dipoles oscillating along each of the orthogonal axes in the Cartesian basis. Although there is quite some similarity between the two sets of tensors, since mathematically both sources can be expressed in terms of each other, both cases should be treated separately. From the proportionality relation of the Green’s tensors to the source term, we can introduce a set of Maxwell equations describing the electromagnetic ﬁeld response, which, for the oscillating electric dipole, results in

∇ · Gee(r, r) =−∇ · δ(r − r)I , (3.1a)

∇ × Gee(r, r) = ıωμGeh(r, r) , (3.1b)

∇ · μGeh(r, r) = 0 , (3.1c)

∇ × Geh(r, r) =−ıωGee(r, r)− ıωδ(r − r)I . (3.1d)

For the set of Green’s tensors used to describe the electromagnetic response caused by the oscillating magnetic dipole, we have

∇ · Ghe(r, r) = 0 , (3.2a)

∇ × Ghe(r, r) = ıωμGhh(r, r) + ıωδ(r− r)I , (3.2b)

∇ · μGhh(r, r) =−∇ · δ(r − r)I , (3.2c)

∇ × Ghh(r, r) =−ıωGhe(r, r) . (3.2d)

Combining the equations for Eqs.(3.1) and Eqs.(3.2) results in two sets of inhomogeneous ten-sorial wave equations, Eqs.(3.3a-3.3b) and Eqs.(3.3c-3.3d),

μ∇ × 1 μ∇ × Gee− κ 2_G ee= ω2μδ(r− r)I , (3.3a) ∇ ×1 ∇ × Geh− κ 2_G eh=−ıω∇ × δ(r− r) I , (3.3b) μ∇ × 1 μ∇ × Ghe− κ 2_G he= ıωμ∇ × δ(r− r) μ I , (3.3c) ∇ ×1 ∇ × Ghh− κ 2_G hh= ω2δ(r− r)I , (3.3d)

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3.1 Green’s tensors 21 with I the identity tensor, and the wave number κ2 = μω2, defined as usual. To distinguish the individual Green’s tensors, we use the first subscript to denote the origin (electric or magnetic) of the inhomogeneity and the second subscript to denote the field component.

Figure 3.2: The electromagnetic ﬁeld emitted by a dipole oscillating in the x-direction in the plane

y = 0, (a) the x-component of the electric ﬁeld stored in the Ge

xx-component, (b) the z-component of the electric ﬁeld stored in the Ge

zx-component, (c) the y-component of the magnetic ﬁeld stored in the

Gh

yx-component, and (d) the total electric ﬁeld.

A derivation of Green’s tensors in a single homogeneous medium is given in section B.1, resulting in the expressions

Gee(r, r) = μGhh(r, r) = κ2 I + 1 κ2∇∇ g(r, r) , (3.4) Geh(r, r) =−Ghe(r, r) =−ıω∇ × g(r, r)I , (3.5)

where g(r, r) is the scalar Green’s function

g(r, r) = e

ıκ|r−r|

4π|r − r| . (3.6)

The point of observation is denoted by r, the origin of the tensor deﬁning the location of the electric or magnetic dipole is denoted by r. In the point r = r, Green’s function is singular and Eq.(3.4) is equal to the principal value L, which only depends on the shape of the chosen

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22 Chapter 3 Scattering structures

exclusion volume, as discussed in section B.3. An explicit expression for the Green’s tensors in a single homogeneous medium is given by

Gee(r, r) = μGhh(r, r) = −L r = r_, κ2 ıκR−1 κ2_R2 I− 3 ˆR ˆR + I− ˆR ˆR g(r, r) r = r, (3.7) Geh(r, r) =−Ghe(r, r) = 0 r = r , −ıωıκR−1 R ˆ R × Ig(r, r) r = r , (3.8)

where we introduced R = r− r, and ˆR ˆR denotes a matrix as defined in section A.1. The first term in between the square brackets in Eq.(3.7) is associated to the near-field, the second is associated to the far-field. In Fig.(3.2)(a) and (b), we have plotted the x- and z-component of the electric field distribution as stored in the Green’s tensor G_ee for a x-oscillating dipole; in Fig.(3.2)(c) the y-component of the magnetic field distribution as stored in the Green’s tensor

