Uniform Asymptotics Applied to Ultrawideband Pulse Propagation

SIAM REVIEW Vol. 49, No. 4, pp. 628–648

c 2007 Society for Industrial and Applied Mathematics

Uniform Asymptotics Applied to Ultrawideband Pulse Propagation∗ Natalie A. Cartwright† Kurt E. Oughstun† Abstract. A canonical problem of central importance in the theory of ultrawideband pulse propagation through temporally dispersive, absorptive materials is the propagation of a Heaviside step-function signal through a medium that exhibits anomalous dispersion. This problem is rich in the use of asymptotic theory. Sommerfeld and Brillouin provided the first (qualitatively accurate but quantitatively inaccurate) closed-form approximations of the dynamic evolution of this waveform through a single-resonance Lorentz model dielectric based upon Debye’s method of steepest descent. An improved approximation has since been provided by Oughstun and Sherman using modern, uniform asymptotic methods that rely upon the saddle-point method. An accurate, uniform asymptotic approximation describing the dynamical evolution of the unit step-function modulated sine wave signal through a single-resonance Lorentz model dielectric is presented here based upon their work. This refined asymptotic description results in a continuous evolution of the propagated field for all space-time points. Key words. asymptotic methods, dispersive attenuative wave propagation AMS subject classifications. 78M35, 78A40, 30E15 DOI. 10.1137/050635833

1. Introduction. Asymptotic analysis is widely used in the study of pulse propagation in electromagnetics, optics, and acoustics. Typical applications include the study of the propagated pulse evolution as either the propagation distance or the wavelength becomes either large or small. Here, we study two-dimensional (space and time) electromagnetic pulse propagation through an unbounded dielectric material and show that many facets of uniform asymptotic theory are drawn upon in order to provide a continuous asymptotic approximation to the propagated pulse. Consider a linearly polarized plane-wave electromagnetic pulse traveling in the positive z-direction. The material through which the pulse travels is a linear dielectric whose relative magnetic permeability µ is unity and whose relative dielectric permittivity (ω) is given by the single-resonance Lorentz [18] dielectric model so that √ the complex index of refraction n(ω) = µ is given by 1/2 ωp2 . (1.1) n(ω) = 1 − 2 ω − ω02 + 2iδω ∗ Received by the editors July 12, 2005; accepted for publication (in revised form) July 28, 2006; published electronically November 1, 2007. This work was supported by the Air Force Office of Scientific Research under grant 9550-04-1-0447. http://www.siam.org/journals/sirev/49-4/63583.html † College of Engineering and Mathematics, University of Vermont, Burlington, VT 05401. Current address: Department of Mathematics, State University of New York at New Paltz, New Paltz, NY 12561 ([email protected]). Questions, comments, or corrections to this document may be directed to this e-mail address. 628

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

UNIFORM ASYMPTOTICS IN WAVE PROPAGATION

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Here, ω denotes the angular frequency, ω0 denotes the undamped resonance frequency, ωp denotes the plasma frequency of the material, and δ is a phenomenological damping constant. Assume that this material occupies all of space and that the time evolution of the electric component of the pulse is known on the plane z = 0 and is given by E(0, t). The electric field component E(z, t) on some plane z > 0 is then given by the Fourier–Laplace integral representation [29] ia+∞ 1 ˜ ˜ ω) exp[i(k(ω)z (1.2) E(z, t) = E(0, − ωt)]dω, 2π ia−∞ ˜ where k(ω) ≡ (ω/c)n(ω) = β(ω) + iα(ω) is the complex wave number, a is greater ˜ ω) is the temporal than the abscissa of absolute convergence [38] for E(0, t), and E(0, ˜ Fourier transform of E(0, t). Here β(ω) ≡ {k(ω)} is the propagation factor and ˜ α(ω) ≡ {k(ω)} is the attenuation factor at the angular frequency ω. Because the complex wave number is dependent upon ω, the phase and amplitude of each frequency component of the wave form E(z, t) change at their own rates as the propagation distance z increases. This dependence is further complicated by causality which requires that {n(ω)} and {n(ω)} be interrelated through the Kramers–Kronig relations [20]. These fundamental physical requirements complicate the study of wave motion through a dispersive dielectric material, particularly for an ultrawideband wave form whose spectrum spans a wide range of frequencies over ˜ which k(ω) varies in a complicated manner that is not amenable to a simple Taylor series approximation about some characteristic frequency of the pulse, as is done in the group velocity approximation [41]. Sommerfeld [37] and Brillouin [5, 6] first addressed this problem in 1914 with an input pulse given by a step-function modulated sine wave, E(0, t) = u(t) sin(ωc t), of fixed carrier frequency ωc , where u(t) denotes the Heaviside step-function. Until that time, it was believed that the rate at which an electromagnetic pulse propagates through a dispersive medium was determined by the classical group velocity [15, 32] −1 ˜ vg (ω) = ∂ k(ω)/∂ω) evaluated at some characteristic angular frequency of the pulse. However, in regions of anomalous dispersion, this group velocity can yield values that range from negative to positive infinity, raising doubts about its physical applicability to either energy or information transfer. By completing the contour in the upper half plane, Sommerfeld proved that E(z, t) identically vanishes for all t < z/c. He also approximated the integral representation of the propagated pulse given in (1.2) for very small values of t−z/c > 0 as a Bessel function which starts at time t = z/c with negligibly small but increasing amplitude and period. In this manner, Sommerfeld proved that the first part of the signal to arrive at any given point in space occurs along the space-time ray z = ct in complete agreement with the special theory of relativity [11]. Brillouin [5, 6] completed the analysis begun by Sommerfeld by considering the evaluation of (1.2) for all subluminal times t > z/c. In this case, the deformation of the contour in the lower half plane is complicated by the branch cuts of the complex wave number. Instead, Brillouin employed the then recently developed method of steepest descents [10] which requires the deformation of the contour through the saddle points of the complex phase function ˜ (1.3) φ(ω, θ) ≡ i(c/z) k(ω)z − ωt = iω [n(ω) − θ] , where θ = ct/z is a dimensionless space-time parameter. The integral is then approximated by contributions from small neighborhoods about these saddle points, the accuracy of this approximation increasing in the sense of Poincaré [31] as z → ∞.


