Vectorial diffraction of extreme ultraviolet light and ultrashort light pulses

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(3) Vectorial Diffraction of Extreme Ultraviolet Light and Ultrashort Light Pulses. 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 17 juni 2008 om 12:30 uur. door Aura Mimosa NUGROWATI. Magister Teknik Institut Teknologi Bandung geboren te Bandung, Indonesië.

(4) Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. J.J.M. Braat. Toegevoegd promotor: Dr. S.F. Pereira.. Samenstelling promotiecommissie: Rector Magnificus, Prof. dr. ir. J.J.M. Braat, Dr. S.F. Pereira, Prof. dr. H.P. Urbach, Prof. A.V. Tishchenko, Prof. G. Leuchs, Dr. J-Y. Robic, Prof. H-P. Herzig, Prof. dr. ir. P.M. van den Berg,. voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft Technische Universiteit Delft University Jean Monnet Saint-Etienne, France University Erlangen-Nürnberg, Germany CEA-Leti Minatec, France University Neuchatel, Switzerland Technische Universiteit Delft, reservelid. The work described in this thesis was performed at the Faculty of Applied Sciences, Department Imaging Science and Technology, Optics Research Group, Technische Universiteit Delft, The Netherlands. This work was supported by the Technische Universiteit Delft and mainly funded by the European Commission in the Sixth Framework Programme (IST-1-507754-IP), the More Moore project.. ISBN-10: 90-78314-09-7 ISBN-13: 978-90-78314-09-7. Copyright © 2008 by A.M. Nugrowati. 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 Printed by PrintPartners IPSKAMP Enschede – http://www.ppi.nl.

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(6) Cover: The frontside of the cover shows a schematic drawing of a conical illumination falling onto a mask structure in the extreme ultraviolet lithography system. The image crossing the spine of the book shows the diffraction of a femtosecond pulsed light by a single slit aperture made of aluminium for the y− component of the magnetic field in the transverse magnetic case. The backside of the cover shows the diffraction of a femtosecond pulsed light by Young’s double-slit aperture made of aluminium for the y− component of the magnetic field in the transverse magnetic case..

(7) Propositions accompanying the PhD thesis: Vectorial Diffraction of Extreme Ultraviolet Light and Ultrashort Light Pulses to be defended on Tuesday June 17, 2008 at 12.30 in Delft by Aura M. Nugrowati.. 1. Numerically obtained solutions are significant for understanding the physics behind complex experiments. 2. General mask types in the extreme ultraviolet region cannot yield a pure amplitude nor a pure phase modulation.. Paragraph 3.3 of this thesis. 3. To produce near-wavelength resolution in extreme ultraviolet lithography, phase mask structures are more crucial to be applied and less complicated to fabricate than initially predicted. 4. The effect of dispersion on the ultrashort pulses transmitted through Young’s double slit arrangement is strongly polarisation dependent, resulting in a noticeable time delay between pulses of two orthogonal polarisation states, even for very thin slit apertures.. Paragraph 4.3 of this thesis. 5. The ubiquitous quote ’To measure is to know’ stimulates incorrect claims on new experimental results, thus the correct expression ’From measurement to knowledge’ of Heike Kamerlingh Onnes should be used. 6. The simplest correct solution of a complex problem - Occam’s razor - is the hardest to find. 7. To procrastinate is essential to optimise the efficiency in solving a problem. 8. The degree of polarisation in opinions on a peculiar taste of unusual edible substances leads to the label délicatesse and prompts the incredulously overrated prices. 9. The average intake of sugar is proportional to that of caffeine or theine. 10. Cunning observations for original ideas with witty presentations at the traditional Sinterklaas surprise are the recipes for creating opposable yet defendable society-oriented propositions.. These propositions are regarded as opposable and defendable and as such have been approved by the supervisor, prof.dr.ir. J.J.M. Braat..

(8) Stellingen behorende bij het proefschrift: Vectorial Diffraction of Extreme Ultraviolet Light and Ultrashort Light Pulses te verdedigen op dinsdag 17 juni 2008 om 12.30 uur te Delft door Aura M. Nugrowati.. 1. Numeriek verkregen oplossingen zijn belangrijk om de natuurkundige verschijnselen bij ingewikkelde experimenten te begrijpen. 2. Algemene maskerstructuren in het extreem ultraviolet gebied kunnen niet tot alleen amplitudeof alleen fasemodulatie leiden.. Paragraaf 3.3 van dit proefschrift. 3. Om bij lithografie met extreem ultraviolet licht resolutie te verkrijgen van de grootteorde van de golflengte van het licht blijkt, in tegenstelling tot eerdere voorspellingen, de toepassing van fasemaskers belangrijker te zijn en is hun fabricage ook minder ingewikkeld. 4. Het effect van dispersie op ultrakorte pulsen die door Young’s twee-spletenopstelling worden doorgelaten is sterk afhankelijk van de polarisatietoestand en geeft aanleiding tot een significante tijdsvertraging tussen pulsen met onderling loodrechte polarisatietoestand, zelfs bij zeer geringe dikte van de twee spleten.. Paragraaf 4.3 van dit proefschrift. 5. Het alomtegenwoordige citaat ’Meten is weten’ suggereert onjuiste claims op nieuwe experimentele resultaten, daarom moet de correcte uitdrukking ’Door meten tot weten’ van Kamerlingh Onnes worden gebruikt. 6. De meest eenvoudige juist oplossing van een complex probleem - Occam’s scheermes - is het lastigst te vinden. 7. Uitstellen is essentieel om de efficiëntie bij het oplossen van een probleem te optimaliseren. 8. Op de spits gedreven meningen over de bijzondere smaak van ongebruikelijke eetbare substanties leiden tot het etiket délicatesse en veroorzaken de ongelooflijk overdreven prijzen. 9. De gemiddelde opname van suiker is evenredig met die van cafeïne of theïne. 10. Geestige observaties voor originele ideeën met grappige presentatie bij de traditionele Sinterklaas surprise zijn de recepten om te komen tot opponeerbare maar toch verdedigbare maatschappijgeoriënteerde stellingen.. Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedgekeurd door de promotor, prof.dr.ir. J.J.M. Braat..

(9) Preface "Optics, developing in us through study, teach us to see" — Paul Cézanne (1839-1906). To my opinion, a general introduction to the applications of optics in our daily lives can already be given at an earlier level of education. This requires supports from the government in terms of education policy, and highly qualified teachers at the school-level that are capable of explaining advanced science in a simple and accurate manner. In many cases, these are not always readily available. Hence, the lack of familiarity with the subject of optics and the under appreciation of its vast amount of applications by simply exploiting the properties of light. Although, by its nature, this thesis is not intended to draw the potential young students to the broad fascinating applications of optics, in the next few paragraphs, I would like to address the general reader (with a limited background in optics) to several examples of applications in optics discussed in my thesis. The first application that will be discussed in the thesis is about creating patterns by using light. When exposing a chemical substance to light with varying intensity, the chemical properties of the substance are changed, proportional to the intensity. This is analogous to the working principle of analogue photography that captures images, or rather the light intensity distribution that exposes the film inside the camera. With a chemical process, the variations (exposed and unexposed area) on the substrate can be transformed into relief patterns, for instance the patterns that can be found in the circuit boards of our computers. Since we desire to have light and compact electronic gadgets, the patterns on these circuit boards that serve as the brain of these gadgets should be made very small. In the thesis, we will discuss the ways to generate small patterns by using light that has an extremely short wavelength (almost similar to the wavelength of X-rays). Another topic that will be discussed in the thesis involves light that oscillates during an extremely short period of time (ultrashort pulses). This special type of light has a broad spectrum, i.e. it involves many wavelengths (many colours). Different colours travel at the same speed in vacuum (and most of the time also in air), but at different speed in a dispersive medium (this is a property of light called dispersion). The interaction between light and matter can be exploited for studying the properties of various substances, since each substance has a unique fingerprint when examined with light consisting of many colours. The shape and dimension of the substance or object under study influence the intensity and the colour distribution. In this thesis, we will investigate several optical phenomena that occur when the ultrashort pulse propagates through holes in metal objects. These phenomena include properties of light such as dispersion, diffraction (the bent trajectory of light), and superposition (interference) of light. With this knowledge, we hope to open the way for other applications that can exploit and process the wealth of information present in these phenomena. A more lengthy explanation of the motivation and the objectives of each topic is readily available for the general reader, in the first chapter. For advanced readers, a summary outlining the work in the thesis is given at the end of the book. I hope that they demonstrate the versatility of optics in our daily lives and in state-ofthe-art technologies. Aura Mimosa Nugrowati, June 2008.. v.

