Query: Effective-medium model for fast evaluation of scatterometric measurements on gratings

Candidate papers:

abstract: Scatterometry is now an accepted technique for linewidth measurement in semiconductor manufacturing. To reduce the evaluation time when using real time optimization procedures, we introduce an effective medium approach to simulate the optical signatures of subwavelength line and space gratings. Such gratings behaves approximately like uniaxial crystals whose optical properties are completely described by two refractive indices. We propose an algebraic method for their calculation up to second order in pitch/wavelength for the extraordinary index, and up to 4th order in pitch/wavelength for the ordinary index. The formulas are valid for any angle of incidence and can be used with standard matrix formalism to calculate the optical properties of any arbitrary layer stack. We deduce the formulas for the indices. The comparison of effective medium calculations to rigorous coupled wave simulations for spectral measurements (polarized reflectometer and spectroscopic ellipsometer) shows excellent agreement. The sensitivity of scatterometry to tilt of the structures is very low for all measurement parameters except phase difference. It is shown that left and right tilt cannot be distinguished at all with spectral measurements with non conical incidence of light.
author_names: A Weidner

Matthias Slodowski

Christian Halm

Claus Schneider

Lothar Pfitzner
title: Effective medium model for fast evaluation of scatterometric measurements on gratings
venue: SPIE Advanced Lithography
year: 2004

abstract: Abstract There is a strategic importance for the steel rolling industry to get a better understanding of the strip roll interaction to improve roll gap models, increase strip quality and decrease roll degradation. This requires roll gap sensors able to measure this interaction under industrial rolling conditions and in real time in order to propose a feed back control of process parameters. To reach these goals, this paper proposes a new roll gap friction sensor based on an inverse method that interprets optical fiber Bragg gratings (FBG) strain measurements under the roll surface (fully embedded) which enables to evaluate contact stresses with very short computation times, compatible with real time interpretation. This elastic inverse method is analytical and relies on plane strain and isothermal assumptions. The experimental apparatus is detailed, technical issues are clearly exposed as well as calibration procedures. Several pilot cold rolling tests have been performed at various rolling speeds and different strip thicknesses in order to demonstrate the industrial feasibility. Resulting evaluations of contact stresses are then compared with numerical simulations. Reasonable agreement is obtained for normal stress (i.e. pressure) but not for shear stress (only an order of magnitude is obtained)
author_names: D Weisz-Patrault

L Maurin

Nicolas Legrand

Anas Ben Salem

Abdelkebir Ait Bengrir
title: Experimental evaluation of contact stress during cold rolling process with optical fiber Bragg gratings sensors measurements and fast inverse method
year: 2015

abstract: Many light field displays are fundamentally different from other displays in that they do not have quantized pixels, quantized angular outputs, or a physical screen position, which can make definitions and characterization problematic. We have determined that it is more appropriate to express the spatial resolution in terms of spatial cutoff frequency rather than a physical distance as in the case of a display with actual quantized pixels. This concept is then extended to also encompass angular resolution. The technique exploits the fact that when spatial resolution of a sinusoidal grating pattern is halved, its contrast ratio is reduced by a known proportion. An improved model, based on an earlier design concept, has been developed. It not only can be used to measure spatial and angular cutoff frequencies, but also can enable comprehensive characterization of the display. This model provides fast, simple measurement with good accuracy. It does not use special equipment or require time consuming subjective evaluations. Using the model to characterize images in a rapid, accurate manner validates the effectiveness of this technique.
author_names: Phil Surman

Shizheng Wang

Junsong Yuan

Yuanjin Zheng
title: One stop measurement model for fast and accurate tensor display characterization.
venue: Journal of the Optical Society of America. A, Optics, image science, and vision
year: 2018

abstract: With decreasing critical dimensions (CD) on lithography masks, increasing demands on CD metrology techniques come along. Already today the results of the three standard methods for CD measurements currently used, atomic force microscopy (AFM) scanning electron microscopy (SEM) and optical microscopy, typically do not yield the same results. This is because of, e.g. incomplete knowledge of the material parameters, insufficient modelling accuracy or especially in the case of optical microscopy insufficient resolution. With decreasing CDs these systematic differences increase. The need on new cross calibration strategies arises. Non imaging metrology methods like scatterometry as non destructive, non diffraction limited, fast optical methods offer access to the geometrical parameters of periodic structures like e.g. top and bottom CD, pitch, side wall angle, line height, or roughness. Therefore, these methods provide independently achieved additional information that can be used for cross calibration. At the PTB two scatterometers are in use (VIS l=633 nm and EUV l=13.55 nm) A third device (DUV l=193 nm) is under construction. It will offer a wide spectrum of measuring principles like scatterometry, ellipsometric scatterometry, reflectometry, polarisation reflectometry, and using a broadband light source spectroscopic ellipsometry and spectroscopic reflectometry. For simulation and modelling of the intensity distribution of the diffraction pattern two programs based on the rigorous coupled wave analysis (RCWA) method and a finite element method (FEM) respectively are used. The program features will be illustrated. A comparison of the simulations with the results achieved with the scatterometers on different types of lithography masks (chrome on glass, EUV masks) will be presented as well.
author_names: Matthias Wurm