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NATALIE A. CARTWRIGHT AND KURT E. OUGHSTUN

Brillouin found two sets of saddle points: the contributions from the distant saddle points are responsible for the first forerunner, which had been evaluated for small θ ≥ 1 by Sommerfeld; the contributions from the near saddle points give rise to a second forerunner which begins after the arrival of the first forerunner. Brillouin also identified the main signal, characterized by its harmonic oscillation at the carrier frequency ωc of the input pulse, which arrives after the second forerunner. Sommerfeld and Brillouin mistakenly concluded that both forerunners are of negligible amplitude in comparison to the amplitude of the main signal. Although Sommerfeld and Brillouin correctly predicted the existence of the first and second forerunners, they did not accurately describe their behavior. It wasn’t until 1975 when Oughstun and Sherman [25, 26] examined Sommerfeld’s and Brillouin’s seminal works that an accurate description of these two forerunners (now known as the Sommerfeld and Brillouin precursors) and the main signal was given based upon uniform asymptotic expansion techniques. Of significant importance, Oughstun [22] proved that the Brillouin precursor can be of significant amplitude when the rise time of the pulse is faster than the relaxation time of the material. This analysis showed that, asymptotically, the Brillouin precursor achieves a peak amplitude at the spacetime point ct/z = n(0), where n(0) is the static refractive index of the material, and that the amplitude of this point decays only algebraically as z −1/2 with propagation distance z, whereas the amplitudes of the Sommerfeld precursor and the main signal decay exponentially. The relatively slow decay rate of the peak amplitude point of the Brillouin precursor has important practical applications to biomedical imaging as well as to ground- and foliage-penetrating radar systems (see the proceedings series titled Ultra-Wideband Short-Pulse Electromagnetics [1, 2, 7, 19, 34, 35]). The exponential appearing in (1.2) may be written in terms of the complex phase function defined in (1.3) so that, with some algebraic manipulation, the integral representation of the propagated pulse may be expressed as [29] ia+∞ z i 1 i exp φ(ω, θ) dω (1.4) E(z, t) = 2π c ia−∞ ω − ωc for the input step-function modulated sine wave pulse with fixed carrier frequency ωc > 0. Because of the exponential appearing in the integrand of (1.4), the Fourier– Laplace representation is ideally suited for analysis by asymptotic expansion techniques that are valid as z → ∞ (i.e., as the propagation distance becomes large), as was originally done by Brillouin. In fact, this problem is much richer in asymptotic theory than the direct application of the method of steepest descent. In order to find a continuous asymptotic approximation to the propagated field E(z, t), three different uniform asymptotic theories are required: a method for two saddle points symmetrically located about the imaginary axis whose real parts are located at plus and minus infinity; a method for two first-order saddle points that coalesce into one second-order saddle point at some fixed space-time point and then separate into firstorder saddle points again; and finally, a method for first-order saddle points in the vicinity of a simple pole. These methods have been used extensively by Oughstun and Sherman to study pulse propagation through Lorentz model dielectric materials [21, 25, 26, 27, 28, 29]; their results form the basis of this paper. The material parameters (ω0 = 4.0×1016 /s, ωp2 = 20.0×1032 /s2 , δ = 0.28×1016 /s) used here are the same as those used by Sommerfeld and Brillouin [6]. The real and imaginary parts of the complex index of refraction n(ω) for the single-resonance Lorentz model dielectric with these parameters are illustrated in Figure 1.1. This choice of medium parameters corresponds to an extremely absorptive medium. Nev-



631

Complex Index of Refraction, n( ω)

3 2.5 2 ℜ{n(ω)}

1 0.5 0

Fig. 1.1

ℑ{n(ω)}

1.5

ω0 1e17 ω (rad/s)

1e16

The real (solid) and imaginary (dashed) parts of n(ω) for the single-resonance Lorentz model dielectric with resonance frequency ω0 = 4.0 × 1016 /s, plasma frequency ωp2 = 20.0 × 1032 /s2 , and damping constant δ = 0.28 × 1016 /s. The frequency domains where d{n(ω)}/dω ≥ 0 are termed normally dispersive, while those for which d{n(ω)}/dω < 0 are said to exhibit anomalous dispersion.

ertheless, the resulting analysis remains qualitatively unchanged when different dielectric medium parameters are chosen that correspond to weaker absorption [23, 24], provided that the propagation distance z > zd , where zd ≡ α−1 (ωc ) is the e−1 absorption depth at the pulse carrier angular frequency ωc . 2. The Saddle-Point Method. Consider the complex integral (2.1) I(z) = f (ω) exp[zg(ω)]dω C

for z real and positive, where f (ω) and g(ω) are assumed to be analytic in a domain D, and where C is some path of integration within D. The most common approaches used to find an asymptotic expansion of the integral (2.1) as z → ∞ are either the method of stationary phase [17] or the method of steepest descents [33]. The method of stationary phase is used when g(ω) is purely imaginary, which, in terms of the complex phase function (1.3), requires the material loss to be constant. Although this may be a valid approximation for a narrowband pulse, it is not, in general, valid for an ultrawideband pulse. The method of steepest descents, as well as its generalization to the saddle-point method [30], is used when the phase function g(ω) is complex and is thus the preferred method to be used when investigating dispersive pulse propagation. Because g(ω) is analytic in D, it possesses neither maxima nor minima in D, only saddle points. Suppose that g(ω) has one saddle point in D located at ω = ωSP . A point ω ∈ D is said to lie in a valley of g(ωSP ) if {g(ω)} < {g(ωSP )}. If it is possible to deform the original contour C such that it crosses ωSP while remaining in the valleys of the saddle point when ω = ωSP , then, along this deformed contour, {g(ω)} has a


