Adaptive Optics for EUV Lithography: Phase Retrieval for Wavefront Metrology

(1)

(2)

Adaptive Optics for EUV Lithography

Phase Retrieval for Wavefront Metrology

Proefschrift

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

op gezag van de Rector Magniﬁcus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 24 februari 2014 om 12:30 uur

door Alessandro POLO

Master of Science in Physics University of Turin, Italy geboren te Gagliano del Capo, Italy

(3)

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. H.P. Urbach

Copromotor: Dr. S.F. Pereira

Samenstelling promotiecommissie:

Rector Magniﬁcus, voorzitter

Prof. dr. H.P. Urbach, Technische Universiteit Delft, promotor Dr. S.F. Pereira, Technische Universiteit Delft, copromotor Prof. dr. ir. M.H.G. Verhaegen, Technische Universiteit Delft

Prof. dr. N.J. Doelman, Universiteit Leiden

Prof. F. Goudail, Institut d’Optique Graduate School

Dr. S.M.B. Bäumer, TNO Delft

Dr. B. Kneer, Carl Zeiss SMT GmbH

Prof. dr. ir. A. Gisolf, Technische Universiteit Delft, reservelid

This work was supported by the Dutch Ministry of the Economic Aﬀairs and the Province of Noord-Brabant and Limburg in the frame of the "Pieken in de Delta" program.

ISBN

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

(4)

(5)

Cover:

The cover of this thesis shows a fictitious mirror reflection of water pouring from the tap into a glass. The turbulence due to the water flow is no longer visible in the reflection of the mirror which acts as a corrective optical element, restoring it to a perfectly standing water state.

(6)

Summary

In the semiconductor industry, optical lithography is presently the most widespread technology used to print a geometrical pattern on a semiconductor wafer. Because of the plans imposed by the International Technology Roadmap for Semiconductors (ITRS) for more powerful and smaller chips, new printing technologies capable of printing smaller features on the wafer need to be developed. Extreme UltraViolet Lithography (EUVL) is a promising candidate for the next-generation of pattern technology beyond the current machines that operate at wavelength of 193 nm based on optical lithography. EUVL uses photons of 13.5 nm wavelength to carry-out the imaging and therefore it is suitable for addressing not only the 22 nm half-pitch nodes but also several nodes beyond that. However, material properties at the EUV range make this tech-nique completely different from present-day lithography. EUV radiation is strongly absorbed in all materials and gases and therefore multilayer (ML) aspherical mirrors have to be used in the projection optics. Moreover, because of the partial reflectivity of a ML mirror (about 70%), part of the radiation is absorbed by the mirrors causing thermally induced aberrations in the optical projection box. Adaptive optics (AO) is an attractive technology that could be implemented in lithography machines as a method to compensate these aberrations and achieve high quality diffraction-limited imaging. Since the aberrations we aim to correct are very small, a significant amount of research has been done to develop metrology techniques that can measure them with extreme accuracy and in high speed.

Most accurate wavefront sensing are usually done using interferometry techniques, but these require extremely stable setups and laborious measurement interpretations. In this thesis, we focus alternatively on another technique, called phase retrieval, which can be a valid substitute for interferometry.

In the first part of this thesis, we show how phase retrieval can be employed as a method to increase the accuracy of an already established wavefront measuring technique based on an Hart-mann wavefront sensor (HWS). The HWS technique relies on the assumption that the wavefront slope can be locally approximated by a linear function which is sensed on each sub-aperture of the sensor. Departures from this linearity assumption is the main limiting factor in the accuracy of the reconstructed wavefront. The implementation of the phase retrieval method presented here aims to recover the nonlinearities in the phase distribution and therefore achieves an improved reconstruction. Simulations showed that the RMS wavefront reconstruction has an accuracy of the order of 10−4λ. This value is typically one order of magnitude lower than the linear slope computational method. Experiments have been carried-out to compare the retrieved results with an independent wavefront measurement showing good agreement with the theoretical predictions. In the second part of this thesis, we focus our attention on phase retrieval from the intensity distributions of the field in the focal region. This method has already been demonstrated to be capable to estimate the phase aberrations of an optical system. However, the conventional implementation requires the measurement of intensity distributions in several different planes in

(11)

2 Summary the focal region in order to obtain a stable phase estimation. As an alternative, we develop here a method where phase retrieval from a single plane of measurement is achieved. Hence, our task is to reduce the measurement time and the amount of data to be analysed. As a result, we show that using a statistical approach, an optimal plane of measurement at one focus depth according to the axial Rayleigh criteria, can be identiﬁed. The analysis of the statistical uncertainties and correlation coeﬃcients shows that the phase retrieval from a single intensity measurement plane at said out-of-focus distance leads to a stable phase estimation with minimum uncertainties and minimum RMS wavefront deviation.

The concept of phase retrieval from a single intensity measurement plane is further investigated. By expanding the exponential term in the exit pupil function in Taylor series up to the first order we are able to describe, with proper approximations, the contribution of the aberration functions to the intensity distribution in the focal region. This allows us to show that aberration func-tions characterised by an even spatial distribution do not contribute to the intensity distribution of the field in the Gaussian focal plane. Hence, the problem of retrieving phase information from intensity measurements in such a plane is impossible. Moreover we show that, also in this approximation, it is possible to identify an optimal plane of measurement. By looking at the con-tribution of the aberrations to the intensity discon-tribution as a function of the out-of-focus distance, we have found analogous results. Namely, the aberrations contribute to the intensity distribu-tion in a minimal manner for distances close to the focal plane. Conversely, the contribudistribu-tion is maximised for a distance of one focus depth (Rayleigh criteria), confirming the previous findings. To exploit these results, a new experimental AO system based on the phase retrieval method has been built. In particular, this has been possible by joining the knowledge of the optimised phase retrieval method together with an innovative type of control strategy for the AO system. These two key elements need to work in perfect symbiosis: phase retrieval has to provide the accurate measurement of the phase, and the control strategy needs to translate the phase information to an electrical signal that drives the compensating element (e.g., a deformable mirror) in a correct and efficient way. For this reason, we also provide a complete analysis of the control algorithm employed in our setup, focusing on the convergence speed, shape quality and voltage saturation. This allows us to demonstrate that aberrations can be corrected in real-time up to λ/100 with a simple optical bench and a standard computer.

Finally, the last chapter of this thesis is dedicated to the feasibility analysis of the phase retrieval method implemented on a patented EUVL optical design. Thermally induced phase aberrations are modelled in the system on the basis of a thermo-elastic Finite Element Model (FEM) sim-ulation. A metrology procedure using different operational wavelengths i.e., 13.5 nm 193 nm and 633 nm has been proposed. Simulations have shown that the RMS wavefront reconstruction scales in general with the wavelength. Nevertheless, wavefront metrology at 193 nm has shown an accuracy below 650 pm RMS wavefront. This result is sufficient to fulfil the requirements and moreover allows relaxed experimental conditions for the metrology procedure.

