Acoustic excitation due to large amplitude oscillating bodies

Transcription

1 Acoustic excitation due to large amplitude oscillating bodies T. Rutjes B.Sc. DCT (b) Masters Thesis Committee: prof.dr. H. Nijmeijer (b) (Supervisor) prof.dr.ir. N.B. Roozen (a,b) (Coach) prof.dr.ir. A. Hirschberg (b) dr.ir. M. van der Wijst (c) (a) Philips Applied Technologies Mechatronics Department Applied System Analysis Group (b) Eindhoven University of Technology (TU/e) Department of Mechanical Engineering Division: Dynamical Systems Design Group: Dynamics and Control (c) ASML Eindhoven, March 28, 28

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5 Preface This master thesis is the result of my graduation project at Eindhoven University of Technology, conducted at Philips Applied Technologies. I would like to use this opportunity to thank several people for supporting me. To prof.dr.ir. N.B. Roozen, my direct coach, for his patience and constructive criticism. To prof.dr. H. Nijmeijer, my supervisor, for giving me the opportunity to work on this challenging and interesting project. Last but not least, i would like to thank my parents and girlfriend for giving me the thoroughness to succeed in finalizing this study.

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7 Abstract High-precision is essential for the ongoing trend in miniaturization of manufacturing and leading-edge research. The disturbances that affect the accuracy of high-precision machinery are of key interest in the development of high-precision machinery. To mention a few: floor vibrations, internally generated vibrations, electromagnetic interference and acoustic excitation due to flow and/or clean-room air-conditioning systems, etc. With the accuracy of these high-precision machinery entering the nanometer-range, acoustic degradation of machine performance has become a real issue. In this work, acoustically induced vibrations due to large amplitude oscillations are studied. In particular, the acoustic excitation problem of wafer steppers is investigated. Wafer steppers are an essential part of the complex process, called photolithography, that creates millions of microscopic circuit elements on the surface of tiny chips of silicon. The high speed movements of the wafer stage and the reticle stage, which are two assemblies of the stepper, generate sound waves that acoustically excite the reduction lens, which is also part of the stepper. It may be clear that these disturbances need to be reduced for the ongoing trend in miniaturization of manufacturing. In the literature of acoustics, source motions are usually considered to have small amplitude, with the result that the radiated acoustic field is linearly related to motion. However, a large amplitude motion of the source introduces an important non-linear relationship between the kinematic motion of the source and the radiated acoustic field. The acoustic excitation problem of the wafer stepper, which is considered in this study, is a typical industrial problem of acoustic excitation due to large amplitude oscillations. In order to evaluate various concepts of high-precision machinery with respect to external disturbances a numerical model that predicts the response of a system to acoustic excitation is indispensable. Therefore, such a model has been developed using COMSOL Multiphysics and MATLAB. The model can roughly be divided in three segments, namely: acoustics, structural-mechanics and mesh movement. The acoustic segment (wave equation) calculates the sound pressure field generated by the movement of a body and the structural-mechanics segment calculates the structural response of an external structure caused by this sound pressure field. The non-linear effects caused by the large amplitude oscillations are taken into account by implementing a moving mesh (the mesh of the acoustic field moves/deforms in time according to the movement of the moving body). In this way, linear acoustics can still be used to model the problem. For validating purposes of the developed model an experiment has been set up and measurements have been carried out on a reticle stage of ASML. By making use of a microphone and two lasers, the sound pressure, the velocity of the stage and the velocity of a polystyrene ball (which represents the external structure) have been measured. The measured structural response of the external structure and the sound pressure mea-

8 iv surement are compared with the predicted structural response and sound pressure of the numerical model. The peaks in the autospectra of the sound pressure measurements correspond with the model prediction (the peaks that are not predicted by the model correspond with the sound pressure measurements of the background noise). Furthermore, the results of the experiment and simulations show that the model that takes into account the non-linear effects caused by large amplitude oscillations gives a good prediction of the structural response and sound pressure, while the linear model does not. The developed numerical model can be used in the design phase of for example wafer steppers to investigate how to reduce acoustic excitation by simulating different mechanical designs and different setpoints of the stage. A few recommendations are given for future research. For better accuracy of the result, the model could be improved by extending the model to 3D and by improving the remesh algorithm so that it can be applied in the model. Another recommendation is to use the developed numerical model to investigate the influence of the cavity walls as in most industrial applications the cavity walls are close to the acoustic excitation problem.

9 Contents Preface Abstract i ii 1 Introduction Motivation Acoustic excitation problem of wafer stepper Aims and scope Contents of this report Literature study; acoustically induced vibrations External disturbances wafer stepper Sound waves Interaction sound waves and solid structures The principle of reciprocity Numerical analyses Non-linear behavior large amplitudes Frequency response identification of non-linear systems Frequency response function Higher Order Sinusoidal Input Describing functions Volterra series Discussion Model description and discussion Physics model description Uncoupled model Acoustics Moving mesh Simulations moving mesh Smart boundary conditions Remesh algorithm Coupled model Acoustics Moving mesh External structure and coupling Time-dependent solver

10 vi CONTENTS 4 Experiment description and results Experimental setup Pressure microphone and Nexus amplifier Scanning laser and single head laser vibrometer SigLab Test experiment (anechoic room) ASML Testrig Location polystyrene ball and microphone Setpoint stage Background measurements Experiment results and observations Higher order sinusoidal input describing functions Simulation data and comparison experimental results Description generation simulation data D representation Simulation data compared with experimental results Frequency contribution Magnitude Force calculation Discussion Conclusions and Recommendations Conclusions Recommendations Appendix A Remesh algorithm 51 Appendix B Charger experiment 59 B.1 Setpoint stage B.2 Background measurements B.3 Experiment results and observations B.4 Simulation data and comparison experimental results B.5 Discussion List of figures 64 Bibliography 66

11 Chapter 1 Introduction Modern technology to a great extent relies on the use of high-precision machinery. On the one hand, high precision is essential for the ongoing trend in miniaturization of manufacturing, a striking example of which is found in the industry of photolithography; integrated circuits patterns of ever decreasing size. On the other hand, high precision is essential for leadingedge research. Modern microscopy techniques for instance enable analysis of materials in ever decreasing detail, even down to the level of individual atoms. 1.1 Motivation The disturbances that affect the accuracy of high-precision machinery are of key interest in the development of high-precision machinery. To mention a few: floor vibrations, internally generated vibrations, electromagnetic interference and acoustic excitation due to flow and/or clean-room air-conditioning systems, etc. With the accuracy of these high-precision machinery entering the nanometer-range, acoustic degradation of machine performance has become a real issue. Two people speaking in close proximity of an electron microscope can for example significantly disturb the quality of the image of the electron microscope. Another example of high-precision machinery that is subject to acoustic excitation is a (wafer) stepper. Steppers are an essential part of the complex process, called photolithography, that creates millions of microscopic circuit elements on the surface of tiny chips of silicon. The high speed movements of the wafer stage and the reticle stage, which are two subassemblies of the stepper, generate sound waves that acoustically excite the reduction lens, which is also part of the stepper. It may be clear that these disturbances need to be reduced for the ongoing trend in miniaturization of manufacturing. The next section describes the acoustic excitation problem of the wafer stepper in more detail. In order to evaluate various concepts of high-precision machinery with respect to external disturbances a numerical model that predicts the response of a system to acoustic excitation is indispensable. 1.2 Acoustic excitation problem of wafer stepper In figure 1.1, a schematic overview of the the stepper process is illustrated. The stepper passes light through the reticle, forming an image on the reticle pattern. The image is focused and

12 2 1.2 Acoustic excitation problem of wafer stepper reduced by a lens and projected onto the surface of a silicon wafer. When the wafer is processed in the stepper, the pattern on the reticle (which may contain a number of individual chip patterns) is exposed repeatedly across the surface on the wafer in a grid. Currently, the most detailed patterns in semiconductor device fabrication are transferred using a type of stepper called a scanner, which moves the wafer stage and reticle stage with respect to each other during exposure, as a way of increasing the size of the exposed area. light source Fixed exposure slit Reticle stage (RS) Direction of movement Reduction projection lens Exposure field on wafer Direction of movement Wafer stage (WS) Figure 1.1: Stepper process. The greatest limitation on the ability to produce increasingly finer lines on the surface of the wafer has been the wavelength of the light used in the exposure system. As the required lines have become narrower, illumination sources producing light with progressively shorter wavelengths have been put into service in steppers and scanners. The ability of an exposure system, such as a stepper, to resolve narrow lines is limited by the wavelength of the light used for illumination, the ability of the lens to capture the light coming at increasingly wider angles and various improvements of the process itself. This is expressed by the following equation: CD = k λ NA, (1.1) where CD is the critical dimension, or finest resolvable, k is a coefficient expressing processrelated factors, λ is the wavelength of the light, and NA is the numerical aperture. Lines as low as 65 nm are being resolved by steppers using argon-fluoride (ArF) lasers that emit light with a wavelength of 193 nm. Although fluoride (F2) lasers are available that produce 157 nm light, they are not practical because of their low power and because they quickly degrade materials used to make the lenses in the stepper. Since practical light sources with wavelengths narrower than these lasers have not been available, manufacturers seek to

13 Chapter 1: Introduction 3 improve resolution by reducing the process coefficient k. In figure 1.2, the most important assemblies of the stepper are shown. It consists of a metroframe on which the production lens is mounted, and which is fully de-coupled from the base frame holding the stage actuators (by means of airmounts). To further suppress vibrations resulting from stage movements, a balance mass principle is adopted. This principle uses an opposite movement of the balance mass to cancel out the acceleration forces caused by the high stage mass (both reticle and wafer stage). Reticle RS balance mass SS Motor RS chuck Lens Metroframe Airmounts Wafer WS chuck Airmounts SS Motor WS balance mass Baseframe Figure 1.2: Stepper. Besides the demands on accuracy, also the demands on production, i.e. the amount of wafers per hour, increase. Therefore, high specs are required for the moving stage(s). The movement of the reticle stage and wafer stage generate sound waves which acoustically excite the reduction lens. The reticle stage needs to displace more (with higher accelerations) than the wafer stage because of lens magnification. Consequently, the reticle stage has more influence on the acoustically induced vibrations of the reduction lens. The large displacements of the reticle stage complicate the acoustic excitation problem, because the problem can no longer be modeled using just linear acoustics. 1.3 Aims and scope The goal of this study is to obtain insight in the phenomenon of acoustic excitation and to create a numerical model that can be used in the design phase of high-precision machinery to predict the response of the system to acoustic excitation. The non-linear effects caused by large amplitude oscillations are of particular interest. The goal is to investigate to what extent linear acoustics can be used to model these non-linear effects and what way of modeling is

14 4 1.4 Contents of this report most appropriate. Another goal is to set up an experiment and carry out measurements that can be used to validate the numerical model. In order to achieve above mentioned goals a literature study has been carried out in the area of acoustically induced vibrations due to large amplitude oscillations. Also, the program that is found most suitable to create the model, COMSOL Multiphysics, has been studied in dept. A numerical model has been developed for uncoupled simulations (no external structure) and coupled simulations. By making use of a moving mesh, the non-linear effects that are caused by the large amplitude oscillations of for example the reticle stage of a stepper, can be modeled using linear acoustics. The model is verified by means of an experiment. 1.4 Contents of this report The next chapter presents the results of the literature study about what is known in the area of acoustic excitation of structures. It includes some basic information about the propagation of sound and the principle of reciprocity. The principle of reciprocity is important because it relates the acoustically induced structural response to sound radiation. The following chapter, chapter 3, presents the numerical models that are created using COMSOL Multiphysics and Matlab. Both the uncoupled model (without external structure) and coupled model are presented and explained. Chapter 4 is used to present the experimental setup and the measurements that are carried out. In chapter 5, the results of the simulations and the experiment are discussed and compared. Finally, chapter 6 provides the conclusions and recommendations.