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maximum at ωSP , and this maximum becomes more pronounced as z → ∞. Hence, the integral appearing in (2.1) may be approximated by the contribution due to a neighborhood of ωSP , the accuracy of this approximation increasing in the sense of Poincaré [31] as z → ∞. This then forms the principle upon which the saddle-point method is based. The saddle-point method may be directly extended to integrals of the type (see Chapter 5 of [29]) (2.2) I(z, θ) = f (ω) exp [zg(ω, θ)] dω, C

in which the position of the saddle point of the exponential function g(ω, θ) is a function of the real-valued parameter θ, as is the case in this paper. This is achieved by choosing a path of integration which moves in a continuous manner as θ varies continuously over some specified space-time domain. 3. Behavior of the Complex Phase Function. In order to apply the saddle-point method to the integral appearing in the integral representation (1.4) of the propagated electric field component of a step-function modulated sine wave signal, the behavior of the complex phase function φ(ω, θ) = iω[n(ω) − θ] must be known in the complex ω-plane [6, 29, 36] as a function of the real parameter θ. For the single-resonance Lorentz model dielectric [18], the complex phase function φ(ω, θ) is analytic in the ω-plane formed by the two branch cuts in the lower half of the ω-plane symmetrically located about the imaginary axis. √In the right half plane, the branch cut extends from ω+ ≡ ω02 − δ 2 − iδ to ω+ ≡ ω02 + ωp2 − δ 2 − iδ. The saddle points of φ are solutions of the equation φω (ω, θ) = 0, which need only be solved for θ > 1 since the field given by (1.4) identically vanishes for all θ ≤ 1. Brillouin [6] first showed that φ possesses two sets of saddle points and provided first-order approximations of the locations of these saddle points; more accurate approximations have since been provided by Oughstun and Sherman [29]. The first set, referred to as the distant saddle points and denoted by ωSP ± (θ), consists of two D first-order saddle points. As θ → 1+ , lim ωSP ± ≈ ±∞ − i2δ,

(3.1)

θ→1+

D

while in the opposite limit as θ → ∞, the distant saddle points approach the outer branch points , (3.2) lim ωSP ± ≈ ± ω02 + ωp2 − δ 2 − iδ = ω± θ→∞

D

√ respectively, so that |ωSP ± (θ)| ≥ ω02 + ωp2 for all θ ≥ 1. D The second set of saddle points, referred to as the near saddle points and denoted by ωSP ± (θ), consists of two first-order saddle points that lie on the imaginary axis N for values of θ < θ1 , coalesce into a single second-order saddle point when θ = θ1 , and separate into two first-order saddle points symmetrically located about the imaginary axis for θ > θ1 , where θ1 ≈ θ0 + 2δ 2 ωp2 /3θ0 ω04 , so that θ1 ≈ 1.502 for the material parameters used here. As θ → ∞, the locations of the near saddle points approach the inner branch points (3.3) lim ωSP ± ≈ ± ω02 − δ 2 − iδ = ω± , θ→∞

N

respectively. Of particular interest in this evolution is the space-time point θ = θ0 ≡ n(0), where θ0 = 1.5 for the material parameters considered here, at which the


633

UNIFORM ASYMPTOTICS IN WAVE PROPAGATION 16

1

x 10

w SP +

ℑ{ω}

0.5

N

0

w'_ -0.5

w SP -

w SP + w SP w SP (q1) w+ N

w_

w'+

N

N

w SP +

D

D

w SP N

-1 -1

Fig. 3.1

-0.5

0 ℜ{ω}

0.5

1 17

x 10

The movement of the saddle points of φ(ω, θ) with increasing θ > 1 in the complex ω-plane formed with the appropriate branch cuts for a single-resonance Lorentz model dielectric.

upper near saddle point ωSP + (θ0 ) is located at the origin, and both φ(ωSP + , θ0 ) and N N φω (ωSP + , θ0 ) identically vanish. The locations of the branch points and saddle points N of φ(ω, θ) are illustrated in Figure 3.1. 3.1. Paths of Integration. In the asymptotic analysis of the propagated field, the contour of integration appearing in (1.4) needs to be deformed into a path P(ω, θ) that is continuous in θ, passes through the accessible saddle points of φ(ω, θ), and lies in the valleys of the accessible saddle points for all θ ≥ 1. In order to determine P(ω, θ), a study of the topography about each saddle point is required; this analysis can be found in [29]. What is found is that the lower near saddle point ωSP − (θ) N of φ(ω, θ) for a Lorentz model dielectric is inadmissible for 1 ≤ θ < θ1 due to the branch cuts of n(ω). Hence, the appropriate path P(ω, θ) to be used here passes through the saddle points ωSP − (θ), ωSP + (θ), ωSP + (θ) for 1 ≤ θ < θ1 (see Figure 3.2 D N D for the case θ = 1.25), through ωSP − (θ), ωSPN , ωSP + (θ) for θ = θ1 (see Figure 3.3), D D and through ωSP − (θ), ωSP − (θ), ωSP + (θ), and ωSP + (θ) for θ > θ1 (see Figure 3.4 for D N N D the case θ = 3). This figure sequence represents the typical θ-evolution of the path P(ω, θ) for the asymptotic analysis of the integral representation given in (1.4). In the deformation of the contour appearing in (1.4) into the path P(ω, θ), it may happen that at some value of θ the path P(ω, θ) crosses the simple pole located at ωc along the positive real frequency axis (see Figure 3.4, which depicts the case when θ = 3 and ωc = 1.25ω0 ); this occurrence gives rise to the pole contribution whose uniform asymptotic description is presented in section 6. Due to linearity, the asymptotic behavior of the propagated electric field of the step-function modulated sine wave signal (1.4) may be expressed as (3.4)

E(z, t) = ES (z, t) + EB (z, t) + Ec (z, t),

where ES (z, t) is the contribution to the integral due to the distant saddle points,


634


16

8

x 10

6 4 2

ω

c

ℑ {ω}

0 −2

P(ω,θ) −4 −6 −8 −10 −1.5

Fig. 3.2

−1

−0.5

0

0.5

ℜ {ω}

1

1.5 17

x 10

Contours of {φ(ω, θ)} for θ = 1.25 in the complex ω-plane along with an acceptable deformed path of integration P(ω, θ). Notice that the original and deformed contours lie on the same side of the pole.

16

5

x 10

4 3 2

ℑ {ω}

1

ωc

0 −1

P(ω,θ)

−2 −3 −4 −5 −1

Fig. 3.3

−0.8

−0.6

−0.4

−0.2

0

ℜ {ω}

0.2

0.4

0.6

0.8

1 17

x 10

Contours of {φ(ω, θ)} for θ = θ1 in the complex ω-plane along with an acceptable deformed path of integration P(ω, θ). Notice that the original and deformed contours lie on the same side of the pole.