(12)

Samenvatting

Optische lithografie is heden ten dage de meest wijdverbreide technologie in de halfgeleiderin-dustrie voor het aanbrengen van geometrische patronen op een halfgeleider ’wafer’. Vanwege de aanhoudende vraag naar krachtigere en kleinere chips dienen nieuwe technologieën voor het printen van kleinere details op de ’wafer’ te worden ontwikkeld. Extreem UltraViolet Lithografie (EUVL) is een veelbelovende kandidaat voor de volgende generatie van technologieën voor het aanbrengen van patronen, waarvan de prestaties uitstijgen boven die van de huidige machines, welke werken bij een golflengte van 193 nm, en gebaseerd zijn op optische lithografie. EUVL maakt gebruik van fotonen met een golflengte van 13,5 nm voor het maken van afbeeldingen en is daardoor niet alleen in staat te voldoen aan de afbeeldingseis van 22 nm voor de halve lijnafstand, maar ook aan nog zwaardere eisen. In verband met materiaaleigenschappen in het EUV bereik echter, wijkt deze techniek sterk af van de huidige lithografie technieken. EUV straling wordt sterk geabsorbeerd in alle materialen en gassen, waardoor asferische spiegels met multi-layer (ML) coating moeten worden toegepast in het optische projectiesysteem. Bovendien wordt als gevolg van de slechts gedeeltelijke reflectie van een ML spiegel (ongeveer 70%) een deel van de straling door de spiegels geabsorbeerd, met als gevolg dat er door opwarming ver-oorzaakte aberraties ontstaan in het optische projectiesysteem. Adaptieve optica (AO) vormt een aantrekkelijke technologie, die zou kunnen worden geïmplementeerd in lithografiemachines teneinde deze aberraties te compenseren en diffractie-begrensde afbeeldingen te verkrijgen van hoge kwaliteit. Aangezien de aberraties waarvoor moet worden gecorrigeerd zeer klein zijn, is in het verleden zeer veel onderzoek uitgevoerd naar meettechnieken die deze aberraties uiterst nauwkeurig en snel kunnen meten.

Nauwkeurige meting van golﬀronten wordt gewoonlijk uitgevoerd met behulp van interferometrie, maar deze techniek vereist een extreem stabiele meetopstelling en vergt bewerkelijke interpretatie van de meetresultaten. Als alternatief richten wij ons in dit proefschrift op een andere techniek, ’phase retrieval’, die een passend alternatief kan bieden voor interferometrie.

In het eerste deel van dit proefschrift laten wij zien hoe ’phase retrieval’ kan worden ingezet ten einde de nauwkeurigheid te vergroten van reeds gevestigde technieken voor het meten van golffronten, zoals die gebaseerd zijn op een Hartmann golffrontsensor (HWS). De HWS techniek steunt op de aanname dat de helling van het golffront kan worden benaderd door een lineaire functie die op iedere sub-apertuur van de sensor kan worden waargenomen. De belangrijkste beperkende factor voor de nauwkeurigheid van het gereconstrueerde golffront wordt gevormd door afwijkingen van deze aangenomen lineariteit. Implementatie van de in dit proefschrift ge-presenteerde methode voor ’phase retrieval’ is erop gericht de niet-lineaire componenten in de faseverdeling te bepalen, met als resultaat een verbeterde reconstructie. Uit simulaties blijkt dat bij reconstructie van een RMS golffront een nauwkeurigheid in de orde van 10−4_λ_kan

wor-den bereikt. Deze waarde is een orde beter dan met de lineaire aanname voor de helling van het golﬀront mogelijk is. Er zijn experimenten uitgevoerd om de verkregen resultaten te kun-nen vergelijken met onafhankelijk uitgevoerde golﬀrontmetingen. De resultaten tokun-nen een goede overeenkomst met de theoretische voorspellingen.

(13)

4 Samenvatting

In het tweede deel van dit proefschrift richten wij ons op ’phase retrieval’ vanuit de intensiteits-verdeling van het veld rond het brandpunt. Er is reeds eerder aangetoond dat het mogelijk is met behulp van deze methode een schatting te maken van de fase-aberraties van een optisch systeem. De conventionele implementatie van deze methode vereist echter dat voor een stabiele schatting van de fase de intensiteitsverdeling moet worden gemeten voor verscheidende vlakken in het gebied rondom het brandpunt. Als alternatief hiervoor hebben wij een methode ontwikkeld waarbij ’phase retrieval’ wordt gerealiseerd op basis van een meting in èè n enkel vlak. Daarbij hebben wij ons als taak gesteld de meettijd en de hoeveelheid te analyseren data te reduceren. Als resultaat hiervan laten wij zien dat op basis van een statistische aanpak een optimale posi-tie van het meetvlak kan worden vastgesteld op een uit-focus afstand u = 4π. De analyse van statistische onzekerheden en correlatiecoëfficiënten laat zien dat ’phase retrieval’ op basis van een intensiteitsmeting in èèn enkel vlak op de genoemde uit-focus afstand resulteert in een sta-biele faseschatting met een minimum aan onzekerheden en een minimale RMS golffront afwijking. Het concept van ’phase retrieval’ op basis van een intensiteitsmeting in èèn enkel vlak is verder onderzocht. Door de exponentiële term in de uittreepupil te expanderen in een Taylor reeks tot aan de eerste orde kunnen we, met de nodige benaderingen, de bijdrage van de aberratiefunc-ties aan de intensiteitsverdeling in het gebied rondom het brandpunt beschrijven. Hiermee kan worden aangetoond dat aberratiefuncties die worden gekenmerkt door een even spatiële verde-ling geen bijdrage leveren aan de intensiteitsverdeverde-ling van het veld in het Gaussische brandvlak. Dientengevolge is het niet mogelijk om fase te herleiden uit intensiteitsmetingen in dat vlak. Bovendien laten wij zien dat het mogelijk is het optimale meetvlak te identificeren, eveneens op basis van deze benadering. Door de bijdrage van de aberraties aan de intensiteitsverdeling te beschouwen als functie van de uit-focus afstand hebben wij overeenkomstige resultaten gevonden. De aberraties dragen namelijk slechts voor een zeer gering deel bij aan de intensiteitsverdelingen dicht bij het brandvlak. Omgekeerd is hun bijdrage maximaal voor een uit-focus afstand van u = 4π, waarmee het eerdere resultaat wordt bevestigd.

Om deze resultaten te exploiteren is een nieuw experimenteel AO systeem gebouwd, gebaseerd op de methode voor ’phase retrieval’. In het bijzonder is dit mogelijk geweest door de geopti-maliseerde methode voor ’phase retrieval’ te koppelen aan een innovatieve besturingsstrategie voor het AO systeem. Deze twee sleutel-elementen dienen in perfecte symbiose samen te werken: ’phase retrieval’ dient een nauwkeurige meting van de fase te verschaﬀen, en de besturingsstra-tegie moet fase-informatie vertalen naar een elektrisch signaal dat het compenserende element (bijv. een vervormbare spiegel) op een correcte en eﬃciënte wijze aanstuurt. Om deze reden voorzien wij ook in een volledige analyse van het algoritme voor de besturing in onze opstel-ling, waarbij de nadruk is gelegd op convergentiesnelheid, kwaliteit van de vorm, en verzadiging van stuurspanningen. Hierdoor hebben wij kunnen aantonen dat aberraties in real-time kunnen worden gecorrigeerd tot op λ/100 voor een eenvoudige optische meetopstelling en een niet zo krachtige computer.

Het laatste hoofdstuk van dit proefschrift tenslotte, is gewijd aan de haalbaarheidsanalyse van de methode voor ’phase retrieval’, geïmplementeerd in een gepatenteerd EUVL optisch ont-werp. Er zijn thermisch geïnduceerde fase-aberraties voor het systeem gemodelleerd, op basis van een simulatie met een thermo-elastische eindige-elementenmethode (FEM). Er is een me-trologieprocedure voorgesteld voor verschillende golflengtes, te weten 13,5 nm, 193 nm en 633 nm. Simulaties tonen aan dat de RMS golffrontreconstructie in het algemeen schaalt met de golflengte. Niettemin levert golffrontmetrologie bij 193 nm een nauwkeurigheid op van 650 pm RMS. Dit resultaat voldoet aan de vereisten en laat bovendien voldoende extra experimentele ruimte voor de metrologieprocedure.