15 Chapter 2 Literature study; acoustically induced vibrations This chapter is the result of a literature study and aims to describe what is known in the area of acoustically induced vibrations. Shortly, some fundamentals of sound waves are reviewed (for a complete overview see [1]). The interaction between sound waves and solid structures is discussed and special attention is paid to the principle of reciprocity. Hereafter, the part of the literature study about what is known about the numerical analysis of acoustically induced vibrations is presented and discussed. Then, the non-linear behavior of large amplitude oscillations is discussed and a new concept for frequency based analysis of non-linear systems is presented. Finally, the relevance of the literature study to this project is discussed. 2.1 External disturbances wafer stepper The disturbances that affect the accuracy of the stepper process are floor vibrations, internally generated vibrations and acoustic excitation. Vibration problems can often be relieved by means of adequate isolation of the equipment from the floor, through which most of the disturbing vibrations enter. In this respect, the design principle concerning vibration isolation, the metroframe (on which the lens is mounted) is resiliently isolated from the floor, both passively and actively, by means of so-called airmounts. However, once the equipment is sufficiently isolated from floor vibrations, another disturbance source becomes dominant: acoustics. It is practically impossible to come up with isolation means for acoustic vibrations, due to for instance the design principle concerning thermal management, which requires a well-conditioned airflow within the machine. The incoming vibrations can not be further reduced, and as a consequence, the mechanical structure needs to be changed, such that it becomes less sensitive to these vibrations [2]. 2.2 Sound waves Acoustics is the physics of sound. Sound is the sensation, as detected by the ear, of very small rapid changes in the air pressure above and below a static value. Sound is a pressure wave that propagates through an elastic medium at some characteristic speed. For this wave motion to exist, the medium has to possess inertia and elasticity. Whilst vibration relates to

16 6 2.3 Interaction sound waves and solid structures such wave motion in structural elements, noise relates to such wave motion in fluids (gases and liquids). There are four variables that are of direct relevance for the study of sound waves: pressure(p), velocity( U), density(ρ) and temperature(t). The wave equation can be set up in terms of any of these four variables. In acoustics, the pressure fluctuations, p( x, t), are of primary concern. Therefore, it is common for acousticians to solve the wave equation in terms of the pressure as a dependent variable. The linearized, homogeneous acoustic wave equation follows from conservation of mass, conservation of momentum and the thermodynamic equation of state. In terms of the pressure as dependent variable it states, 2 p = 1 c 2 2 p t 2, (2.1) where is the divergence operator and c is the constant velocity propagation of the wave (the speed of sound). 2.3 Interaction sound waves and solid structures The first part of the described problem is the radiation of sound from a vibrating body (in case of the acoustic excitation problem of the stepper, the reticle stage). At its most fundamental level, the radiation of sound from an arbitrary vibrating body can be formulated in terms of an integral equation involving Green s function (Green s functions present solutions to the wave equation). The integral equation is attributable to Kirchhoff, although Helmholtz modified it for single frequency (harmonic) applications. The integral equation, sometimes referred to as the Kirchhoff-Helmholtz integral equation, relates harmonic surface vibrational motion on any arbitrary body to the radiated sound pressure field in the surrounding fluid. It is, { p( r) = p( r ) G } ω( r,ω r,ω) + iωρ u n ( r )G ω ( r,ω r,ω) ds, n (2.2) S where r is a position vector at some receiver point in the sound field, r is a position vector on the vibrating body, n is the unit normal vector, p( r ) is the surface pressure on the body and iω u n ( r ) is the normal surface acceleration, see figure 2.1. G ω is the frequency domain Green s function - it is a solution to the wave equation for a harmonic source. It should be noted that the acoustic pressure fluctuations are a function of both space and time, thus p( r,t) = p( r)e iωt (in (2.2) both the surface pressure, p( r ), and the normal surface acceleration, iω u n ( r ), are a function of time). The integral equation can thus be interpreted as representing the radiating sound pressure field of a vibrating body by a distribution of point sources and forces on the surface of the body. Generally, no analytical solutions are possible for complicated three-dimensional bodies. The usual procedure is either to use numerical techniques to solve the integral equation, or to use experimental techniques to establish the Green s function. The Green s function is symmetrical, i.e. the source and receiver positions can be interchanged. This property is commonly referred to as the principle of reciprocity. 2.4 The principle of reciprocity Loosely speaking, the principle of reciprocity states that the response of a linear system to a disturbance, which is applied at some point by an external agent, is equal in magnitude if the points of input and observed response are reversed [3].

17 Chapter 2: Literature study; acoustically induced vibrations 7 p(r ) r r p( r ) ω n i u ( r ) Figure 2.1: Visualization of variables Kirchhoff-Helmholtz equation. For acoustically induced vibrations, the reciprocity relationship, formulated by Lyamshev et al. [4], states that the ratio of applied force to sound pressure is equal in magnitude and opposite in sign to the ratio of the source strength to the acoustically induced structural velocity in the force direction. In mathematical terms, F bl ( r s ) p( r ) = Q( r ) v( r s ), (2.3) where the left part is related to the sound radiation problem, Fbl ( r s ) being the force at position r s on the structure and p( r ) being the resulting pressure due to this force at some point r in space. The right part is related to the sound irradiation problem, Q( r ) being the source strength of the point source at that same point r in space and v( r s ) the resulting structural velocity due to this point source at position r s on the structure. See figure 2.2 for an illustration. This relationship is of great practical value because it obviates the need to simulate vibrational forces in operating systems, for the purpose of determining the resulting sound, by replacing direct measurement with an observation of the vibrational response to a point source placed at the sound field observation point. 2.5 Numerical analyses The majority of systems of practical engineering interest exhibit such complexity of geometry and structural detail that their vibrational and acoustic behavior cannot be accurately predicted by analytical mathematical methods. Some noise sources that are of concern to engineers (e.g. vehicles, construction equipment, industrial machinery, appliances, flow-duct systems, etc.) can, however, be modeled in terms of simple sources such as spheres, pistons in an infinite baffle, cylinders or combinations thereof (monopoles, dipoles and quadrupoles). The reader is referred to [1] for an overview of these source models. Modern methods for

18 8 2.5 Numerical analyses p( r ) Q( r ) r r r s v( r s ) r s F r ) bl ( s Figure 2.2: Lyamshev principle of reciprocity. numerical analyses are available; these are either finite element methods or boundary element methods or combinations of thereof [5]. The boundary element method (BEM) is a numerical technique for calculating the sound radiated by a vibrating body or for predicting the sound field inside of a cavity such as a vehicle interior. The BEM is becoming a popular numerical technique for acoustical modeling in industry [6]. Keunen [7] developed a procedure to calculate sound induced vibrations, which is based on the reciprocity relationship and requires use of a FEM package and a BEM package. The procedure works as follows. A force is applied at some point on the geometry that is put into a FEM package. The FEM analysis consists of a modal analysis and a harmonic response analysis using modal superposition. These analyses result in a frequency dependent nodal velocity field, which is put into a BEM package, in which the geometry is also modeled. The BEM package then calculates the radiated sound power. Using the reciprocity relationship a transfer function is then calculated which can be used to determine an acceleration at that specific point on the geometry due to a certain pressure. For further explanation, see [7]. A drawback of this procedure is that the calculations, which are very time consuming, have to be carried out for each point of interest. Also, it is not very convenient that two packages are required. Many papers/books have been published on the use of FEM and BEM and the use of modal analysis in the analysis of acoustically induced vibrations, see for example: Moosrainer [8], Coyette [9] and Bokil [1]. Very little to nothing, however, is published on what to do with large amplitude oscillations. The acoustic excitation problem of the wafer stepper as depicted in the introduction is a typical example of an industrial problem of acoustic excitation with large amplitude oscillations. In the next section, the non-linear behavior caused by the large amplitudes oscillations is discussed.

19 Chapter 2: Literature study; acoustically induced vibrations Non-linear behavior large amplitudes In the literature of acoustics, source motions are usually considered to have small amplitude, with the result that the radiated acoustic field is linearly related to motion. Thus, if the motion is sinusoidal with frequency f, the radiated field (as described by an ordinary wave equation) has a temporal variation that is sinusoidal at frequency f. However, a large amplitude motion of the source introduces an important non-linear relationship between the kinematic motion of the source and the radiated acoustic field. In general, when the source motion is sinusoidal and large in amplitude, a harmonically rich radiated field will result (harmonic distortion). Large amplitude refers to the fact that the motion cannot be adequately be specified as a small perturbation from a nominal configuration. Large amplitude motion, in the absence of fluid non-linearities is a source of harmonic distortion. Frost [11] illustrates this concept with two examples involving the harmonic motion of a spherical surface. Two cases of harmonic distortion from a rigid sphere, oscillating sinusoidally with large amplitude and low Mach number are considered. To a given order of approximation, which depends on the body motion, these oscillations produce periodic fields which are solutions of the linear wave equation and which exhibit a rich spectral and spatial structure. In particular, the radiated overtones are ordered in powers of the fundamental wave number appropriate to the harmonic motion of the sound-generating body. 2.7 Frequency response identification of non-linear systems In the analysis and synthesis of dynamic systems, frequency domain-based concepts like the frequency response function (FRF), as described in subsection 2.7.1, presume linear system behavior. Since, in non-linear systems, harmonic distortion leads to response at frequencies that are not in the input signal, the value of the frequency response (FRF) at these frequencies is questionable. The FRF measurement results of non-linear systems always have to be treated with caution because the characteristics of the excitation signal can have significant influence on the measured system characteristics [12]. Other concepts of frequency response identification of non-linear systems are therefore investigated. The Sinusoidal Input Describing Function concept, Gelb [13] and Taylor [14], extends the frequency response function in a way that identification of amplitude dependency becomes possible by replacing the non-linear element with a quasi-linear descriptor which gain is a function of input amplitude. However, it ignores the excitation frequency transformation property of many non-linear systems. Nuij [15] developed an alternative concept for frequency based non-linear systems analysis. This new concept is the generalization of the Sinusoidal Input Describing Function to Higher Order Sinusoidal Input Describing Functions (HOSIDF) as it yields the magnitude and phase relations between individual higher harmonics in the response signal and the sinusoidal excitation signal, both as a function of magnitude and frequency of the excitation signal. This concept is described in subsection Another approach of frequency domain analysis of non-linear systems is based on the Volterra functional series, Bedrosian [16], Chau [17] and Bendat [18]. This approach is described in subsection