635

UNIFORM ASYMPTOTICS IN WAVE PROPAGATION 16

2

x 10

1.5 1 0.5

ω ℑ {ω}

c

0

P(ω,θ)

−0.5 −1 −1.5 −2 −2.5 −1

Fig. 3.4

−0.8

−0.6

−0.4

−0.2

0

ℜ {ω}

0.2

0.4

0.6

0.8

1 17

x 10

Contours of {φ(ω, θ)} for θ = 3 in the complex ω-plane along with an acceptable path of integration. Notice the deformation of P(ω, θ) about the simple pole at ω = ωc .

called the Sommerfeld precursor ; EB (z, t) is the contribution to the integral due to the near saddle points, called the Brillouin precursor ; and Ec (z, t) is the contribution to the integral due to the simple pole located at ωc , called the pole or signal contribution. ˜ ω) has This pole contribution is nonzero only when the initial pulse spectrum E(0, ˜ ω) pole singularities. If the initial field E(0, t) is bounded for all time t, then E(0, can have poles only if the amplitude of E(0, t) does not tend to zero too fast as t → ∞. Consequently, the implication of a nonzero pole contribution Ec (z, t) is that the field E(z, t) has a steady-state oscillation at the input plane z = 0 and will do the same for z > 0 for sufficiently large t. Because the asymptotic approximation increases in accuracy as z → ∞, there exists a minimum propagation distance for which the asymptotic approximation gives an acceptable level of accuracy. Numerical results [41] suggest that this minimum distance is bounded below by the absorption ˜ depth zd = α−1 (ωc ) at the input signal carrier frequency, where α(ω) ≡ {k(ω)}. 4. The Sommerfeld Precursor. The contour appearing in (1.4) may be completed in the upper half plane, and a direct application of Jordan’s lemma [39] proves that the propagated step-function modulated signal E(z, t) identically vanishes for all θ ≤ 1, as originally shown by Sommerfeld [37]. For values of θ > 1, the asymptotic expansion of E(z, t) as z → ∞ about the distant saddle points ωSP ± (θ) yields the D dynamical evolution of the Sommerfeld precursor, ES (z, t). These first-order saddle points are symmetrically located about the imaginary axis and approach the values ωSP ± → ±∞ − i2δ as θ → 1+ , at which point they are infinite-order saddle points. D The saddle-point method cannot be applied to the distant saddle points for values of θ → 1+ because the complex phase function does not have a Taylor series valid about these saddle points. A uniform expansion of the Sommerfeld precursor, valid for all θ ≥ 1, is then found through use of the theorem due to Handelsman and Bleistein [16]. As with all uniform asymptotic methods, Handelsman and Bleistein


636


introduced a change of variable such that the saddle points of the transformed phase function retain the limiting values of the original saddle points and then expanded the amplitude function as a finite number of terms plus a remainder term, the remainder term being regular and equal to zero at the saddle points of the transformed phase function. Here, the transformation that retains the essential behavior of the distant saddle points is given by (4.1)

φ(ω, θ) = αd2 (θ)s + βd (θ) +

1 , 4s

where αd (θ) and βd (θ) are functions to be determined. The amplitude function is then written as

i dω 1 = γ0 (θ) + γ1 (θ)s + αd2 (θ) − 2 H0 (s, θ), (4.2) ω − ωc ds 4s where H0 (s, θ) is regular and equal to zero at s = (±1/αd ). Substitution of (4.1) and (4.2) into (1.4) results in the uniform asymptotic description of the Sommerfeld precursor ES (z, t) for the step-function modulated sine wave [29],

z z αd (θ) ES (z, t) ∼ exp −i βd (θ) γ0 (θ)J0 c c

z (4.3) + R(z, θ) + 2αd (θ)e−iπ/2 γ1 (θ)J1 αd (θ) c as z → ∞ for all θ ≥ 1, where Jn (ξ) denotes the Bessel function of the first kind of integer order n. The magnitude of the remainder term is bounded as 2c|αd (θ)| z z (4.4) |R(z, θ)| ≤ K J1 αd (θ) + J2 αd (θ) z c c for z ≥ Z > 0 and θ ≥ 1, where K > 0 is a constant independent of θ and z. The coefficients appearing in (4.3) are given by i αd (θ) = φ(ωSP + , θ) − φ(ωSP − , θ) = − φ(ωSP + , θ) , (4.5a) D D D 2 i φ(ωSP + , θ) + φ(ωSP − , θ) = i φ(ωSP + , θ) , (4.5b) βd (θ) = D D D 2 1/2 3 i 4αd (θ) 1 1 γ0 (θ) = 2 ωSP + − ωc 2αd (θ) iφ(2) (ωSP + , θ) D D 1/2  4αd3 (θ) 1 i , − (2) (4.5c) − + ωSP − − ωc 2αd (θ) iφ (ωSP − , θ) D D  1/2 i 4αd3 (θ) 1 1  γ1 (θ) = 4αd (θ) ωSP + − ωc 2αd (θ) iφ(2) (ωSP + , θ) D D 1/2  4αd3 (θ) 1 i . − (2) (4.5d) − − 2αd (θ) ωSP − − ωc iφ (ωSP − , θ) D

D

For values of θ bounded away from 1 such that (z/c)|αd (θ)| 1, the large argument asymptotic expansion of the Bessel function may be substituted into (4.3). This


637

UNIFORM ASYMPTOTICS IN WAVE PROPAGATION -3

8

x 10

6 4

S

E (z,t)

2 0

-2 -4 -6 -8

Fig. 4.1

1

1.05

1.1

θ = ct/z

1.15

1.2

1.25

Dynamic behavior of the Sommerfeld precursor ES (z, t) for the step-function modulated sine wave with below-resonance applied signal frequency ωc = 0.5 ω0 rad/s at a relative observation distance z/zd ≈ 3.8 into the single-resonance Lorentz dielectric.

substitution then reduces the uniform expansion given above to the nonuniform result obtained by direct application of the saddle-point method for θ bounded above 1 for finite z. The Sommerfeld precursor field ES (z, t) for the step-function modulated sine wave E(z, t) with below-resonance applied signal frequency ωc = 0.5 ω0 at an observation distance of z = 1.2 × 10−6 m ≈ 3.8 zd into the single-resonance Lorentz dielectric is illustrated in Figure 4.1. Characteristic of the Sommerfeld precursor is an amplitude that starts at zero when θ = 1, increases rapidly to a peak occurring shortly after θ = 1, and then monotonically decreases with increasing θ. The instantaneous angular frequency of oscillation of the Sommerfeld precursor, defined as the time derivative of the oscillatory phase [5, 6, 29], begins at infinity (when the field amplitude is zero) and √ then rapidly decreases with increasing θ, approaching the angular frequency value ω02 + ωp2 − δ 2 associated with the upper edge of the material absorption band as θ → ∞ [29, 40]. 5. The Brillouin Precursor. The asymptotic expansion of E(z, t) as z → ∞ about the upper first-order near saddle point ωSP + (θ) for 1 < θ < θ1 , about the N second-order near saddle point ωSPN for θ = θ1 , and about both first-order near saddle points ωSP ± (θ) for θ > θ1 yields the dynamical evolution of the Brillouin N precursor, EB (z, t). Direct application of the saddle-point method does not result in a uniform approximation to the propagated field because the order of the saddle point changes discontinuously at θ = θ1 . In order to see this, consider applying the saddle-point method to two first-order saddle points ωi , ωj that coalesce into one second-order saddle point at some value θ = θ1 . The remainder term of the asymptotic approximation


