2008---Numerical simulation and parametric study on new type of high temperature

LEOPOLD-FRANZENS UNIVERSITYChair of Engineering Mechanicso.Univ.-Prof.Dr.-Ing.habil.G.I.Schu¨e ller,Ph.D.G.I.Schueller@uibk.ac.at Technikerstrasse13,A-6020Innsbruck,Austria,EU Tel.:+435125076841Fax.:+435125072905 mechanik@uibk.ac.at,http://mechanik.uibk.ac.atIfM-Publication2-407G.I.Schu¨e ller.Developments in stochastic structural mechanics.Archive of Applied Mechanics,published online,2006.Archive of Applied Mechanics manuscript No.(will be inserted by the editor)Developments in Stochastic Structural MechanicsG.I.Schu¨e llerInstitute of Engineering Mechanics,Leopold-Franzens University,Innsbruck,Aus-tria,EUReceived:date/Revised version:dateAbstract Uncertainties are a central element in structural analysis and design.But even today they are frequently dealt with in an intuitive or qualitative way only.However,as already suggested80years ago,these uncertainties may be quantified by statistical and stochastic procedures. In this contribution it is attempted to shed light on some of the recent advances in the now establishedfield of stochastic structural mechanics and also solicit ideas on possible future developments.1IntroductionThe realistic modeling of structures and the expected loading conditions as well as the mechanisms of their possible deterioration with time are un-doubtedly one of the major goals of structural and engineering mechanics2G.I.Schu¨e ller respectively.It has been recognized that this should also include the quan-titative consideration of the statistical uncertainties of the models and the parameters involved[56].There is also a general agreement that probabilis-tic methods should be strongly rooted in the basic theories of structural en-gineering and engineering mechanics and hence represent the natural next step in the development of thesefields.It is well known that modern methods leading to a quantification of un-certainties of stochastic systems require computational procedures.The de-velopment of these procedures goes in line with the computational methods in current traditional(deterministic)analysis for the solution of problems required by the engineering practice,where certainly computational pro-cedures dominate.Hence,their further development within computational stochastic structural analysis is a most important requirement for dissemi-nation of stochastic concepts into engineering practice.Most naturally,pro-cedures to deal with stochastic systems are computationally considerably more involved than their deterministic counterparts,because the parameter set assumes a(finite or infinite)number of values in contrast to a single point in the parameter space.Hence,in order to be competitive and tractable in practical applications,the computational efficiency of procedures utilized is a crucial issue.Its significance should not be underestimated.Improvements on efficiency can be attributed to two main factors,i.e.by improved hard-ware in terms of ever faster computers and improved software,which means to improve the efficiency of computational algorithms,which also includesDevelopments in Stochastic Structural Mechanics3 utilizing parallel processing and computer farming respectively.For a con-tinuous increase of their efficiency by software developments,computational procedure of stochastic analysis should follow a similar way as it was gone in the seventieth and eighties developing the deterministic FE approach. One important aspect in this fast development was the focus on numerical methods adjusted to the strength and weakness of numerical computational algorithms.In other words,traditional ways of structural analysis devel-oped before the computer age have been dropped,redesigned and adjusted respectively to meet the new requirements posed by the computational fa-cilities.Two main streams of computational procedures in Stochastic Structural Analysis can be observed.Thefirst of this main class is the generation of sample functions by Monte Carlo simulation(MCS).These procedures might be categorized further according to their purpose:–Realizations of prescribed statistical information:samples must be com-patible with prescribed stochastic information such as spectral density, correlation,distribution,etc.,applications are:(1)Unconditional simula-tion of stochastic processes,fields and waves.(2)Conditional simulation compatible with observations and a priori statistical information.–Assessment of the stochastic response for a mathematical model with prescribed statistics(random loading/system parameters)of the param-eters,applications are:(1)Representative sample for the estimation of the overall distribution.4G.I.Schu¨e ller Indiscriminate(blind)generation of samples.Numerical integration of SDE’s.(2)Representative sample for the reliability assessment by gen-erating adverse rare events with positive probability,i.e.by:(a)variance reduction techniques controlling the realizations of RV’s,(b)controlling the evolution in time of sampling functions.The other main class provides numerical solutions to analytical proce-dures.Grouping again according to the respective purpose the following classification can be made:Numerical solutions of Kolmogorov equations(Galerkin’s method,Finite El-ement method,Path Integral method),Moment Closure Schemes,Compu-tation of the Evolution of Moments,Maximum Entropy Procedures,Asymp-totic Stability of Diffusion Processes.In the following,some of the outlined topics will be addressed stressing new developments.These topics are described within the next six subject areas,each focusing on a different issue,i.e.representation of stochastic processes andfields,structural response,stochastic FE methods and parallel processing,structural reliability and optimization,and stochastic dynamics. In this context it should be mentioned that aside from the MIT-Conference series the USNCCM,ECCM and WCCM’s do have a larger part of sessions addressing computational stochastic issues.Developments in Stochastic Structural Mechanics5 2Representation of Stochastic ProcessesMany quantities involving randomfluctuations in time and space might be adequately described by stochastic processes,fields and waves.Typical ex-amples of engineering interest are earthquake ground motion,sea waves, wind turbulence,road roughness,imperfection of shells,fluctuating prop-erties in random media,etc.For this setup,probabilistic characteristics of the process are known from various measurements and investigations in the past.In structural engineering,the available probabilistic characteristics of random quantities affecting the loading or the mechanical system can be often not utilized directly to account for the randomness of the structural response due to its complexity.For example,in the common case of strong earthquake motion,the structural response will be in general non-linear and it might be too difficult to compute the probabilistic characteristics of the response by other means than Monte Carlo simulation.For the purpose of Monte Carlo simulation sample functions of the involved stochastic pro-cess must be generated.These sample functions should