20 1 2.7 Frequency response identification of non-linear systems Frequency response function The dynamic behavior of a casual, stable, time-invariant linear system (output does not depend explicitly on time) can be described by its impulse response function h(τ). The output y(t) of the system is related to the input u(t) through the convolution integral, y(t) = h(τ) u(t τ)dτ, u(t) = for t <. (2.4) The Fourier transform of this integral results in the Frequency Response Function H(jω) which describes the system behavior in the frequency domain, Y (jω) = H(jω)U(jω). (2.5) Higher Order Sinusoidal Input Describing functions Consider a stable, non-linear time invariant system (output does not depend explicitly on time), which has a harmonic response to a sinusoidal excitation. Let u(t) = â cos(ω + ϕ) be the input signal. The system response y(t) is considered to consist exclusively of harmonics of the fundamental frequency ω of the input signal u(t), i.e. the transient behavior has vanished. Response y(t) can be written as a summation of harmonics of the input signal u(t), each with an amplitude and phase, which can depend on the amplitude â, phase ϕ and frequency ω of the input signal, y(t) = A n (â,ω)cos (n(ωt + ϕ) + ϕ n (â,ω)). (2.6) n= The describing function H(â,ω) of the system is defined as the complex ratio of the fundamental component ỹ(t) = A 1 (â,ω)cos (ωt+ϕ+ϕ 1 (â,ω)) of the system response and the input sinusoid u(t), H(â,ω) = A 1(â,ω)e j(ωt+ϕ+ϕ 1(â,ω)) âe jωt+ϕ = 1 â (b 1(â,ω) + ja 1 (â,ω)). (2.7) Herein the Fourier components a 1 and b 1 are calculated by, a 1 = 2 T t +T t y(t)cos(ωt)dt, (2.8) b 1 = 2 T t +T t y(t)sin(ωt)dt, (2.9) where T = 2π ω. The describing function H(â,ω) can be interpreted as the first element of a set of higher order describing functions H k (â,ω). These functions can be defined as the complex ratio of the k th harmonic component in the output signal to the virtual k th harmonic signal derived from the excitation signal. Like the first order describing function, the higher order describing functions are calculated from the corresponding Fourier coefficients, H k (â,ω) = A k(â,ω)e j(ϕ k(â,ω)) â = 1 â (b k(â,ω) + ja k (â,ω)). (2.1)

21 Chapter 2: Literature study; acoustically induced vibrations 11 The HOSIDF, H k (â,ω), can be interpreted as a descriptor of the individual harmonic distortion components in the output of a time invariant non-linear system with a harmonic response as a function of the amplitude and frequency of the driving sinusoid. Both the input signal u(t) and output signal y(t) are Fourier transformed. The resulting single sided spectra contain frequency lines each with Hz in frequency line. The frequency spacing is f = 1 T b, with T b the length of the data block. T b is chosen a multiple p times the period T = 2π ω of the excitation signal. This assures that all the power of the excitation signal is fully concentrated in the frequency line p. The power of the response signal is fully concentrated in the frequency lines n p, with n N, so leakage is absent. In the frequency spectrum of u(t) the frequency line p with complex value a p + jb p represents the input signal. The square root of the power in this frequency line is the amplitude, â, â = a 2 p + b 2 p, (2.11) and the phase angle of this frequency line equals ϕ, ( ) bp ϕ = arctan a p if a p, ( ) bp ϕ = arctan a p + π if a p <. (2.12) In the spectrum of the system output y(t), the frequency line with number k p and complex value a pk + jb kp represents the output of the subsystem H k (â,ω). The square root of the power in this frequency line is the amplitude, A k (â,ω) = a 2 kp + b2 kp. (2.13) Using (2.1), (2.11) and (2.13) the magnitude of the k th -order HOSIDF can be calculated as, a 2 kp + b2 kp H k (â,ω) =. (2.14) a 2 p + b 2 p The phase ϕ k (â,ω) of the HOSIDF can be calculated as, ( ) [ ( bp bp ϕ k (â,ω) = arctan a p k arctan a p if a p,a kp )]mod 2π ( ) [ { ( ) }] bp bp = arctan a p k arctan a p + π if a p <,a kp mod 2π ( ) [ ( bp bp = arctan a p + π k arctan a p if a p,a kp < )]mod 2π ( ) [ { ( ) }] bp bp = arctan a p + π k arctan a p + π if a p <,a kp <, mod 2π (2.15) where mod 2π stands for the modulo operation, which finds the remainder of division by 2π.

22 Discussion Volterra series Volterra series have been described as power series with memory which express the output of a non-linear system in powers of the input x(t). A substantial number of the non-linear systems can be represented by Volterra series, y(t) = n=1 1 n! du n g n (u 1,...,u n ) n x(t u r ), (2.16) where y(t) is the output, x(t) the input, and the kernels g n (u 1,...,u n ) describe the system. Note that the first-order kernel, g 1 (u 1 ), is simply the familiar impulse response of a linear system. The higher order kernels can be viewed as higher order impulse responses which serve to characterize the various orders of non-linearity. The n-fold Fourier transform plays an important role in the analysis, G n (f 1,...,f n ) = du 1... du n g n (u 1,...,u n )e j(ω 1u ω nu n), (2.17) where ω i = 2πf i. G is identically zero because the Volterra series starts with n = 1. Also, G 1 (f 1 ) will be recognized as the familiar transfer function of a linear system. Thus the transform of the nth-order Volterra kernel is seen to be analogous to an nth-order transfer function. Suppose that the G n, n = 1,2,..., for a particular system are known. Suppose further that the input x(t) to the system of (2.16) consists of one or more sine waves, gaussian noise, a sine wave plus gaussian noise or a random pulse train. Then one can obtain expressions for a number of items of interest regarding the output y(t) by substituting the G n in formulas derived from the the Volterra series for y(t). r=1 2.8 Discussion In this section, the relevance of the literature study is discussed. The first sections on the fundamentals of sound waves and the interaction with solid structures are only used to obtain insight in the acoustic excitation problem. Because the required numerical model for the acoustic excitation problem should work without the need of direct measurements, the principle of reciprocity is not directly applicable. The principle does, however, give a lot of insight in the acoustic excitation problem, as it relates sound radiation to structural response and vice versa. The available numerical analyses for acoustically induced vibrations are discussed in section 2.5. The promising MATLAB environment program, COMSOL Multiphysics, is found most suitable to model the acoustic excitation problem as it can be used to model both acoustics and structural mechanics in one model. The section on large amplitude oscillations is important for the acoustic excitation problem of the wafer stepper because the reticle stage oscillates with large amplitudes. The large amplitude oscillations make the problem non-linear. In non-linear systems, harmonic distortion leads to response at frequencies that are not in the input signal (a linear model obviously fails to predict these contributions). This necessitates the development of a numerical model that takes into account the non-linear effects caused by large amplitude oscillations.

23 Chapter 2: Literature study; acoustically induced vibrations 13 For the frequency response identification of non-linear systems the well-known frequency response function can not be used because it is based on the imperative assumption of linearity of the system behavior. Section 2.7 addresses three other concepts for frequency response identification of non-linear systems, namely the Sinusoidal Input Describing Function, the Higher Order Sinusoidal Input Describing Function and the Volterra series, of which the latter two are described in more detail. In this study, the Higher Order Sinusoidal Input Describing Function is used to analyse the response of the system. The next chapter presents and discusses the numerical model that is developed to predict the response of a system to acoustic excitation due to large amplitude oscillations.

24 Discussion

25 Chapter 3 Model description and discussion This chapter is used to present and discuss the numerical model that is developed to predict the response of a system to acoustic excitation. The program that is used to create the model is COMSOL Multiphysics. COMSOL Multiphysics is a simulation environment that facilitates all steps in the modeling process: defining the geometry, specifying physics, meshing and solving and then post-processing results. The acoustic excitation problem of the wafer stepper, as described in section 1.2, is used as the basis for the numerical model. The problem can be generalized by considering a moving body and an external structure in some cavity, see figure 3.1. Figure 3.1: Generalization of the acoustic excitation problem of the stepper; a moving body that acoustically excites an external structure in some cavity. 3.1 Physics model description The problem is modeled using the wave equation (which is a simplification of the full Navier- Stokes equation). With the aid of dimensionless parameters this assumption can be verified.

26 Uncoupled model For these dimensionless parameters characteristic length- and time- scales (L [m] and V [m/s]) are required. Beltman [19] showed that the complete parameter range is covered by three classes of models: the standard wave equation model, the low reduced frequency model and the full linearized Navier-Stokes model. The following assumptions are used: no internal heat generation, homogenous medium, no mean flow and laminair flow. The assumption of laminair flow can be verified by calculating the Strouhal number, Sr = fl V, (3.1) where f [s 1 ] is the frequency. For the acoustic excitation problem of the wafer stepper, the characteristic parameters, f, the frequency of the stage, ω, the radian frequency of the stage, V, the velocity of the stage and L the round off radius of the stage, are typically of the order, f = O(), ω = O(), V = O() and L = O( 3). For these order of parameters the order of the Strouhal number is, Sr = O( 3). A Strouhal number, Sr 1, indicates the negligibility of non-stationary flow. The two important parameters that are used for the classification of Beltman are the shear wave number and the reduced frequency. The shear wave number is, ρ ω s = l µ, (3.2) where l [m] is a length scale (this length scale represents the boundary layer thickness), ρ [kg/m 3 ] is the density of air (1.25 kg/m 3 ), ω [rad/s] is the angular frequency and µ [N s/m 2 ] is the viscosity or air ( N s/m 2 ). The reduced frequency is, k = ωl c s, (3.3) where c s [m/s] is the speed of sound (344 m/s). The boundary layer thickness can be determined using the Blasius-boundary layer thickness solution, νx δ.99 = 4.9 V, (3.4) wherein ν [m 2 /s] is the kinematic fluid viscosity (ν air = m 2 /s). For x = O() and V = O(), δ.99 = O( 2). When this is filled in for l in (3.2) and (3.3), the order of s and k are, s = O(1) and k = O( 4). For s 1, which is the case, the general wave equation can be used. This is also illustrated in figure 3.2. Since k/s 1, even the modified wave equation can be used (Low-wave). 3.2 Uncoupled model First consider the moving body in some cavity. Of key interest is the sound field generated by the body when it moves with large displacements. The non-linear effects caused by the large displacements are taken into account by implementing a mesh that deforms in time. This, so-called moving mesh, is described in subsection The major advantage of the moving mesh is that linear acoustics can still be used to model the problem.