(10) vi. Preface.

(11) Contents Preface. v. 1 Observation and manipulation of light at near-wavelength dimensions 1.1 Frontiers in optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Improving the information density with an extremely short wavelength 1.1.2 Improving the temporal resolution with extremely short pulses . . . . . 1.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Modal expansion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. 1 2 2 3 4 5 5. 2 Fourier mode expansion for diffraction in optics 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 History of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Motivation to implement the Fourier modal method . . . . . . . . . . . . . . . . 2.3 Wave theory of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Mode decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 1-d grating in conical mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 1-d grating in classical mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Fields in multilayered media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Fields matching at the interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Field reconstruction by the recursion of matrix propagation . . . . . . . . . . . . 2.6 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Maxwell’s equations in a complex coordinate system . . . . . . . . . . . . . . . . 2.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Periodic grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Multilayered periodic structure illuminated with extreme ultraviolet wavelength 2.7.3 Groove-slit structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 7 8 13 13 16 18 19 20 20 20 25 25 27 27 28 32 35. 3 Near-wavelength resolution with extreme ultraviolet lithography 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Extreme ultraviolet optics . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Optical material properties . . . . . . . . . . . . . . . . . 3.2.2 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Imaging algorithm . . . . . . . . . . . . . . . . . . . . . . 3.3 Mask structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Phase mask concept . . . . . . . . . . . . . . . . . . . . . 3.3.2 Existing structures . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Design categories . . . . . . . . . . . . . . . . . . . . . . . 3.4 Optimisation method . . . . . . . . . . . . . . . . . . . . . . . .. 43 43 44 44 45 47 48 49 50 51 53. vii. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . . . . ..

(12) Contents. viii. 3.5. 3.6. 3.7. 3.8. 3.4.1 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Computational effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Brute force method for 1-d scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Levenberg-Marquardt algorithm for 2-d scheme . . . . . . . . . . . . . . . . . . . 3.4.5 Legendre-Gauss quadrature and orthogonal polynomials for efficient sampling 3.4.6 Quantifications and requirements of image quality . . . . . . . . . . . . . . . . . Optimising multilayer thickness ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Attenuated phase mask optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Chromeless phase mask optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimising pitch and layer thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Attenuated phase mask optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Chromeless phase mask optimisation . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected phase mask designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Near-field distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Aerial images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Ultrashort pulse propagation through small apertures 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modelling ultrashort pulse propagation . . . . . . . . . . . . . . . . 4.2.1 Spectral function of the pulse . . . . . . . . . . . . . . . . . . 4.2.2 Theory of dispersion effects . . . . . . . . . . . . . . . . . . . 4.2.3 Geometry of structures . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Strong polarisation-dependent dispersion in the transmitted field 4.3.1 Dispersion by a single slit . . . . . . . . . . . . . . . . . . . . . 4.3.2 Dispersion by a circular aperture . . . . . . . . . . . . . . . . 4.3.3 Dispersion by finite thickness apertures . . . . . . . . . . . . 4.3.4 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Diffraction through slit structures . . . . . . . . . . . . . . . . . . . . 4.4.1 Diffraction by a single slit . . . . . . . . . . . . . . . . . . . . . 4.4.2 Double-slit configuration . . . . . . . . . . . . . . . . . . . . . 4.4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 53 53 53 54 55 55 56 56 57 57 58 58 59 59 60 60 65 67 69 69 70 70 71 74 74 76 77 79 80 81 82 82 87 90. 5 Concluding remarks. 91. Nomenclature. 95. A Toeplitz matrix. 101. B Eigenproblems in the Fourier mode expansion of grating structures 103 B.1 2-d geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 B.2 1-d geometry with conical mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.3 1-d geometry in classical mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 C The multiplication of the interface matrix s. 109. Bibliography. 113. Summary. 123.

(13) Contents. ix. Samenvatting. 125. Acknowledgements. 127. Curriculum vitae. 129.

(14) x. Contents.

(15) Chapter 1. Observation and manipulation of light at near-wavelength dimensions One of the natural properties of a human-being is curiosity, a cognitive pleasure to understand. We do not limit our hypotheses and theories to explain the mechanism of things that can only be observed by our unaided eyes — those in the scale of visible dimension. We have also built the technology to aid us to ’see’ what lies beyond. Light, being one of the most important aid for us to see, is continuously explored and manipulated for observations, as well as for designing tools that operate at a continuously decreasing dimension. The most intriguing property of light is the particle-wave duality. Introduced more than a century earlier, the wave theory of light is considered as the classical approach. The behaviour that classifies light as wave is the bent trajectory of light when interacting with objects — the diffraction of light. That is to say, light that hits an opaque object seems to bend around the object, such that at a plane located further behind the object, we can see light in the area that is not directly illuminated.. λ. Δx d. Δz. Figure 1.1: A simplified illustration of the bending of light with wavelength λ around an opaque sphere with diameter d . At the plane Δz from the sphere, the intensity of the light is modulated. The distance between the maxima is Δx. This figure is taken from Reference [1].. In the simplified illustration of diffraction given in Figure 1.1, the obstruction of an incoming plane wave with a wavelength λ by a sphere with a diameter d , results in two points at the edges of the illuminated side of the sphere that act as secondary sources, emanating waves of a cylindrical shape. At a distance Δz behind the sphere, the superposition of the two waves becomes apparent from a modulated dark and light area — corresponding to a destructive or a constructive interference, respectively — with a distance Δx between two maxima. The interaction of light with matter, described in terms of the relation between the properties of a known light distribution (wavelength, polarisation state, amplitude, phase, temporal dependence, etc.) and the properties of a known structure (geometry, refractive index, etc.), determines the dis1.