G_ehfor the x-oscillating electric dipole is shown. The total intensity emitted by the x-orientated dipole is plotted in Fig.(3.2)(d). The general expressions for the Green’s tensors in stratiﬁed media can be found in section B.5, where the tensors are obtained in the Fourier domain. Like in section 2.5, there exists a reciprocity relation for the Green’s tensors, describing the eﬀect of interchanging the coordinates of the observation and source point, as derived in section B.2,

Gee(r, r) = GTee(r, r) , (3.9a) Geh(r, r) = GTeh(r, r) , (3.9b) Ghe(r, r) = GThe(r, r) , (3.9c) Ghh(r, r) = GThh(r, r) . (3.9d) Δ_ε(r),Δ_μ(r) Eu,Hu Es,Hs εu(r),μu(r)

Figure 3.3: Scattered field distribution (Es,Hs) as obtained by illuminating an arbitrary shaped ma-terial difference (Δ(r),Δμ(r)) with the field distribution (Eu,Hu).

Next, we can obtain a solution of the homogeneous wave equations for a scatterer with arbitrary shape by treating the difference with the known initial situation without the scatterer as an externally applied virtual source, where the configuration is shown in Fig.(3.3). This way the wave equations become inhomogeneous, and the solution can be obtained by applying Green’s theorem, where the previously introduced Green’s tensors are required. As a starting point we use the solution of the homogeneous wave equations for the incident field in the initial configuration denoted by subscript u

μu∇ × 1 μu∇ × E u− κ2uEu = 0 , (3.10a) u∇ × 1 u∇ × Hu− κ 2 uHu = 0 . (3.10b)

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3.2 Solving the volume integral 23 By introducing a scatterer of arbitrary shape in the initial situation, we obtain a diﬀerent set of position-dependent material properties t and μt. The solution of the homogeneous wave

equations with the new set of material properties is given by Et and Ht, satisfying

μt∇ × 1 μt∇ × Et− κ 2 tEt= 0 , (3.11a) t∇ × 1 t∇ × Ht− κ 2 tHt= 0 . (3.11b)

Separating the scattered ﬁeld from the incident ﬁeld Et= Eu+Esyields a set of inhomogeneous

wave equations for the scattered ﬁeld in terms of the initial material properties, as discussed in more detail in section C.1,

∇ × ∇ × Es− κ2uEs= (κ2t − κ2u)Et, (3.12a) u∇ × 1 u∇ × Hs− κ 2 uHs=−ıωu∇ × t− u u Et, (3.12b)

where we assumed μu = μt = μ0; in the remaining part of this chapter we concentrate on

non-magnetic materials, as this is generally true for materials in the optical domain.

Remains to solve these inhomogeneous wave equations for a virtual electric source, which can be done using a corollary of Green’s theorem as can be found in full detail in section C.2. In general, the solution is given in terms of a volume and a surface integral, Eq.(C.6) and Eq.(C.8). Note, that the transpose of the Green’s tensors can be taken into account by using Eqs.(3.9). We concentrate on two specific solutions. In the first situation we choose the integration volume equal to the complete spaceR. Now, using the radiation condition for Green’s tensors, Eq.(B.21), to show that the surface integral closing at infinity does not contribute, the general solution reduces to a single volume integral

Et(r) = Eu(r) + t(r)− u(r) Gee(r, r)· Et(r)dr , (3.13a) Ht(r) = Hu(r) + t(r)− u(r) Geh(r, r)· Et(r)dr . (3.13b)

Note once more that in the point r = r the principal value should be used to avoid the singularity of the Green’s function, as discussed in section B.3. In the second situation, we choose the integration volume in such a way that it excludes the virtual source. This way, the volume integral does not contribute and, using the radiation condition of Eq.(B.21), it is easy to see that the surface integral closing at inﬁnity still does not contribute either, and the general solution reduces to a single surface integral with a chosen integration surface.

Et(r) = Eu(r)− ı ω Geh(r, r)× Es(r) + Gee(r, r)× Hs(r) · ˆndAr , (3.14a) Ht(r) = Hs(r)− ı ω Geh(r, r)× Hs(r)− u(r) μ0 Gee(r, r ₎_{× E} s(r) · ˆndAr , (3.14b)

where ˆn denotes the surface normal pointing outward.