represent accurately the characteristics of the underlying stochastic process orfields and might be stationary and non-stationary,homogeneous or non-homogeneous,one-dimensional or multi-dimensional,uni-variate or multi-variate,Gaussian or non-Gaussian,depending very much on the requirements of accuracy of re-alistic representation of the physical behavior and on the available statistical data.6G.I.Schu¨e ller The main requirement on the sample function is its accurate represen-tation of the available stochastic information of the process.The associ-ated mathematical model can be selected in any convenient manner as long it reproduces the required stochastic properties.Therefore,quite different representations have been developed and might be utilized for this purpose. The most common representations are e.g.:ARMA and AR models,Filtered White Noise(SDE),Shot Noise and Filtered Poisson White Noise,Covari-ance Decomposition,Karhunen-Lo`e ve and Polynomial Chaos Expansion, Spectral Representation,Wavelets Representation.Among the various methods listed above,the spectral representation methods appear to be most widely used(see e.g.[71,86]).According to this procedure,samples with specified power spectral density information are generated.For the stationary or homogeneous case the Fast Fourier Transform(FFT)techniques is utilized for a dramatic improvements of its computational efficiency(see e.g.[104,105]).Advances in thisfield provide efficient procedures for the generation of2D and3D homogeneous Gaus-sian stochasticfields using the FFT technique(see e.g.[87]).The spectral representation method generates ergodic sample functions of which each ful-fills exactly the requirements of a target power spectrum.These procedures can be extended to the non-stationary case,to the generation of stochastic waves and to incorporate non-Gaussian stochasticfields by a memoryless nonlinear transformation together with an iterative procedure to meet the target spectral density.Developments in Stochastic Structural Mechanics7 The above spectral representation procedures for an unconditional simula-tion of stochastic processes andfields can also be extended for Conditional simulations techniques for Gaussianfields(see e.g.[43,44])employing the conditional probability density method.The aim of this procedure is the generation of Gaussian random variates U n under the condition that(n−1) realizations u i of U i,i=1,2,...,(n−1)are specified and the a priori known covariances are satisfied.An alternative procedure is based on the so called Kriging method used in geostatistical application and applied also to con-ditional simulation problems in earthquake engineering(see e.g.[98]).The Kriging method has been improved significantly(see e.g.[36])that has made this method theoretically clearer and computationally more efficient.The differences and similarities of the conditional probability density methods and(modified)Kriging methods are discussed in[37]showing the equiva-lence of both procedures if the process is Gaussian with zero mean.A quite general spectral representation utilized for Gaussian random pro-cesses andfields is the Karhunen-Lo`e ve expansion of the covariance function (see e.g.[54,33]).This representation is applicable for stationary(homoge-neous)as well as for non-stationary(inhomogeneous)stochastic processes (fields).The expansion of a stochastic process(field)u(x,θ)takes the formu(x,θ)=¯u(x)+∞i=1ξ(θ) λiφi(x)(1)where the symbolθindicates the random nature of the corresponding quan-tity and where¯u(x)denotes the mean,φi(x)are the eigenfunctions andλi the eigenvalues of the covariance function.The set{ξi(θ)}forms a set of8G.I.Schu¨e ller orthogonal(uncorrelated)zero mean random variables with unit variance.The Karhunen-Lo`e ve expansion is mean square convergent irrespective of its probabilistic nature provided it possesses afinite variance.For the im-portant special case of a Gaussian process orfield the random variables{ξi(θ)}are independent standard normal random variables.In many prac-tical applications where the random quantities vary smoothly with respectto time or space,only few terms are necessary to capture the major part of the randomfluctuation of the process.Its major advantage is the reduction from a large number of correlated random variables to few most important uncorrelated ones.Hence this representation is especially suitable for band limited colored excitation and stochastic FE representation of random me-dia where random variables are usually strongly correlated.It might also be utilized to represent the correlated stochastic response of MDOF-systems by few most important variables and hence achieving a space reduction.A generalization of the above Karhunen-Lo`e ve expansion has been proposed for application where the covariance function is not known a priori(see[16, 33,32]).The stochastic process(field)u(x,θ)takes the formu(x,θ)=a0(x)Γ0+∞i1=1a i1(x)Γ1(ξi1(θ))+∞i1=1i1i2=1a i1i2(x)Γ2(ξi1(θ),ξi2(θ))+ (2)which is denoted as the Polynomial Chaos Expansion.Introducing a one-to-one mapping to a set with ordered indices{Ψi(θ)}and truncating eqn.2Developments in Stochastic Structural Mechanics9 after the p th term,the above representations reads,u(x,θ)=pj=ou j(x)Ψj(θ)(3)where the symbolΓn(ξi1,...,ξin)denotes the Polynomial Chaos of order nin the independent standard normal random variables.These polynomialsare orthogonal so that the expectation(or inner product)<ΨiΨj>=δij beingδij the Kronecker symbol.For the special case of a Gaussian random process the above representation coincides with the Karhunen-Lo`e ve expan-sion.The Polynomial Chaos expansion is adjustable in two ways:Increasingthe number of random variables{ξi}results in a refinement of the random fluctuations,while an increase of the maximum order of the polynomialcaptures non-linear(non-Gaussian)behavior of the process.However,the relation between accuracy and numerical efforts,still remains to be shown. The spectral representation by Fourier analysis is not well suited to describe local feature in the time or space domain.This disadvantage is overcome in wavelets analysis which provides an alternative of breaking a signal down into its constituent parts.For more details on this approach,it is referred to[24,60].In some cases of applications the physics or data might be inconsistent with the Gaussian distribution.For such cases,non-Gaussian models have been developed employing various concepts to meet the desired target dis-tribution as well as the target correlation structure(spectral density).Cer-tainly the most straight forward procedures is the above mentioned memo-ryless non-linear transformation of Gaussian processes utilizing the spectralrepresentation.An alternative approach utilizes linear and non-linearfil-ters to represent normal and non-Gaussian processes andfields excited by Gaussian white noise.Linearfilters excited by polynomial forms of Poisson white noise have been developed in[59]and[34].These procedures allow the evaluation of moments of arbitrary order without having to resort to closure techniques. Non-linearfilters are utilized to generate a stationary non-Gaussian