Bernd Bodermann

Werner Mirande
title: Investigation and evaluation of scatterometric CD metrology methods
venue: SPIE Optical Metrology
year: 2005

abstract: The design and physical verification of contemporary integrated circuits is a challenging task due to their complexity. System in Package is an example of generally congested electronic components and interconnects which in the initial design process rely on computationally intensive electromagnetic simulations. Hence the available computer memory capacity and computational speed become meaningful limitations. An alternative method which allows the designer to overcome or reduce the limits is desired. This work represents the first demonstration of the application of effective medium theory to the analysis of those segments of the entire integrated system where the interconnect networks are more dense. The presented approach takes advantage of the deep subwavelength characteristic of interconnect structures. In order to achieve the aim of defining the homogeneous equivalent for the interconnect grating structure a few steps were followed towards proving the homogenisation concept and finally presenting it by an analytical formulation. A set of parameters (metal fill factor, aspect ratio, dielectric background and period to wavelength ratio) with values related to typical design rules were considered. Relating these parameters allows the empirical models to be defined. In order to show the relationship between existing effective medium theories and those developed in this Thesis, the presented empirical models are defined in terms of the Maxwell Garnett mixing rule with an additional scaling factor. The distribution of the scaling factor was analysed in terms of the calculated reflection and transmission coefficients of the homogenised structures that are equivalent to a given grating geometry. Finally the scaling factor, for each empirical model, was expressed by an analytical formula and the models validated by their application to the numerical analysis of grating structures. The numerical validation was carried out by comparing the reflection and transmission coefficients obtained for the detailed and homogenised structures. In order to ensure the empirical models can be broadly employed, the performance of the model in the presence of non normally incident plane wave was evaluated. For the range of angles 30o the model is accurate to 5% The impact of the shape of the grating, specifically the case of a tapered profile, typical of actual fabricated interconnects was also considered, with sidewall tapers of up to 5o giving the same error not higher than 5% Experimental validation of the application of the homogenisation concept to the analysis of interconnects is desired for two main applications: for the reflectivity estimation of a whole chip in a System in Package and for the performance estimation of interconnects on lower metal layers in an interconnect stack. For the first, free space measurements are taken of a grating plate with copper rods aligned in parallel illuminated by a plane wave in the X band (8.2 12.4 GHz) For the second, S parameters are measured for microstrip waveguides with a number of metal rods embedded in the substrate between the signal line and ground plane. The good agreement with the simulations validates the homogenisation approach for the analysis of interconnects.
author_names: Sonia M Holik
title: Application of effective medium theory to the analysis of integrated circuit interconnects
year: 2010

abstract: Diffraction based overlay (DBO) technologies have been developed to address the overlay metrology challenges for 22nm technology node and beyond. Most DBO technologies require specially designed targets that consist of multiple measurement pads, which consume too much space and increase measurement time. The traditional empirical approach (eDBO) using normal incidence spectroscopic reflectometry (NISR) relies on linear response of the reflectance with respect to overlay displacement within a small range. It offers convenience of quick recipe setup since there is no need to establish a model. However it requires three or four pads per direction (x or y) which adds burden to throughput and target size. Recent advances in modeling capability and computation power enabled mDBO, which allows overlay measurement with reduced number of pads, thus reducing measurement time and DBO target space. In this paper we evaluate the performance of single pad mDBO measurements using two 3D targets that have different grating shapes: squares in boxes and L shapes in boxes. Good overlay sensitivities are observed for both targets. The correlation to programmed shifts and image based overlay (IBO) is excellent. Despite the difference in shapes, the mDBO results are comparable for square and L shape targets. The impact of process variations on overlay measurements is studied using a focus and exposure matrix (FEM) wafer. Although the FEM wafer has larger process variations, the correlation of mDBO results with IBO measurements is as good as the normal process wafer. We demonstrate the feasibility of single pad DBO measurements with faster throughput and smaller target size, which is particularly important in high volume manufacturing environment.
author_names: Jie Li

Oleg Anton Kritsun

Yongdong Liu

Prasad Dasari

C Brooks Volkman

Jiangtao Hu
title: Faster diffraction based overlay measurements with smaller targets using 3D gratings
venue: Advanced Lithography
year: 2012

abstract: We present effective medium approximation (EMA) that provides an homogeneous equivalent for the top layer of on chip interconnects in an integrated circuit. An empirical mixing model for straightforward and fast extraction of the effective dielectric constant of a slab of integrated circuit wiring and its analytical formulation based on Maxwell Garnett rule is presented for a range of aspect ratios, dielectric materials, metal fills, and frequencies. We expect this approach to find use in the electromagnetic modeling of System in Package. Experimental validation of proposed approach is demonstrated by free space measurement of a grating structure illuminated by a pair of lens horn antennas.
author_names: Sonia M Holik

