Wave Propagation in Linear Media | re-examined

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8 Application of the quadrature routine OscInt[fneg[a, 0,-c,0,w,k], QuadZero[a, -c,0,o12],1,FunctionType->ZeroList,opts]) + (OscInt[fpos[a, b, c,d,w,k], QuadZero[a, b +c,d,o21],1,FunctionType->ZeroList,opts] + OscInt[fneg[a,-b,-c,d,w,k], QuadZero[a,-b -c,d,o22],1,FunctionType->ZeroList,opts]) ) RectConst[w,k] ]; The second possibility to compute this integral is not to use the approximate partitioning strategy but employ the exact zeros for this purpose. Since we do not need this module for any other function than the one inside the tunnel, we need not write it as a template. So we pass only the parameters of the wave to the module but keep the names of the integrands xed. This time we call the quadrature routine without a value for the option FunctionType because it defaults to Argument which is exactly what we want in order to compute the zeros numerically. The rest of the module is the same as the previously discussed version. RectTransExact[x_, t_, w_, k_, opts___Rule] := Module[fa,b,c,d, wp = WorkingPrecision/.foptsg/.Options[OscInt]g, a = -t; b = N[2 Pi k/Sqrt[w],wp+2]; c = x; d = N[-2 Pi k,wp+2]; (-(OscInt[RectTransPos[a, 0, c,0,w,k], PhiArg[a, 0, c,0],1,opts] + OscInt[RectTransNeg[a, 0,-c,0,w,k], PhiArg[a, 0,-c,0],1,opts]) + (OscInt[RectTransPos[a, b, c,d,w,k], PhiArg[a, b, c,d],1,opts] + OscInt[RectTransNeg[a,-b,-c,d,w,k], PhiArg[a,-b,-c,d],1,opts]) ) RectConst[w,k] ]; The template for the calculation of the evanescent part of the wave is di erent. First, we do not split the integrand and thus have only one call of the integration routine. Second, as we use the built-in Mathematica function NIntegrate, we must set a number of options according to the values given by the user. As method for the evaluation of this de nite integral we choose again DoubleExponential. A third di erence is that this module can be used for all waveforms and therefore the waveform-dependent constant must be passed to the module as an argument. EvanTemp[f_, const_, opts___Rule] := Module[fwp = WorkingPrecision/.foptsg/.Options[OscInt], ag = AccuracyGoal/. foptsg/.Options[OscInt], 202
8.3 Implementation of the quadrature modules pg = PrecisionGoal/. foptsg/.Options[OscInt], mi = MinRecursion/. foptsg/.Options[OscInt], ma = MaxRecursion/. foptsg/.Options[OscInt]g, NIntegrate[f[xi],fxi,-1,1g, Method->DoubleExponential, WorkingPrecision->wp, AccuracyGoal->ag, PrecisionGoal->pg, MinRecursion->mi, MaxRecursion->ma] const ]; The functions discussed so far are internal only and not visible to the user | in contrast to the driver functions. These are used to provide a consistent user interface that is easy to control. For the transmitted part of the wave function in the tunnel, the module PhiTrans, for example, looks at the user speci ed values of the options Shape and PPoints or their default values, respectively, to call the appropriate quadrature module. Furthermore, warning and error messages are generated if an option has an unkown or inappropriate value. The argument n is a geometrical parameter that applies only to the Gaussian waveform. Thus we must provide an alternative way to call the module where n is not speci ed. This is the second rule given for the function name, which is applicable for rectangular and triangular waves, but issues an error when called with Shape->Gauss. In the other cases, it calls the main function with n set to the dummy value one. The message bodies are declared in the package header. PhiTrans[x_, t_, w_, k_, n_?Positive, opts___Rule] := Module[fsh = Shape/. foptsg/.Options[Phi], pt = PPoints/.foptsg/.Options[Phi]g, Switch[sh, Rect, Switch[pt, Approximate, RectTransTemp[RectTransPos, RectTransNeg,x,t,w,k,opts], Zeros, RectTransExact[x,t,w,k,opts], _, Message[Phi::invalidpart,pt]], Tria, Switch[pt, Approximate, TriaTransTemp[TriaTransPos, TriaTransNeg,x,t,w,k,opts], Zeros, TriaTransExactTemp[TriaTransPos, TriaTransNeg,x,t,w,k,opts], _, Message[Phi::invalidpart,pt]], Gauss, Switch[pt, Approximate, GaussTransTemp[GaussTransPos, GaussTransNeg,x,t,w,k,n,opts], Truncate, GaussTransTruncTemp[GaussTransPos, 203
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DISSERTATION Wave Propagation in Li
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Kurzfassung Seit der Entdeckung des
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Nullum est iam dictum, quod non sit
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Preface Our popular writers and rep
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the quest for superluminality and t
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Contents Part I Wave propagation ph
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7.3.4 PartitionPoints . . . . . . .