(16) 2. Chapter 1 Observation and manipulation of light at near-wavelength dimensions. tribution of light everywhere around and inside the structure. This information can be used to study the properties of an unknown structure (e.g. in spectroscopy), to manipulate the illumination properties for other applications (e.g. microscopy), or to modulate the light intensity for creating patterns that store information (e.g. optical data storage, lithography).. 1.1 Frontiers in optics Now, our exploration to understand and to manipulate the diffraction of light goes to a smaller dimension. The spatial modulation of light can be increased by modifying the properties of light, for example by using a smaller wavelength. Alternatively, a structure with dimensions close to the wavelength also yields a higher spatial modulation of light. If sufficient amount of intensity can be measured from the interaction between light and the structure, then the higher the spatial modulation, the more information we can perceive. This leads to many new interesting applications.. 1.1.1 Improving the information density with an extremely short wavelength Lithography, a well-known technique for printing patterns, is commonly applied to art as well as to science. In principle, lithography is a chemical process that creates patterns by using chemical selectivity. A pattern is imprinted on a substrate, and the separated areas between the patterned and the non-patterned substrate yields the final print when the substrate is chemically developed. In semiconductor technology, the lithography technique is applied on a silicon substrate to print integrated circuits. The challenge is to increase the density of the patterns and, therefore to shrink the size of the circuits that results in, for example, lighter and faster digital electronics. There are many ways to transfer the pattern to the substrate. For artwork and text-print, it is common to apply the so-called, contact printing. An example is given in Figure 1.2, where a copy of an artwork (left box) is transferred to a substrate by a fine tip, resulting in a dark tone of the print when the substrate is chemically developed (right box) [2].. Figure 1.2: A copy of Pablo Picasso’s "Don Quixote" (left box), printed with contact lithography technique into a 5 μm-width substrate. The image resolution is in the order of a couple of nanometers. The researchers have used an atomic force microscope cantilever to transfer the pattern onto the substrate. These figures are taken from Reference [2].. While the cost of production is also important in the semiconductor technology, an optical projection technique to transfer a pattern (illuminating and imaging the pattern) into the substrate is considered as the most efficient way to have a high-volume production. Currently, mass-produced.

(17) 1.1 Frontiers in optics. 3. integrated circuits have printed structures as small as 65 nm. To go below this number, many leading semiconductor companies are looking at the option of using a smaller wavelength, 13.5 nm. It is far below the ultraviolet wavelength that ranges in between 200 and 400 nm. Since ’ultra’ has a connotation of beyond the limit (of visible light), scientists have to add another (similar) superlative adjective word ’extreme’, hence the name: extreme ultraviolet. The competing researchers (and companies) in the optical projection lithography technique that use extreme ultraviolet sources have been able to print features below 40 nm, shown in Figure 1.3. Having the technology to generate the extreme ultraviolet source at hand, groups from Lawrence Livermore National Laboratory, Sandia National Laboratories, and Berkeley Laboratory were able to print elbow structures at 39 nm [3], (left box). Two years later, in 2004, a larger consortium (the partner names were imprinted as shown in the middle box) has produced 35 nm line patterns [4]. In 2007, a research organisation in Japan (New energy and industrial technology development organization - NEDO) claimed the printed size of 26 nm [5] (right box). The potential of the technology has been demonstrated, but the question whether patterns with dimensions near and smaller than the wavelength can be printed and economically mass produced, is still under investigation.. 100nm. 117 nm 35 nm. 39 nm. 70 nm. 26 nm 50 nm. (a). (b). (c). Figure 1.3: Records for extreme ultraviolet lithography nodes: (a) 39 nm nodes by the scientist from Lawrence Livermore National Laboratory, Sandia National Laboratories, and Berkeley Laboratory [3], (b) 35 nm nodes by the consortium of several partners (the names given in the left top picture) [4], and (c) 26 nm nodes by the Japanese organisation NEDO [5].. Part of this thesis will discuss the principle of imaging a pattern with the optical projection system operating in the extreme ultraviolet wavelength range. Our aim is to have a higher spatial modulation (higher resolution) of extreme ultraviolet light by imaging phase objects (phase mask structure). We will present our study on the interaction of extreme ultraviolet light with the phase mask structures to yield high spatial image resolution, that is aimed at a dimension of around 20 nm [6].. 1.1.2 Improving the temporal resolution with extremely short pulses Not only that our eyes have a limit to spatially resolve a small object, they also have a limit to temporally resolve fast movements. The latter prevents us to have a direct observation on interesting fast dynamics. Given a normal illumination, an average person sees dynamic movements with frequencies above 50 Hz as stationary movements. Therefore, instruments are essential for high temporal resolution measurements and are continuously improved to help us to resolve an almost instantaneous effects in dynamical processes. Pulsed lasers are one promising way to achieve a high temporal resolution measurement. The first working laser in 1960 was a pulsed laser at a monochromatic wavelength of 694 nm. Due to the.

(18) Chapter 1 Observation and manipulation of light at near-wavelength dimensions. 4. development of pulsed laser generation techniques that now contains a broad spectral range, currently a laser with a single pulse duration of 130 × 10−18 seconds is available [7]. In a more popular term, if this single pulse duration is analogous to a second, then our one second is analogous to about 250 million years (or 1/20 of the age of our earth). These extremely short pulses are labelled as ultrashort pulses — up to now, the ultrashort pulse duration ranges from pico (10−12 ), femto (10−15 ), to atto (10−18 ) seconds, with different wavelength ranges. In the year 1999, Zewail who used femtosecond laser pulses to study rapid chemical reactions was awarded a Nobel prize. An even shorter duration of the pulsed laser helps us to see the oscillations of visible light with a frequency of about one thousand trillion, i.e. 1 000 000 000 000 000, times per second, faster than a femtosecond pulse. For the first time, Krausz and co-workers (2004) showed the oscillation of visible (red) light (λ = 750 nm) with a 250-attosecond laser pulse, yielding a temporal resolution of 100 attoseconds [8], as shown in the left box of Figure 1.4. Another application of a high temporal resolution measurement is for the stroboscope technique. To capture an image of a hovering hummingbird with wings flipping at a rate of 80 times per second, we can use a camera equipped with a strobe flashlight at the same frequency, yielding sharp images of the wings, see the right box in Figure 1.4. A similar technique, but by slightly detuning the frequency, can be used to resolve fast dynamic processes by extracting the information from different frequency components. Recently, this technique has been used to the detect the dynamic distribution of electrons [9]. eV. 80. 70. 60 -6. -4. 0. 4 time [s]. 8. 12 x10-15. Figure 1.4: Left box: An image of oscillating visible (red) light at λ = 750 nm measured with a 250 attosecond pulse, the figure is taken from Reference [8]. Right box: A stroboscope technique is applied to capture a clean image of a hovering hummingbird, the picture is taken from Reference [10] (right box).. Thus, we have shown that ultrashort pulses are versatile tools to obtain a high temporal resolution measurements. By combining the temporal resolution of the pulse and the spatial resolution when illuminating a structure with a dimension close to the wavelength of the source, we can have a wealth of information to perceive. Therefore, in another part of this thesis, we will present the temporal and spatial evolution of diffraction when illuminating small apertures with a single femtosecond pulse. Our aim is to provide a fundamental insight about the interaction the pulse with small apertures.. 1.2 Research objectives In the paragraphs above, we have presented two applications at the frontiers of optics that will be discussed in two separate chapters. Our objectives are: (i) to investigate the role and the importance of phase mask structures, as compared to the existing amplitude mask, in generating high resolution.