(14)

Chapter 1

Introduction

"The number of transistors that can be in-expensively placed on an integrated circuit is increasing exponentially, doubling approx-imately every two years."

Gordon Moore (1965)

Adaptive optics (AO) as well as the principle of Optical Lithography (OL) are brieﬂy intro-duced in this chapter.

1.1 Adaptive optics

The field of adaptive optics has been developed to correct for the phase disturbances of an optical signal by understanding the medium between object and image. Initially proposed by Babcock in 1953 [1], it has found its first application in the field of astronomy (but also in the military environment, where most of the information have been kept secret) as a method to continuously compensate for wavefront deformations created by the atmosphere. As predicted by Newton (1730) [2], the atmospheric disturbances caused by wind eddies, thermal fluctuations, density differences, and molecular absorption, lead to small changes in the index of refraction. This causes the atmosphere to act as if it were an array of tiny lenses, bending light rays as they pass to an observer on Earth. As shown in Fig. 1.1, if we assume that the light from some distance object (e.g., a star) leaves as a spherical wavefront, the waves reaching an observer on Earth would theoretically be described by a plane wave. The atmospherical turbulence has the effect of changing the phase of the plane wave such that are distorted when it reaches the observer . The purpose of AO is to correct these distortion. To be able to do this detailed knowledge of these errors is crucial. Once these distortions are corrected, the telescope (in theory) can once again become diffraction limited. Therefore, we can think of an AO system like a dynamically reconfigurable, high resolution "contact lens" for imaging system.

In the simple description, an AO system consists of three key elements: a wavefront sensor (WFS) used to measure the distortion of the wavefront coming from an object; a wavefront cor-rection active device, which in most cases consists of a deformable mirror (DM)(though other types of active elements are nowadays also available [3–5]); and a control computer, which trans-lates the wavefront information from the WFS in an appropriate signal for the DM to obtain the desired correction. The residual error in the corrected aberration is directly dependent on the type of wavefront sensor, i.e., the complexity and amplitude of the aberrations that can be sensed and the number of actuators in the deformable mirror, i.e., the complexity and ampli-tude of the aberrations that can be reproduced by the DM [6]. In general, both WFS and DM

(15)

6 Introduction

Figure 1.1: Principle of an adaptive optics system: The AO system tries to correct the distorted wavefront by measuring it using a wavefront sensor, calculating an appropriate correction, and applying this correction to a deformable mirror. This feedback loop is carried out several hundred times per second in order to comply with the temporal bandwidth requirement

can only sense and reproduce a ﬁnite number of shapes (later in this thesis, we will represent this "shape" in terms of the orthonormal Zernike polynomial expansion [7]), so there are always residual aberrations that are not corrected by the AO system.

Nowadays, due to the cost reduction of these devices and the progress in computer tech-nology, wavefront correction systems have widened their applications to other research areas than astronomy such as microscopy [8], ophthalmology [9], optical communication [10] and more recently lithography.

In particular, lithography has been considerably changed in the last 15 years with the imple-mentation of an AO system in the lithography lenses [11]. Originally limited to only magniﬁcation adjustment, modern scanners have been upgraded with complex manipulators that are able to minimise the total lens aberrations in a dynamic manner.

1.2 Semiconductor lithography

Lithography refers to an old technique invented in 1796 by the German author and actor Alois Senefelder as a cheap method of publishing theatrical works [12]. The word originates from Greek lithos and graphia which means literally writing on stone. In the lithographic printing process, ink is applied to a grease-treated image on the ﬂat printing stone surface; non-image (blank) areas, which hold moisture, repel the lithographic ink. This inked surface is then printed directly on paper by means of a special press or onto a rubber cylinder. This basic idea can be transferred in the case of semiconductor lithography (often called photo-lithography or optical-lithography) which refers to the process used in the micro (nowadays, nano) fabrication of an integrated circuit (IC). In semiconductor lithography the stone is "replaced" by a silicon wafer

(16)

1.2 Semiconductor lithography 7 and the image, i.e. the mask, is printed with a photo-sensitive polymer called photoresist. To build a complete IC which can consist of millions (or even billions) of transistors, several steps are needed in repeated sequences. The most common photo-lithography procedure is depicted in Fig. 1.2 and consists of the following steps:

(a) deposition: a layer of new material (e.g., silicon oxide) is added over the substrate; (b) coating: an uniform coating of photoresist material, sensitive to the exposing light, is

deposited onto the wafer;

(c) exposure: the photoresist is exposed to light. The exposure causes chemical local changes in the photoresist where it has been exposed, creating an image of the mask in the photoresist material;

(d) developing: the exposed regions of the photoresist are removed by a special solution called "developer";

(e) etching: the deposited material is chemically and mechanically removed in the regions that are not protected by the photoresist;

(f) ion implantation: the deposited material is exposed to a speciﬁc dopant impurity to form the foundational structure of the semiconductor device;

(g) photoresist removal: the remaining part of the photoresist is removed from the substrate with a solution caller "resist stripper".

Figure 1.2: Photo-lithography basic procedure: a) deposition; b) coating; c) exposure; d) developing; e) etching; f) ion implantation; g) photoresist removal.

The combination of exposing different masks and select different dopings on several layers allows the buildup of the structure of the required IC. The typical time for producing a complete chip starting from a bare silicon wafer is typically 30 to 60 days. The evolution of semiconductor technology is strictly related to smaller feature sizes which can be printed on a semiconductor wafer. In general, the smaller the size of the transistor the faster is its performance and the smaller is the package. The progress of the semiconductor industry was first analysed by Gordon

(17)

8 Introduction Moore. In his paper dated 1965 [13], he noted that the number of components in an integrated circuit had doubled every year from 1958 until 1965 and he predicted that the trend would continue for at least ten years. His prediction was proved to be amazingly accurate and for this reason we refer to it as Moore’s law (Fig.1.3).

Figure 1.3: Plot of CPU transistor counts against dates of introduction. Note the logarithmic vertical scale; the line corresponds to exponential growth with transistor count doubling every two years. Source: "Wikipedia".

In order to anticipate the evolution of the market and to plan and control the technological needs of IC production, lithography machines should evolve with an equivalent progress. The performance of a lithography machine is determined by the smallest feature (i.e., the critical dimension CD) that can be printed on the wafer. The CD is related to the resolution of the system deﬁned by:

RES = k1

λ

NA, (1.1)

where k1 is a process parameter (typical values for k1 are 0.2 < k1 < 0.6, depending on the

illumination conditions [14]), λ is the wavelength of the exposure light and NA is the numerical aperture of the projection system. Hence, from Eq. 1.1 it is straightforward to understand that a continuous shrinkage of the illumination wavelength and/or an increase of the NA is needed to meet the market requirements. Lithography machines have progressed over the time from UV light (365 nm) to deep-UV light (248 nm) to the current 193 nm. At the same time, also the NA of the optical system has risen from 0.16 to 0.93 NA. Nowadays diﬀerent techniques are used to decrease the achievable CD with the 193 nm source, such as immersion lithography (which increases the NA around 1.35) and double patterning, but such technology is now close to the physical limit for producing small feature sizes [15].

(18)

1.3 The Pieken in de Delta program 9

Figure 1.4: Schematic of a EUV lithography system powered by a laser produced plasma source. The picture is a representation from [16].