stochas-tic process in agreement with a givenfirst-order probability density function and the spectral density[48,15].In the Kontorovich-Lyandres procedure as used in[48],the drift and diffusion coefficients are selected such that the solutionfits the target probability density,and the parameters in the solu-tion form are then adjusted to approximate the target spectral density.The approach by Cai and Lin[15]simplifies this procedure by matching the spec-tral density by adjusting only the drift coefficients,which is the followed by adjusting the diffusion coefficient to approximate the distribution of the pro-cess.The latter approach is especially suitable and computationally highly efficient for a long term simulation of stationary stochastic processes since the computational expense increases only linearly with the number n of dis-crete sample points while the spectral approach has a growth rate of n ln n when applying the efficient FFT technique.For generating samples of the non-linearfilter represented by a stochastic differential equations(SDE), well developed numerical procedures are available(see e.g.[47]).3Response of Stochastic SystemsThe assessment of the stochastic response is the main theme in stochastic mechanics.Contrary to the representation of of stochastic processes and fields designed tofit available statistical data and information,the output of the mathematical model is not prescribed and needs to be determined in some stochastic sense.Hence the mathematical model can not be selected freely but is specified a priori.The model involves for stochastic systems ei-ther random system parameters or/and random loading.Please note,due to space limitations,the question of model validation cannot be treated here. For the characterization of available numerical procedures some classifi-cations with regard to the structural model,loading and the description of the stochastic response is most instrumental.Concerning the structural model,a distinction between the properties,i.e.whether it is determinis-tic or stochastic,linear or non-linear,as well as the number of degrees of freedom(DOF)involved,is essential.As a criterion for the feasibility of a particular numerical procedure,the number of DOF’s of the structural system is one of the most crucial parameters.Therefore,a distinction be-tween dynamical-system-models and general FE-discretizations is suggested where dynamical systems are associated with a low state space dimension of the structural model.FE-discretization has no essential restriction re-garding its number of DOF’s.The stochastic loading can be grouped into static and dynamic loading.Stochastic dynamic loading might be charac-terized further by its distribution and correlation and its independence ordependence on the response,resulting in categorization such as Gaussian and non-Gaussian,stationary and non-stationary,white noise or colored, additive and multiplicative(parametric)excitation properties.Apart from the mathematical model,the required terms in which the stochastic re-sponse should be evaluated play an essential role ranging from assessing thefirst two moments of the response to reliability assessments and stabil-ity analysis.The large number of possibilities for evaluating the stochas-tic response as outlined above does not allow for a discussion of the en-tire subject.Therefore only some selected advances and new directions will be addressed.As already mentioned above,one could distinguish between two main categories of computational procedures treating the response of stochastic systems.Thefirst is based on Monte Carlo simulation and the second provides numerical solutions of analytical procedures for obtaining quantitative results.Regarding the numerical solutions of analytical proce-dures,a clear distinction between dynamical-system-models and FE-models should be made.Current research efforts in stochastic dynamics focus to a large extent on dynamical-system-models while there are few new numerical approaches concerning the evaluation of the stochastic dynamic response of e.g.FE-models.Numerical solutions of the Kolmogorov equations are typical examples of belonging to dynamical-system-models where available approaches are computationally feasible only for state space dimensions one to three and in exceptional cases for dimension four.Galerkin’s,Finite El-ement(FE)and Path Integral methods respectively are generally used tosolve numerically the forward(Fokker-Planck)and backward Kolmogorov equations.For example,in[8,92]the FE approach is employed for stationary and transient solutions respectively of the mentioned forward and backward equations for second order systems.First passage probabilities have been ob-tained employing a Petrov-Galerkin FE method to solve the backward and the related Pontryagin-Vitt equations.An instructive comparison between the computational efforts using Monte Carlo simulation and the FE-method is given e.g.in an earlier IASSAR report[85].The Path Integral method follows the evolution of the(transition)prob-ability function over short time intervals,exploiting the fact that short time transition probabilities for normal white noise excitations are locally Gaus-sian distributed.All existing path integration procedures utilize certain in-terpolation schemes where the probability density function(PDF)is rep-resented by values at discrete grid points.In a wider sense,cell mapping methods(see e.g.[38,39])can be regarded as special setups of the path integral procedure.As documented in[9],cumulant neglect closure described in section7.3 has been automated putational procedures for the automated generation and solutions of the closed set of moment equations have been developed.The method can be employed for an arbitrary number of states and closed at arbitrary levels.The approach,however,is limited by available computational resources,since the computational cost grows exponentially with respect to the number of states and the selected closurelevel.The above discussed developments of numerical procedures deal with low dimensional dynamical systems which are employed for investigating strong non-linear behavior subjected to(Gaussian)white noise excitation. Although dynamical system formulations are quite general and extendible to treat non-Gaussian and colored(filtered)excitation of larger systems,the computational expense is growing exponentially rendering most numerical approaches unfeasible for larger systems.This so called”curse of dimen-sionality”is not overcome yet and it is questionable whether it ever will be, despite the fast developing computational possibilities.For this reason,the alternative approach based on Monte Carlo simu-lation(MCS)gains importance.Several aspects favor procedures based on MCS in engineering applications:(1)Considerably smaller growth rate of the computational effort with dimensionality than analytical procedures.(2) Generally applicable,well suited for parallel processing(see section5.1)and computationally straight forward.(3)Non-linear complex behavior does not complicate the basic procedure.