638


would be bounded by (5.1)

|RN | ≤ A|z|−(2N +λ)/2 + B exp [{z[φ(ωj −, θ) − g(ωi , θ)]}] ,

where A, B are some constants independent of z, and where ωi is the dominant saddle point while ωj is the other saddle point, i.e., {ωi } < {ωj }. Because ωi is the dominant saddle point, the second term in (5.1) is negligible in comparison to the first, for large enough |z|. However, as θ → θ1 , {ωi } → {ωj } and |z| must increase without bound in order for the second term in (5.1) to remain negligible compared to the first. A uniform expansion of the Brillouin precursor, valid for all θ > 1, is obtained through use of the theorem [9, 12, 29] originally due to Chester, Friedman, and Ursell [9]. Here, the transformation that retains the behavior of the saddle points about θ = θ1 is given by 1 3 ν − α1 (θ)ν − α0 (θ) + φ(ω, θ) = 0, 3 where α0 , α1 are functions to be determined. Note that there are three possible branches of the inverse function. Chester, Friedman, and Ursell [9] proved that only one branch of the transformation defines a conformal mapping of some disc that contains both saddle points. It is this branch which must be used, as shown by Bleistein and Handelsman [4]. As before, the amplitude function is expanded as a finite number of terms plus a remainder that is regular and equal to zero at the transformed saddle points. This is obtained using the expansion (5.2)

(5.3)

i dω ≡ G0 (ν, θ) = h1 (θ) + h2 (θ)ν + (ν 2 − α1 (θ))H0 (ν, θ), ω − ωc dν

where h1 , h2 , and H0 are to be determined. Substitution of (5.2) and (5.3) into (1.4) results in the uniform asymptotic expansion of the Brillouin precursor EB (z, t) for the step-function modulated sine wave [29]

c 2/3 z c 1/3 EB (z, t) = − exp α0 (θ) e−i2π/3 · Ai α1 (θ) e−i2π/3 c z z 1 i i 1 · h+ (θ) + h− (θ) + O 2 ωSP + − ωc ωSP − − ωc z N N

c 2/3

c 2/3 + e−i2π/3 Ai(1) α1 (θ) e−i2π/3 z z 1 i 1 i · (5.4) , h+ (θ) − h− (θ) + O 1/2 + − ωc − − ωc ω ω z 2α (θ) SP SP 1

N

N

where Ai(ξ) denotes the Airy function and the coefficients appearing in (5.4) are given by 1 α0 (θ) = φ(ωSP + , θ) + φ(ωSP − , θ) , (5.5a) N N 2 1/3 3 1/2 α1 (θ) = φ(ωSP + , θ) − φ(ωSP − , θ) (5.5b) , N N 4 1/2 1/2 2α1 (θ) h± (θ) = ∓ (2) (5.5c) φ (ωSP + , θ) N


639


0.2

0.15

EB(z,t)

0.1

0.05

0

-0.05

-0.1

Fig. 5.1

1

1.2

1.4

1.6 θ = ct/z

1.8

2

2.2

Dynamic behavior of the Brillouin precursor field EB (z, t) for the step-function modulated sine wave with below-resonance applied signal frequency ωc = 0.5 ω0 at a relative observation distance z ≈ 3.8zd into the single-resonance Lorentz dielectric.

for θ = θ1 . At the critical value θ1 when the two near saddle points coalesce into a single second-order saddle point, these coefficients take on the limiting values (5.6a)

lim h± (θ) =

θ→θ1

−

1/3

2

≡ hs (θ1 ),

φ(3) (ωSPN , θ1 )

(5.6b)

i 1 i i h+ (θ) + h− (θ) = hs (θ1 ), lim θ→θ1 2 ωSP + − ωc ωSP − − ωc ωSPN − ωc N

(5.6c) lim

1

θ→θ1 2α1/2 (θ) 1

N

i ωSP + − ωc N

h+ (θ) −

i ωSP − − ωc N

h− (θ) = h2s (θ1 )

i d , dω ωSPN − ωc

and α0 (θ1 ) remains as given by (5.5a). For values of θ bounded away from θ1 such that (z/c)2/3 |α1 (θ)| 1, the large argument asymptotic expansion of the Airy function and its derivative may be substituted into (5.4). This substitution then reduces the uniform expansion to that obtained by a direct application of the saddle-point method for θ bounded away from θ1 . The Brillouin precursor field EB (z, t) for the step-function modulated sine wave E(z, t) with below-resonance applied signal frequency ωc = 0.5 ω0 at an observation distance of z = 1.2 × 10−6 m ≈ 3.8zd into the dispersive material is depicted in Figure 5.1. Characteristic of this Brillouin precursor is an amplitude that peaks near the space-time point θ = θ0 = n(0). Because it decays only as z −1/2 with increasing propagation distance, this peak amplitude point becomes more pronounced in the


640


total field evolution as z → ∞. The instantaneous angular frequency of oscillation of the Brillouin precursor starts at zero when θ = θ1 and monotonically increases to approach the value ω02 − δ 2 with increasing θ. Previous work [29, 36] exhibited a discontinuity in the uniform asymptotic approximation of the Brillouin precursor about the space-time point θ = θ1 , which the authors attributed to numerical instabilities. These numerical instabilities are now known to be caused by the use of an unnecessary approximation of the lower near saddle-point location ωSP − (θ) and incorrect phase values for the coefficients h± (θ) for N all θ > θ1 . Here, all saddle-point locations and coefficients are numerically determined so that this approximation error is avoided, resulting in a continuous evolution of the Brillouin precursor for all θ > 1. 6. The Signal Contribution. The signal contribution to the propagated field of the step-function modulated sine wave is due to the simple pole singularity at ω = ωc appearing in the integrand of (1.4). It is assumed here that ωc is real, positive, and finite (as all physical frequencies must be). The simple pole located at ωc may influence the value of E(z, t) if the path P(ω, θ) crosses the pole singularity, or if either of the saddle points ωSP + (θ) comes within close proximity of the pole, or both. D,N Let the path P(ω, θ) comprise the portions of the steepest descent paths emanating from ωSP + (θ) and ωSP + (θ) and continuing into the upper half plane. In N D the deformation of the original path of integration appearing in (1.4) into the path P(ω, θ), the simple pole located at ωc is crossed at some space-time point θ = θs ≥ 1. Hence, the original integral and the integral z i 1 i (6.1) ESDP (z, t) = exp φ(ω, θ) dω 2π c P(ω,θ) ω − ωc along the deformed contour are related by (6.2a)