In Extreme-ultraviolet (EUV) lithography (Fig. 1.4) the wavelength of 13.5 nm is used, and thus it represents the next significant step in the reduction of feature sizes on integrated circuits. However, moving from 193 nm to 13.5 nm has been demonstrated to be far from straightforward, for a number of reasons. The most important one is that EUV radiation is strongly absorbed by almost all materials, and the refractive index of the left-over materials is close to 1. EUV radiation is absorbed even by air at nominal pressure. Thus, the entire optical system including mask and wafer stage must be kept in an environment close to vacuum during the lithography process. Due to the above two reasons, all optical elements in the imaging system must be reflective rather than refractive. The most important one is that EUV radiation is strongly absorbed by all materials since their indices of refraction is close to unity. Multilayer mirrors made of several layers of Silicon/Molybdenum (Si/Mo) are employed as optical reflectors. The maximum theoretical reflectivity of these mirrors is 72% around 13.5 nm [17]. This factor leads to an important consequence: considering a typical EUV system with 6 mirrors in addition to the optics for the illumination, the residual power on the wafer stage is only 2%. In order to achieve the required throughput, a bright light source is needed. But when the source power increases, the energy absorbed by the multilayer mirrors increases as well, and consequently its thermal load, increases. Hence, dynamical thermal deformations occur in the optical system, causing unavoidable phase aberrations which can have serious impact on its performance [18]. For an EUV system, the allowable wavefront aberration should be less than λ/20 which at the wavelength of 13.5 nm equals to a total deviation of less than 650 pm with even tighter constraints for the individual mirrors [19]. Therefore, to guarantee a diffraction limited imaging on the silicon wafer, an AO system, able to monitor and at the same time correct for the thermally induced wavefront aberrations, could be attractive.

1.3 The Pieken in de Delta program

This PhD project is part of a bigger research collaboration in which a total number of four PhD students, specialised in diﬀerent ﬁelds have been involved. In particular, the whole project has been branched in the following way:

• overall system design (R. Saathof, Department of Precision and Microsystems Engineering (PME), Faculty 3mE, TU Delft);

(19)

10 Introduction 3mE, TU Delft );

• active mirror design and methodology (S. Ravensbergen, Control Systems Technology Group (CST), Faculty of Mechanical Engineering, TU/e);

• wavefront metrology (the author of this thesis).

This research is supported by the Dutch Ministry of Economic Aﬀairs and the Provinces of Noord-Brabant and Limburg in the Pieken in de Delta program.

1.4 Goal and outline of this thesis

As briefly explained before, our study will mainly focus on the development of a wavefront measurement technique which can be used in situ in an EUV lithography scanner. In this thesis, we address the problem of measuring the phase distribution on the exit pupil of an optical system exploring different approaches. In Chapter 2, a short overview of the state-of-the-art for EUV optical testing is given. We consider interferometric and non-interferometric wavefront measuring techniques, pointing out their pros and cons in EUV metrology. Moreover, we introduce the concept of phase retrieval by a non-linear optimisation algorithm, which will be intensively used throughout the thesis. In Chapter 3, we present how, by means of phase retrieval algorithms, the performance of a conventional non-interferometric technique such as a Hartmann Wavefront Sensor, can be improved in order to meet the EUV metrology requirements. In Chapter 4, an alternative wavefront measurement technique, based on the phase retrieval from focused field, is illustrated. In particular, an investigation of the performance of the algorithm as a function of the axial distance of the measurement plane to the Gaussian focal plane is carried-out. This is attained by a topological study of the merit function landscape and a sensitivity analysis of the algorithm. In Chapter 5, we introduce an analytical model developed in the limit of small aberrations which gives direct insight of the phase retrieval problem. The model allows us to simplify the involved equations and derive a solution for the identification of an unique measurement plane which is optimal for the phase retrieval. In Chapter 6, we present a linear phase retrieval model for focused field based on the approximation of small aberrations. The linear model together with the results discussed in Chapter 4 and 5, allow us to implement a real-time adaptive optics system based on the measurement of the phase retrieval algorithm. This overcomes the issues which has prevented the use of non-linear phase retrieval in real-time applications. Chapter 7 is completely dedicated to the optical characterisation and the control strategy of the active optical element which has been used extensively in all the experiments described in this thesis. This is the result of the fruitful collaboration with the other partners involved in this project. In Chapter 8, using the results of the overall system analysis carried-out by the Department of PME, simulations based on the founding of Chapter 4 and 5 together with a patented EUV lithography design have been performed using a ray tracing software package. This work shows the feasibility of using a phase retrieval method as a tool to characterise the thermally induced wavefront aberration occurring in the EUV optical system.

(20)

Chapter 2

EUV optical testing state-of-the-art

and phase retrieval

"Door meten tot weten" (Knowledge through measurement).

Heike Kamerlingh Onnes (1853-1926)

2.1 Introduction

An essential step in enhancing the performance of an optical system, is the ability to measure the optical quality of the system components. As already mentioned in Chapter 1 of this thesis, in an Extreme UltraViolet Lithography (EUVL) system, these components consist of multilayer aspherical mirrors which are the most complex components to manufacture. The optical quality of these mirrors can be threatened by the increasing EUV source power, which is planned in the near future, to meet the customer’s performance requirements [18,20]. For EUVL projection optics, aberrations on the scale of λ/20 are known to adversely aﬀect both imaging performance and distortion. At the wavelength of 13.5 nm, the λ/20 constraint equals a deviation of less than 650 pm with even tighter requirements for the individual mirrors1_{. For this reason, over the past}

several decades, researchers have been pushed to develop extremely accurate wavefront aberra-tions measurement techniques. However, because of the recognised trade-off between complexity, costs, and necessity, no common conclusion has been yet reached in the EUVL community on the choice of a metrology technique. It is generally acknowledged that some progress is still needed to meet the demands of commercial lithography machines. Over the past several decades, in-terferometry has become the pillar of metrology technique for high-accuracy, diffraction limited optical system. Interferometry includes a family of techniques in which electromagnetic waves are superimposed in order to have, as a result, an intensity fluctuation that is used to recover the original state of the wave. A number of successful techniques have been developed specifically to meet the challenge presented by EUVL. They include both at-wavelength and visible light interferometry (the most commonly applied EUV interferometric techniques belong to the class of common path interferometers) as well as non interferometric methods (Hartmann test and aerial image monitoring). Here, we would like to give an overview of the most typical techniques with a brief discussion over their merits and weaknesses.

(21)

12 EUV optical testing state-of-the-art and phase retrieval

2.2 Phase-shift point diffraction interferometer

The phase-shift point diffraction interferometer (PS/PDI) belongs to the class of common path interferometers [7]. Contrarily to the general type of interferometers, such as the Michelson inter-ferometer (where the reference and test beam follow separated paths) in the so-called common-path interferometer, the reference and test beams shear the same general optical common-path. First pro-posed by Medecki et al. [21] and later improved by Goldberg and Naulleau [22, 23] the PS/PDI is an improvement of the conventional point diffraction interferometer [24]. In the PS/PDI (Fig. 2.1), the optical system under test is coherently illuminated by a spherical wave generated by diffraction from a pinhole (i.e., an aperture smaller than the resolution of the optical system) which is placed in the object plane. A grating, placed either before or after the optical system, is used for two reasons. First, it splits the illuminating beam creating the required test and reference beams. Second, it adds phase-shifting capability to the interferometer. A mask with two apertures is placed in the image plane of the optical system. The first aperture consist of a window that blocks unwanted diffracted orders generated by the grating and lets the test beam propagate undisturbed to the detector. The second aperture consists of a reference pinhole that spatially filters the reference beam, removing the aberrations introduced by the optical sys-tem. The test and reference beams propagate and they overlap to create an interference pattern recorded by the detector.