27 Chapter 3: Model description and discussion 17 Figure 3.2: Beltman s parameter overview of models Acoustics Acoustic wave propagation is modeled by equations from linearized fluid dynamics and solid dynamics. The full equations are time dependent, but noting that a harmonic excitation with a time dependence of the form f = ˆfe iωt gives rise to an equally harmonic response with the same frequency, the time can be eliminated completely from the equations. Instead the angular frequency, ω = 2πf, enters as parameter. From the mathematical point of view, the time-harmonic equation is a Fourier transform of the original time-dependent equations and its solution as a function of ω is the Fourier transform of a full transient solution. Therefore, a time-dependent solution can be synthesized from a frequency-domain by applying an inverse Fourier transform. The acoustic domain is the space around the moving body inside the cavity. It is modeled as air, having a density of ρ = 1.25 kg/m 3 and the speed of sound c s = 344 m/s. The pressure, p(x,y), is calculated using the time harmonic formulation based on the inhomogeneous Helmholtz equation, ( 1 ) ( p q) ω2 p ρ ρ c 2 = Q, (3.5) s where q [N/m 3 ] is a dipole source and Q [1/s 2 ] a monopole source (both zero in case of the acoustic excitation problem of the stepper). In 2D, the independent variables are the Cartesian coordinates x and y, while the model is assumed to be uniform in the perpendicular z-direction. Under this assumption, any wave motion in the latter direction is accounted for by specifying the out of plane wave number, k z, defined by the equation, p(x,y,z,t) = p(x,y)e i(ωt kzz). (3.6) Using (3.6) and expanding the 3D operators in (3.5), the equation for the pressure, p(x, y), becomes, ( 1 ) ( ( ) ) ω 2 ( p q) k 2 p z = Q. (3.7) ρ ρ c s

28 Uncoupled model The boundary conditions are of two different types. At all the walls of the cavity and the top and bottom boundary of the moving body, sound hard (wall) boundary conditions are used. A sound-hard boundary is a boundary at which the normal component of the acceleration is zero, ( n 1 ) ( p q) =. (3.8) ρ At the left and right boundary of the moving body a normal acceleration is specified, ( n 1 ) ( p q) = a n, (3.9) ρ which is of course the acceleration of the moving body itself Moving mesh The technique for mesh movement is called an arbitrary Lagrangian-Eulerian (ALE) method. In the special case of a Lagrangian method, the mesh movement follows the movement of the physical material. Such a method is often used in solid mechanics, where the displacements often are relatively small. When the material motion is more complicated, like in a fluid flow model, the Lagrangian method is not appropriate. For such models, an Eulerian method, where the mesh is fixed, is often used (expect that this method cannot account for moving boundaries). The ALE method is an intermediate between the Lagrangian and the Eulerian method, and it combines the best features of both, it allows moving boundaries without the need for the mesh movement to follow the material. Let the spatial coordinates denote (x,y) and let (X,Y ) be the coordinates of a mesh node in the initial, undeformed configuration. To describe the spatial coordinates (x, y) of the same mesh node in the deformed configuration, the following functions are used x = x(x,y,t), y = y(x,y,t) (3.1) where t is time. This coordinate transformation relates two frames (see figure 3.3): - The spatial frame is the usual, fixed coordinate system with the spatial coordinates (x,y). In this coordinate system the mesh is moving, that is, the coordinates (x,y) of a mesh node are functions of time. - The reference frame is the coordinate system defined by the reference coordinates (X, Y ). In this coordinate system the mesh is fixed to its initial position. One can view the reference frame as a curvilinear coordinate system that follows the mesh. In the case of the stepper, the mesh displacement is constrained by the following boundary conditions: the mesh displacement at the surrounding walls of the cavity is fixed and the mesh displacement of the mesh around the body is constrained by the movement of the body (setpoint stage), see figure 3.4. The mesh displacement inside the cavity is obtained by solving a partial differential equation (PDE). The PDE smoothly deforms the mesh given the constraints that are placed on the boundaries. There are two type of smoothing methods, namely Laplace smoothing and Winslow smoothing. To see how these smoothing methods differ, let again x and y be the spatial coordinates of the frame which the application mode defines, and let X and Y be

29 Chapter 3: Model description and discussion 19 (a) An undeformed mesh, frames coincide. (b) A deformed mesh Figure 3.3: Example of mesh with spatial frame (x,y) and reference frame (X,Y ). Figure 3.4: Boundary conditions for the mesh displacement. Herein, dx is the displacement in x-direction, dy is the displacement in y-direction and setdisp(t) is the prescribed displacement defined by the setpoint. the reference coordinates of the reference frame. With Laplace smoothing, the software introduces deformed mesh positions x and y as degrees of freedom in the model. It solves the equations, 2 x X 2 t + 2 x Y 2 t =, 2 y X 2 t + 2 y Y 2 =. (3.11) t With Winslow smoothing, the software solves the equations, 2 X x X y 2 =, 2 Y x Y =. (3.12) y2 Both smoothing methods have no physical meaning. If the mesh displacement becomes large, the mesh elements eventually have a very bad quality or even become inverted. Once a mesh element becomes inverted, it is no longer possible to solve any other equation on that frame. As is the case with the large displacements of the stepper problem, the smoothing methods alone fail to withhold a good quality of the mesh. Inevitably, some of the mesh elements become inverted Simulations moving mesh In order to achieve more insight into the process of the moving mesh and the smoothing methods some simulations are carried out with coarse meshes. One of the things that became

30 2 3.3 Coupled model clear from these simulations was that the mesh around the moving body was experiencing a lot of shear, i.e. the mesh in close proximity of the moving body was moving along with the moving body, but a little further away from the moving body the mesh was practically standing still. This caused the mesh to become inverted rather quickly since the elements were sort of sliding apart. Figure 3.5(a) and figure 3.5(b) illustrate this phenomenon Smart boundary conditions A solution to this problem is the use of smart boundary conditions. By splitting up the upper and lower boundary wall of the cavity in three different parts, different constraints can be set on these parts. In this way, the mesh movement can be guided more, resulting in a good quality mesh, even at times when the moving body is far away from its initial position. Basically, the parts of the mesh left and right of the moving body now linearly contract and expand, while the part of the mesh above and below the moving body stays in tact and moves along with the body, see figure 3.6. In figure 3.5(c) the result of the use of the smart boundary conditions is visualized. The mesh withholds a good quality and the mesh elements do not become inverted Remesh algorithm Another, more complicated, solution to the problem is a remesh algorithm. Before the mesh becomes inverted, a new geometry is created from the moved mesh and a new mesh is generated in this new geometry. In this way, the mesh withholds a good quality. Two remesh algorithms are developed. One is based on a stop condition, which compares the mesh quality with a certain limit, the other is based on a constant time step. Both algorithms are described and discussed in Appendix A. Unfortunately, the algorithms occasionally give some problems in mapping the solution from one mesh to another. The algorithms are therefore not applied in the model. 3.3 Coupled model Now consider also an external structure in the cavity. In the uncoupled model (without the external structure), the mesh deformation of the acoustic field is guided by the smart boundary conditions of subsection Now that the external body is put inside the acoustic domain, it is not possible to let the entire mesh move along with the moving body. As a solution to this problem, the model is divided in subdomains, in which different boundary conditions for the moving mesh are specified. Figure 3.7 shows how the model is divided in subdomains Acoustics Basically, the acoustics as described in subsection 3.2.1, remain the same for the coupled model. The model is now, however, divided in subdomains. By making use of a continuity boundary condition between subdomain 1 and 2, and between subdomain 2 and 3, the subdomains form one acoustic pressure field. The continuity condition between subdomain 1 and 2 is, [( n 1 ) ( ( p q) 1 ) ] ( p q) =, (3.13) ρ 1 ρ 2

31 Chapter 3: Model description and discussion 21 (a) Initial mesh. (b) Deformed mesh at time t using original boundary conditions. (c) Deformed mesh at time t using smart boundary conditions. Figure 3.5: Mesh deformation simulation results. where the dipole source q [N/m 3 ] is again left zero Moving mesh Figure 3.8 shows how the mesh displacement is constrained in the coupled model with subdomains. For the intelligibility of the figure, not all the constraints are visualized. When the external body is close to the moving body, i.e. the height of subdomain 2 is small, subdomain 2 is experiencing a lot of shear, which decreases the quality of the elements and thereby the accuracy of the result. By refining the mesh a good accuracy can be maintained External structure and coupling The external structure is modeled as a rigid body, having a density, ρ. The equation that is used to calculate the global displacements of the external structure is, ρ 2 u t 2 c s u = F, (3.14)

32 Coupled model Figure 3.6: Smart boundary conditions for the mesh displacement. Herein, dx is the displacement in x-direction, dy is the displacement in y-direction and setdisp(t) is the prescribed displacement defined by the setpoint. The curve parameter s runs linearly from to 1 along each boundary. Applying a velocity of s or (1 s) times that of the moving body thus makes a line contract or expand linearly. cavity external body moving body Figure 3.7: Subdomains for the realization of the moving mesh in the coupled model. Figure 3.8: Boundary conditions for the mesh displacement of subdomain 2. The mesh displacement in domain 3 is fixed and the mesh displacement in domain 1 is constrained according to figure 3.6.

33 Chapter 3: Model description and discussion 23 where u = (u,v) are the global displacements in the x- and y-directions. To prevent the external structure from drifting away a very weak spring is modeled. The spring is modeled by specifying a body load in x-direction, F x [N/m 2 ], F x = ku A, (3.15) where k [N/m] is the spring stiffness and A [m 2 ] is the surface area of the external structure. The coupling of the acoustic field to the external structure is modeled by specifying an edge load on the boundaries of the external structure. Because the pressure, p, works in normal direction on the boundaries of the external structure, a tangent and normal coordinate system (t,n) is used to specify the edge load. This edge load, F n [N/m 2 ], then becomes, wherein p is the calculated pressure. 3.4 Time-dependent solver F n = p, (3.16) The solution is dependent upon the acceleration and position of the stage. Because the large amplitude oscillations of the position of the stage require a simulation in the time-domain, a time-dependent solver is chosen. Boundary conditions and initial conditions together with the PDE fully define the problem. It is essential to set absolute and relative tolerance parameters for the time-dependent solver. The absolute and relative tolerances control the error in each integration step. Because the displacements in the structural-mechanics module are small, it is important that the absolute tolerance is smaller than the displacements. For ease of understanding, a schematic overview of the coupled model is visualized in figure 3.9. Besides the solver settings, the position and acceleration of the moving body are put into the model. From the position of the moving body, the moving mesh application mode calculates the position of the mesh of the surrounding air. This is put into the acoustic module together with the acceleration of the moving body at that location and time. The acoustic module then calculates the generated pressure field p(x, y, t). This pressure field is used in the structural-mechanics application mode to calculate the displacements of the external structure.

34 Time-dependent solver INPUT COUPLED MODEL OUTPUT s(t) position moving body Moving mesh (x,y) coordinates mesh (of air) (x,y) coordinates mesh (of air) a(t) acceleration moving body Acoustics aaaa aaaa aaaa p(x,y,t) pressure field p(x,y,t) Solversettings Structural-mechanics aa aa Initial geometry, material properties and constants, constraints, etc. u(t), v(t) displacements in x-direction and y-direction of external structure Figure 3.9: Schematic overview coupled model.