Timothy D Drysdale
title: Simplified model for on chip interconnects in electromagnetic modelling of System in Package
venue: 2010 International Conference on Electromagnetics in Advanced Applications
year: 2010

abstract: Abstract Ships often operate under challenging conditions, considering that marine environment can cause failures of the structure related to overloads, fatigue, corrosion and erosion. As a consequence, advanced methods and procedures are under development for the evaluation of the on site structural performance for both traditionally and newly designed ships. One of the main challenges in this field is the live monitoring of the loads acting on the ship hull; the load data processing can lead, through suited algorithms, to a real time control of ship trim and, as a consequence, to the development of automatic or semi automatic trim control systems. The presented procedure for load reconstruction requires a well suited sensing network. In this kind of application, a high resolution, large sampling frequencies and low sensibility to possible noise factors such as moisture, electromagnetic fields or vibrations are required. These requirements lead to the choice of Fiber Bragg Grating sensors. In this paper, an experimental methodology is proposed to reconstruct the characteristics of loads acting on a fast ship, starting from a finite number of local strain data obtained with FBG sensors. Sensors positions have been defined considering the ship hull as a beam subjected to a set of standard loads acting on a fast ship and taking into account the maximum strain positions. The development of the FE model of the ship hull, obtained from a three dimensional CAD of a real powerboat obtained with a 3D scan, allows the calculation of the strain field related to a set of standard loads applied on the ship. The above mentioned data are used as input for a fast computational algorithm, in which standard and actual strain fields provided by a network of FBG sensors are compared to reach the reconstruction of global loads acting on the structure. The algorithm has been effectively applied in sailing condition to the powerboat, detecting the acting loads on the hull in real time.
author_names: Pierluigi Fanelli

Alessandro Mercuri

Simone Trupiano

Francesco Vivio

Giacomo Falcucci

Elio Jannelli
title: Live reconstruction of global loads on a powerboat using local strain FBG measurements
year: 2019

abstract: Rigorous simulations using combined finite/boundary element methods (FEM/BEM) are often too time consuming for the rapid evaluation of long surface acoustic wave (SAW) structures, and soon become impractical as device length increases. This situation is complicated by the need of subsequent design modifications and analyses. Scalar models, and in particular, transmission line (TL) based network descriptions, enable fast evaluation of arbitrary device topologies comprised of inter digital transducers (IDTs) and reflector/grating structures. For example, simulating a resonator structure using a network model reduces computation time from days or hours to minutes, allowing the identification of optimal designs, which may then be further analyzed using rigorous FEM/BEM techniques. The traditional network model (NM) consists of mismatched TLs, shunt susceptances, and transformers, and has been successfully used to simulate transduction and propagation of pure Rayleigh type and generalized SAWs in periodic structures. In these cases, it is often acceptable to neglect bulk acoustic wave (BAW) radiation, since the operational frequencies of the SAW devices are far away from the BAW radiation frequency. However, in the case of pure shear horizontal (SH) SAW orientations significant BAW scattering is frequently observed because the BAW radiation frequency takes place near the SAW bandwidth, and thus the traditional NM fails to account for the energy radiated to the bulk. In this work a modification of the traditional NM is introduced that accounts for energy loss near the upper stopband edge for a long periodic structure, thus allowing the use of the NM to adequately simulate structures along pure SH SAW orientations. The BAW radiation is modeled by a frequency dependent shunt conductance located at electrode edge discontinuities. A parameter extraction technique is presented that allows the rapid determination of frequency dependent parameters. In particular, the parameters are fit to FEM/BEM computed dispersion and harmonic admittance relations. The response of a SH SAW resonator was computed using the NM and compared to measurements of devices fabricated on langasite, Euler Angles (0deg, 22deg, 90deg) Good agreement is observed with a slight variation in center frequency ~0.25% thus verifying the modified model and its applicability to rapid design of pure shear horizontal SAW devices
author_names: T B Pollard