The recorded interferograms yields information on the deviation of the test beam from the nominally spherical reference beam.

Figure 2.1: Principle used in the phase-shift point diffraction interferometer

The PS/PDI is an eﬀective metrology technique with an accuracy that can be λ/350 [25]. However, the two primary sources of measurement error that limit the accuracy of the PS/PDI are imperfections in the reference wave generated by diﬀraction from the image-plane pinhole and systematic errors that arise from the challenging alignment of the system due to the extreme small size of the pinhole used (at 0.3 NA the pinhole size are below 40 nm [26]).

Another signiﬁcant limitation of PS/PDI is its dynamic range. If large aberrations occur in the system, the intensity at the pinhole, described in a relative unit by the Strehl ratio, decreases considerably. This limits the dynamic range of the system to less than one wave or even smaller. Furthermore, PS/PDI has the drawback of being limited to highly coherent EUV sources such as undulator radiation [27].

(22)

2.3 Lateral shearing interferometer 13

2.3 Lateral shearing interferometer

The method of lateral shearing interferometer LSI (Fig.2.2) consists of displacing the beam under test by a small amount and obtaining the interference pattern between the original and the displaced wavefront. In this way the resultant phase measurement approximates the derivative of the wavefront in the direction of the shear. Performing two orthogonal measurement allows to unambiguously reconstruct a wavefront without rotational symmetry.

The EUV version of the shearing interferometer uses a cross-grating as a low-angle beam splitter [26]. The spatial scale of the aberration that can be measured (i.e., the spatial frequency response) is determined by the displacement (shear), which is determined by the period of the grating. In general, large shear increases sensitivity to low-frequency aberrations at the expense of the high spatial frequency cut-oﬀ. Simultaneously, with large shear, the system is not able to measure aberrations in the areas close to the edges of the pupil. However, it has been proven that shearing interferometry has accuracy that is comparable to PS/PDI [24, 28, 29]

Compared to PS/PDI, shearing interferometer has relaxed coherence requirements, oﬀers much higher eﬃciency and higher dynamic range but at the expense of more complex wavefront analysis and reduced sensitivity near the edge of the pupil.

Figure 2.2: Principle used in the shearing interferferometer

2.4 Hartmann sensor

Until now, we have described two interferometric methods that could be used to characterise the optical box of an EUVL system. However, as accurate as interferometry can be, its sensitivity to external factors limits its use. It usually fails, for example if the aberrations in the system are changing rapidly or if vibrations of the system under test, in relation to the test station, are present. Nevertheless, even if the phase information can be retrieved from multiple recordings, such retrieval can be laborious and diﬃcult.

When speed, reproducibility, flexibility, and accuracy of the wavefront measurement become essential, the implementation of theoretically less accurate and lees elaborate methods such as the Hartmann wavefront sensor (HWS) are usually preferred [30]. The HWS belongs to the class of non-interferometric techniques with low coherence requirement and high overall efficiency. Its first development dates back to 1900 when it was proposed by the German astrophysicist Johannes Hartmann (1865 -1936) [31]. During its history, a wide variety of applications have been developed thanks to the simple and elegant way to measure the shape of the wavefront.

In the HWS (Fig. 2.3) a beam passes through a hole array and is projected onto a CCD camera that detects the beamlet sampled by each hole. The positions of the individual spot

(23)

14 EUV optical testing state-of-the-art and phase retrieval centroids are measured and compared with reference positions. This enables the measurement of the slope of the wavefront at a large number of points within the beam, from which the wavefront can be reconstructed [32].

Figure 2.3: Principle used in the Hartmann wavefront sensor

The mask (i.e., the hole size and distance between two adjacent holes) and the CCD distance, are designed in a way that adjacent spots in the projected pattern do not overlap, resulting in clearly isolated spots. That is the reason why the HWS is technically not an interferometer: the diffracted beam do not overlap and do not interfere at the detector. In general, a longer mask-CCD distance results in a more accurate wavefront measurement but at the expenses of a maximum spatial frequency of the aberration that can be detected. Therefore, different masks are usually designed in order to be used in different measurement conditions. An evolution of the HWS is the Shack-Hartmann wavefront sensor (S-H WFS), in which the hole array is replaced by a lens array. By adding lenses, the light passing through the apertures is concentrated into a focal spot. This concentration would aid in boosting the photon density and allowing the spot to be recorded. However, this improved technique can not be applied at the EUV wavelength because the refractive lens array have to be replaced by a diffractive system (i.e., Fresnel zone plates) where diffraction orders would contribute significantly to the noise on the detector. Another alternative would be to use a micro mirror array to produce focused spots on a detector. However, also in this case, the mirror array would experience the same thermal issues of the optics under test. Moreover, such an array would not be easy to manufacture due to the fact that each micro mirror should have a parabolic profile and should be polished with a nanometer accuracy.

The HWS has been successfully applied to characterise wavefront at the EUV wavelength obtained from a synchrotron beam line [33–36]. Nevertheless it still has has yet to be proven to be sensitive enough to meet the requirement of EUVL. However, the HWS presents some important advantages over interferometry. With the HWS, both intensity and phase are measured at the same time. It can work with spatially and temporally partially incoherent beams, and any kind of optics, focusing or otherwise, with large or small aberrations, can be measured. Finally, HWS is compact, inexpensive, and easy to build.

Chapter 3 will be fully dedicated to the investigation of the use of a HWS as a possible candidate for wavefront metrology tool in EUVL system. We will show that the performance of this device can be boosted when it is combined with a particular non-linear algorithm called phase retrieval. Before describing the improved setup we will discuss in the next section the principle on which the phase retrieval algorithm is based.

(24)

2.5 Introduction to phase retrieval 15

2.5 Introduction to phase retrieval

Phase retrieval (PR) refers to a class of algorithms that aim to determine the phase information of a complex-valued function, from the information about the measured intensity and some a priori knowledge about the function or its transform. The complexity of the problem is due to the fact that the equations relating the phase and the intensity of an electromagnetic field are non-linear, which makes the phase retrieval problem falling in the class of inverse ill-posed problem. This problem is common in several fields besides wavefront sensing, such as astronomical imaging by interferometry [37], and x-ray crystallography [38,39]. We give credit to Gerchberg and Saxton [40] for the first works on the phase retrieval method in 1972 in the electron microscopy context. Phase retrieval methods were successfully implemented for reconstructing optical images a few years later (1978) with important results accomplished by Fienup [41] and later for characterising the aberrations of the Hubble Space Telescope primary mirror [42]. Also Gonsalves must be mentioned for his contributions to expanding the phase retrieval methods to even the case of an extended and unknown object [43, 44] as well as the contributions by Braat and Janssen in providing a semi-analytical solution to this problem [45–48].

Conversely to the optical techniques described in the previous section, which belong to the class of direct sensor, PR is classiﬁed as indirect sensor in which the information about the wavefront is obtained from the analysis of an image of a given object. An advantage of PR is that it uses the absolute minimum of optical components (basically just a detector that measures the ﬁeld intensity). Furthermore it is the only method that is sensitive to all aberrations that occur in the complete optical system.