(19) 1.3 Modal expansion method. 5. aerial images when projected by an extreme ultraviolet lithographic imaging system, and (ii) to understand the fundamental behaviour and the dynamics of the light, when illuminating femtosecond pulses through small aperture.. 1.3 Modal expansion method To understand the behaviour of light propagation that involves different wavelength ranges as well as different geometries of structures, we model the problems governed by the Maxwell’s wave theory of light and solve them numerically by using a modal expansion method. In principle, light waves can be modelled by an expansion (linear combination) of modes. When light interacts with a structure, the distribution of light in and around this structure can also be represented by these modes. In our approach, we decompose the incoming wave into a set of Fourier modes. By finding the solutions of the Maxwell’s equations for this set of modes and the corresponding contribution of each mode (taking into account the geometry and material of the structure), we can obtain the total light distribution. The analogy of finding the contribution of these modes for a certain geometry under a certain type of illumination, is similar to finding the modes of vibration for acoustic waves on surfaces. Chladni (1787) invented the technique to excite a certain mode by bowing a surface that is slightly covered by sand, such that at resonance (for a certain mode) the sand will form a pattern at the nodes, see the left box of Figure 1.5. In modern days, the bow is replaced by a loudspeaker. This method is used for designing musical instruments, such as a guitar (right box of Figure 1.5).. Figure 1.5: Left box: Chladni’s method to find the vibration modes of acoustical waves on musical instruments, consisting of a bow to excite the mode on the surface (of a violin) that is slightly covered by sand or salt; the picture is taken from [11]. Right box: The patterns at the nodes found for a guitar at different frequencies; the figure is taken from [12].. In the beginning of the thesis, we will explain the implementations of the method to obtain the solution of light distributions by finding the contributing modes, and we will also give a detailed discussion about the physical interpretation of the obtained results.. 1.4 Structure of this thesis The main elements of this thesis are the following three chapters that cover three different topics. In Chapter 2, we present the Fourier modal expansion method, including the history and the implementation of the method with several given examples to facilitate the demonstration of its working principle. For readers who are interested in the more detailed derivations, these can be found in the Appendices A, B, and C..

(20) 6. Chapter 1 Observation and manipulation of light at near-wavelength dimensions. Following the description of the method, we have two chapters covering the two research objectives. In Chapter 3, we apply the numerical method to model the imaging of phase mask structures under extreme ultraviolet illumination. Further, we explain the specification and assumptions used to model the illumination and optical projection system of extreme ultraviolet lithography. Several optimisation schemes are applied to find the phase mask structures that can yield the desired spatial resolution with the optimum modulation in the image plane. In Chapter 4, we discuss the propagation of ultrashort light pulses through apertures with dimensions close to the wavelength, by evaluating the light distribution and evolution in the spatial and temporal domain. This study is carried out by also applying the numerical method mentioned in the previous section. The transmitted pulse exhibits characteristics such as dispersion and spectral shaping that are not observed when using a (monochromatic) continuous-wave source. Since the thesis covers a broad range of topics, we have written three stand-alone chapters, each of them is provided with an introductory section to give ample background information on the subject. Thus, readers will be able to pick a chapter that suits their interest best. However, we would like to note that Chapter 2 and its appendices are provided also as a complementary chapter to the other two main chapters, such that the reader can have detailed information on the used method. Additionally, a nomenclature that consists of lists of notations and abbreviations is given before the appendices to help the readers keeping track on the used symbols and their definitions. Final remarks and conclusions on the main chapters are presented in Chapter 5..

(21) Chapter 2. Fourier mode expansion for diffraction in optics "In the discovery of secret things, and in the investigation of hidden causes, stronger reasons are obtained from sure experiments and demonstrated arguments than from probable conjectures and the opinions of philosophical speculators" — William Gilbert. 2.1 Introduction In science, theory and experiments are entwined and mutually advancing each other. That is to say, that there is an iterative process of observing phenomena to refine the theory, and analysing the phenomena to improve the experimental conditions for the following observation. William Gilbert was amongst the first scientists in the Renaissance era (a few decades earlier than Galileo’s period of work) to have introduced and conducted meticulous scientific methods [13] — he explained the nature of magnetism and investigated the property of electricity. His explanations were thorough that not until two centuries after, his results were studied and improved after a series of separate observations (amongst others by Cavendish, Coulomb, Volta, Faraday, and Hertz) and then unified in the wave theory of electromagnetic fields by Maxwell (1864). The diffraction of light, originated from the Latin word diffringere that means ’to break into pieces’, was first carefully observed and characterized by Grimaldi in mid 17th century. This unexpected light inside the geometrical shadow was also separately observed by Robert Hooke (1672). By taking the established Snell’s law of refraction and Fermat’s principle on the minimum optical path of light, Huygens (1690) introduced the wave theory of light. Later on, Young’s double slit experiment (1801) and Fresnel’s theory of diffraction (1821), firmed the experimental and mathematical footing of the wave theory of light. Thus, the corpuscular theory of light supported by Newton was abandoned. The particle theory of light was picked up in the early 1900’s, during the development of the quantum theory that treats light as dual particle-wave entity. Chronologically, the wave theory of light is considered as the classical approach. On the other side, the quantum electrodynamics theory is more general with the relativistic treatment that is applicable to any dimensional scale, and particularly useful when dealing with small numbers of photons. Studies on and applications of diffraction of light generally involve apertures, most commonly periodic apertures known as gratings. Recently, the research interest goes to a smaller dimension of these apertures, in which new(-ly observed) phenomena of the diffracted light are enhanced, for instance the surface plasmons generation and the enhanced transmission of light through small 7.

(22) 8. Chapter 2 Fourier mode expansion for diffraction in optics. apertures, the tight focusing of light with a high numerical aperture, etc. Considering the sufficient amount of photons involved when observing these phenomena, we can treat them in the classical way, i.e. using the wave theory of light. However, theory and hypotheses to analyse these phenomena cannot simply be taken from scaling down the scalar diffraction theory since the properties and geometries of materials affect the propagation of light differently at smaller scales. With the advances of computational science, the effort for testing hypotheses or theory through complex (sometimes expensive) experiments can be gradually reduced. It provides the ability to model a problem — a representation of the physical nature of a system based on theory or mathematical descriptions — and to simulate the behaviour of the system, for a better insight of its operation under the assumed parameters and experimental conditions. For the case of diffraction of light by smaller structures, it is often too difficult to find the analytical solutions. Hence, numerical computation becomes an important feature in science that can simulate the outcome of an experiment. However, due to the discrete nature of numerical computation, the solutions are often not exact. Moreover, efficient numerical techniques often need approximations to reduce the computational burden. Therefore, clever approximations have to be chosen, such that the problem is still solved in a sufficiently rigorous way. Now, to start evaluating an exhaustive list of existing numerical methods that can handle diffraction problems would be a very lengthy procedure. However, one may refer to commonly referred benchmark articles that compare many different numerical methods, such as Reference [14]. Each numerical method has its own advantages and drawbacks, depending on the geometry of the structure and the domain where the result is of interest (near or far field). In this thesis, we have chosen the Fourier mode expansion method to solve the diffraction problem by rectangular-shaped structures. The brief history of the method and our motivation to choose this particular method will be discussed in Section 2.2. To the readers who are more interested in the implementation of the method, the following subsequent sections are dedicated to explain the matter in detail. We will start in Section 2.3 with the formulation of the electromagnetic field behaviour given by the wellknown Maxwell’s equations for the wave theory of light. In Section 2.4, the method of Fourier mode expansion will be given for 2-d geometry, for 1-d geometry in conical mounting and for 1-d geometry in classical mounting. Next, in Section 2.5, we will discuss the boundary conditions and apply them for multilayered media with stable transfer matrices. For non-periodic structures, a coordinate transformation should be applied to the method in order to have a more efficient computation, which will be described in Section 2.6. Last, in Section 2.7, several numerical examples will be given to understand the working principle of the method and to have a physical interpretation of the results.. 2.2 History of the method The motivation for developing the numerical method described here, came from an observation made by Wood [15], as early as the year 1902. He reported the possible polarisation dependence from the extraordinary distribution of light by a grating, known as Wood’s anomaly. Lord Rayleigh tried to explain this anomaly by analytically calculating the amplitude of different diffraction orders [16]. In his model, Lord Rayleigh used a mode expansion to define the fields in the media above and below the grating structure — henceforth, the so-called Rayleigh’s expansion in the modal expansion method. However, he limited his analysis to the transverse electric case only. Note that most of the diffraction theory models at that time considered scalar approximations by using the Fresnel-Kirchoff analytic expression that is only applicable for a thin grating and not for small apertures where polarisation effects are more noticeable..