(4)Manageable for complex systems.Contrary to numerical solutions of analytical procedures,the employed structural model and the type of stochastic loading does for MCS not play a deceive role.For this reason,MCS procedures might be structured ac-cording to their purpose i.e.where sample functions are generated either for the estimation of the overall distribution or for generating rare adverse events for an efficient reliability assessment.In the former case,the prob-ability space is covered uniformly by an indiscriminate(blind)generationof sample functions representing the random quantities.Basically,at set of random variables will be generated by a pseudo random number generator followed by a deterministic structural analysis.Based on generated random numbers realizations of random processes,fields and waves addressed in section2,are constructed and utilized without any further modification in the following structural analysis.The situation may not be considered to be straight forward,however,in case of a discriminate MCS for the reliability estimation of structures,where rare events contributing considerably to the failure probability should be gener-ated.Since the effectiveness of direct indiscriminate MCS is not satisfactory for producing a statistically relevant number of low probability realizations in the failure domain,the generation of samples is restricted or guided in some way.The most important class are the variance reduction techniques which operate on the probability of realizations of random variables.The most widely used representative of this class in structural reliability assess-ment is Importance Sampling where a suitable sampling distribution con-trols the generation of realizations in the probability space.The challenge in Importance Sampling is the construction of a suitable sampling distribu-tion which depends in general on the specific structural system and on the failure domain(see e.g.[84]).Hence,the generation of sample functions is no longer independent from the structural system and failure criterion as for indiscriminate direct MCS.Due to these dependencies,computational procedures for an automated establishment of sampling distributions areurgently needed.Adaptive numerical strategies utilizing Importance Direc-tional sampling(e.g.[11])are steps in this direction.The effectiveness of the Importance sampling approach depends crucially on the complexity of the system response as well as an the number of random variables(see also section5.2).Static problems(linear and nonlinear)with few random vari-ables might be treated effectively by this approach.Linear systems where the randomness is represented by a large number of RVs can also be treated efficiently employingfirst order reliability methods(see e.g.[27]).This ap-proach,however,is questionable for the case of non-linear stochastic dynam-ics involving a large set of random variables,where the computational effort required for establishing a suitable sampling distribution might exceed the effort needed for indiscriminate direct MCS.Instead of controlling the realization of random variables,alternatively the evolution of the generated sampling can be controlled[68].This ap-proach is limited to stochastic processes andfields with Markovian prop-erties and utilizes an evolutionary programming technique for the genera-tion of more”important”realization in the low probability domain.This approach is especially suitable for white noise excitation and non-linear systems where Importance sampling is rather difficult to apply.Although the approach cannot deal with spectral representations of the stochastic processes,it is capable to make use of linearly and non-linearlyfiltered ex-citation.Again,this is just contrary to Importance sampling which can be applied to spectral representations but not to white noisefiltered excitation.4Stochastic Finite ElementsAs its name suggests,Stochastic Finite Elements are structural models rep-resented by Finite Elements the properties of which involve randomness.In static analysis,the stiffness matrix might be random due to unpredictable variation of some material properties,random coupling strength between structural components,uncertain boundary conditions,etc.For buckling analysis,shape imperfections of the structures have an additional impor-tant effect on the buckling load[76].Considering structural dynamics,in addition to the stiffness matrix,the damping properties and sometimes also the mass matrix might not be predictable with certainty.Discussing numerical Stochastic Finite Elements procedures,two cat-egories should be distinguished clearly.Thefirst is the representation of Stochastic Finite Elements and their global assemblage as random structural matrices.The second category addresses the evaluation of the stochastic re-sponse of the FE-model due to its randomness.Focusingfirst on the Stochastic FE representation,several representa-tions such as the midpoint method[35],the interpolation method[53],the local average method[97],as well as the Weighted-Integral-Method[94,25, 26]have been developed to describe spatial randomfluctuations within the element.As a tendency,the midpoint methods leads to an overestimation of the variance of the response,the local average method to an underestima-tion and the Weighted-Integral-Method leads to the most accurate results. Moreover,the so called mesh-size problem can be resolved utilizing thisrepresentation.After assembling all Finite Elements,the random structural stiffness matrix K,taken as representative example,assumes the form,K(α)=¯K+ni=1K Iiαi+ni=1nj=1K IIijαiαj+ (4)where¯K is the mean of the matrix,K I i and K II ij denote the determinis-ticfirst and second rate of change with respect to the zero mean random variablesαi andαj and n is the total number of random variables.For normally distributed sets of random variables{α},the correlated set can be represented advantageously by the Karhunen-Lo`e ve expansion[33]and for non-Gaussian distributed random variables by its Polynomial chaos ex-pansion[32],K(θ)=¯K+Mi=0ˆKiΨi(θ)(5)where M denotes the total number of chaos polynomials,ˆK i the associated deterministicfluctuation of the matrix andΨi(θ)a polynomial of standard normal random variablesξj(θ)whereθindicates the random nature of the associated variable.In a second step,the random response of the stochastic structural system is determined.The most widely used procedure for evaluating the stochastic response is the well established perturbation approach(see e.g.[53]).It is well adapted to the FE-formulation and capable to evaluatefirst and second moment properties of the response in an efficient manner.The approach, however,is justified only for small deviations from the center value.Since this assumption is satisfied in most practical applications,the obtainedfirst two moment properties are evaluated satisfactorily.However,the tails of the。