Figure 2.4: Schematic model of the phase retrieval formulation

Mathematically, the PR problem is formulated as follows. Consider Fig. 2.4 and let A(ξ, η) be the amplitude of the complex field distribution at the exit pupil of an optical system and let Φ(ξ, η) be its phase distribution. Here and in the reminder of this thesis we do not consider polarisation effects. The NA of the system that we study is sufficiently low (i.e., NA < 0.6) to justify this. Hence, the field in the pupil plane is given by:

P (ξ, η; Φ) = A(ξ, η) exp[ikΦ(ξ, η)]. (2.1)

Here, k = 2π/λ is the wavenumber corresponding to the wavelength λ and i is the imaginary unit. The ﬁeld is then propagated to the image plane (rx, ry) orthogonal to the optical axis, usually

using a 2D Fourier transform. We will see later in this thesis that phase retrieval algorithms normally use through-focus image planes. The ﬁeld in focus is then propagated over a given distance by propagating its angular spectrum [49]. In the general case, called Gz[·] the operator

(25)

16 EUV optical testing state-of-the-art and phase retrieval that maps the exit pupil to the measurement plane at a distance z beyond the Gaussian focus2_,

the intensity distribution I(rx, ry; Φ)in that plane is given by:

I(rx, ry; Φ) = |Gz[P (ξ, η; Φ)]|2. (2.2)

The task of the PR is to recover Φ(ξ, η) given the measured I(rx, ry; Φ) and the constraint of

existence of the field in the pupil plane i.e, we restrict our study to a function which impulse response is band-limited. It is important to note here that such band-limited functions have remarkable analyticity properties which have been applied extensively in optics. In particular when the field in the pupil plane is non-zero over a finite domain, then the modulus of its Fourier Transform must be a smooth and analytic function. This implies that there is a limited number of functions having a limited support and whose Fourier transforms have the same modulus on the real axis (Paley-Wiener theorem, see Appendix A).

As mentioned above, several methods have been proposed to solve this problem. In general, they can be distinguished in two classes: iterative methods [38, 40] and non-linear optimisation methods [42, 44]. All of them share the common feature of using multiple plane images to improve stability and to remove a sign ambiguity [43] (i.e., non-unicity of the solution), that occurs speciﬁcally in the recovered phase from the image taken at the geometrical focus of the system.

In this thesis, we will focus our attention on the non-linear optimisation methods and we will explore the possibility to solve the PR problem by using intensity data for only a single measurement plane and in addition some a priori assumptions on the pupil intensity distribution. Our goal is to retrieve the phase information using a minimum set of intensity data at the advantages of speed and computational complexity.

In Table 2.1 the pros and cons of the previously considered wavefront measurement techniques are summarised.

Table 2.1: Comparison of different wavefront measurement techniques for EUV radiation.

PS/PDI LSI HWS PR

Pros high accuracy high accuracy high dynamic range high accuracy high dynamic range easy setup minimum of optics coherence requirement coherence requirement sensitive to all aberrations Cons dynamic range sensitivity on pupil edge accuracy computation complexity

calibration wavefront reconstruction slow measurement coherence requirement

2.6 Phase retrieval by non-linear optimisation

The class of iterative phase retrieval methods although simple and straightforward in its im-plementation has been proved to be prone to stagnation, very slow in convergence (typically thousands of iterations) and the incorporation of experimental factors is difficult. Therefore the method is not suitable for real application [50]. For these reasons, a preferred choice to solve the PR problem is based on performing a non-linear optimisation on the phase distribution in order to find the one that combined with the a priori known amplitude in the exit pupil, will result in the measured intensity distribution. Essentially, we wish to fit our M data points at the coordinate (rx, ry) to a non-linear function described by Eq. 2.2 with a set of N parameters.

2_{Here we use the scalar angular spectrum propagation: G}

z[·] = F−1 n Fh_ei2fkr⊥iF_{[·] e}ikzzo_{/iλf where k} z = q k2 −k2

(26)

2.6 Phase retrieval by non-linear optimisation 17 Because of the non-linear character of the involved equations, all minimisation techniques are based on improving a trial solution by applying iteratively an update rule to it until a point can be accepted as solution. Among the minimisation techniques, there are several diﬀerent approaches that can be adopted to obtain the optimal solution. An exhaustive description of them can be found in [51]. In this thesis we follow a speciﬁc class of non-linear minimisation algorithm called "trust-region method" [52] which is based on a simple yet powerful concept of optimisation. The idea behind it will be introduced in the next section.

2.6.1 Trust-region algorithm for non-linear optimisation

In order to understand the trust-region approach to optimisation, we consider the unconstrained problem of minimising a merit function which express, in our case, the mismatch between the measured intensity distribution and the computed one in the focal region of an optical system. The merit function takes a vector argument (i.e., the exit pupil phase distribution) and return a scalar that must be minimised. While many merit functions can be used here, the most suitable is (in case the noise of the system is characterised by a Gaussian distribution) the dimensionless quantity χ2 [53] deﬁned as the sum of the square diﬀerences between the calculated I(rx, ry; Φ)

and measured Imeas intensities on the detector, which is aﬀected by the noise uncertainties with

variance σ2_: 3 χ2(Φ) =X r⊥ Imeas− I(rx, ry; Φ) σ 2 . (2.3)

The basic idea of the trust-region approach is to approximate the non-linear optimisation problem by a quadratic approximation. When χ2 _{is twice continuously diﬀerentiable, Taylor’s}

theorem gives: χ2_{(Φ + ∆Φ) ∼ ˜}χ2(∆Φ) = χ2(Φ0) + ∆ΦT∇χ2(Φ0) + 1 2∆Φ T_H(Φ 0)∆Φ, (2.4)

where ∆Φ = (Φ − Φ0) describes the size and the direction of the step towards the estimate

solution, ∇χ2_(Φ

0) is the gradient of χ2 at the current estimate Φ0 and H(Φ0) is the second

order partial derivative (Hessian) matrix of χ2 _{in Φ}

0. From this approximated model, at each

iteration k we seek a solution ∆Φk of the approximated problem Eq. 2.4 in a subdomain called

"trust-region": min ∆Φ∈Rn n χ2(Φ0) + ∆ΦT∇χ2(Φ0) + 1 2∆Φ T_H(Φ 0)∆Φ o subject to _k∆Φk2 ≤ ∆k (2.5)

where k · k is a 2-norm. The approximated model is only "trusted" in a region near the current iteration. This seems reasonable because, for a general non-linear function, the model can only ﬁt the original function locally, therefore it is called the trust-region. This sub-domain is normally centered at the current estimate and its dimension is adjusted from iteration to iteration. In general, if the computation indicates that the approximate model ﬁts the original problem well, the trust-region ∆kcan be enlarged, otherwise ∆kwill be reduced. In particular, once a step ∆Φ

is chosen, the function is evaluated at the new point, and the actual function value is checked against the value predicted by the quadratic model. What is actually computed is the ratio of actual to predicted reduction ρk:

ρk= χ2_(Φ 0) − χ2(Φ0+ ∆Φ) ˜ χ2_{(0) − ˜}_χ2_(∆Φ) = actual reduction of χ2

predicted model reduction of χ2 (2.6)

(27)

18 EUV optical testing state-of-the-art and phase retrieval If ρk is close to 1, then the quadratic model is quite a good predictor and the region can be

increased in size. On the other hand, if ρk is too small, the region is decreased in size. When ρk

is below a threshold, η, the step is rejected and recomputed. For further details on the size of ∆k, we refer to [54, 55].