35 Chapter 4 Experiment description and results This chapter is used to present the experiment that is set up for validation purposes of the developed model. In the first section, the experimental setup is explained. In the second section, a test experiment that is carried out in the anechoic room of Philips Applied Technologies (Apptech) is presented. A first necessity of the experiment is a moving body. Since the acoustic excitation problem of the wafer stepper of section 1.2 is used as the basis of the model, it obviates the use of a (reticle) stage for the experiment. Apptech and ASML have both made measurement time available on a working stage. The experiment on the Charger stage of Apptech appeared not so useful to the project and the reader is therefore referred to appendix B. In the third section, the experiment on ASML s Testrig is presented and discussed. In the last section, the Higher Order Sinusoidal Input Describing Function of section 2.7 is used to analyze the system response of a particular measurement. 4.1 Experimental setup On the one hand, a microphone is placed in close proximity to the moving stage to measure the sound pressure. On the other hand, a polystyrene ball is placed above the moving stage, which is attached to the ceiling by a thin rope, to represent the external structure. The polystyrene ball is used because it has a relatively low mass and high volume, making it sensitive to acoustic excitation. Moreover, the dynamic behavior is predictable since the polystyrene ball is considered as a rigid body. Both the velocity of the polystyrene ball and the velocity of the moving stage are measured by lasers. The scanning laser and single laser head, which have been made available at Apptech for the experiment, are used for the polystyrene ball and moving stage respectively, as the scanning laser is more sensitive to small velocity changes and the single laser is capable of measuring higher velocities. A SigLab unit and laptop are used for efficient signal processing. The microphone, the single laser head and scanning laser are attached to SigLab through a Nexus amplifier, a laservibrometer controller and a scanning laser controller respectively (all analogue signals). For a schematic overview of the experimental setup, see figure 4.1. In table 4.1, the brand/type of the measurement equipment is given. The following subsections describe the measurement equipment in more detail.

36 Experimental setup Nexus amplifier Figure 4.1: Experimental setup Pressure microphone and Nexus amplifier The microphone that is used is a Brüel & Kjaer 1/2 condenser microphone. The condenser microphone is accepted as the standard acoustical transducer of all sound and noise measurement because of its very high degree of accuracy; an accuracy which is higher than what is possible with any other acoustical transducer. The reason for the high degree of acceptance is that the condenser microphone has the following properties which are essential for a standard transducer: high stability under various environmental conditions flat frequency response over a wide frequency range low distortion Table 4.1: Measurement equipment Measurement equipment Brand/type Microphone Brüel & Kjaer 1/2 condenser microphone Amplifier Nexus Scanning laser Polytec PSV-3-H Single laser head Polytec OFV 33 Laservibrometer controller Polytec OFV 31 SigLab MATLAB appl.

37 Chapter 4: Experiment description and results 27 very low internal noise wide dynamic range high sensitivity The condenser microphone converts the acoustical pressure variations into an electric signal which thereafter is amplified in a preamplifier. The preamplifier must be connected close to the microphone since its main purpose is to convert the very high impedance of the microphone into a low output impedance permitting use of long cables and connection to instruments with a relatively low input impedance. The low impedance ensures very little pick up of external electrical noise and this is especially important when using long cables. The microphone consists of a thin metallic diaphragm in close proximity to a rigid backplate. This forms an air dielectric capacitor whose capacitance is variable since the diaphragm moves when excited by external forces such as a sound wave. A change in distance between diaphragm and backplate, d, is converted into a change in voltage, V = Q/C = (Q/ǫA) d, (4.1) where Q is the charge on the backplate, C is the microphone capacitance and A is the area of the microphone. The microphone will work according to the theory provided that Q is held constant. This required constant charge can be provided by connecting a DC voltage through a very high impedance charger resistor. This is combination with the capacitance of the microphone gives a long time constant compared to the period of the sound waves. Hence, practically no current will flow through the charging resistor - the criterion for conversation of charge. By connection of the condenser microphone to an amplifier (in this case a Nexus amplifier) via a coupling capacitor the DC voltage is removed, leaving the AC voltage which is an electrical replica of the sound pressure variations Scanning laser and single head laser vibrometer The laser that is used for the velocity of the stage is a single head laser of Polytec, type OFV 33, which is connected to a laservibrometer controller of Polytec, type OFV 31. The velocity of the ball is measured by a scanning vibrometer of Polytec, type PSV-3-H. Both lasers are based on the Doppler principle. Laser Doppler Vibrometers work according to the principle of laser interferometry. The laser beam, with a certain frequency f, strikes a point on the vibrating object. Light reflected from that point travels back to the sensor head. The backscattered light is shifted in frequency (Doppler effect). This frequency shift, f D, is proportional to the velocity of the vibrating object, v, f D = 2 v λ, (4.2) where λ is the wavelength of the HeNe laser utilized in the Laser Doppler Vibrometer. To distinguish between movement towards and away from the sensor and offset frequency, f B, is added onto the backscattered light. The resulting frequency seen by the photo detector becomes, f = f B + f D = f B + 2v λ, (4.3)

38 Test experiment (anechoic room) where the sign in this equation depends upon the direction of movement of the object. Unlike laser diodes that are sensitive to ambient conditions, the wavelength of ta HeNe laser is a highly stable physical constant (.6328 µm). This is why no calibration of the laser interferometer itself is necessary SigLab For applications like DSP Design verification, Noise/Vibration Analysis, Modal Analysis and Control System design and more, SigLab is an ideal tool for characterizing signals and systems in the lab or field. Set up and control of all measurements are through point and click Virtual Instruments (VIs) coded in MATLAB. Results are viewed in real-time using MATLAB s graphics visualization capabilities. The acquired data is saved in MATLAB data format readily available for post-processing. 4.2 Test experiment (anechoic room) In order to test the equipment that is used for the experiment, an experiment is carried out in the anechoic room of Apptech. The polystyrene ball is acoustically excited by a loudspeaker. The velocity of the ball is measured by a laservibrometer. See figure 4.2 for a schematic overview of this setup. In figure 4.3, the measured frequency response of the laservibrometer to the microphone is shown. laservibrometer controller single laserhead polystyrene ball laser microphone speaker SigLab Figure 4.2: Schematic overview of experimental setup anechoic room. The theoretical sensitivity of the polystyrene ball derived by Roozen et al. [2] is expressed in the radius, a, the speed of sound, c, and the mass, m, of the ball. For frequencies below the critical radiation frequency, f = c 2 2πa, (4.4) which is about 516 Hz for the polystyrene ball, the sensitivity is, v(r ) p (r ) = 2πa3 cm. (4.5)

39 Chapter 4: Experiment description and results Frequency response (laservibro, microphone) anechoic room Measured frequency response Theoretical sensitivity of 5.2 db [db, ref 1 (mm/s)/pa] Frequency [Hz] Figure 4.3: Structural response of the polystyrene ball to acoustic excitation. The measured frequency response of the laservibrometer to the microphone and the theoretical sensitivity according to Roozen et al.[2]. Herein, p (r ) is pressure in the absence of the rigid body. For a radius of.15 m and a mass of.114 kg, equation 4.5 gives a sensitivity of -5.2 [db ref 1 (mm/s)/pa]. In figure 4.3, the theoretical sensitivity of -5.2 [db ref 1 (mm/s)/pa] is shown. The figure indicates that the equipment correctly measures the structural response. 4.3 ASML Testrig For the final experiment, ASML s Testrig XTIV, see figure 4.4(a), is used. As described in section 4.1, the polystyrene ball, of m =.2674 kg (higher mass polystyrene ball as test experiment), is attached to the ceiling by a thin rope and the microphone is placed in close proximity to the moving stage, see figure 4.4(b). The lasers need a reflecting surface in order to acquire a good signal. To this end, a retro-reflecting piece of tape is pasted on the polystyrene ball and on the moving stage, see figure 4.5(a). Figure 4.5(b) shows the rest of the equipment (the Nexus amplifier is outside the frame) Location polystyrene ball and microphone Since the experiment is set up for validation purposes of the numerical model that takes into account the geometric non-linear effects caused by the large displacements, it is important to position the polystyrene ball such that the contribution of the non-linear effects is high. Simulations are carried out for different positions of the ball with two models, the coupled model that takes into account the geometric non-linear effects and a coupled model that does not take into account the geometric non-linear effects. The output of these simulations is used to determine a good location for the ball, for which the non-linear behavior is high. In close proximity of the end of the stage (both left and right), the influence of the non-linear effects appeared to be the highest. Therefore, the polystyrene ball is positioned close to the end of the moving stage, see figure 4.6. The stage is shown in its initial position in each setup.

40 3 4.3 ASML Testrig (a) ASML Testrig XTIV. (b) Polystyrene ball (left), microphone (right). Figure 4.4: Photos experiment setup ASML Testrig XTIV (1) Setpoint stage As described in the introduction, modern wafer steppers do the exposure scan wise, where both reticle and wafer move and the light is passing through a narrow slit. Scanning is using the lens more effectively than static exposure of the entire area. A typical scan setpoint of the reticle stage, as shown in figure 4.7(a), is used for the experiment. This synthesized setpoint gives a clear autospectrum for the velocity as shown in figure 4.7(b), where only a contribution of the uneven harmonics is visible (as should be the case). The measured

41 Chapter 4: Experiment description and results 31 (a) Scanning laser (left), single laser head (just outside frame). (b) Scanning laser and laservibrometer controllers, SigLab unit and laptop. Figure 4.5: Photos experiment setup ASML Testrig XTIV (2). autospectrum of the velocity of the stage by the single laser head is shown in figure 4.8(b). This autospectrum is the same for each setup and is therefore not repeated in the results in section The little contribution of the 6 th and 8 th harmonic can be explained by the fact that the synthesized setpoint was not precisely met (while tracking the setpoint of the stage, an error in the acceleration was visible).

42 ASML Testrig polystyrene ball 15 mm 8 mm 335 mm 3 mm microphone stage (a) Setup 1. polystyrene ball 15 mm 115 mm 11 mm 35 mm microphone polystyrene ball (b) Setup 2. stage microphone 15 mm 1 mm 8 mm 18 mm (c) Setup 3. stage Figure 4.6: Different locations experimental setup polystyrene ball and microphone. The stage is 55 by 975 mm and the polystyrene ball has a radius of 15 mm. The dashed line shows the outer left and outer right position of the stage (of the Scansetpoint of 4.7(a)) Background measurements For the microphone measurements of the experiment some background measurements are carried out to find out if the predicted pressure values are generally higher than the background noise. In figure 4.8(a) the autospectrum of the background noise of the clean room of ASML is shown. The background measurement can be used to identify certain frequency contribution which are not caused by the movement of the stage. The figure shows a clear peak at 12.4 Hz, and a smaller peak at 16.4 Hz and 27.6 Hz.