Numerical Simulation of Optical Wave PropagationWith examples in MATLAB®Library of Congress Cataloging-in-Publication DataSchmidt, Jason Daniel, 1975-Numerical simulation of optical wave propagation with examples in MATLAB / Jason D. Schmidt.p. cm. -- (Press monograph ; 199)Includes bibliographical references and index.ISBN 978-0-8194-8326-31. Optics--Mathematics.2. Wave-motion, Theory of--Mathematical models.3. MATLAB. I. Title.QC383.S36 2010535'.42015118--dc222010015089Published bySPIEP.O. Box 10Bellingham, Washington 98227-0010 USAPhone: +1 360.676.3290Fax: +1 360.647.1445Email: Books@Web: Copyright © 2010 Society of Photo-Optical Instrumentation Engineers (SPIE)All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher.The content of this book reflects the work and thoughts of the author(s).Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.Printed in the United States of America.About the cover: 50-watt laser for generating mesospheric sodium guide stars over 90 km above the ground. In operation at the Air Force Research Laboratory's 3.5-m telescope at the Starfire Optical Range, Kirtland AFB, NM. (Robert Q. Fugate, © 2005, Albuquerque, NM).Numerical Simulation of Optical Wave PropagationWith examples in MATLAB®Jason D. SchmidtBellingham, Washington USAContentsPreface (ix)Chapter 1 Foundations of Scalar Diffraction Theory (1)1.1 Basics of Classical Electrodynamics (1)1.1.1 Sources of electric and magnetic fields (2)1.1.2 Electric and magnetic fields (2)1.2 Simple Traveling-Wave Solutions to Maxwell's Equations (5)1.2.1 Obtaining a wave equation (5)1.2.2 Simple traveling-wave fields (7)1.3 Scalar Diffraction Theory (9)12............................................................................................Problems1.4Chapter 2 Digital Fourier Transforms (15)2.1 Basics of Digital Fourier Transforms (15)2.1.1 Fourier transforms: from analytic to numerical (15)2.1.2 Inverse Fourier transforms: from analytic to numerical (17)2.1.3 Performing discrete Fourier transforms in software (18)2.2 Sampling Pure-Frequency Functions (21)2.3 Discrete vs Continuous Fourier Transforms (23)2.4 Alleviating Effects of Discretization (26)2.5 Three Case Studies in Transforming Signals (30)Sincsignals (30)2.5.12.5.2 Gaussian signals (31)2.5.3 Gaussian signals with quadratic phase (33)2.6 Two-Dimensional Discrete Fourier Transforms (35)37............................................................................................Problems2.7Chapter 3 Simple Computations Using Fourier Transforms (39)39......................................................................................Convolution3.143........................................................................................3.2Correlation3.347Functions............................................................................Structure50........................................................................................Derivatives3.43.5 Problems (53)Chapter 4 Fraunhofer Diffraction and Lenses (55)4.1 Fraunhofer Diffraction (55)4.2 Fourier-Transforming Properties of Lenses (58)4.2.1 Object against the lens (59)4.2.2 Object before the lens (59)4.2.3 Object behind the lens (61)4.3 Problems (64)Chapter 5 Imaging Systems and Aberrations (65)65........................................................................................5.1Aberrations5.1.1 Seidel aberrations (66)5.1.2 Zernike circle polynomials (66)5.1.2.1 Decomposition and mode removal (73)5.1.2.2 RMS wavefront aberration (75)5.2 Impulse Response and Transfer Function of Imaging Systems (77)5.2.1 Coherent imaging (77)5.2.2 Incoherent imaging (79)5.2.3 Strehl ratio (82)............................................................................................84Problems5.3Chapter 6 Fresnel Diffraction in Vacuum (87)6.1 Different Forms of the Fresnel Diffraction Integral (88)6.2 Operator Notation (89)6.3 Fresnel-Integral Computation (90)6.3.1 One-step propagation (90)6.3.2 Two-step propagation (92)6.4 Angular-Spectrum Propagation (95)6.5 Simple Optical Systems (102)6.6 Point Sources (107)..........................................................................................113 Problems6.7Chapter 7 Sampling Requirements for Fresnel Diffraction (115)7.1 Imposing a Band Limit (115)7.2 Propagation Geometry (117)7.3 Validity of Propagation Methods (120)7.3.1 Fresnel-integral propagation (120)7.3.1.1 One step, fixed observation-plane grid spacing (120)7.3.1.2 Avoiding aliasing (121)7.3.2 Angular-spectrum propagation (124)7.3.3 General guidelines (128)7.4 Problems (130)Chapter 8 Relaxed Sampling Constraints with PartialPropagations (133)8.1 Absorbing Boundaries (134)8.2 Two Partial Propagations (135)8.3 Arbitrary Number of Partial Propagations (138)8.4 Sampling for Multiple Partial Propagations (139)8.5 Problems (146)Chapter 9 Propagation through Atmospheric Turbulence (149)9.1 Split-Step Beam Propagation Method (149)9.2 Refractive Properties of Atmospheric Turbulence (150)9.2.1 Kolmogorov Theory of turbulence (152)9.2.2 Optical propagation through turbulence (156)9.2.3 Optical parameters of the atmosphere (157)9.2.4 Layered atmosphere model (164)9.2.5 Theory (164)9.3 Monte-Carlo Phase Screens (166)9.4 Sampling Constraints (172)9.5 Executing Properly Sampled Simulation (174)9.5.1 Determine propagation geometry and turbulenceconditions (174)9.5.2 Analyze the sampling constraints (176)9.5.3 Perform a vacuum simulation (178)9.5.4 Perform the turbulent simulations (179)9.5.5 Verify the output (180)9.6 Conclusion (182)9.7 Problems (183)Appendix A Function Definitions (185)Appendix B MATLAB Code Listings (187)References (189)Index (195)PrefaceDiffraction is a very interesting and active area of optical research.Unfortunately, analytic solutions are rare in many practical problems,particularly when optical waves propagate through randomlyfluctuating media.For many of these problems, researchers must resort to numerical solutions.Still,simulations in optical diffrac-tion are ually,these simulations take advantage of discrete Fourier transforms,which means using discretely spaced samples on afinite-sized grid. This leads to a few tradeoffs in speed and memory versus accuracy.Thus,the pa-rameters of the sampling grids must be chosen very carefully.Some people seek to fully automate those choices,but this cannot be done automatically in every case. To determine grid properties,one must carefully consider computational speed, available computer memory,the Nyquist sampling criterion,geometry,accurate representation of source apertures,and impact on the propagatedfield’s quantities of interest.This book grew out of an independent study I did while I was a doctoral student at University of Dayton.The study was directed by LtCol Matthew Goda,then a professor at the Air Force Institute of Technology(AFIT).After the independent study was over,Goda then created a course at AFIT on wave-optics simulations. When I graduated,I became a professor at AFIT while Goda moved on to a new military assignment.When I began teaching the wave-optics simulation course, there was no book written to the level of detail required for a graduate course fo-cused on wave-optics simulations and sampling requirements.The course was al-ways taught out of the professor’s notes,originally compiled by piling these notes was no small feat,and Goda did a tremendous job combining material from books on discrete Fourier transforms,optics journal articles and conference proceedings,technical reports from companies like