(23) 2.2 History of the method. 9. Early modal expansion methods One of the earliest attempts to use rigorous numerical calculation to solve diffraction problems for both polarisation states was carried out six decades later by Burckhardt [17]. His motivation was to model a thick hologram with a periodic grating structure for 3-d storage media. To simplify the problem, he assumed that the grating had non-absorptive dielectric properties and a sinusoidal profile such that it could be approximated by a cosine function. He used a Fourier series expansion to represent the electromagnetic field in the grating slab, solved the transverse electric and transverse magnetic case separately, and matched the field at the interfaces to obtain the electromagnetic distribution in each layer. Kogelnik (1969) also tried to model, in a less rigorous way, a thick grating but for a low-contrast material, by using a coupled wave analysis method [18]. It is numerically faster, but with more approximations. In 1973, Kaspar tried to give a more realistic model of the grating, assuming a rectangular shape and a complex refractive index [19]. He used Fourier series expansion and Floquet’s theorem to model the periodic refractive index and to decompose the field with respect to the grating vector, respectively. A few years later, Peng (1975) established a more rigorous, but somewhat cumbersome way to solve different types of grating profile, i.e. sinusoidal, rectangular, or curvy [20]. He also increased the complexity of the problem by introducing a multilayered grating and discussed several types of matrix propagation methods for matching the fields at the interfaces. Separately, Knop (1978) made an effort to define a more general model that can cope with various types of zeroth order gratings and provided numerical results for benchmarking [21].. The branching point in the developments of modal expansion methods: coupled wave analysis vs. modal method With many available approaches in solving electromagnetic field problems by the modal decomposition methods (some were not mentioned here), Magnusson and Gaylord (1977) made an exhaustive list to compare the assumptions and limitations of each method [22]. Based on this knowledge, they extended the coupled wave method introduced by Kogelnik to more general conditions for describing diffraction properties of thick gratings. In another paper [23], they categorized the different approaches into the two most common methods (coupled wave analysis and modal approach) and proved that the coupled wave method is equivalent to the modal method when including all diffraction orders and retaining the second derivatives of the electromagnetic fields. So far, the above mentioned modal expansion methods use the Fourier expansion technique to represent the permittivity function of the grating profile. The electromagnetic field is decomposed into a set of linear equations (Fourier series) that has to be solved, i.e. by defining the appropriate eigenproblem. Therefore, the obtained eigenmodes and the decomposed field possess the same continuous and periodic properties. Hence, to calculate the overlap integral that links between the mode of the grating structure and the order of the diffracted light is straightforward. In his PhD thesis, Botten (1978) with other co-workers presented a method that is more suitable for rectangular (discontinuous) profiles. Instead of representing the rectangular profile by the Fourier series, he tried to find the exact solutions for each homogeneous region inside the grating [24]. Note that, in the case of perfect electric conductor materials, the analytical solution of this modal problem is known, expressed in a set of sine and cosine functions. An exact solution can also be found for real metals, although the overlap integral will be more complicated, especially in the case of multilayered gratings. Based on the nature of the analytical approach, this method has been referred later to the modal method. Henceforth, we use this italicised term when referring to this method. Around the same period, McPhedran (one of Botten’s co-workers) and Nevière implemented a similar modal method for a 2-d grating..

(24) 10. Chapter 2 Fourier mode expansion for diffraction in optics. Meanwhile, continuing the previous development, Gaylord (1981) concentrated on the improvement of the coupled wave analysis. Together with Moharam [25], he presented a more rigorous coupled wave analysis (RCWA) method and coined the abbreviation. In this improved method, they incorporated the role of high order modes and the second partial derivatives of the electromagnetic field to formulate the eigenproblem, yielding a better accuracy for the calculated diffraction efficiency. Later on, they also extended the method to various profile shapes, such as trapezoidal gratings [26], and for an arbitrary complex permittivity, such as for real metals [27]. So far, when mentioning the diffraction by a grating, we implicitly assume a plane of incidence that is parallel to the modulation direction of the grating (the grating vector), known as the classical mounting. To have a more general description of the problem, Moharam and Gaylord (1983) implemented the method for a 1-d grating in conical mounting, that is, to have an angle between the plane of incidence and the grating vector [28]. Similar to the previous step by Nevière, Moharam (1988) implemented the RCWA method for 2-d profile gratings [29, 30]. In the 1990’s, the modal expansion methods have drawn an active discussion amongst the users, especially those of the RCWA method, who felt the necessity to improve the method. This episode began when Li (1993) started his contribution to the modal method by improving the implemented algorithm for a conical mounting [31] that was done earlier by Peng (1989) [32]. Li proved the completeness and the orthogonality of the eigenfunctions in the problem formulated by Peng. He extended further the modal method by discretising an arbitrary grating profile with a step function, creating a multilayer system [33]. Being interested in the existing modal expansion methods and the popularity of the RCWA method, Li compared the RCWA method and the modal method. In particular, he discussed the consequences of using the Fourier expansion to model the discontinuous permittivity function in the RCWA method, as opposed to the modal method which tries to find the exact solution of the field inside the grating (that was referred by Li as the lamellar-grating) [34]: "As the truncation order N in the Fourier expansion increases, the permittivity distribution approaches the original surface-relief profile, and, at the same time, the eigenvalues and the eigenfunctions of the modal fields approach their true values. In contrast, in the modal method, the lamellar-grating geometry is not changed and the modal fields are determined individually and exactly". The basic difference of these two methods is illustrated in Figure 2.1, where the permittivity function of the RCWA method has oscillations after being decomposed into the Fourier series.. Major numerical improvements to the RCWA method A major breakthrough for the improvement of the RCWA method occured after Li pointed out the convergence problem of the transverse magnetic case — Li addressed the method as the modal method by Fourier expansion, abbreviated into MMFE [34]. He blamed the use of Fourier expansion for the permittivity function as the cause of the slow convergence: "We further demonstrate that these expansions give rise to the slow convergence of the eigenvalues and the eigenfunctions of the modal fields". This remark was taken as a stimulus by the people who implemented the RCWA method, and was followed by a series of publications that showed the improvements of the method. For instance, Chateau and Hugonin (1994) introduced a more numerically efficient characteristic-matrix in the RCWA method that is applicable for a multilayered grating profile [35]. Moharam and his co-workers (1995) also took a step by formulating the RCWA method in a more stable and efficient way for both binary and arbitrary profile gratings [36, 37]. The following year, Song Peng (different from the previous Peng) and Morris improved the RCWA method by reducing the eigenvalue problem for the 1-d grating in conical mounting from a [4 × 4] matrix system to two [2 × 2] matrices [38]..