The trust-region algorithm can be related to the well-known Levenberg-Marquardt method in which the step ∆Φ at the k iteration is given by:

∆Φ = −(H(Φ0) + λI)−1∇χ2(Φ0), (2.7)

where λ is a regularisation parameter, updated at each iteration, which prevents k∆Φk2becoming

too large [56]. It can be shown that Eq. 2.7 is also a solution of the following problem min

∆Φ∈Rnkχ

2_(Φ

0) + ∇χ2(Φ0)k22 (2.8)

subject to _k∆Φk2≤ ∆k (2.9)

Hence, the Levenberg-Marquardt method can be regarded as a trust-region method. The main diﬀerence between these two methods is that the region method updates the trust-region radius ∆k directly, whereas the Levenberg-Marquardt method updates the parameter λ,

which in turn modiﬁes the value of ∆k implicitly.

Thus, the trust-region method forms a respected class of algorithms for solving minimisation problems. It is characterised by strong convergence properties, and recently development of reliable and eﬃcient software is available. An implementation of this algorithm can be found in the Matlab®_{optimisation toolbox function lsqnonlin [57].}

2.6.2 Parametrisation of the phase distribution

The minimisation algorithm we have described above minimises the merit function with respect to a set of phase value sampled at discrete point at the exit pupil coordinate (ξ, η). This can lead to a set of optimisation parameters that may be more than 104 or 105 over which we are optimising. A more useful choice is to parametrise the phase in terms of coeﬃcients of a modal basis set. The Zernike polynomials [7] form an orthonormal basis on a disk and thus make a convenient basis for the expansion of the phase at the exit pupil coordinates due to the fact that many optical systems are characterised by a circular aperture. The expansion of the phase Φ on this basis reads:

Φ(ξ, η; αn) = N

X

n=1

αnZn(ξ, η), (2.10)

where Zn(ξ, η)is the nth Zernike polynomial (assuming a single index numbering scheme4) with

coeﬃcient αn. In this case we optimise with the coeﬃcient as variables (normally up the 36),

making the problem less computationally demanding. We write α = {αn}Nn=1, then Eq. 2.3 and

Eq. 2.5 can be rewritten as follows: χ2(α) =X r⊥ Imeas− I(rx, ry; α) σ 2 , (2.11) min ∆α∈Rn n χ2(α0) + ∆αT∇χ2(α0) +1 2∆α T_H(α 0)∆α o subject to k∆αk2≤ ∆k (2.12)

4_{also common is a double index scheme that reads Φ(ξ, η; α}m n) =PNn

Pn m=0α

m

nZmn(ξ, η). See Appendix B for

(28)

2.6 Phase retrieval by non-linear optimisation 19 2.6.3 Uncertainties in the optimisation parameters

Assuming that our mathematical model describes the experiment satisfactory, we need to assess whether or not the final solution of Eq. 2.12 represents the best estimate. Moreover, since our data set are affected by unavoidable noise, it is also interesting to evaluate what are the uncertainties associated in the solution of the optimisation problem Eq. 2.12. This analysis will provide us with the information about the sensitivity of our model to the fit parameters, or in other words, if the optimal solution is stable enough to be accepted as a reliable solution of the problem.

The uncertainties on the best ﬁt parameters can be obtained by investigating the shape of the merit function in the proximity of the minimum value. Let us consider for simplicity a two-parameter function (say a and b). In this case, Eq. 2.11 will be a two-dimensional surface (Fig. 2.5). Let (A, B) in the (a −b) space be point where the merit function χ2_{(a, b)}_{is minimum.}

Figure 2.5: Contour of a general 2 variables χ2

function. The minimum value shown by a red dot is at the coordinates (A,B). The figure is a reproduction from [53].

The shape of the contour of constant χ2 in the proximity of the minimum value deﬁnes how sensitive the optimisation problem is to a variation of the optimisation variables. A high density of contours along a given parameter axis indicates high sensitivity to that parameter. Conversely a sparse contour density shows that the ﬁt is insensitive to that parameter. In general χ2 _{is a}

function of N parameters, therefore it will be described by a hyper-surface. Extending this way of thinking, the shape of χ2_(α) _{function with N parameters in the proximity of the minimum}

α0 can be evaluated by performing a Taylor expansion up to the second order (see Eq. 2.4). At

the location of the minimum ∇χ2(α0) = 0, therefore close to a minimum, χ2 is expected to have

a quadratic behaviour which is described by: χ2_{(α + ∆α) ∼ χ}2(α0) +

1 2∆α

T_H(α

0)∆α. (2.13)

The quantity deﬁned by one-half of the Hessian matrix is deﬁned as curvature matrix Aij [58]:

Aij = 1 2H = 1 2 ∂χ2 ∂αi∂αj with i, j = 1 . . . N, (2.14)

and it describes the local curvature of a function of many variables. Its inverse is called the covariance matrix Cij:

(29)

20 EUV optical testing state-of-the-art and phase retrieval It can be shown that, under the assumption of normally distributed uncertainties [59], a relation between the elements of the covariance matrix and the statistical errors of the best ﬁt parameters can be established. The uncertainties (i.e., the standard deviations) σn associated to the best ﬁt

parameters αn are obtained from the elements on the diagonal of the covariance matrix:

σn=

p

Cnn with n = 1 . . . N. (2.16)

Thus, we see that a large (small) curvature leads to a small (large) standard deviation of the parameters αn. Another important result is given by the analysis of the oﬀ-diagonal terms. These

terms are related to the degree of correlation between the uncertainties of the parameters. The geometrical interpretation of this correlation is that the parameters αnare not in general in the

direction of the eigenvectors of the Hessian matrix. It is convenient to quantify the correlation of the variables αi and αj by the following dimensionless quantity:

ρij =

Cij

CiiCjj

, (2.17)

which has the well-known property

−1 ≤ ρij ≤ +1. (2.18)

Therefore, if two variables are uncorrelated, then ρij ≃ 0. On the other hand, if ρij ≃ ±1

the variables are highly correlated. A strong correlation of the uncertainties means that the two parameters are not independently resolved by the data set and that only some linear combinations of the parameters are resolved.

In conclusion, we have seen that an analysis of the elements of the covariance matrix is crucial to check the sensitivity of the phase retrieval with respect to the optimisation parameters. Large uncertainties and correlation coeﬃcients close to ±1 reveal that the system is highly sensitive to the variation of the optimisation parameters making the retrieval unpredictable. On the contrary, small uncertainties and correlation coeﬃcients close to 0 indicate that a good and reliable estimation of the unknown parameters can be obtained.

2.6.4 Analytical expression for the merit function Hessian matrix

In the previous section we have seen that both the optimisation algorithm and the sensitivity analysis rely on computing and inverting the (exact) Hessian matrix, which contains the second-order derivatives of the χ2_(α

n) surface for any αn. However, computing this matrix may be

a tough task, especially because of its computation time cost and/or because of its memory requirements. In order to built an eﬃcient algorithm, it is common rule to implement in the optimisation a simpler form that approximates the Hessian [60] which derivation is recalled below. We start by diﬀerentiating χ2_(α)_{with respect to the parameters α}

n: ∂χ2 ∂αn = 1 σ2 ∂ ∂αn X r⊥ [I(r⊥; α) − Imeas]2 (2.19) = 2 σ2 X r⊥ ∂ [I(r⊥; α) − Imeas] ∂αn [I(r⊥; α) − Imeas] (2.20)

(30)

2.7 Conclusion 21 Consequently the Hessian matrix is given by:

∂χ2 ∂αn∂αm = 1 σ2 ∂ ∂αn∂αm X r_⊥ [I(r⊥; α) − Imeas]2 (2.21) = 2 σ2 X r⊥ ∂ [I(r⊥; α) − Imeas] ∂αn ∂ [I(r⊥; α) − Imeas] ∂αm + (2.22) + 2 σ2 X r_⊥ [I(r⊥; α) − Imeas]∂ 2_[I(r ⊥; α) − Imeas] ∂αn∂αm . (2.23)