3.2 Solving the volume integral

Since, in Eqs.(3.13) the total electric ﬁeld inside the scatterer appears both on the left and the right hand side of the equation, we are dealing with an inverse problem, known as a Fredholm integral equation of the second kind. A solution of the inverse problem can be found in various

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24 Chapter 3 Scattering structures

Δ_ε,i,Δ_μ,i

Eu,Hu

Es,Hs εu,i,μu,i

Figure 3.4: Scattered field distribution (Es,Hs) as obtained by illuminating a discretised scattering element consisting of refractive index differences (Δ(r),Δμ(r)) with the field distribution (Eu,Hu). ways; we will follow the iterative solution as proposed in [14, 15]. In the iterative algorithm, we subdivide the problem into smaller problems, where we take the solution of one of the sub-problems as the starting point of the next sub-problem. To be able to adapt the system to incorporate the result of a previous sub-problem, we need an additional set of equations, similar to Eqs.(3.13), but now for the Green’s tensors, which can be derived in the same way as in appendix C, yielding

Gee,t(r, r) = Gee,u(r, r) +

t(r)− u(r)

Gee,u(r, r)· Gee,t(r, r)dr , (3.15a) Geh,t(r, r) = Geh,u(r, r) +

t(r)− u(r)

Geh,u(r, r)· Gee,t(r, r)dr , (3.15b)

where Gee,t= Gee,u+Gee,swith the subscripts t, u and s denote the total, incident and scattered

response tensors, respectively. Since Δ(r) = [t(r)−u(r)]/u(r) is only not equal to zero within

the scattering element, we can restrict the integration volume to the volume occupied by the scatterer. For the iterative algorithm, we will subdivide the scattering element in piecewise constant elements, where the size of each element is determined by the volume over which the electric ﬁeld Etin Eqs.(3.13) or the electric Green’s tensor Gee,tin Eqs.(3.15) can be considered

constant. The discretised scatterer is shown in Fig.(3.4), where we denote the discretised total electric ﬁeld in cell i as Et(r) = Eti, and the discretised total electric Green’s tensor in cell

i caused by the dipole in cell j as Gee,t(r, r) = Gee,ti,j . While the algorithm iterates over all

individual perturbating cells, the superscripts for the initial u and ﬁnal situation t are each time associated with the system where N and N + 1 scattering cells have been incorporated. After each iteration the ﬁeld and tensor containing the information of the already calculated cells are updated. The discretised version of Eq.(3.13a) and Eq.(3.15a) are given by

Et

i = Eui + N

k=1,k=i

Δ,kVkGee,u_i,k · Etk+ Δ,iMi· Eti− Δ,iLi· Eti , (3.16)

Gee,t_i,j = Gee,u_i,j +

N

k=1,k=i

Δ,kVkGee,u_i,k · Gee,t_k,j + Δ,iMi· Gee,t_i,j − Δ,iLi· Gee,t_i,j , (3.17)

where Vk represents the volume of cell k. Because of the special nature of the point r = r

its contribution is included explicitly, only depending on the shape of the integration exclusion volume. However, note that the chosen exclusion volume can be diﬀerent depending on the location of the singularity. For a spherical or a cubic form of the exclusion volume, as shown in section B.3, the principal value Li is given by,

L(r) = lim Sδ→0 1 4π Sδ ˆ Rˆn R2 dAr = 1 3 ⎛ ⎝10 01 00 0 0 1 ⎞ ⎠ , (3.18)

Rigorous electromagnetic field calculations for advanced optical systems

Propositions

Stellingen

Rigorous Electromagnetic Field Calculations

for Advanced Optical Systems

Contents

Chapter 1

Introduction

a

b

c

1.1

Diﬀraction limit

1.2

Improving the information density

1.3

Scope of this thesis

Chapter 2

Theoretical background

2.1

Maxwell’s equations

2.2

The conservation laws

2.3

The electric and magnetic dipole

2.4

Radiation and boundary conditions

x

y

z

ε

,

μ

ε

,

μ

E

H

E

H

|

r|→∞

E,H

x

z

y

r

2.5

Reciprocity theorem

Chapter 3

Scattering structures

3.1

Green’s tensors

3.2

Solving the volume integral