E(z, t) = ESDP (z, t)

(6.2b)

E(z, t) = ESDP (z, t) − πiγe

(6.2c)

E(z, t) = ESDP (z, t) − 2πiγe

for θ < θs , (z/c)φ(ωc ,θs ) (z/c)φ(ωc ,θ)

for θ = θs , for θ > θs ,

where (6.3)

γ = lim (ω − ωc ) ω→ωc

i =i ω − ωc

is the residue of the amplitude function at the simple pole ωc . Direct application of the saddle-point method to ESP (z, t) yields a uniform expansion. However, it does not provide a uniform expansion for E(z, t). In order to see this, let θc denote the space-time point at which the pole becomes dominant over the saddle point ωSP in the sense that {φ(ωc , θ)} > {φ(ωSP , θ)}. Because the path P(ω, θ) lies in the valleys of ωSP , it necessarily follows that θs < θc . From (6.2), it is obvious that E(z, t) changes discontinuously about θ = θs when the saddle-point method is applied to ESDP (z, t). However, because the simple pole lies in the valley of the saddle point, the residue contributions appearing in (6.2) are exponentially smaller than ESDP (z, t) and pose no problem in obtaining a uniform approximation to E(z, t). This is not the case at θ = θc when the simple pole ωc becomes dominant over the saddle point ωSP . Here, {φ(ωSP , θ)} < {φ(ωc , θ)} for θs < θ < θc , and the


641


asymptotic expansion of E(z, t) is equivalent to that of ESDP (z, t). However, for θ > θc > θs , {φ(ωSP , θ)} < {φ(ωc , θ)} and the asymptotic expansion of E(z, t) is equivalent to the residue term appearing in (6.2c). That is, the asymptotic behavior of E(z, t) changes discontinuously about the space-time point θ = θs . A uniform asymptotic expansion is provided by the theorem due to Felsen and Marcuvitz [13, 14] and later generalized by Bleistein [3]. In this case, the usual transformation of the complex phase function is used, (6.4)

φ(ω, θ) = φ(ωSP , θ) − s2 ,

where s is real along the path of steepest descent, while the amplitude function is expanded as (6.5)

γ i dω = + T (s), ω − ωc ds s−b

where γ is the residue given in (6.3), b is the location of the transformed pole, and T (s) is regular at both s = 0 (the transformed saddle point) and at s = b. Substitution of (6.4) and (6.5) into the integral representation of the propagated field (1.4) then yields (to first order) ∞ 2 z i −2 1 ,θ) φ(ω −(z/c)s SP e ds E(z, t) = ie c 2π ωSP − ωc φ (ωSP , θ) −∞ ∞

z γ −(z/c)s2 1 γ φ(ωSP ,θ) c ie + e + (6.6) ds . 2π b −∞ s − b The asymptotic approximation of the integral appearing in the first term of (6.6) is obtained using the method of steepest descents applied to a first-order saddle point located at ωSP . Because the individual contributions from the distant and near saddle points have already been accounted for with either the Sommerfeld or Brillouin precursor, respectively, this term need not be considered further. The second term in (6.6) accounts for the effect of the pole located at ωc on the saddle point ωSP and may be evaluated in terms of the complementary error function. This term, together with the appropriate residue contribution, yields the pole contribution Ec (z, t) to the propagated field. The particular saddle point used in the evaluation of (6.6) depends upon the value of the input carrier frequency ωc . If ωc is below resonance, then all saddle points except the near saddle point ωSP + (θ) are sufficiently removed from the pole so N that ωSP = ωSP + (θ) is used in (6.6). If ωc is above resonance, then all saddle points N except the distant saddle point ωSP + (θ) are sufficiently removed from the pole so that D ωSP = ωSP + (θ) is used in (6.6). These results have been given previously [29] with a D minor mistake in the above-resonance case which was π-out-of-phase with the correct results. 2 2 For √ carrier frequencies within the absorption band of the material, ω0 − δ ≤ ωc ≤ ω02 − δ 2 + ωp2 , previous work [29] hypothesized that all saddle points are sufficiently removed from ωc so that Ec (z, t) is solely composed of the residue term which begins abruptly at θ = θs . We have found that this hypothesis is not accurate. Instead, for carrier frequencies within the absorption band of the material, both saddle points ωSP + (θ) and ωSP + (θ) are influenced by the pole located at ωc . Hence, the inN D tegral (6.6) must be evaluated for both saddle points ωSP = ωSP + and ωSP = ωSP + , N


D

642


while θs and the appropriate residue contribution are determined by the saddle point whose path of steepest descent crosses the simple pole located at ωc . The space-time point θs at which the path P(ω, θ) crosses the simple pole located at ωc is determined by the equation (6.7)

Y (ωc , θs ) = Y (ωSP , θs ),

where Y ≡ {φ} and ωSP denotes either ωSP + (θ) or ωSP + (θ). We have found (by D N inspection) that the deciding factor as to which saddle point determines the value of θs is the value of Y (ωc , θ) at θ = 1. If Y (ωc , 1) > 0, the steepest descent path emanating from the near saddle point ωSP + (θ) is used to determine the value θs . N If Y (ωc , 1) < 0, the steepest descent path emanating from the distant saddle point ωSP + (θ) is used to determine the value θs . If Y (ωc , 1) = 0, then θs = 1. The value of D ωc = ωY which separates the two cases is given approximately by ωY ≈ 4.2925 × 1016 rad/s for the material parameters considered here. As a consequence, we have applied the uniform theory to account for two first-order saddle points with a nearby simple pole singularity. As an example, for angular frequencies ωc satisfying Y (ωc , 1) > 0, the uniform signal contribution is given by

z z 1 iγu −iπ erfc i∆D (θ) exp φ(ωc , θ) Ec (z, t) = 2π c c z πc 1 exp φ(ωSP + , θ) + D ∆D (θ) z c

z z exp φ(ωc , θ) + iγu −iπ erfc i∆N (θ) c c z 1 πc exp φ(ωSP + , θ) + (6.8a) , θ < θs , N ∆N (θ) z c

z z 1 iγu −iπ erfc i∆D (θ) exp φ(ωc , θ) Ec (z, t) = 2π c c 1 πc z + exp φ(ωSP + , θ) D c ∆D (θ) z

z z exp φ(ωc , θ) + iγu iπ erfc −i∆N (θ) c c z 1 πc z + (6.8b) + i exp φ(ωc , θ) , θ ≥ θs , exp φ(ωSP + , θ) N ∆N (θ) z c c as z → ∞. Here, γu = i is the residue of the amplitude function i(ω − ωc ) at ωc , erfc(ξ) is the complementary error function, and (6.9)