(25) 2.2 History of the method. 11. a) structure. c) field expansion. x b) refractive index. x. RCWA uses the decomposed refractive index profile to obtain the mode expansion of the field.. x. x the exact profile used in the modal method Fourier series representing the profile used in the RCWA. The modal method applies the mode expansion in each homogeneous domain, and the fields still have to be matched at the boundaries.. Figure 2.1: An illustration of the basic difference between the approaches taken by the RCWA method and the modal method. An arbitrary structure with varying refractive index (white and shaded areas) along the x-axis (a), and the corresponding refractive index profile (b). The RCWA method represents the profile with a truncated Fourier series (b), grey-line, yielding an oscillating refractive index profile (known as the Gibbs phenomenon). The field expansion is obtain from the Fourier representation of the permittivity profile (c), top. In the case of the modal method, the field expansion is determined for each homogeneous medium by finding all the contributing modes, and then matched at the boundaries (c), bottom.. The improvement of the RCWA method culminated, when Lalanne and Morris [39], and separately Granet and Guizal [40], reformulated the Fourier representation of the permittivity for the transverse magnetic case, yielding a faster convergence. Li demonstrated in mathematical terms the reason why the proposed numerical implementations in the previously mentioned two papers improves the convergence problem [41]: "The success of the new formulation (RCWA)... is due to the fact that it uniformly preserves the continuity of the electromagnetic-field quantities that should be continuous across permittivity discontinuities." Furthermore, he added the lesson to learn when using Fourier mode decomposition: " ..., in converting Maxwell’s equations in spatial variables to equations in the discrete Fourier space, one cannot blindly substitute the Fourier series of every term and every factor into the spatial equations; appropriate factorisation rules must be applied when discontinuities are present in the factors of products.". Further developments of the modal expansion methods After this major improvement for the RCWA method was made, Li shared more interest in the implementation of the RCWA method, and together with his co-workers (Granet, amongst others), converted the method name to a more well-describing one: Fourier modal method (FMM) [42]. Henceforth, we use this abbreviation when referring to this method. Recently, Li made an extension to the FMM by modelling anisotropic gratings [43] and efficiently formulating 2-d symmetric grating problems [44]. In contrast to the development of the RCWA method, that culminated only in the mid 1990’s, the modal method development was steadily progressing under a few leading researchers already in mid 1980’s. McPhedran, who had previously worked on the modal method with Nevière and Botten, improved the method with Roberts (1986) mostly for the perfect electrically conducting gratings, grids, and different shapes of apertures [45]. A lot of contributions have been made to extend the modal method to a more general type of material (several other important contributions for the im-.

(26) 12. Chapter 2 Fourier mode expansion for diffraction in optics. provement of modal method are not mentioned here). Recently, Tishchenko and co-workers (2005) picked up again the work of Botten and explained the straightforwardness of the method to interpret the physical meaning of the obtained results by giving some examples on short periodic dielectric gratings that involve only a few diffraction orders [46]. This method is best suited for grating profiles when the analytic solution is well-known, for instance in the case of perfect electric conductor gratings with hard wall boundaries. Although the preceding discussion has been largely devoted to the development of the Fourier modal method, and consequently the story of the modal method is discussed with much less emphasis, it is not our intention to demote the use of the modal method. As will be discussed at the last paragraph of this section, our interest focuses on the implementation of FMM because of its natural match with the relevant geometries studied in this thesis.. Propagation matrix algorithms for mode expansion methods Conjoined to the development of the modal expansion method, the development of the propagation matrix algorithm for multilayered media also took place. This propagation matrix is needed when matching the fields at interfaces, after obtaining the solutions in each grating layer. It is particularly important for multilayered gratings or discretised profiles of arbitrary gratings. Li once more took an important role in overviewing the existing matrices and describing the applicability of each [47]. The classical interface matrix is denoted as the t-matrix, linking adjacent media based on the order of the medium layer. The system transfer T-matrix constitutes one or more matrices t that describes the total transmission or reflection through the system. The T-matrix is prone to numerical instability since it creates increasing exponential terms (see Section 2.5.2). One alternative to overcome this numerical problem is to reorder the matrix in such a way that it links the inward and outward propagating waves at the interface. It is inherently stable due to the matrix inversion for all increasing exponential terms. This method was first addressed as Bremmer series by Pai and Awada [48] in 1991. Li used a similar matrix for his implemented modal method, and realised that his matrix is equivalent to that of the Bremmer series [49]. At that point, he briefly mistook the name of the matrix with the known R-matrix algorithm (discussed in the following paragraph). He found out later that Maystre and Cotter [50] also used the same matrix for the modal method but with a different name. Cotter addressed the matrix as a scattering matrix approach, and henceforth the matrix name is the S-matrix. The previously mentioned R-matrix is another alternative for a more stable matrix propagation algorithm. It has a longer history than the S-matrix, dating back to 1947 [51]. This matrix links the variables and its derivatives in the first and the second layer, respectively, by using trigonometric relations [47]. With the classical order of the adjacent layers, this matrix is called a variant of the Tmatrix. When it is ordered similarly to the S-matrix, it becomes the so-called R-matrix. The R-matrix was originally used for studying atom-molecule reactions in chemistry [52], and it has been used in the modal method by DeSandre [53], Montiel and Nevière [54] for grating problems. Li pointed out in his review paper [47] that the R-matrix is naturally numerically suitable for the differential method, whereas the S-matrix is most efficient for modal expansion methods.. Coordinate transformation for the RCWA method or FMM A few extensions of the FMM were made by means of a coordinate transformation along the grating direction. Granet (1999) introduced an adaptive spatial resolution such that more points are taken close to the discontinuity of the permittivity function [55]. Clearly, this was an attempt to silence the critics over FMM which always approximates discontinuous functions by Fourier series such that it.

(27) 2.3 Wave theory of light. 13. potentially reduces the exactness of the solution when only a few modes are retained. Later on, Vallius and Honkanen (2002) improved the adaptive coordinate transformation by making sure that the transformation function has no discontinuities [56]. Although the computational time is longer, it improves the convergence rate and gives higher resolution in presence of resonance peaks. With confidence in the numerical versatility of the FMM, another extension was applied such that it can be used to simulate non-periodic structures, for instance waveguide and photonic bandgap problems. This extension was following the introduced technique of the perfectly matched layer (PML) boundary by Bérenger (1994): the truncation of the domain of calculation without causing reflections. Lalanne and Silberstein (2000) have applied absorbing boundaries to each cell of a periodic structure, to model an isolated waveguide problem [57]. Bientsman and Baets (2002) have discussed various types of boundary conditions and concluded that the PML gives the lowest reflectivity from the boundaries and can even be combined with that of transparent boundary conditions, yielding extremely absorbing walls [58]. Hugonin and Lalanne (2005) compared the effect of a PML boundary that has a complex permittivity function, with that of a continuous complex coordinate transformation [59]. With the extended features of FMM, we have a more general method to model various types of geometry problems.. 2.2.1 Motivation to implement the Fourier modal method So far, we have discussed the development of the modal expansion method, in particular the Fourier modal method. Of course, other rigorous numerical methods are available to solve diffraction and scattering problems in optics. Moreover, it is shown that under certain circumstances, the Fourier modal method does not give an exact solution in contrast with that of the modal method. Our motivations, however, were based on the intended applications of rigorous diffraction theory to the imaging of mask structures at extreme ultraviolet wavelengths. A brief introduction on the subject can be found in Chapter 3.1. The structures under consideration are 80 multilayers with a total thickness of around 20 wavelengths. The profile of the modulation itself can contain from 40 up to 80 multilayers, corresponding to 10 to 20 wavelengths. Clearly, rigorous vectorial methods are needed to solve the diffraction problem of such a considerably thick structure. Moreover, we need a fast and efficient method that can solve the electromagnetic field problems in multilayered systems. In this study, the structures are assumed to be infinitely periodic, and to have only a 1-d profile in conical mounting. Taking into account all these aspects, the Fourier modal method is seen as the most suitable method to treat our problem. The discussion of the implemented method is given in Sections 2.4 and 2.5. Although later on we have used the method only for a 1-d grating profile in conical mounting, we start the discussion with a 2-d grating profile for the sake of completeness. After having implemented and applied the FMM to the multilayer structures in the extreme ultraviolet region, we have extended it to solve diffraction problems of one and two slit apertures when illuminated with ultrashort femtosecond pulses in the visible light spectral region as discussed in Chapter 4. For this purpose, we have modified the FMM to treat non-periodic structure illuminated by using a useful coordinate transformation that will be discussed in Section 2.6.. 2.3 Wave theory of light In most textbooks on classical electrodynamics [60, 61] and classical optics [62, 63], we can find the standard model describing the behaviour of electromagnetic fields. However, for the sake of.