the Optical Sciences Company and MZA Associates Corporation,and private communication with researchers.Until this book,simulations have always been an afterthought in just a few books on image processing and nonlinear optics.Clearly there was a gap between the practical knowledge required to perform wave-optics simulations and the the-oretical material covered in great Fourier-optics textbooks like those by Joseph Goodman and Jack Gaskill.I have heard professors across the U.S.talk about how they include material on simulations in their graduate Fourier-optics courses.I ap-plaud them for that effort because it is challenging to teach students both the the-ory and practical simulation of Fourier optics in one course.However,if the stu-ixdents are to become capable enough to write wave-optics simulations for thesis or dissertation research and beyond,they cannot get enough detail in a one-term Fourier-optics course.This is why AFIT has separate courses on Fourier optics and wave-optics simulations.This book is intended for graduate students in programs like physics,electrical engineering,electro-optics,or optical science.The book gives all of the relevant equations from Fourier optics,but to fully understand and appreciate the material, it is important to have a thorough understanding of Fourier optics before reading this book.I believe that part of the benefit of this book is the use of specific code examples, rather than just pseudo-code.However,the programming or scripting language for the examples needs to be one that is widely used and easy to understand by those who do not already use it.For those reasons,I have used M ATLAB in all of the examples throughout this book.It is heavily used in engineering both at universities and research institutions.Further,it is easy to read because of its simple language and because many numerical algorithms,such as discrete Fourier transforms and convolution,are part of its basic library.If I used other languages like C,C++, FORTRAN,Java,and Python,I would need to pick a particular external library of numerical routines or write my own algorithms and include them in the book.I believe that using M ATLAB in this book allows readers to focus on the wave propagation,rather than the most basic numerical algorithms like discrete Fourier transforms.Further,any user with access to the M ATLAB interpreter can execute the code examples as shown.No additional libraries need to be acquired and installed. Moreover,my examples rarely use M ATLAB’s toolboxes,relying heavily on its basic functionality.Readers should note that the code examples used throughout the book are designed for conceptual simplicity,rather than optimized for speed or memory usage.I encourage readers to rework my M ATLAB examples to achieve greater performance or even implement them in other languages.I offer my thanks and appreciation to all those who have paved the way for this work,particularly Glenn Tyler,David Fried,and Phillip Roberts at the Optical Sciences Company and Steve Coy at MZA Associates Corporation.In1982,Fried and Tyler wrote a technical report describing methods of simulating optical wave propagation and related sampling constraints.A few years later,Roberts wrote a follow-on report giving another clear,nicely detailed description of one-step,two-step,and angular spectrum propagation methods.More recently,Coy wrote a tech-nical report that gives a very nice description of the relationship between sampling requirements propagation geometry.These reports formed the beginnings of Goda’s notes and eventually this book.Also,thanks to those who answered my questions about wave-optics simula-tions while I was a student at UD and then while I taught the wave-optics simula-tion course as a professor at AFIT:Jeffrey Barchers,Troy Rhoadarmer,Terry Bren-nan,and Don Link.These gentlemen are experienced and accomplished researcherswhose advice was very much appreciated.Additionally,thanks to Michael Havrilla for his help with the basic electrodynamics in Ch.1.Special thanks to Matthew Goda for his foundational work in the course and its notes.Without him,this book would not be possible.He made much of the material in this book accessible to dozens of students who went on to do great things for the U.S.Air Force.Finally,I’d like to thank all those students who helpedfind errors in the drafts of this book and whose inquisitive nature caused me to refine and add material along the way.Jason SchmidtJune,2010Chapter1Foundations of Scalar Diffraction TheoryLight can be described by two very different approaches:classical electrodynam-ics and quantum electrodynamics.In the classical treatment,electric and magnetic fields are continuous functions of space and time,and light comprises co-oscillating electric and magnetic wavefields.In the quantum treatment,photons are elemen-tary particles with no mass nor charge,and light comprises one or more photons. There is rigorous theory behind each approach,and there is experimental evidence supporting both.Neither approach can be dismissed,which leads to the wave-particle duality of light.Generally,classical methods are used for macroscopic properties of light,while quantum methods are used for submicroscopic proper-ties of light.This book describes macroscopic properties,so it deals entirely with classi-cal electrodynamics.When the wavelengthλof an electromagnetic wave is very small,approaching zero,the waves travel in straight lines with no bending around the edges of objects.That is realm of geometric optics.However,this book treats many situations in which geometric optics are inadequate to describe observed phe-nomena like diffraction.Therefore,the starting point is classical electrodynamics with solutions provided by scalar diffraction theory.Geometric optics is treated briefly in Sec.6.5.1.1Basics of Classical ElectrodynamicsClassical electrodynamics deals with relationships between electricfields,magnetic fields,static charge,and moving charge(i.e.,current)in space and time based on the macroscopic properties of the materials in which thefields exist.We define each quantity here along with some basic relationships.This introduces the reader to the quantities in Maxwell’s equations,which describe how electrically charged particles and objects give rise to electric and magneticfields.Maxwell’s equations are introduced here in their most general form,and then the discussion focuses on a specific case and solutions for oscillating electric and magneticfields,which light comprises.12Chapter 11.1.1Sources of electric and magneticfieldsElectric charge,measured in coulombs,is a fundamental property of elementary particles and bulk materials.Classically,charge may be positive,negative,or zero. Further,charge is quantized,specifically the smallest possible nonzero amount of charge is the elementary charge e=1.602×10−19C.All nonzero amounts of charge are integer multiples of e.For bulk materials,the integer may be very large so that total charge can be treated as continuous rather than discrete.We denote the volume density of free charge,measured in coulombs per cubic meter,byρ(r,t), where r is a three-dimensional spatial vector,and t is time.Moving charge density is called free volume current density J(r,t).V olume current density is measured in Ampères per square meter(1A=1C/s).This represents the time rate at which charge passes through a surface of unit area.Finally,charge is conserved,meaning that the total charge of any system is constant.This is mathematically stated by the continuity equation=0.