43 Chapter 4: Experiment description and results ScanSetpoint ASML Testrig XTIV position * 1 [m] velocity * 1 [m/s] acceleration [m/s 2 ] 8 6 Autospectrum velocity stage (setpoint input) [db, ref 1 (mm/s) 2 ] Time [s] (a) A maximum acceleration of 69 m/s 2, a maximum velocity of 2.12 m/s and a maximum displacement of.117 m. The period time, T, of this setpoint is.284 s, which corresponds to a frequency, f, of 3.52 Hz Frequency [Hz] (b) Autospectrum velocity stage (setpoint input). The fundamental frequency (1 st harmonic) of 3.52 Hz and the 3 following uneven harmonics of 1.56 Hz, Hz and Hz (3 th, 5 th and 7 th harmonic). Figure 4.7: Scansetpoint ASML Testrig XTIV. 7 Autospectrum velocity stage (single head laser) 6 5 [db, ref 1 (mm/s) 2 ] (a) Autospectrum microphone background noise clean room ASML. A clear peak at 12.4 Hz, a small peak at 16.4 Hz and 27.6 Hz Frequency [Hz] (b) Autospectrum velocity stage (single head laser). Figure 4.8: Autospectra microphone and velocity stage Experiment results and observations Using the setpoint, as described in section 4.3.2, and the different locations, as described in section 4.3.1, several measurements are carried out. For each setup (1-3), as indicated in figure 4.6, the result of one measurement is shown in figure For these measurements a bandwidth of 5 Hz is chosen, 1 averages are taken (8192 lines) and the resolution frequency is, f,.156 Hz. Quite some observations can be made from the figures. First look at the results of the microphone measurements of each setup (figure 4.9(a), 4.1(a) and 4.11(a)). Besides the

44 ASML Testrig 1 Autospectrum microphone ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg 3 2 Autospectrum velocity ball (scanning laser) ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg 2 1 [db, ref 1 (Pa) 2 ] [db, ref 1 (mm/s) 2 ] [db, ref 1 Pa/(mm/s)] Frequency [Hz] 4 6 (a) Frequency response (microphone, velocity stage) ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg Phase margin 5 Phase [rad] Coherence 1 [db, ref 1 (mm/s)/(mm/s)] Frequency [Hz] 4 5 (b) Frequency response (velocity ball, velocity stage) ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg Phase margin 5 Phase [rad] Coherence 1 [ ].5 [ ] Frequency [Hz] Frequency [Hz] (c) (d) Figure 4.9: Autospectra, frequency response, phase and coherence of setup 1. contribution of the uneven harmonics of the setpoint, the even harmonics are also visible because of the geometric non-linear effect. Also, the peaks at 12.4 Hz and smaller at 16.4 Hz and 27.6 Hz of the background noise are visible. As expected, the magnitude of the microphone measurement of setup 2 is much larger than the magnitude of setup 1, because the microphone is placed closer to the moving stage. Now look at the results of the autospectra of the velocity of the ball (figure 4.9(b), 4.1(b) and 4.11(b)). In these figures, only one frequency contribution can not be explained by the movement of the stage, namely, the contribution at.343 Hz. Because the polystyrene ball is attached to the ceiling with a thin rope, it has a certain pendulum frequency, f pendulum, which is determined by the height of the ball, l [m], and the gravitational constant, g [m/s 2 ]. It is, T = 2π l g = 2π = 2.9 s,f pendulum = 1 =.343 Hz. (4.6) T Again, as expected, the magnitude is generally larger when the ball is closer to the stage.

45 Chapter 4: Experiment description and results 35 Another interesting observation is that the contributions of the 6 th and 8 th harmonic are not clearly present in figure 4.9(b) and 4.1(b), while they are in figure 4.11(b) of setup 3. This can be explained by the fact that the ball is the closest to the moving stage in setup 3 (greater influence geometric non-linear effect). The frequency response, phase and coherence of the velocity of the ball to the velocity of the stage for each setup are shown in figure 4.9(d), 4.1(d) and 4.11(d). The phase between the velocity of the ball and the velocity of the stage is exactly π at the fundamental frequency (1 st harmonic). The coherence of the measurements is 1 at the uneven harmonics and only about.5 at the even harmonics. The value of the frequency response and phase at these even harmonics is therefore questionable. The new concept of frequency response identification of non-linear systems as discussed in section 2.7 is applied and the result is presented in the next section. 5 Autospectrum microphone ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg 2 Autospectrum velocity ball (scanning laser) ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg [db, ref 1 (Pa) 2 ] [db, ref 1 (mm/s) 2 ] [db, ref 1 Pa/(mm/s)] Frequency [Hz] 5 (a) Frequency response (microphone, velocity stage) ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg Phase margin 5 Phase [rad] Coherence 1 [db, ref 1 (mm/s)/(mm/s)] Frequency [Hz] Phase [rad] 4 6 (b) Frequency response (velocity ball, velocity stage) ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg Phase margin Coherence 1 [ ].5 [ ] Frequency [Hz] Frequency [Hz] (c) (d) Figure 4.1: Autospectra, frequency response, phase and coherence of setup 2.

46 Higher order sinusoidal input describing functions 1 Autospectrum microphone ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg 3 Autospectrum velocity ball (scanning laser) ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg 2 2 [db, ref 1 (Pa) 2 ] [db, ref 1 (mm/s) 2 ] [db, ref 1 Pa/(mm/s)] Frequency [Hz] 4 6 (a) Frequency response (microphone, velocity stage) ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg Phase margin 5 Phase [rad] Coherence 1 [db, ref 1 (mm/s)/(mm/s)] Frequency [Hz] 2 4 (b) Frequency response (velocity ball, velocity stage) ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg Phase margin 5 Phase [rad] Coherence 1 [ ].5 [ ] Frequency [Hz] Frequency [Hz] (c) (d) Figure 4.11: Autospectra, frequency response, phase and coherence of setup Higher order sinusoidal input describing functions As described in section 2.7, in case of a non-linear system, the frequency response function is not a sufficient description of the system behavior. The Higher Order Sinusoidal Input Describing Function (HOSIDF), of subsection 2.7.2, is used in this section to analyze the frequency response. In figure 4.12(b), the HOSIDF frequency response for a measurement of setup 1 is shown. For the calculation of the magnitude and phase (2.14) and (2.15) are used. Now that the frequency response and phase of the harmonics are related to the fundamental frequency, the frequency response and phase of the higher harmonics are about zero.

47 Chapter 4: Experiment description and results 37 [db, ref 1 (mm/s)/(mm/s)] Frequency response (velocity ball, velocity stage) ASML, Setup 1, meas2, 5Hz, 496 lines, 2 avg Phase margin Phase [rad] 2 2 [db, ref 1 (mm/s)/(mm/s)] Frequency [Hz] (a) Frequency response. Frequency response HOSIDF (velocity ball, velocity stage) ASML, Setup 1, meas2, 5Hz, 496 lines, 2 avg Phase k (HOSIDF) Phase [rad] Frequency [Hz] (b) HOSIDF. Figure 4.12: Frequency response function and HOSIDF frequency response.

48 Higher order sinusoidal input describing functions

49 Chapter 5 Simulation data and comparison experimental results This chapter is used to present and discuss the simulation data. A comparison is made with the experimental results of Chapter 4. The first section explains how the simulation data is acquired and the second section explains the 2D representation of the problem. The third section is used for the comparison of the simulation data and experimental results. Hereafter, a force calculation is presented in section 5.4 and the results of the chapter are discussed in section Description generation simulation data For the solver settings a begin time, t, a step time, t step, and a stop time, t end, need to be specified. In the uncoupled model, it suffices to only simulate one period of the input setpoint, because the response will be the same for every period. In the coupled model, however, it takes time for the start-up effects of the external structure to disappear. A total simulation time of about 2 periods of the input signal is therefore chosen, of which the first 2 seconds are not used for analysis, see figure Velocity ball COMSOL prediction full data 6 Velocity ball COMSOL prediction del ini Velocity [mm/s] 2 2 Velocity [mm/s] Time [s] (a) Time [s] (b) Figure 5.1: Simulation data velocity ball, (a) with and (b) without start-up effects.

50 D representation For the simulation data that is presented in this chapter, a time step, t step = s, is chosen with a relative tolerance of and an absolute tolerance of Furthermore, the real traced setpoint is used (including acceleration errors). The acquired simulation data is in the time domain. The time data is Fourier transformed in Matlab for comparison with the measurements D representation The developed numerical model is in 2D. The moving stage is uniform in the third dimension. The polystyrene ball, however, is not. In figure 5.2, the 2D representation of the polystyrene ball is shown. In order for the cylinder to have the same mass as the polystyrene ball, the 2r 2r y z x Figure 5.2: 2D representation of polystyrene ball. density needs to be changed (volume cylinder > volume ball). The density of the cylinder, ρ cyl [kg/m 3 ], is chosen, ρ cyl = m ball πr 2 2r, (5.1) wherein m ball [kg] is the mass of the polystyrene ball (.2674 kg) and πr 2 2r [m 3 ] is the volume of the cylinder. 5.3 Simulation data compared with experimental results In figure , the autospectra of the measurements, the COMSOL predictions of the linear model and the COMSOL predictions of the large amplitude model are plotted in blue, red and green respectively. The plotted graphs are compared on frequency contribution and magnitude Frequency contribution First look at the autospectra of the pressure (at microphone) of each setup (5.3(a), 5.4(a) and 5.5(a)). The peak at 12.4 Hz in the autospectra of the microphone is explained by the background noise, as indicated in figure 4.8(a). The contribution of the even harmonics are not predicted in the linear model, but are indeed predicted by the model that takes into account the non-linear effects caused by the large amplitude oscillations. Now look at the autospectra of the velocity of the polystyrene ball of each setup (5.3(b), 5.4(b) and 5.5(b)). The peak at.343 Hz in the autospectra of the velocity of the ball is explained by the pendulum frequency of the polystyrene ball. This behavior is incorporated in the model by the modeling of a spring, which is given a spring stiffness such that the

51 Chapter 5: Simulation data and comparison experimental results 41 response frequency corresponds to the pendulum frequency. The spring stiffness is calculated by, k 2πf = m, k = m(2πf)2 = N/m. (5.2) Again, the contribution of the even harmonics are not predicted in the linear model, but are indeed predicted by the model that takes into account the non-linear effects caused by the large amplitude oscillations Magnitude The magnitude of the predictions of COMSOL appear to be slightly higher than the magnitude of the measurements. This can be explained by the fact that the peaks of the predictions are narrower (only a value for one frequency) than the peaks of the measurements (a value for a few frequencies). In stead of comparing the magnitudes at one frequency by, db(f) = 1log 1 (aspec(f)), (5.3) it makes more sense to compare the amount of energy in a frequency band, f 1 f 2, by, ) db(f 1 f 2 ) = 1log 1 ( i aspec(f i ) for f 1 < f i < f 2. (5.4) Look for example at the autospectrum of the velocity of the ball of a measurement of setup 1, with a bandwidth of 2 Hz, see figure 5.6. In particular, look at the first and third harmonic, see figure 5.7(a) and 5.7(b). Between 3.4 and 3.6 Hz (the first harmonic) the measurement shows a peak of 8 and 1 db. Using equation 5.4 this gives, db( ) = 1log 1 ( 1 8/ /1) = 12.2, (5.5) which corresponds with the predicted magnitude of 11.5 db. Between 1.4 and 1.7 Hz (third harmonic) the measurement shows a peak of -13.5, and db. Using equation 5.4 this gives, db( ) = 1log 1 ( / / /1) = 8.2, (5.6) which is now only 2 db off the predicted value of -6.2 db. 5.4 Force calculation In order to obtain insight in the forces that are acting on the polystyrene ball the calculated pressure is integrated over the boundary of the ball. In figure 5.8, the polystyrene ball is shown with center (a,b). The force in x-direction per unit of length, F x [N/m], is calculated by, F x = p( n x)dϕ, (5.7) wherein p is the pressure, n = e n, x = e x and dϕ the angle step. The dot product in (5.7) is simply, n x = n x cos(ϕ), (5.8)

52 Force calculation 1 1 Autospectrum pressure ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg Microphone measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (Pa) 2 ] Frequency [Hz] (a) Autospectrum pressure at microphone Autospectrum velocity ball (scanning laser) ASML, Setup 1, meas8, 5Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (mm/s) 2 ] Frequency [Hz] (b) Autospectrum velocity ball. Figure 5.3: Autospectra measurement vs COMSOL prediction, setup 1.