(28) 14. Chapter 2 Fourier mode expansion for diffraction in optics. completeness and consistency of the following notations, we closely follow the discussion in [64] and start approaching the problem by defining the well known set of Maxwell’s equations: ∇ · D = f ,. (2.1a). ∂B , ∂t ∇·B = 0 ,. ∇×E = −. ∇ × H = Jf +. ∂D , ∂t. (2.1b) (2.1c) (2.1d). where D and E describe the electric displacement and the electric field, respectively, and B and H describe the magnetic field and the magnetic induction, respectively. These notations and namesgiving were introduced in the beginning of the chapters in references [60, 61]. However, the most commonly accepted conventions are to define B as the magnetic induction (magnetic flux density) and H as the magnetic field. We will use the latter notations in the thesis. The free current density J f describes the flow of the free charge density f through the medium. The difference between the electric field and the electric displacement (electric flux density) becomes clear when an interaction with a medium takes place. For linear media, the electric displacement is described by D = 0 E + P e = 0 (I + χe )E = E ,. (2.2). with 0 the electric permittivity of vacuum, P e the electric polarisation induced in the medium, χe the electric susceptibility tensor and the electric permittivity tensor of the medium. Therefore, we can replace the electric displacement by the medium-dependent electric permittivity tensor times the electric field. Correspondingly, the difference between the magnetic field and the magnetic induction can be clearly seen in linear media, where the magnetic field and the magnetic induction are described by 1 B − Pm μ0 1 = B − χh H , μ0. H =. B = μ0 (I + χh )H = μH ,. (2.3). with μ0 the magnetic permeability of vacuum, P m the magnetic polarisation induced in the medium, χh the magnetic susceptibility tensor and μ the magnetic permeability tensor of the medium. Therefore, we can replace the magnetic induction by the medium-dependent magnetic permeability tensor times the magnetic field. 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 + Je × 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 Je = v interacting with a magnetic induction. 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.

(29) 2.3 Wave theory of light. 15. between the free current in a medium and the electromagnetic field J f = σe Fn = σe E + μv × H ≈ σe E ,. (2.5). with Fn = F / the normalised electromagnetic force, σe 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 assume a time harmonic dependence with oscillation frequency ω for the electromagnetic field, E (r , t ) = Re E (r ) e −i ωt 1 = E (r ) e −i ωt + E ∗ (r ) e i ωt , 2 H (r , t ) = Re H (r ) e −i ωt 1 = H (r ) e −i ωt + H ∗ (r ) e i ωt . 2. (2.6). (2.7). The electric and magnetic fields are physical quantities, therefore we only have to consider the real part. Any frequency-dependent, or similarly, wavelength-dependent effect, will be included in the calculation by discretising the wavelength-dependent function, calculating the solution for each wavelength independently, and then superposing the solutions to obtain the combined electromagnetic field. Since the propagation speed of the electromagnetic field in vacuum is given by c = (μ0 0 )−1/2 , we can associate a wavelength in vacuum to the path travelled by the light in a single oscillation λ = 2πc/ω. If we assume no local charges or externally applied currents, i.e. ρ = 0 and J e = 0, we can write the time harmonic Maxwell’s equations as ∇ · E = 0 ,. (2.8a). ∇ × E = i ωμH ,. (2.8b). ∇ · μH = 0 ,. (2.8c). ∇ × H = −i ωE .. (2.8d). We can manipulate equation (2.8b) by taking the permeability μ in equation (2.8b) to the left-hand side of the equation, taking the cross product on both sides, substitute equation (2.8d) and multiply both sides with permeability μ, yielding the homogeneous electric field μ∇ ×. 1 ∇ × E − ω2 μE = 0 , μ. (2.9). and similarly, we manipulate equation (2.8d) by taking the permittivity in equation (2.8d) to the left-hand side of the equation, taking the cross product on both sides, substitute equation (2.8b) and multiply both sides with permittivity , yielding the homogeneous magnetic field 1 ∇ × ∇ × H − ω2 μH = 0 . . (2.10). In a homogeneous medium with a complex refractive index n = , the electromagnetic consists of waves travels with a speed of c/n. The wavenumber is equal to k 2 = |k|2 = ω2 μ, with the direction of propagation given by the wave vector k. In this thesis, the optical axis is chosen to be along the.

(30) 16. Chapter 2 Fourier mode expansion for diffraction in optics. z-axis, and, hence, the main propagation direction of the wave. The projection of the wave vector onto the Cartesian axes in a homogeneous medium is described by. 2π n sin θ cos φ , λ 2π ky = n sin θ sin φ , λ k z = k 2 − k x2 − k y2 , kx =. (2.11a) (2.11b) (2.11c). with the choice for the argument of k z such that the wave amplitude decreases in the propagation direction. The angle θ is the angle between the wave vector k of incoming light and the z-axis, and the angle φ is the angle between the plane of incidence and the x− axis, depicted in Figure 2.2. These angle definitions are chosen for convenience when implementing the Fourier mode expansion in the Cartesian coordinate system. Following from Maxwell’s equations, it is clear that for every plane waves, the electric and magnetic field components are both perpendicular to each other and to the propagation direction.. 2.4 Mode decomposition The name of Fourier modal method (FMM) originates from the mode expansion technique by using the Fourier series and is best suited for periodic structures. In principle, this method can be implemented for different coordinate systems, according to the direction of periodicity that can be arbitrary. For our main interest of applications, we will only consider structures with spatial frequency vectors that are orthogonal to each other in the Cartesian coordinate system. Important derivations for the numerical implementations of FMM in the Cartesian systems can be found in references [31, 36, 39]. The structure discussed in these references is usually a 1-d grating structure, although in separate publications, there have been also discussions to model 2-d grating structures [29, 44]. For a cylindrical profile, the readers can refer to other references, such as [65]. In this section, we start by deriving the FMM formulation for a 2-d grating and simplify further the model for a 1-d grating. We consider only grating profiles with rectangular shapes. The material under study can be dielectric or real metal — the optical properties of perfect metals causes numerical instability due to high absorption coefficient (high optical contrast). For a structure with the so-called 2-d grating profiles, we have a 3-d geometrical case. Thus, the permittivity and the permeability are assumed to vary in the x−y plane with period Λx along the x- and Λ y along the y-axis, and with a piecewise constant dependence in the z-direction. An example of a 2-d rectangular grating is given in Figure 2.2. To each layer which contains a homogeneous medium along the z−axis, we assign an index j . Therefore, to account for the possible layer dependence, each variable implicitly has an index j in this section. The x- and y-component of the wave vector in equation (2.11), for each layer j , can now be represented by the Floquet’s theorem for a periodic structure that also includes the incident angles θ and φ in the first medium with refractive index n 1 . We use 2N + 1 modes, comprising N modes for each positive and negative frequencies. For mathematical convenience in the numerical computation, the components of the wave vector are represented by a [(2Nm x + 1) × (2Nm x + 1)] diagonal matrix Kx , a [(2Nm y + 1) × (2Nm y + 1)] diagonal matrix K y , and a [N j × (2Nm x + 1) × (2Nm y + 1)] matrix Kz ,.