(1.1)∇·J(r,t)+∂ρ(r,t)∂tAlmost every material we encounter in life is composed of many,many atoms each with many positive and negative ually,the numbers of positive and negative charges are equal or nearly equal so that the whole material is electrically neutral.Still,such a material can give rise to electric or magneticfields when the to-tal charge and free current are zero.If the distribution of charge is not homogeneous or if the charges are circulating in tiny current loops,fields could be present.The separation of charge is described by the electric dipole moment,which is the amount of separated charge times the separation distance.If a bulk material has its charge arranged in many tiny dipoles,it is said to be electrically polarized.The volume polarization density P(r,t)is the density of electric dipole moments per unit volume,measured in coulombs per square meter.Magnetization is a similar concept for moving charge.Charge circulating in a tiny current loop is described by magnetic dipole moment,which is the circulat-ing current times the area of the loop.When a bulk material has internal current arranged in many tiny loops,it is said to be magnetized.The volume magnetiza-tion density M(r,t)is the density of magnetic dipole moments per unit volume, measured in Ampères per meter.1.1.2Electric and magneticfieldsWhen a hypothetical charge,called a test charge,passes near a bulk material that has non-zeroρ,J,P,or M,the charge experiences a force.This interaction is char-acterized by two vectors E and B.The electromagnetic force F on a test particle at a given point and time is a function of these vectorfields and the particle’s charge q and velocity v.The Lorentz force law describes this interaction asF=q(E+v×B).(1.2)If this empirical statement is valid(and,of course,countless experiments over the course of centuries have shown that it is),then two vectorfields E and B are thereby defined throughout space and time,and these are called the“electricfield”and “magnetic induction.”1Eq.(1.2)can be examined in a little more detail to provide more intuitive defini-tions of thesefields.The electricfield is the amount of force per unit of test charge when the test charge is stationary,given byE=limq→0+Fq v=0.(1.3)This is called a push-and-pull force because the force is in either the same or op-posite direction as thefield,depending on the sign of the charge.Electricfield is measured in units of volts per meter(1V=1N m/C).The magneticfield is related to the amount of force per unit test charge given byv×B=limq→0+F−q Eq v=0.(1.4)The force due to a magneticfield is called deflective because it is perpendicular to the particle’s velocity,which deflects its trajectory.Magneticfield is measured in units of Tesla[1T=1N s/(C m)].With this understanding of thefields,they now need to be related to the sources. This was accomplished through centuries of experimental measurements and theo-retical and intuitive insight,resulting in∇×E+∂B∂t=0(1.5)∇×B−µ0 0∂E∂t=µ0 J+∂P∂t+∇×M .(1.6)These are two of Maxwell’s equations,the former being Faraday’s law and the latter being Ampère’s law with Maxwell’s correction.In Eq.(1.6),the sources on the right hand side include the free current J and two terms due to bound currents. These are the polarization current∂P/∂t and the magnetization current∇×M.These equations can be written in a more functionally useful form.Eq.(1.6) can be rewritten as∇× Bµ0−M =J+∂∂t( 0E+P).(1.7) Making the definitionsD= 0E+P(1.8)H=Bµ0−M(1.9)Foundations of Scalar Diffraction Theory3introduces the concepts of electric displacement D and magnetic field H ,which are fields that account for the medium’s response to the applied fields.Now,the working form of these Maxwell equations becomes∇×E =−∂B∂t (1.10)∇×H =J +∂D ∂t .(1.11)Further,when these are combined with conservation of charge expressed in Eq.(1.1),this leads to∇·∇×H =∇·J +∂∂t ∇·D (1.12)=−∂ρ∂t +∂∂t∇·D (1.13)=0.(1.14)Focusing on the right-hand side,∂∂t(∇·D −ρ)=0(1.15)∇·D −ρ=f (r ),(1.16)where f (r )is an unspecified function of space but not time.Causality requires that f (r )=0before the source is turned on,yielding Coulomb’s law:∇·D =ρ.(1.17)Similar manipulations yield∇·B =0.(1.18)This indicates that magnetic monopole charges do not exist.Finally,Eqs.(1.10),(1.11),(1.17),and (1.18)constitute Maxwell’s equations.1In this model of macroscopic electrodynamics,Eqs.(1.10)and (1.11)are two independent vector equations.With three scalar components each,these are six in-dependent scalar equations.Unfortunately,given knowledge of the sources,there are four unknown vector fields D ,B ,H ,and E .Each has three scalar components for a total of twelve unknown scalars.With so many more unknown field compo-nents than equations,this is a poorly posed problem.The key is to understand the medium in which the fields exist.This produces a means of relating P to E and M to H ,which amount to six more scalar equations.For example,in simple media (linear,homogeneous,and isotropic),P = 0χe E(1.19)M =χm H ,(1.20)4Chapter 1Foundations of Scalar Diffraction Theory5 whereχe is the electric susceptibility of the medium andχm is its magnetic sus-ceptibility.Substituting these into Eqs.(1.8)and(1.9)yieldsD= 0E+P(1.21)= 0(1+χm)E(1.22)= E(1.23) andB=µ0(H+M)(1.24)=µ0(1+χm)H(1.25)=µH,(1.26) where =(1+χe) 0is the electric permittivity andµ=(1+χm)µ0is the magnetic permeability of the medium.Now,this simplifies Eqs.(1.10)and(1.11) so that∇×E=−µ∂H∂t(1.27)∇×H=J+ ∂E∂t.(1.28) Now,there are still six equations but only six unknowns(as long as the free current density J is known).Finally,with a proper understanding of the materials,this is a well posed problem.1.2Simple Traveling-Wave Solutions to Maxwell’s Equations There are many solutions to Maxwell’s equations,but there are only a few that can be written in closed form without an integral.This section begins with transforming Maxwell’s four equations into two uncoupled wave equations.It continues with a few specific simple solutions such as the infinite-extent plane wave.A more general solution is left to the next section.1.2.1Obtaining a wave equationThis book deals with optical wave propagation through linear,isotropic,homoge-neous,nondispersive,dielectric media in the absence of source charges and cur-rents.In this case,the media discussed throughout the remainder of this book have=a scalar,independent ofλ,r,t(1.29)µ=µ0(1.30)ρ=0(1.31)J=0.(1.32)Taking the curl of Eq.(1.27)yields∇×(∇×E )=−µ0∂∂t (∇×H ).(1.33)Then,substituting in Eq.(1.28)gives∇×(∇×E )=−µ0 ∂2∂t 2E .(1.34)Now,applying the vector identity ∇×(∇×E )=∇(∇·E )−∇2E leads to∇(∇·E )−∇2E =−µ0 ∂2∂t 2E .(1.35)Finally,substituting in Eqs.(1.17)and (1.23),and keeping in mind that is inde-pendent of position results in a wave differential equation:∇2E −µ0 ∂2∂t 2E =0.