1/2 ∆D,N (θ) ≡ φ(ωSP + , θ) − φ(ωc , θ) . D,N

A similar expression may be given for the uniform signal contribution for carrier frequencies ωc satisfying Y (ωc , 1) > 0 [8]. Questions about the accuracy of this uniform expansion as it applies to the classical problem of a step-function modulated sine wave signal have persisted because, until now, it has been difficult to isolate the leading edge of the pole contribution from the remainder of the field. For all values of ωc , either the Sommerfeld or the


643

UNIFORM ASYMPTOTICS IN WAVE PROPAGATION 0.015

0.015

(a)

(b)

0.01

0.01

0.005

0.005

0

0

-0.005

-0.005

-0.01

1

Fig. 6.1

2

3

θ = ct/z

4

5

6

-0.01

1

2

3

θ = ct/z

4

5

6

Comparison of asymptotic (dashed curve) and numerical (solid curve) pole contributions for a step-function modulated sine wave with resonant frequency ωc = ω0 at an observation distance of z ≈ 21.3zd . The asymptotic approximation is made (a) utilizing only the near saddle point ωSP + (θ), and (b) utilizing both saddle points ωSP + (θ) and ωSP + (θ). N

N

D

Brillouin precursor, or both, have an amplitude that is comparable with the pole contribution about this space-time region, which then masks its leading edge. With an asymptotic code which provides an accurate uniform asymptotic approximation of both the Sommerfeld and Brillouin precursors, we have now been able to compare the uniform asymptotic approximation of the pole contribution Ec (z, t) to a numerically determined pole contribution Ecn (z, t) that is found by subtracting the asymptotic Sommerfeld and Brillouin precursors from a numerical computation of the total field E n (z, t), as given by Ecn (z, t) = E n (z, t) − ES (z, t) − EB (z, t).

(6.10)

The numerical evaluation of the Fourier–Laplace integral representation of the propagated pulse given in (1.4) is obtained here through a straightforward application of the fast Fourier transform algorithm. Consider the case of an on-resonant carrier frequency ωc = ω0 which satisfies Y (ωc , 1) > 0, as illustrated in Figure 6.1. Part (a) of the figure compares the asymptotic approximation of the pole contribution calculated using only the near saddle point ωSP + (θ) (dashed curve) along with the numerical pole contribution (solid curve) N for the resonant case ωc = ω0 at a propagation distance of z ≈ 21.3zd . In part (b) of the figure, the asymptotic approximation of the pole contribution (dashed curve) is calculated using both ωSP + (θ) and ωSP + (θ), as given in (6.8). The open circle N D denotes the space-time value θs when the path P(ω, θ) crosses ωc . A comparison of parts (a) and (b) of this figure shows that the contribution from the distant saddle point ωSP + (θ) contributes to the high-frequency oscillations observed in the numerD ical pole contribution. As a consequence, the inclusion of both saddle points does indeed provide a better approximation to the numerical pole contribution than the near saddle point ωSP + alone or by assuming only a residue contribution as previously N hypothesized. Next, consider the intra-absorption band case of ωc = 1.25ω0 which satisfies Y (ωc , a) < 0, as presented in Figure 6.2. Part (a) of the figure compares the asymptotic approximation of the pole contribution calculated using only the distant saddle point ωSP + (θ) (dashed curve) along with the numerical pole contribution (solid curve) D


644


0.04

0.04

(a)

0.03 0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

Fig. 6.2

2

4

6

8 θ = ct/z

(b)

0.03

10

12

14

-0.04

2

4

6

8 θ = ct/z

10

12

14

Comparison of asymptotic (dashed curve) and numerical (solid curve) pole contributions for a step-function modulated sine wave with intra-absorption band frequency ωc = 1.25ω0 at an observation distance of z ≈ 5.3zd . The asymptotic approximation is made (a) utilizing only the distant saddle point ωSP + (θ), and (b) utilizing both saddle points ωSP + (θ) D D and ωSP + (θ). N

at a propagation distance of z ≈ 5.3zd . In Figure 6.2 (b), the asymptotic approximation of the pole contribution (dashed curve) is calculated using both ωSP + (θ) and N ωSP + (θ). The open circle denotes the space-time value θs when the path P(ω, θ) D crosses ωc . A comparison of the two figures shows that the contribution from the near saddle point ωSP + (θ) is minimal. However, these figures again confirm that the uniN form asymptotic theory provides the correct approximation to the pole contribution. This is in contrast to the previous hypothesis [29] that both saddle points ωSP + (θ) D and ωSP + (θ) are sufficiently removed from ωc when ωc is within the absorption band N of the material so that Ec (z, t) is composed solely of the residue term which abruptly begins at θ = θs . 7. Conclusions. The propagated electric field component of a linearly polarized, plane-wave, step-function modulated sine wave signal of applied carrier frequency ωc traveling through a single-resonance Lorentz dielectric has been studied in this paper. An analytic, asymptotic approximation of the propagated wave field requires the application of three uniform theories in the subject of asymptotic expansions of integrals. The results presented here provide the first correct description of the pole contribution of the field (1.4) for applied carrier frequencies that lie within and above the absorption band of the material. Previous attempts to isolate the pole contribution from the rest of the field were unsuccessful due to inaccurate asymptotic results for the Brillouin precursor about the space-time point θ = θ1 and an incorrect asymptotic representation of the pole contribution for applied carrier frequencies in √ the range ωc > ω02 + ωp2 − δ 2 . Four different carrier frequencies of the step-function modulated sine wave signal were considered in this paper: the below-resonance carrier frequency ωc = 0.5ω0 , the intra-absorption band carrier frequencies ωc = ω0 and ωc = 1.25ω0 , and the aboveresonance carrier frequency ωc = 2.5ω0 . The total asymptotic field, which is the sum of the three asymptotic components, E(z, t) = ES (z, t) + EB (z, t) + Ec (z, t), is presented for each applied carrier frequency as the dashed curves in the upper plots of Figures 7.1–7.4. The solid curves appearing in these figures represent the