(31) 2.4 Mode decomposition. 17. plane of incidence E. z. H. Λx. θ k0. k0 x cosφ. θ. Λy. φ. y. x z. layer j (εj, μj) layer j+1 (εj+1, μj+1). Figure 2.2: Scheme of a two layered rectangular grating with a periodicity along the x− (Λx ) and along the y−axis (Λ y ) illuminated at an angle θ with and angle φ between the plane of incidence and the x−axis. Each layer has a homogeneous permittivity and permeability along the z−axis.. with the matrix elements 2π 2π mx + n 1 sin θ cos φ , Λx λ 2π 2π n 1 sin θ sin φ , = my + Λy λ 2 2 = k 2j − k x,m − k y,m , x y. k x,m x =. (2.12a). k y,m y. (2.12b). k z, j ,m x ,m y. (2.12c). where m x = −Nm x , −Nm x + 1, . . . , Nm x − 1, Nm x and m y = −Nm y , −Nm y + 1, . . . , Nm y − 1, Nm y . Subsequently, we decompose the scalar permittivity function and permeability function μ in each layer into 4Nm x + 1 modes with running index m x , and 4Nm y + 1 modes with running index m y (x, y) =. μ(x, y) =. 2N mx . 2Nm y. . m x =−2Nm x m y =−2Nm y 2N mx . 2Nm y. . m x =−2Nm x m y =−2Nm y. . 2π 2π m x ,m y exp i mx x + my y , Λx Λy. (2.13a). . 2π 2π μm x ,m y exp i mx x + my y , Λx Λy. (2.13b). and arrange the matrix elements into [(2Nm x + 1) × (2Nm x + 1)] Toeplitz matrices and μ, each element having the dimensions [(2Nm y + 1) × (2Nm y + 1)], as described in Appendix A. Therefore, when number Nm of modes is retained in the FMM calculation, the number of modes is intrinsically doubled to cover both the negative and positive spatial frequencies, and quadrupled to establish the Toeplitz permittivity and permeability matrix. This way, we have an equal dimension of the matrices and the correct order of the modes that are necessary when multiplying these matrices. Naturally, the electric and magnetic field components inside the medium with varying permittivity and permeability are decomposed also into 2Nm +1 modes with index m x and 2Nm +1 modes with index m y , resulting in the following definition of equations (2.9) and (2.10) in each layer j ,. μ∇ × μ−1 ∇ × E − ω2 μE = 0 , −1. ∇ × ∇ × H − ω μH = 0 . 2. (2.14a) (2.14b). Note the different order of the matrix multiplication between and μ in the electric and magnetic field equations. As for the layers with a corrugated medium along the x − y plane, we can obtain the electromagnetic field in a certain layer by solving Maxwell’s equations. For the 3-d case with permeability μ0 in.

(32) Chapter 2 Fourier mode expansion for diffraction in optics. 18. vacuum, the system has been derived in Appendix B.1 and reads as

(33).

(34). ∂2 ˇ − Kx −1 Kx ˇ − Ky Ky E ω2 μ0 ∂z 2 x = − −1 ∂2 ˇ + Kx K y −K y Kx E ∂z 2 y.

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(36) 2 ∂ ˇ − ˇ K y −1 K y − Kx Kx H ω2 μ0 ∂z 2 x = − ∂2 ˇ Kx −1 K y − K y Kx H ∂z 2 y. Ex , Ey

(37) Hx ˇ K y −1 Kx − Kx K y , 2 −1 ˇ − ˇ Kx Kx − K y K y H y ω μ0 ˇ + K y Kx −Kx −1 K y 2 −1 ˇ − Ky Ky ˇ − Kx Kx ω μ0 .

(38). (2.15a). (2.15b). ˇ being the double inverse of for computational efficiency, see Appendix B. The eigenwith modes in this particular layer are then found by obtaining the components of the eigenvectors U and the corresponding eigenvalue γ that represents the propagation constant in layer j along the z-direction,. 2 ˇ − Kx −1 Kx ˇ − Ky Ky Uex ω μ0 e = −1 ˇ + Kx K y Uy −K y Kx h 2 2 ˇ − ˇ K y −1 K y − Kx Kx U ω μ0 γ h hx = Uy ˇ Kx −1 K y − K y Kx γ. 2. e. Uex , Uey. h. Ux ˇ K y −1 Kx − Kx K y , 2 −1 ˇ − ˇ Kx Kx − K y K y Uhy ω μ0 ˇ + K y Kx −Kx −1 K y 2 −1 ˇ − Ky Ky ˇ − Kx Kx ω μ0 . (2.16a) (2.16b). with superscripts e and h denoting the variables corresponding to the electric and magnetic field, respectively. The z−components of the electromagnetic field is obtained from the following relations, 1 Uez = − −1 Kx Uhy − K y Uhx , ω 1 h Uz = Kx Uey − K y Uex . ωμ0. (2.17a) (2.17b). Hence, for any electromagnetic component in layer j , we obtain U (x, y) =. N mx. Nm y. . m x =−Nm x m y =−Nm y. Um x ,m y exp i k x,m x x + k y,m y y ,. (2.18). where Um x ,m y is an element of an eigenvector (or of the derived function from an eigenvector, e.g. in the case of the z-component). The electromagnetic field distribution in each layer j is obtained from a linear combination of eigenmodes. The contribution of each eigenmode in the linear combination still has to be found by matching the field at the interfaces (see Section 2.5.2). Formulations of the eigenproblems to find the eigenmodes for simplified structures are given in the following paragraphs.. 2.4.1 1-d grating in conical mounting Assuming a 1-d profile of a grating along the x−axis and therefore a constant wavenumber along the y-axis k y , we simplify the system as derived in Appendix B.2 to

(39).

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(41) ˇ − Kx −1 Kx ˇ − k y2 I ω2 μ0 0 Ex −1 =− , 2 2 ˇ − Kx ˇ − k y I − Kx Kx E y ω μ0 −k y Kx .

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(44) 2 2 2 ∂ ω μ − k I − K K 0 H Hx 0 x x x 2 y ∂z =− −1 2 −1 2 ∂2 ˇ Kx − Kx ˇ − ˇ Kx Kx − k y I H y ω μ0 ky 2 Hy ∂2 E ∂z 2 x ∂2 E ∂z 2 y. ∂z. (2.19a). (2.19b).