Wave Propagation in Linear Media | re-examined

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0.75 0.5 0.25 -0.25 -0.5 -0.75 U/U0 1 50 100 150 200 T 3Wave propagation in electromagnetic transmission lines |A| 4 3 2 1 -2 -1 1 2 η Figure 3.24: Initial pulse and its spectrum jA( )j for the parameter values T0 = 100 and = 17. The carrier frequency is = 0:5 in these particular graphs. Unfortunately, a pole turns up at ! =0,preventing straightforward integration. Taking the Cauchy principal value of the integral would solve the problem, but as we do not need the current to gain some general insight into the propagation of a Gaussian wave, we leave it aside. The expression for the voltage (3.103) is valid in both the stop and pass bands. We can thus compute the voltage inside the plasma for a given pulse with varying carrier frequencies. The initial shape and its spectrum are shown in g. 3.24. If the cuto frequency of the plasma is assumed to be !p = 10 10 9 s ,1 (a value roughly corresponding to Nimtz' microwave experiments), then the pulse width is about 6 ns, which is fairly short. Accordingly, the spectrum is quite wide, and so we must expect a small portion of the pulse to be classically transmitted even though the major part of the eld originates from the evanescent region. The rst case we examine is the base-band signal with =0. The result given in g. 3.25 is not very spectacular since the spectrum is practically con ned to the range below cuto . Thus there are no propagating waves, and all we can see is the exponential decay along the spatial coordinate. In contrast to the following gures, this one shows no modulation, and so the vanishing voltage outside the actual pulse also denotes the least possible value of the eld. Therefore the area surrounding the contour of the pulse is shaded in black. When we use a carrier with = 0:5, the behaviour of the wave is much more interesting. Fig. 3.26 shows rst of all a dominating evanescent signal that resembles the unmodulated case treated before. It is particularly noteworthy that the carrier frequency increases gradually with X. This is manifested in the graph by the narrowing of the black and white stripes and can be explained as a pulse reshaping e ect. Since the dispersion in the plasma causes higher frequency components to be less attenuated than lower ones, the centre frequency of the spectrum is slightly raised. Note that although it looks as if the pulse became shorter with increasing X, this is not the case. It is only the attenuation in combination with the limited plotting range that creates this deceptive impression. The overall shape of the pulse remains more or less the same, albeit with reduced amplitude. Direct comparison of the unmodulated and the present results reveal another detail: the plot ranges of the two graphs are identical, 66
3.9 A Gaussian pulse in plasma 250 200 150 100 50 0 0 10 20 30 40 50 Figure 3.25: Evolution of the voltage for = 0. The abscissa denotes X, the ordinate T . 250 200 150 100 50 0 0 10 20 30 40 50 Figure 3.26: Evolution of the voltage for = 0:5. The abscissa denotes X, the ordinate T . 67
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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
Page 31 and 32: 1.4 Energy velocity For electromagn
Page 33 and 34: 1.5 Other velocity de nitions For n
Page 35 and 36: 1.5 Other velocity de nitions evide
Page 37 and 38: 2.1 Superluminal wave propagation 2
Page 39 and 40: 2.1 Superluminal wave propagation a
Page 41 and 42: 2.1 Superluminal wave propagation t
Page 43 and 44: 2.2 Quantum mechanical tunnelling e
Page 45 and 46: 2.2 Quantum mechanical tunnelling R
Page 47 and 48: Chapter 3 Wave propagation in elect
Page 49 and 50: 3.1 Model of a transmission line th
Page 51 and 52: 3.2 Excursion: a delay line section
Page 53 and 54: 3.3 Re ection due to termination mi
Page 55 and 56: 3.4 A simple thought experiment I0
Page 57 and 58: 3.5 A dispersive system: the lossle
Page 63 and 64: 3.6 Inhomogeneous transmission line
Page 65 and 66: 3.7 Turn-on e ects in a lossless pl
Page 77 and 78: 3.8 Turn-on e ects in a wave guide
Page 79 and 80: 3.8 Turn-on e ects in a wave guide
Page 81: 3.9 A Gaussian pulse in plasma Like
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Page 87 and 88: 3.9 A Gaussian pulse in plasma Note
Page 89 and 90: 4.1 The potential step 4.1 The pote
Page 91 and 92: 4.1 The potential step Inside the b
Page 93 and 94: 4.2 Initial wave forms -60 -50 -40
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Page 97 and 98: 4.3 Examples of scattering processe
Page 109 and 110: 4.4 The square barrier they have va
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Page 113 and 114: 4.5 Tunnelling time de nitions for
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Page 117 and 118: 4.6 Examples of tunnelling events P
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Page 123 and 124: 4.6 Examples of tunnelling events t
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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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6.6 Controlling the accuracy of the
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Chapter 7 Mathematica implementatio
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7.1 User interface of the function
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7.3 Implementation of OscInt and re
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7.4 Auxiliary functions While[itera
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7.4 Auxiliary functions Linear appr
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7.4 Auxiliary functions For the sak
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7.4 Auxiliary functions Out[8]= 3 f
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7.5 Test of the quadrature routine
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8.1 Preparation of the wave integra
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8.2 Outline of the program structur
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8.2 Outline of the program structur
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8.3 Implementation of the quadratur
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8.3 Implementation of the quadratur
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8.4 Test of the package 8.4 Test of
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8.4 Test of the package -200 -150 -
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8.4 Test of the package 0.4 0.2 -0.
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8.4 Test of the package 8.4.2 Forma
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8.4 Test of the package Out[24]= 0
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A.1 Numerical quadrature Asymptotic
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A.1 Numerical quadrature WynnDegree
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A.1 Numerical quadrature Which[ft =
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A.2 Solutions for the step potentia
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A.3 Solutions for the square barrie
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A.3 Solutions for the square barrie
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A.4 Electromagnetic waves function
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A.5 Utilities for displaying points
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Bibliography Bibliography [1] James
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Bibliography [29] Kurt Edmund Oughs
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Bibliography [59] Ch. Spielmann, R.
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Bibliography [91] C. R. Leavens and
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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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