53 Chapter 5: Simulation data and comparison experimental results Autospectrum pressure ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg Microphone measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (Pa) 2 ] Frequency [Hz] (a) Autospectrum pressure at microphone. 2 1 Autospectrum velocity ball (scanning laser) ASML, Setup 2, meas11, 5Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (mm/s) 2 ] Frequency [Hz] (b) Autospectrum velocity ball. Figure 5.4: Autospectra measurement vs COMSOL prediction, setup 2.

54 Force calculation 1 1 Autospectrum pressure ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg Microphone measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (Pa) 2 ] Frequency [Hz] (a) Autospectrum pressure at microphone Autospectrum velocity ball (scanning laser) ASML, Setup 3, meas13, 5Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (mm/s) 2 ] Frequency [Hz] (b) Autospectrum velocity ball. Figure 5.5: Autospectra measurement vs COMSOL prediction, setup 3.

55 Chapter 5: Simulation data and comparison experimental results Autospectrum velocity ball (scanning laser) ASML, Setup 1, meas5, 2Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (mm/s) 2 ] Frequency [Hz] Figure 5.6: Autospectrum vs COMSOL prediction, setup 1, 2 Hz bandwidth. [db, ref 1 (mm/s) 2 ] Autospectrum velocity ball (scanning laser) ASML, Setup 1, meas5, 2Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model [db, ref 1 (mm/s) 2 ] Autospectrum velocity ball (scanning laser) ASML, Setup 1, meas5, 2Hz, 8192 lines, 1 avg Scanning laser measurement COMSOL linear model COMSOL large amplitude model Frequency [Hz] (a) First harmonic Frequency [Hz] (b) Third harmonic Figure 5.7: Zoom in on first and third harmonic of figure 5.6. wherein n = x = 1. By expressing cos(ϕ) in x and y, (5.7) becomes, ( (a x) F x = p )dϕ. (5.9) (x a) 2 + (y b) 2 Since, r = (x a) 2 + (y b) 2, (5.9) simplifies to, F x = 1 p(a x)dϕ. (5.1) r

56 Discussion In figure 5.9 and figure 5.1, the force in x-direction and y-direction of one period of the setpoint is shown (position ball, setup 1). The maximum force per unit length on the polystyrene ball in x-direction is.3 N/m. For a cylinder of length 2r, the maximum force in x-direction on the polystyrene ball is.1 N. Using the calculated force per unit of length, the mass, m, per unit of length [kg/m] and Newton s second law, F = ma, the acceleration and of course also the velocity of the ball can be calculated. ( x, y) n x ( a, b) y x Figure 5.8: Polystyrene ball with center (a,b)..4 Force in x direction (per unit of length) of polystyrene ball setup Force [N/m] Time [s] Figure 5.9: Force in x-direction (per unit of length) of the polystyrene ball (setup 1). 5.5 Discussion The results of this chapter show that the linear model fails to predict the response at some frequencies (the even harmonics caused by harmonic distortion). This indicates the need for a model that takes into account the large amplitude oscillations. The developed model that takes into account the non-linear effects caused by the large amplitude oscillations does indeed predict the response at the even harmonics. The slightly larger amplitudes of the prediction are explained by the narrowness of the prediction peaks. By looking at the energy

57 Chapter 5: Simulation data and comparison experimental results 47.5 Force in y direction (per unit of length) of polystyrene ball setup Force [N/m] Time [s] Figure 5.1: Force in y-direction (per unit of length) of the polystyrene ball (setup 1). in a frequency band, the amplitudes of the prediction correspond to the amplitudes of the measurements. Thus, it may be concluded that, the model that takes into account the nonlinear effects caused by large amplitude oscillations gives a good prediction of the system response. Now that the numerical model is validated it can be be used in the design phase of the wafer stepper to change the mechanical design such that the acoustic excitation of the reduction lens is minimized. Also, different setpoints of the stage can be simulated such that the acoustic excitation of the reduction lens is minimized. In this way, the reduction lens will vibrate less and higher accuracies can be met.

58 Discussion

59 Chapter 6 Conclusions and Recommendations High-precision is essential for the ongoing trend in miniaturization of manufacturing and leading-edge research. In the development of high-precision machinery quite some disturbances that affect the accuracy have to be considered. The disturbance that is investigated in this study is acoustic excitation (in particular acoustic excitation due to large amplitude oscillations). The first section summarizes the result of this study and gives some conclusions. The second section presents some recommendations for future research. 6.1 Conclusions A literature study has been carried out in the area of acoustically induced vibrations due to large amplitude oscillations. The literature study provides insight in the propagation of sound and the interaction of sound waves with solid structures and it presents an important relationship, the so-called principle of reciprocity, which relates sound radiation to the acoustically induced structural response. In the literature of acoustics, source motions are usually considered to have small amplitude, with the result that the radiated acoustic field is linearly related to motion. However, a large amplitude motion of the source introduces an important non-linear relationship between the kinematic motion of the source and the radiated acoustic field. The acoustic excitation problem of the wafer stepper, which is considered in this study, is a typical industrial problem of acoustic excitation due to large amplitude oscillations. The literature study, therefore, also includes a new concept of frequency response identification for non-linear systems. A numerical model has been developed for the prediction of the response of a system to acoustic excitation using COMSOL Multiphysics and MATLAB. The model can roughly be divided in three segments: acoustics, structural-mechanics and mesh movement. The acoustic segment (wave equation) calculates the sound pressure field generated by the movement of a moving body and the structural-mechanics segment calculates the structural response of an external structure caused by this sound pressure field. The non-linear effects caused by the large amplitude oscillations are taken into account by implementing a moving mesh (the mesh of the acoustic field moves/deforms in time according to the movement of the moving body). In this way, linear acoustics can still be used to model the problem. For validating purposes of the developed model an experiment has been set up and measurements have been carried out on a reticle stage of ASML. By making use of a microphone and two lasers, the sound pressure, the velocity of the stage and the velocity of a polystyrene

60 5 6.2 Recommendations ball are measured. The measured structural response of the polystyrene ball and the sound pressure measurement are compared with the predicted structural response and sound pressure of the numerical model. he peaks in the autospectra of the sound pressure measurements correspond with the model prediction (the peaks that are not predicted by the model correspond with the sound pressure measurements of the background noise). Furthermore, the results of the experiment and simulations show that the model that takes into account the non-linear effects caused by large amplitude oscillations gives a good prediction of the structural response and sound pressure, while the linear model does not (the linear model fails to predict the contribution of the even harmonics). The magnitude of the predictions of the numerical model appear to be slightly higher than the magnitude of the measurements. This is explained by the fact that the peaks of the predictions are more narrow than the peaks of the measurements (by looking at the energy in a frequency band around the peak, the magnitudes of the contributions correspond). The developed numerical model can be used in the design phase of for example wafer steppers to investigate how to reduce acoustic excitation by simulating different mechanical designs and different setpoints of the stage. 6.2 Recommendations Based on the study described in this report, recommendations are given for further research. The areas of future work are divided in two categories: model improvement and simulations. The model can be improved on a few aspects. A possible improvement could be to model the problem in 3D. However, the computational time will thereby increase. A deliberation needs to be made wether the improvement in accuracy is worth the extra computational time. Mainly, this will depend on how uniform the geometry is in the third dimension. Another improvement could be to improve the remesh algorithm so that it can be applied in the model for better accuracy of the result. The developed numerical model is used to simulate the structural response of the polystyrene ball. More specific simulations could be carried out for structures as the reduction lens of a wafer stepper to investigate the influence of the geometric non-linear effects caused by the large amplitude oscillations for that specific structure. Also, the influence of the cavity walls could be further investigated, as in most industrial applications the cavity walls are close to the acoustic excitation problem.

61 Appendix A Remesh algorithm In this appendix, the developed remesh algorithms are presented. The remesh algorithms are designed as a solution to the problem of an inverted mesh. Both algorithms do basically the same thing, namely: they create a new geometry from the moved mesh and then create a new mesh in the new geometry. The only difference is the time of remesh. The first designed algorithm is based on a stop condition: as soon as the quality of the mesh is lower than a specified limit, the solver is stopped and a new geometry and mesh are created. The second designed algorithm is based on a constant time step: after a constant time step the solver is stopped and a new geometry and mesh are created. The algorithm based on constant time step is attached below with some explanation. Both algorithms are implemented and various simulations are carried out. Unfortunately, both the algorithms occasionally resulted in an inverted mesh. The reason for this is that when a new geometry is created the new mesh can contain a different amount of elements as the mesh before, giving problems with mapping the solution. 1 % This m f i l e is written to automate the remesh process ( based on constant % time step ). At f i r s t some user defined settings concerning the time dependent % solver and number of remehes are defined for easy adjusting. Then the 4 % geometry, mesh, application modes ( structural mechanics, moving mesh, % acoustics ) and setpoint data of the fem structure are defined. The model i s % solved for one time step before i t goes into the loop. 7 % fem total i s the c o llection of fem structures of each time period from % remesh to remesh. 1 clear a l l ; close a l l ; clc ; 13 % Define time settings for time dependent solver total time =.284; step time = 1e 3; 16 r e l ative t o lerance = 1e 4; absolute tolerance = 1e 5; 19 no of meshes = 2; remesh time step = total time /no of meshes ; 22 remesh stop = remesh time step ; % Define i n i t i t a l time of remesh % COMSOL Multiphysics Model M f i l e 25 % Generated by COMSOL 3.4 (COMSOL , $Date : 27/1/1 16:7:51 $)