Mauricio Pereira da Cunha
title: 6A 3 Pure Shear Horizontal SAW Network Model for Periodic Structures Including Bulk Scattering
venue: 2006 IEEE Ultrasonics Symposium
year: 2006

abstract: A quantum random number generator (QRNG) generates genuine randomness from the intrinsic probabilistic nature of quantum mechanics. The central problems for most QRNGs are estimating the entropy of the genuine randomness and producing such randomness at high rates. Here we propose and demonstrate a fast semi self testing QRNG at a rate of 24 Mbit/s based on a high dimensional entanglement system, in which the user monitors the entropy in real time via the observation of a nonlocal quantum interference. Our work provides a practical approach to a robust QRNG with trusted but error prone devices. Introduction. Randomness is indispensable for information processing. Quantum random number generators (QRNGs) can generate true randomness by exploiting the fundamental indeterminism of quantum mechanics [1] Most current QRNGs are photonic systems built with trusted and calibrated devices [2] and provide Gbit/s generation speeds at relatively low cost. However, a central issue in these QRNGs is how to certify and quantify the entropy of the genuine randomness, i.e. the randomness that originates from the intrinsic unpredictability of measurements in quantum mechanics. Entropy estimates for specific setups were recently proposed using sophisticated theoretical models [3] Nevertheless, these techniques require parameter characterization which may be difficult to accurately assess in practice. An elegant solution to estimating the entropy is the device independent (DI) or self testing QRNG [1, 4, 5] but its practical implementation is challenging because it requires loophole free violation of Bell's inequality, resulting in low generation rates of ~1 bit/s [1, 5] Recently, Lunghi et al. proposed a more practical solution without the need for Bell violation [6] which can be termed semi self testing, in which the randomness can be guaranteed based on a few general assumptions that do not require detailed device characterization. This scheme is highly desirable as it focuses on real world implementations with trusted but error prone devices, although its implementation to date still suffers from low generation rates of tens of bits/s [6] In this work, based on high dimensional entanglement, we propose and experimentally demonstrate a simple, practical and fast semi self testing QRNG at a rate over 24 Mbits/s. The amount of genuine quantum randomness is quantified directly from observation of a nonlocal interference and a pair of incompatible quantum measurements, and it is separated from other sources of randomness such as technical noise with a randomness extractor. We achieved the high generation rate by virtue of a number of experimental features: a high dimensional time energy entangled photon source capable of producing multiple random bits per photon, a high visibility Franson interferometer for evaluating entanglement, and high efficiency superconducting nanowire single photon detectors (SNSPDs) Protocol. An entanglement source is used to generate high dimensional entangled photon pairs. In the ideal case, the Nd dimensional entangled state can be written as |ps> 1 Nd Nd 1 i=0 |i> A |i> B. (1) This state is observed by two measurement systems A and B. The randomness is generated from A, while B is used to test the states. To model imperfections, we assume that the state rA of system A is not pure and is correlated with environmental noise. Using the same concept as the semi self testing QRNG [6] we assume the devices in the protocol are not deliberately designed to fool the user, but the implementation may be imperfect. The central task is to estimate the amount of genuine randomness based only on measurements. This is a nontrivial task as the observed randomness can have different origins. If the state rA is a superposition of high dimensional states, then the outcome Ri cannot be predicted with certainty, even if the internal state is known, thus resulting in genuine quantum randomness. On the other hand, the randomness may be due to technical imperfections such as detector noise and temperature fluctuations, whose randomness clearly contains no quantum randomness, since the outcome Ri can be perfectly guessed if the imperfections were well quantified. In our approach, the amount of genuine randomness is quantified via the uncertainty relation [7, 8] from a pair of incompatible quantum measurements, namely time measurement T and frequency measurement W. T is realized by directly measuring the photon's arrival time, while W is performed by Franson interferometry [9] In particular, the Franson visibility V is used to bound the randomness. Classical fields result in V that is no greater than 50% For a maximally entangled state, V would be unity in the ideal case [9, 10] Conceptually, V 50% guarantees that the source's output is entangled and thus contains genuine randomness. More rigorously, Ref. [11] has proved that V provides an explicit bound for the correlations in W, under the assumption that the quantum state is Gaussian, which in turns bounds the conditional maximum entropy (given system B) in W via the theory developed in [8] By using the entropic uncertainty relation for smooth entropies [7] we can determine the conditional min entropy given the environmental noise and thus the guessing probability, i.e. the amount of genuine randomness. Our protocol provides self monitoring because measurements of V directly quantify the amount of genuine randomness in the observed data. A threshold value V0 is pre selected and the randomness can be generated only when the observation satisfies V V0. Two particular advantages of this approach are: (i) the observation of V does not rely on detailed models of the devices that are employed; and (ii) no loophole free Bell inequality violation is required. Experiment. Figure 1 shows our experimental setup. It uses a high dimensional time energy entanglement source based on spontaneous parametric down conversion (SPDC) in a PPKTP waveguide [12] which supports multiple spatial modes at telecom wavelengths. The 46.1 mm grating period was designed for type II quasi phase matched wavelength degenerate outputs at 1560 nm in the fundamental modes of the signal C HWP PBS Diode Laser
author_names: Feihu Xu

Jeffrey H Shapiro

Franco N C Wong
title: Experimental fast quantum random number generation using high dimensional entanglement with semi self testing
year: 2016