645


0.2 0.15

E(z,t)

0.1 0.05 0 -0.05 -0.1

1 -3 x 10 2

1.5

2

2.5

2

2.5

0 -2

Fig. 7.1

1

1.5

θ = ct/z

The total asymptotic (dashed curve) and numerical (solid curve) fields of the step-function modulated sine wave signal with below-resonance applied carrier frequency ωc = 0.5ω0 at a distance of z = 1.2 × 10−16 m ≈ 3.8zd into the single-resonance Lorentz dielectric. The lower figure shows the difference between the two results.

0.2

E(z,t)

0.1 0 -0.1 -0.2

1

2

3

2

3

4

5

6

4

5

6

0.01 0 -0.01 1

Fig. 7.2

θ = ct/z

The total asymptotic (dashed curve) and numerical (solid curve) fields of the step-function modulated sine wave signal with resonant applied carrier frequency ωc = ω0 at a distance of z = 8 × 10−8 m ≈ 21.3zd into the single-resonance Lorentz dielectric. The lower figure shows the difference between the two results.


646


0.3

E(z,t)

0.2 0.1 0 -0.1 -0.2

2

4

6

8

10

12

14

2

4

6

8 θ = ct/z

10

12

14

0.05 0 -0.05

Fig. 7.3

The total asymptotic (dashed curve) and numerical (solid curve) fields of the stepfunction modulated sine wave signal with intra-absorption band applied carrier frequency ωc = 1.25ω0 rad/s at a distance of z = 3 × 10−8 m ≈ 5.3zd into the single-resonance Lorentz dielectric. The lower figure shows the difference between the two results.

0.1

E(z,t)

0.05

0

-0.05

-0.1

1 -3 x 10

1.2

1.4

1.2

1.4

1.6

1.8

2

1.6

1.8

2

5 0 -5 1

Fig. 7.4

θ = ct/z

The total asymptotic (dashed curve) and numerical (solid curve) fields of the step-function modulated sine wave signal with above-absorption band applied carrier frequency ωc = 2.5ω0 at a distance of z = 9 × 10−7 m ≈ 2.7zd into the single-resonance Lorentz dielectric. The lower figure shows the difference between the two results.



647

numerical total field E n (z, t). In all cases considered, the asymptotic representation of the total field is almost indistinguishable from the numerical total field; both fields exhibit the same oscillatory characteristics, the only apparent differences being the peak amplitude values. The differences between the asymptotic and numerical pole contributions for the intra-absorption band frequencies ωc = ω0 and ωc = 1.25ω0 for relatively small values of θ (see Figures 6.1 and 6.2) are not apparent in the total field figures (Figures 7.2 and 7.3) because the amplitude of the pole contribution is dwarfed by the amplitude of the Brillouin precursor for these values of θ. However, these differences are evident in the lower plots of Figures 7.1–7.4 which depict the differences between the total asymptotic and numerically determined fields. As the propagation distance increases in each frequency case, this error decreases to zero in the sense of Poincaré [31]. Ultrawideband pulses play an increasingly important role in the world today, and so it is important to fully understand the dynamic behavior of these pulse types in dispersive media. The step-function modulated sine wave signal is a canonical ultrawideband pulse of central importance to understanding this. For example, a rectangular pulse may be represented as the time-delayed difference between two step-function pulses [28]. The results presented here have shown that the uniform asymptotic theory does indeed provide an accurate description of the propagated step-function modulated sine wave signal as the propagation distance exceeds a single absorption depth at the input pulse carrier frequency. REFERENCES [1] C. Baum, L. Carin, and A. Stone, eds., Ultra-Wideband, Short-Pulse Electromagnetics 3, Springer, New York, 1997. [2] H. Bertoni, L. Carin, and L. Felsen, eds., Ultra-Wideband, Short-Pulse Electromagnetics, Springer, New York, 1993. [3] N. Bleistein, Uniform asymptotic expansions of integrals with stationary point near algebraic singularity, Comm. Pure Appl. Math., 19 (1966), pp. 353–370. [4] N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1975. ¨ [5] L. Brillouin, Uber die Fortpflanzung des Lichtes in dispergierenden Medien, Ann. Phys., 349 (1914), pp. 203–240. [6] L. Brillouin, Wave Propagation and Group Velocity, Academic Press, New York, 1960. [7] L. Carin and L. Felsen, eds., Ultra-Wideband, Short-Pulse Electromagnetics 2, Springer, New York, 1995. [8] N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal, Ph.D. dissertation, University of Vermont, 2004. [9] C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Cambridge Phil. Soc., 53 (1957), pp. 599–611. [10] P. Debye, N¨ aherungsformeln f¨ ur die Zylinderfunktionen f¨ ur große Werte des Arguments und unbeschr¨ ankt ver¨ anderliche Werte des Index, Math Ann., 67 (1909), pp. 535–558. [11] A. Einstein, Zur Elektrodynamic bewegter K¨ orper, Ann. Phys., 322 (1905), pp. 891–921. [12] L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Englewood Cliffs, NJ, 1973. [13] L. B. Felsen, Radiation from a uniaxially anisotropic plasma half-space, IEEE Trans. Antennas Propagation, AP-11 (1963), pp. 469–484. [14] L. B. Felsen and N. Marcuvitz, Modal Analysis and Synthesis of Electromagnetic Fields, Polytechnic Inst. Brooklyn, Microwave Res. Inst. Research Rep., 1959. [15] W. R. Hamilton, Researches respecting vibration, connected with the theory of light, Proc. Royal Irish Academy, 1 (1839), pp. 341–349. [16] R. A. Handelsman and N. Bleistein, Uniform asymptotic expansions of integrals that arise in the analysis of precursors, Arch. Ration. Mech. Anal., 35 (1969), pp. 267–283. [17] L. Kelvin, On the waves produced by a single impulse in water of any depth, or in a dispersive medium, Proc. Roy. Soc., 42 (1887), p. 80.


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