62 52 f l c l e a r fem 28 % COMSOL version clear vrsn 31 vrsn.name = COMSOL 3.4 ; vrsn. ext = ; vrsn. major = ; 34 vrsn. build = 248; vrsn. rcs = $Name: $ ; vrsn. date = $Date : 27/1/1 16:7:51 $ ; 37 fem. version = vrsn ; % Geometry 4 g1=rect2 ( 4, 3, base, corner, pos,{ 2, 1 }, rot, ) ; g2=rect2 (.98,.55, base, corner, pos,{.49,.275 }, rot, ) ; g3=geomcomp({g1, g2 }, ns, { R1, R2 }, sf, R1 R2, edge, none ) ; 43 g4=geomcomp({g3 }, ns, { CO1 }, sf, CO1, edge, none ) ; g5=geomcoerce ( solid, { g4 }) ; 46 parr={point2 (.49, 1) }; g6=geomcoerce ( point, parr ) ; parr={point2 (.49, 1) }; 49 g7=geomcoerce ( point, parr ) ; parr={point2 ( 2,.275) }; g8=geomcoerce ( point, parr ) ; 52 parr={point2 (.49,.275) }; g9=geomcoerce ( point, parr ) ; parr={point2 (.49,.275) }; 55 g1=geomcoerce ( point, parr ) ; parr={point2 (2,.275) }; g11=geomcoerce ( point, parr ) ; 58 carr={curve2 ([ 2,2],[.275,.275],[1,1]) }; g17=geomcoerce ( curve, carr ) ; 61 g18=e l l i p 2 (.15,.15, base, center, pos,{ , }, rot, ) ; g19=geomcoerce ( solid, { g18 }) ; 64 % Analyzed geometry clear p c s p. objs={g6, g1, g14, g9, g12, g7, g15, g8, g11, g13 }; 67 p. name={ PT1, PT5, PT9, PT4, PT7, PT2, PT1, PT3, PT6, PT8 } ; p. tags ={ g6, g1, g14, g9, g12, g7, g15, g8, g11, g13 } ; 7 c. objs={g17, g16 }; c. name={ B2, B1 } ; c. tags ={ g17, g16 } ; 73 s. objs={g5, g19 }; s. name={ CO1, CO2 } ; 76 s. tags ={ g5, g19 } ; fem. draw=struct ( p, p, c, c, s, s ) ; 79 fem. geom=geomcsg(fem ) ; % I n i t i a l i z e mesh

63 Chapter A: Remesh algorithm fem. mesh=meshinit (fem,... hauto,4) ; 85 % Application mode 1 clear appl appl.mode. c lass = AcoPlaneStrain ; 88 appl. sdim = { X, Y, Z } ; appl. module = ACO ; appl. gporder = 4; 91 appl. cporder = 2; appl. a ssignsuffix = acpn ; clear prop 94 prop. analysis = time ; clear weakconstr weakconstr. value = off ; 97 weakconstr. dim = { lm4, lm5 } ; prop. weakconstr = weakconstr ; appl. prop = prop ; 1 clear bnd bnd. loadcoord = { global, local } ; bnd.fy = {, p } ; 13 bnd. ind = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2]; appl. bnd = bnd ; clear equ 16 equ. rho = {785,.2674/(.15ˆ2 pi.3) }; equ.fx = {, 1.247/(.15ˆ2 pi ) u } ; equ. usage = {,1}; 19 equ.e = {2,69}; equ. ind = [ 1, 1,2]; appl. equ = equ ; 112 fem. appl {1} = appl ; % Application mode clear appl appl.mode. c lass = MovingMesh ; appl. sdim = { X, Y, Z } ; 118 appl. shape = { shlag (2, lm2 ), shlag (2, lm3 ), shlag (2, x ), shlag (2, y ) }; appl. gporder = {4,4}; appl. cporder = 2; 121 appl. border = on ; appl. a ssignsuffix = ale ; clear prop 124 prop. smoothing= winslow ; prop. analysis = transient ; clear weakconstr 127 weakconstr. value = on ; weakconstr. dim = { lm2, lm3 } ; prop. weakconstr = weakconstr ; 13 appl. prop = prop ; clear bnd bnd. defflag = { {1;1}}; 133 bnd. deform = { {;}, { s setpoint pos ASML ( t ) ;}, { setpoint pos ASML ( t ) ;... },{ (1 s ) setpoint pos ASML ( t ) ;}}; bnd. wcshape = [ 1 ; 2 ] ; 136 bnd.name = { fixed, s set, set, (1 s ) set } ; bnd. ind = [1,2,1,2,1,3,3,3,3,4,3,4,1,1,1,1,1,1];

64 54 appl. bnd = bnd ; 139 clear equ equ. gporder = 2; equ. type = { free, none } ; 142 equ. shape = [ 3 ; 4 ] ; equ. ind = [ 1, 1,2]; appl. equ = equ ; 145 fem. appl {2} = appl ; % Application mode clear appl appl.mode. c lass = AcoPressure ; appl. module = ACO ; 151 appl. border = on ; appl. a ssignsuffix = acpr ; clear bnd 154 bnd. nacc = {,, setpoint acc ASML ( t ), setpoint acc ASML ( t ) }; bnd. type = { SH, cont, NA, NA } ; bnd. ind = [1,1,1,2,1,1,3,1,1,1,4,2,1,1,1,1,1,1]; 157 appl. bnd = bnd ; clear equ equ. usage = {1,}; 16 equ. ind = [ 1, 1,2]; appl. equ = equ ; appl. var = { freq, }; 163 fem. appl {3} = appl ; fem. sdim = {{ X, Y }, { x, y } }; fem. frame = { xy, ale } ; 166 fem. border = 1; % Functions 169 clear fcns fcns {1}. type= interp ; fcns {1}.name= setpoint acc ASML ; 172 fcns {1}.method= linear ; fcns {1}. extmethod= extrap ; fcns {1}. filename = /home/nly26446/data/matlab/ ASML finalsetpoint acc 2setp. txt ; 175 fcns {2}. type= interp ; fcns {2}.name= setpoint pos ASML ; fcns {2}.method= linear ; 178 fcns {2}. extmethod= extrap ; fcns {2}. filename = /home/nly26446/data/matlab/ ASML finalsetpoint pos 2setp. txt ; fem. functions = fcns ; 181 % Multiphysics fem=multiphysics (fem ) ; 184 % Extend mesh fem. xmesh=meshextend (fem, linshape, [ ] ) ; % Solve problem 19 fem. sol=femtime (fem,... solcomp, { lm3, u, y, lm2, p, x, v },... outcomp, { lm3, u, y, lm2, p, x, v },...

65 Chapter A: Remesh algorithm t l i s t, [ : step time : remesh stop ],... atol, absolute tolerance,... rtol, relative tolerance, maxorder,2,... tout, t l i s t ) ; 199 % Store solution fem total (1) = fem ; 22 % Generate geom from mesh fem = mesh2geom (fem, srcdata, deformed, frame, ale ) ; 25 % Loop solver stop time = remesh time step ; remesh stop = remesh stop + remesh time step ; 28 i = 2; while stop time < total time 211 % I n i t i a l i z e mesh fem. mesh=meshinit (fem, hauto,4) ; % Application mode clear appl appl. mode. c lass = AcoPlaneStrain ; appl. sdim = { X, Y, Z } ; 22 appl. module = ACO ; appl. gporder = 4; appl. cporder = 2; 223 appl. assignsuffix = acpn ; clear prop prop. analysis = time ; 226 clear weakconstr weakconstr. value = off ; weakconstr. dim = { lm4, lm5 } ; 229 prop. weakconstr = weakconstr ; appl. prop = prop ; clear bnd 232 bnd. loadcoord = { global, local } ; bnd.fy = {, p } ; bnd. ind = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2]; 235 appl. bnd = bnd ; clear equ equ. rho = {785,.2674/(.15ˆ2 pi.3) }; 238 equ.fx = {, 1.247/(.15ˆ2 pi ) u } ; equ. usage = {,1}; equ.e = {2,69}; 241 equ. ind = [ 1, 1, 2]; appl. equ = equ ; fem. appl {1} = appl ; 244 % Application mode 2 clear appl 247 appl. mode. c lass = MovingMesh ; appl. sdim = { X, Y, Z } ; appl. shape = { shlag (2, lm2 ), shlag (2, lm3 ), shlag (2, x ), shlag

66 56 (2, y ) }; 25 appl. gporder = {4,4}; appl. cporder = 2; appl. border = on ; 253 appl. assignsuffix = ale ; clear prop prop. smoothing= winslow ; 256 prop. analysis = transient ; clear weakconstr weakconstr. value = on ; 259 weakconstr. dim = { lm2, lm3 } ; prop. weakconstr = weakconstr ; appl. prop = prop ; 262 clear bnd bnd. defflag = { {1;1}}; bnd. deform = { {;}, { s setpoint pos ASML ( t ) ;}, { setpoint pos ASML ( t ) ; },{ (1 s ) setpoint pos ASML ( t ) ;}}; bnd. wcshape = [ 1 ; 2 ] ; bnd. name = { fixed, s set, set, (1 s ) set } ; 268 bnd. ind = [1,2,1,2,1,3,3,3,3,4,3,4,1,1,1,1,1,1]; appl. bnd = bnd ; clear equ 271 equ. gporder = 2; equ. type = { free, none } ; equ. shape = [ 3 ; 4 ] ; 274 equ. ind = [ 1, 1, 2]; appl. equ = equ ; fem. appl {2} = appl ; 277 % Application mode 3 clear appl 28 appl. mode. c lass = AcoPressure ; appl. module = ACO ; appl. border = on ; 283 appl. assignsuffix = acpr ; clear bnd bnd. nacc = {,, setpoint acc ASML ( t ), setpoint acc ASML ( t ) }; 286 bnd. type = { SH, cont, NA, NA } ; bnd. ind = [1,1,1,2,1,1,3,1,1,1,4,2,1,1,1,1,1,1]; appl. bnd = bnd ; 289 clear equ equ. usage = {1,}; equ. ind = [ 1, 1, 2]; 292 appl. equ = equ ; appl. var = { freq, }; fem. appl {3} = appl ; 295 fem. sdim = {{ X, Y }, { x, y } } ; fem. frame = { xy, ale } ; fem. border = 1; 298 % Multiphysics fem=multiphysics (fem ) ; 31 % Extend mesh fem. xmesh=meshextend (fem ) ; 34

67 Chapter A: Remesh algorithm 57 % Mapping current solution to extended mesh i n i t = asseminit (fem, init, fem total ( i 1). sol, xmesh, fem total ( i 1). xmesh, framesrc, ale, domwise, on ) ; 37 % Solve problem fem. sol=femtime (fem, init, init,... solcomp, { lm3, u, y, p, lm2, x, v },... outcomp, { lm3, u, y, p, lm2, x, v }, t l i s t, [ stop time : step time : remesh stop ],... atol, absolute tolerance,... rtol, relative tolerance, maxorder,2,... tout, t l i s t,... tsteps, strict ) ; 319 % Update remesh stop remesh stop = stop time + remesh time step ; 322 % Store solutions fem total ( i ) = fem ; 325 % Generate geom from mesh fem = mesh2geom (fem, srcdata, deformed, frame, ale ) ; 328 % Update counter i = i + 1; 331 end

68 58

69 Appendix B Charger experiment This appendix explains the original experiment that was setup within Apptech using the Charger stage. The experimental setup of section 4.1 is used. A photo of the experiment setup is shown in figure B.1. The setpoint, some experimental results and simulation data are presented. Finally, the results are discussed. In particular, the reason is addressed why this experiment appeared to be not so useful to the project. Figure B.1: Photo experiment setup Apptech Charger stage. From left to right: microphone, Charger stage, polystyrene ball, single laser head and scanning laser. B.1 Setpoint stage The setpoint that is used for the experiment is shown in figure B.2(a) (a maximum acceleration of 4.2 m/s 2, a maximum velocity of.178 m/s and a maximum displacement of.15 m). At the time of the experiment, the stage could not move with higher accelerations or larger displacements.