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Part I Wave propagation phenomena S
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1.1 Phase and group velocity 1.1 Ph
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1.1 Phase and group velocity ! !c v
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1.2 A few notes on dispersion He co
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1.2 A few notes on dispersion v/c 6
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1.3 Signal velocity dipoles with a
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1.3 Signal velocity The arbitrarine
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1.4 Energy velocity For electromagn
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1.5 Other velocity de nitions For n
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1.5 Other velocity de nitions evide
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2.1 Superluminal wave propagation 2
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2.1 Superluminal wave propagation a
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2.1 Superluminal wave propagation t
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2.2 Quantum mechanical tunnelling e
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2.2 Quantum mechanical tunnelling R
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Chapter 3 Wave propagation in elect
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3.1 Model of a transmission line th
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3.2 Excursion: a delay line section
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3.3 Re ection due to termination mi
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3.4 A simple thought experiment I0
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3.5 A dispersive system: the lossle
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3.6 Inhomogeneous transmission line
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3.7 Turn-on e ects in a lossless pl
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3.8 Turn-on e ects in a wave guide
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3.8 Turn-on e ects in a wave guide
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3.9 A Gaussian pulse in plasma Like
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma 250
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3.9 A Gaussian pulse in plasma Note
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4.1 The potential step 4.1 The pote
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4.1 The potential step Inside the b
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4.2 Initial wave forms -60 -50 -40
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4.2 Initial wave forms -60 -50 -40
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4.3 Examples of scattering processe
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4.4 The square barrier they have va
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4.4 The square barrier 1 0.8 0.6 0.
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4.5 Tunnelling time de nitions for
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4.5 Tunnelling time de nitions for
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events 2
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4.6 Examples of tunnelling events -
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4.6 Examples of tunnelling events t
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4.6 Examples of tunnelling events P
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4.6 Examples of tunnelling events T
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Interlude Wave functions in graphic
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Wave functions in graphical represe
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Part II Numerical aspects of wave e
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5.1 Univariate numerical quadrature
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5.1 Univariate numerical quadrature
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5.2 Convergence acceleration one, t
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5.2 Convergence acceleration (1) ,1
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6.1 Partitioning the integration in
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6.2 Choosing the rst partition poin
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6.3 How to compute the rst integral
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6.3 How to compute the rst integral
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6.4 Asymptotic partition consuming
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6.4 Asymptotic partition of the int
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6.5 Considerations for a Mathematic
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Page 171 and 172: 6.6 Controlling the accuracy of the
Page 177 and 178: Chapter 7 Mathematica implementatio
Page 179 and 180: 7.1 User interface of the function
Page 181 and 182: 7.3 Implementation of OscInt and re
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Page 197 and 198: 7.5 Test of the quadrature routine
Page 207 and 208: 8.1 Preparation of the wave integra
Page 213 and 214: 8.2 Outline of the program structur
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Page 221 and 222: 8.4 Test of the package 8.4 Test of
Page 223 and 224: 8.4 Test of the package -200 -150 -
Page 225 and 226: 8.4 Test of the package 0.4 0.2 -0.
Page 227 and 228: 8.4 Test of the package 8.4.2 Forma
Page 229 and 230: 8.4 Test of the package Out[24]= 0
Page 231 and 232: A.1 Numerical quadrature Asymptotic
Page 233 and 234: A.1 Numerical quadrature WynnDegree
Page 235 and 236: A.1 Numerical quadrature Which[ft =
Page 237 and 238: A.2 Solutions for the step potentia
Page 245 and 246: A.3 Solutions for the square barrie
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Page 249 and 250: A.4 Electromagnetic waves function
Page 251 and 252: A.5 Utilities for displaying points
Page 253 and 254: Bibliography Bibliography [1] James
Page 255 and 256: Bibliography [29] Kurt Edmund Oughs
Page 257 and 258: Bibliography [59] Ch. Spielmann, R.
Page 259 and 260: Bibliography [91] C. R. Leavens and
Page 261 and 262: Bibliography [123] T. O. Espelid an
Page 263 and 264: Index Index absorption, 7, 9 accura
Page 265 and 266: Index saddle point integration, 11,
Page 267: Curriculum vitae Dipl.-Ing. Thilo S
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