Wave Propagation in Linear Media | re-examined

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Likewise, we collect the de nitions for a Gaussian wave, p 2 k Cgauss = p p n 2 s + , k n gauss( )=e s , k gauss ( )=e, n 1 p ,1 1 , p ,1 to write the nal form of the integrals p Z 1 l = Cgauss s + gauss( ) e j(,T 2 + k 1 p +X , k) d + inc,gauss ref,gauss trans,gauss evan,gauss + Cgauss + Cgauss p l = Cgauss + Cgauss + Cgauss p l = Cgauss + Cgauss p l = Cgauss 1 Z 1 1 Z 1 ,1 Z 1 1 Z 1 1 Z 1 ,1 Z 1 1 Z 1 1 Z 1 ,1 s , gauss ( ) ej(,T 2 , k 1 p ,X , k) d + s + gauss ( ) ej(,T 2 + k 1 p +X , k) d 8 Application of the quadrature routine s + gauss ( ) c ref( ) e j(,T 2 + k 1 p ,X , k) d + s , gauss( ) c ref( ) e j(,T 2 , k 1 p +X , k) d + s + gauss( ) c ref( ) e j(,T 2 + k 1 p ,X , k) d 2 2 (8.32) (8.33) ; (8.34) s + gauss ( ) ctun( ) e j(,T 2 + k 1 p p +X 2,1, k) d + s , gauss ( ) ctun( ) e j(,T 2 , k 1 p p ,X 2,1, k) d (8.35) (8.36) (8.37) s + gauss ( ) ctun( ) e j(,T 2 + k 1 p p +X 2,1, k) d : (8.38) The computation of the inde nite integrals for a Gaussian wave requires only two calls of the quadrature routine. Furthermore, in this special case, there is an alternative to the extrapolating quadrature strategy: as the integrands decay exponentially, we can safely truncate them at appropriate values of the integration variable and integrate the remaining de nite integral in a conventional way. A safe value for the truncation is where the shaping functions Cgauss sgauss( ) become smaller than the working precision p, for example such that the truncation points are trunc = p Cgauss sgauss 10 ,(p+1) ; (8.39) 1 r 196 ln Cgauss 10 ,(p+1) n k ! (8.40)
8.2 Outline of the program structure 8.2 Outline of the program structure Since we have to cope with a large variety of similar functions, we must adopt a modular approach for the individual subroutines. This means that the integrands are combined from several function modules following the aforementioned structure, which is the only possibility to ensure consistency within the package and to provide comparatively easy extendability. A reasonable implementation strategy involves three distinct hierarchical levels. Driver modules provide the basic interface to the user. The most user-friendly of them expects nothing more than the parameters required to compute the wave function and an option de ning the shape of the initial waveform. Depending on the spatial coordinate it then returns either the wave function inside the tunnel or the sum of the incident and re ected parts on the outside. These partial solutions are computed in turn by other driver modules that can also be accessed directly. All modules must take into account that di erent waveforms may need di erent numbers of parameters (such as the Gaussian wave packet, which requires an additional geometry parameter for the variance of the distribution). Quadrature modules are distinct for all waveforms. The reason for this is that the number of exponential functions in the sum of equation (8.15) as well as the respective coe cients as;i are di erent for each possible waveform. Furthermore, some of the in nite integrals can be computed in di erent ways. So these modules are typically implemented as waveform-dependent templates that need the more general part ss( ) cr( ) of the integrand as input parameter. The they add the speci c functions P m i=1 as;i e j(gs;i( )+pr( )) and call the actual quadrature routine with the appropriate options. The only exception to this scheme is the computation of the evanescent part of the integrals. As this involves only a single de nite integral, there is no point in dividing the integrands like we do for the in nite integrals in order to meet the formal requirements of our quadrature strategy. Hence there is only one template for this task, and the whole integrand function is passed to this routine from the calling module. Because all these routines are highly specialised, they are not made available to the user. Function modules are the actual building blocks of the integrands. There are atomic functions describing the shape- and region-dependent coe cients ss( ), cr( )aswell as the constants Cs. These elements are then combined to form the integrands for the two inde nite transmissive parts and the evanescent section of the integration range. These functions, too, are internal to the package and cannot be accessed by the user. For the triangular and Gaussian wave packet, the gradient of the wave function inside the tunnel is also of interest for the computation of the energy density and nally the energy velocity. These functions have not been included in the general framework, in that they cannot be accessed from the same driver routine as the wave functions themselves. Although the gradient functions share the quadrature modules of the plain wave computation, they use separate driver and, of course, function modules. 197
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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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Page 165 and 166: 6.5 Considerations for a Mathematic
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
Page 189 and 190: 7.4 Auxiliary functions While[itera
Page 191 and 192: 7.4 Auxiliary functions Linear appr
Page 193 and 194: 7.4 Auxiliary functions For the sak
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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
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Page 215 and 216: 8.2 Outline of the program structur
Page 217 and 218: 8.3 Implementation of the quadratur
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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
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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
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Index Index absorption, 7, 9 accura
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Index saddle point integration, 11,
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Curriculum vitae Dipl.-Ing. Thilo S
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