Therefore we see that the Hessian contains two summations. The ﬁrst over the product of the gradient while the second over the product of second order derivatives and the residual [I(r⊥; α) − Imeas]. Typically, we can assume that this residual is suﬃciently close to the

mini-mum to neglect the second term of the Hessian compared to the first term. Hence, by dropping the second term we can approximate the Hessian, for the benefit of efficiency and simplicity in the optimisation, as the product of gradients:

∂2χ2 ∂αn∂αn ≃ 2 σ2 X r⊥ ∂ [I(r⊥; α) − Imeas] ∂αn ∂ [I(r⊥; α) − Imeas] ∂αm . (2.24)

However, the sensitivity analysis can be performed with an absolute accuracy using an ana-lytical expression for the Hessian that is developed in the Appendix D. The anaana-lytical expression reads:

∂2χ2

∂αn∂αm =Re {Gz

[ikZn(ξ)A(ξ) exp(ikΦ)] Gz[A(ξ) exp(ikΦ)]∗} ×

×_σ8₂Re {Gz[ikZm(ξ)A(ξ) exp(ikΦ)] Gz[A(ξ) exp(ikΦ)]∗} +

+[Imeas− I(r⊥; α)]Re

Gz[A(ξ) exp (ikΦ)]∗Gz

−k2Zn(ξ)Zm(ξ)A(ξ) exp (ikΦ)

+ + Gz[ikZnA(ξ) exp (ikΦ)] Gz[ikZmA(ξ) exp (ikΦ)]∗} . (2.25)

2.7 Conclusion

In this chapter, we have discussed the importance of wavefront metrology to assess the quality of, an arbitrary optical system. In particular, concerning a EUV lithography system, we have brieﬂy described the pros and cons of the state-of-the-art of the wavefront measurement techniques that have been developed in the past decades. Particular attention has been given to the Hartmann wavefront sensor technique as a possible candidate for wavefront metrology in the EUV system. Moreover, we have introduced the concept of phase retrieval which will be used throughout this thesis. The optimisation algorithm that was used in the phase retrieval has been described and the uncertainties of the retrieved parameters are investigated. This latter point will be important for assessing the accuracy of the retrieved parameters. An analytical expression for the Hessian matrix has been also given in this chapter that will constitute the basis for computing the correct uncertainties of the optimisation result.

(31)

(32)

Chapter 3

Phase retrieval applied to a Hartmann

Wavefront Sensor

"Prediction is very difficult, especially if it’s about the future."

Niels Bohr (1885 - 1962)

A conventional Hartmann Wavefront Sensor (HWS) is proposed as a possible wavefront mea-surement technique for the EUVL system. The wavefront in the exit pupil of an optical system is determined from the wavefront slope measured at the position of the mask’s sub-apertures. However, the conventional implementation leads to rather low accuracy in the reconstructed aberrations. To overcome this, a non-linear phase retrieval algorithm is applied to the mea-sured intensity pattern from the HWS. We show that the RMS wavefront error achieved with this method is one order of magnitude smaller than the one obtained with the conventional implementation. Experimental results are consistent with phase measurements performed inde-pendently using a Shack-Hartmann wavefront sensor.1

3.1 Introduction

In the previous chapter we have presented an overview of several possible methods for EUVL wavefront metrology. In particular, we have focused our attention on the Hartmann wavefront sensor emphasising its advantages over interferometry techniques. Those advantages pushed us to investigate its performances and, eventually, improve it by implementing a diﬀerent method to reconstruct the phase information from the HWS’s intensity measurement. In this chapter we show that, by applying a phase retrieval algorithm (as described in Section 2.5) the performance of a HWS can be enhanced in order to meet the requirements imposed by EUVL systems.

This chapter is divided into ﬁve parts. In Section 3.2, a brief description of a HWS is presented explaining the limitations of the device. In Section 3.3, the mathematical description of the phase retrieval algorithm is presented. In Section 3.4, a numerical demonstration of phase measurements by the phase retrieval procedure is shown. In Section 3.5 , the experimental setup is discussed and the phase measurements are presented. Finally, in Section 3.6, conclusions are given.

1_{The contents of this chapter are based on Ref. [61]: A. Polo, V. Kutchoukov, F. Bociort, S. F. Pereira, and}

H.P. Urbach, "Determination of wavefront structure for a Hartmann Wavefront Sensor using a phase-retrieval method" Opt. Exp. Vol. 20, No. 7 (2012) and on Ref. [62]: A. Polo, N. van Marrewijk, S.F. Pereira and H.P. Urbach, "Sub-aperture phase reconstruction from a Hartmann Wavefront Sensor by phase retrieval method for application in EUV adaptive optics" Proc. of SPIE Vol. 8322 832219-1 (2012)

(33)

24 Phase retrieval applied to a Hartmann Wavefront Sensor

3.2 Hartmann wavefront sensor and its limits

The Hartmann Wavefront Sensor (Fig. 3.1(a)) has been widely used for several decades for beam characterisation since it provides intensity and phase information in real time [63]. Its eﬀective and simple hardware consists of an opaque screen containing an array of holes placed in the beam path. The beam is sampled by the hole array and the transmitted beamlets are collected by a detector placed at a certain distance from the screen. Aberrations in the beam cause a measurable shift of the beamlets at the detector plane as compared to the aberration-free case. By measuring the shifts, the local slope of the wavefront at the sampling points (i.e., the hole array) can be determined. In this way, a discrete map of the wavefront gradient components in the x and y directions is given by the linear relation:

∂Φ ∂x(xi, yj) ≃ ∆xi,j L ∂Φ ∂y(xi, yj) ≃ ∆yi,j L , (3.1)

where L is the distance between the sampling and detection planes, Φ is the wavefront function and ∆xi,j and ∆yi,j are the measured shifts along the x and y directions of the spot associated

with the sub-aperture (i, j) in the hole array. The displacement of the beamlets is usually calculated in a fast and eﬃcient manner by a centroid algorithm [64, 65]. From the obtained gradient map, the original wavefront can be recovered by solving a least-squared problem [32]. Consequently, the accuracy on the reconstructed wavefront performed by this method is directly related to the smallest beamlet deviation that can be distinguished on the detector.

(a) (b)

Figure 3.1: (a) graphical representation of a HWS. (b) sampled wavefront approximated by a flat wavefront in the wavefront measurement with a HWS

In the characterisation of syncrotron beams with λEU V = 13.5 nm, it has already been

demonstrated that Hartmann sensors can reach a repeatability of λ/120 [33–36]. Nevertheless, approximations are made with this approach. The first approximation is replacing the phase distribution at the sampling points by a tilted plane wave with an average x and y tilt component (Fig. 3.1(b)). If extraneous aberrations components rather than tilt are present within the sub-aperture, the relation between the centroid intensity and the local slope of the wavefront becomes non-linear. This issue is more critical as the distance L increases, affecting consequently the sensitivity range on which the HWS can be operated. Hence, the inability of the centroid algorithm to estimate the non-linear function from the intensity spot distribution leads to a loss of information about the local curvature inside the sub-aperture of the Hartmann screen. Secondly, Eq.(3.1) is only valid under the assumption of small deflection angles (Fig. 3.1(a)), i.e.

Adaptive Optics for EUV Lithography: Phase Retrieval for Wavefront Metrology