(1.36)Similar manipulations beginning with the curl of Eq.(1.28)yield∇2B −µ0 ∂2∂t 2B =0.(1.37)When the Laplacian is used on the Cartesian components of E and B ,the result is six uncoupled but identical equations of the form∇2−µ0 ∂2∂t2 U (x,y,z )=0,(1.38)where the scalar U (x,y,z )stands for any of the x -,y -or z -directed components of the vector fields E and B .At this point,we can define index of refraction n = 0(1.39)and the vacuum speed of light c =1√µ0 0(1.40)so that∇2−n 2c 2∂2∂t 2 U (x,y,z )=0.(1.41)The electric and magnetic fields that compose light are traveling wave fields.There-fore,fields with harmonic time dependence exp (−i 2πνt )(where νis the wave 6Chapter 1frequency)are the types of solutions sought for the purposes of this book.When this is substituted into Eq.(1.41),the result is∇2+ 2πnνc 2 U=0.(1.42) Typically,the wavelength is given byλ=c/ν,and the wavenumber is defined as k=2π/λso that ∇2+k2n2 U=0.(1.43) This is the Helmholtz equation,and it appears in many other branches of physics including thermodynamics and quantum mechanics.At this point,we can dispense with the time dependence since it is the same for all solutions of the Helmholtz equation.From this point forward,thefield U(x,y,z)refers to the phasor por-tion of the opticalfield(i.e,no time dependence).Further,we define the units of U(x,y,z)to be square-root watts per meter(1W=1J/s=1N m/s)so that optical irradiance I=|U|2is in units of watts per meter squared.The value of the electric field or magnetic induction can always be obtained by a simple conversion of units.1.2.2Simple traveling-wavefieldsThere are several simple traveling-wavefields that are useful in this book.These are planar,spherical,and Gaussian-beam waves.With each of these solutions,thefield at all points always maintains its planar,spherical,or Gaussian-beam form,and pa-rameters like radius of curvature change in a simple manner as the wave propagates. The next section on scalar diffraction theory handles more general cases.A planar wave is the simplest possible traveling wave.It has uniform amplitude and phase in any plane perpendicular to its direction of propagation.More gener-ally,when the optical axis is not along the direction of propagation,a planar wave field is given byU P(r)=A exp(i k·r),(1.44) where A is the amplitude of the wave andk=2πλ(αˆx+βˆy+γˆz)(1.45)is the wavevector with direction cosines given byα,β,andγ.Then,making the direction cosines more explicit,U P(r)=A exp i2πλ(αx+βy+γz) .(1.46)This wave travels at an angle cos−1αfrom the x-axis and cos−1βfrom the y-axis as shown in Fig.1.1.Foundations of Scalar Diffraction Theory7yzFigure 1.1Depiction of direction cosines α,β,and γ.A spherical wave is the next simplest wave field.It has a wavefront that is spherical in shape,and it is either diverging or converging.The energy of the wave is spread uniformly over a spherical surface with area given by 4πR 2,where R is the wavefront radius of curvature.Conservation of energy requires that the ampli-tude is accordingly proportional to R −1.A spherical wave is given byU S (r )=A exp [ikR (r )]R (r ).(1.47)If the center of the sphere is located at r c =(x c ,y c ,z c ),then at an observation point r =(x,y,z ),the radius of curvature is given by R (r )= (x −x c )2+(y −y c )2+(z −z c )2.(1.48)Often in optics,attention is restricted to regions of space that are very close to the optical axis.This is called the paraxial approximation,and assuming propagation in the positive z direction,this approximation is mathematically written ascos −1α 1(1.49)cos −1β 1.(1.50)With this approximation,we eliminate the square root by expanding it as a Taylor series and keeping only the first two terms,yielding R (r ) ∆z 1+12 x −x c ∆z 2+12 y −y c ∆z2 ,(1.51)where we have defined ∆z =|z −z c |.With the paraxial approximation,a spherical wave is approximatelyU S (r ) A e ik ∆z ∆z e i k 2∆z [(x −x c )2+(y −y c )2].(1.52)8Chapter 1Onefinal simple traveling wave often encountered in optics is the Gaussian-beam wave.It has a Gaussian amplitude profile and“paraxially spherical”wave-front.The full derivation of the Gaussian-beam solution invokes the paraxial ap-proximation along the way.Such a derivation can be found in common laser text-books like Refs.2–3.This solution is given byU G(r)=Aq(z)exp ik x2+y22q(z) ,(1.53)where1 q(z)=1R(z)+iλπW2(z)(1.54)and the beam radius and wavefront radius of curvature are given byW2(z)=W20 1+ λzπW20 2 (1.55)R(z)=z 1+ πW20λz 2 ,(1.56)where W0is the minimum spot radius.At any point along the z axis,W(z)is the 1/e radius of thefield amplitude.Also,by this convention,W(0)=W0so that the minimum spot radius is located at z=0.1.3Scalar Diffraction TheoryOften,the optical source is not a simple planar,spherical,nor Gaussian-beam wave. For more general cases,we must use more sophisticated means to solve the scalar Helmholtz equation.This means taking advantage of Green’s theorem with clever use of boundary conditions.This process is not discussed in detail here,but the interested reader should consult books like Refs.4–5for a detailed treatment.The geometry for this more general case is shown in Fig.1.2.In thisfigure, the coordinates are r1=(x1,y1)in the source plane and r2=(x2,y2)in the observation plane.The distance between the two planes is∆z.Thefigure illus-trates the basic problem:given the source-plane opticalfield U(x1,y1),what is the observation-planefield U(x2,y2)?The solution is given by the Fresnel diffraction integralU(x2,y2)=e ik∆ziλ∆z∞−∞∞−∞U(x1,y1)e i k2∆z[(x1−x2)2+(y1−y2)2]dx1dy1.(1.57)Note that this is not the most general solution.In fact,it is a paraxial approximation, but it is general enough and accurate enough for the purposes of this book. Foundations of Scalar Diffraction Theory9。

１．９１ ２．０８
１．７４ ｌ－６１
１．５６ １．７０
１．１６
１．１
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１．１６
２．１．２盲孔半径影响 增加盲孔的半径，由１５．８７５ｍｍ增加到２０ｒａｍ，
得到的开孔及毛刺情况如图４所示，重新计量孔边 毛刺的高度，如表３所示。
表３盲孔半径为２０ｍｍ时的最大毛刺高度 Ｔａｂ．３ Ｔｈｅ ｍａｘｉｍｕｍ ｆｌａｓｈｅｓ ｗｈｅｎ Ｒ一２０ｍｍ
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表２射流速度对毛刺高度的影响 Ｔｈｅ ｅｆｆｅｃｔ ｏｆ ｅｆｆｌｕｘ ｖｅｌｏｃｉｔｙ ｏｎ ｆｌａｓｈ ｈｅｉｇｈｔ
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模拟不同盲孔直径、不同孔深、不同射流速度等情 况下的碰撞过程，研究不同情况下射流开孔的情况， 并判断开孔形貌与射孔枪参数的关系。
表１ ４０Ｃ’ＮｉＭｏ材料性能 Ｔａｂ．１ Ｍｅｃｈａｎｉｃａｌ ｐｒｏｐｅｒｔｉｅｓ ｏｆ ４０ＣｒＮｉＭｏ
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4.3 Type B approaches.................................................................................................... 7
4.3.1 Item B1............................................................................................................................ 8 4.3.2 Item B2............................................................................................................................ 9 4.3.3 Derivative of algorithm of Type B................................................................................. 10
Next, we considered comprehensively the pros and cons of the long term development of Sudoku games and defined metrics of five difficulty levels with vague boundary standards. Within the levels we established, we have provided numerical and quantitative descriptions for the difficulties of the puzzles in the same level.