The Dirac Delta Function

Appendix A The Dirac Delta Function

A.1 The One-Dimensional Dirac Delta Function

The Dirac delta function [1] in one-dimensional space may be deﬁned by the pair of equations

δ(x) = 0; x = 0, (A.1) ∞ δ(x)dx = 1. (A.2) −∞

It is clear from this definition that δ(x) is not a function in the ordinary mathematical sense, because if a function is zero everywhere except at a single point and the integral of this function over its entire domain of definition exists, then the value of this integral is necessarily also equal to zero. Because of this, it is more appropriate to regard δ(x) as a functional quantity with a certain well-defined symbolic meaning. For example, one can consider a sequence of functions δ(x,ε) that, with increasing values of the parameter ε, differ appreciably from zero only over a decreasing x-interval about the origin and which are such that

∞ δ(x,ε)dx = 1(A.3) −∞ for all values of ε. Although it may be tempting to try to interpret the Dirac delta function as the limit of such a sequence of well-deﬁned functions δ(x,ε) as ε →∞, it must be recognized that this limit need not exist for all values of the independent variable x. However, the limit

∞ lim δ(x,ε)dx = 1(A.4) ε→∞ −∞

© Springer Nature Switzerland AG 2019 641 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 642 A The Dirac Delta Function must exist. As a consequence, one may interpret any operation that involves the delta function δ(x) as implying that this operation is to be performed with a function δ(x,ε) of a suitable sequence and that the limit as ε →∞is to be taken at the conclusion of the calculation. The particular choice of the sequence of functions δ(x,ε) is immaterial, provided that any oscillations near the origin x = 0 are not too violent [2]. For example, each of the following functions forms a sequence with respect to the parameter ε that satisﬁes the required properties:

ε −ε2x2 ε δ(x,ε) = √ e & δ(x,ε) = rect /ε(x) & δ(x,ε) = sinc(εx) π 1 π in Cartesian coordinates, and ε δ(r,ε) = circ (r) & δ(r,ε) = J (2πεr) 1/ε r 1 in polar coordinates, where rect1/ε(x) ≡ ε/2 when |x| < 1/ε and is zero otherwise, 2 circ1/ε(r) ≡ ε /π when r<1/ε and is zero otherwise, and sinc(x) ≡ sin (x)/x when x = 0 and sinc(0) ≡ limx→0 sin (x)/x = 1. Let f(x) be a continuous and sufﬁciently well-behaved function of x ∈ (−∞, ∞) and consider the value of the deﬁnite integral

∞ ∞ f(x)δ(x− a)dx = lim f(x)δ(x− a,ε)dx. −∞ ε→∞ −∞

When the parameter ε is large, the value of the integral appearing on the right- hand side of this equation depends essentially on the behavior of f(x) in the immediate neighborhood of the point x = a alone, and the error that results from the replacement of f(x) by f(a) may be made as small as desired by taking ε sufﬁciently large. Hence

∞ ∞ lim f(x)δ(x− a,ε)dx = f(a) lim δ(x − a,ε)dx, ε→∞ −∞ ε→∞ −∞ so that ∞ f(x)δ(x− a)dx = f(a). (A.5) −∞

This result is referred to as the sifting property of the delta function. Notice that, for this result to hold, the domain of integration need not be extended over all x ∈ (−∞, ∞); it is only necessary that the domain of integration contain the point x = a A.1 The One-Dimensional Dirac Delta Function 643 in its interior, so that a+Δ2 f(x)δ(x− a)dx = f(a), (A.6) a−Δ1 where Δ1,2 > 0. It is then seen that f(x)need only be continuous at x = a. The above results may be written symbolically as

f(x)δ(x− a) = f(a)δ(x− a), (A.7) the meaning of such a statement being that the two sides yield the same result when integrated over any domain containing the point x = a. For the special case when f(x)= xk with k>0 and a = 0, Eq. (A.7) yields

xkδ(x) = 0, ∀k>0. (A.8)

Theorem A.1 (Similarity Relationship (Scaling Law)) For all a = 0

1 δ(ax) = δ(x). (A.9) |a|

Proof In order to prove this relationship one need only compare the integrals of f(x)δ(ax)and f(x)δ(x)/|a| for any sufﬁciently well-behaved continuous function f(x). For the ﬁrst integral one has (for any a = 0) ∞ 1 ∞ 1 f(x)δ(ax)dx =± f(y/a)δ(y)dy = f(0), −∞ a −∞ |a| where the upper or lower sign choice is taken accordingly as a>0ora<0, respectively, and for the second integral one obtains ∞ 1 1 f(x) δ(x)dx = f(0). −∞ |a| |a|

Comparison of these two results then shows that δ(ax) = δ(x)/|a|, as was to be proved. For the special case a =−1, Eq. (A.9) yields

δ(−x) = δ(x), (A.10) so that the delta function is an even function of its argument. 644 A The Dirac Delta Function

Theorem A.2 (Composite Function Theorem) If y = f(x) is any continuous function of x with simple zeroes at the points xi [i.e., y = 0 at x = xi and f (xi) = 0] and no other zeroes, then

1 δ(f(x)) = δ(x − x ). (A.11) |f (x )| i i i

Proof Let g(x) be any sufﬁciently well-behaved continuous function and let {xi} denote the set of points at which y = 0. Under the change of variable x = f −1(y) one has that ∞ − 1 g(x)δ(f(x))dx = g f 1(y) δ(y) dy −1 −∞ R |f f (y) | − 1 1 = g f 1(0) = g(x ) , −1 i |f f (0) | |f (xi| xi i where R denotes the range of f(x). In addition, ∞ ∞ 1 − = 1 − g(x) δ(x xi) dx g(x)δ(x xi)dx −∞ |f (x )| |f (x )| −∞ i i i i 1 = g(x ) . i |f (x | i i

Comparison of these two expressions then proves the theorem. As an example, consider the function f(x) = x2 − a2 which has simple zeroes at x =±a. Then |f (±a)|=2|a| so that, for a = 0,

1 δ(x2 − a2) = (δ(x − a) + δ(x + a)) . 2|a|

An additional relationship of interest that employs the Dirac delta function is

∞ δ(ξ − x)δ(x − η)dx = δ(ξ − η), (A.12) −∞ which is seen to be an extension of the sifting property to the delta function itself. This equation then implies that if both sides are multiplied by a continuous function of either ξ or η and the result integrated over all values of either ξ or η, respectively, an identity is obtained. That is, because

∞ f(ξ)δ(ξ − η)dξ = f(η), −∞ A.1 The One-Dimensional Dirac Delta Function 645 and ∞ ∞ f(ξ) δ(ξ − x)δ(x − η)dx dξ −∞ −∞ ∞ ∞ = f(ξ)δ(ξ − x)dξ δ(x − η)dx −∞ −∞ ∞ = f(x)δ(x− η)dx = f(η) −∞ then the expression in Eq. (A.12) follows. In a similar manner, because

∞ ∞ g(η) δ(ξ − x)δ(x − η)dx dη −∞ −∞ ∞ ∞ = g(η)δ(x − η)dη δ(ξ − x)dx −∞ −∞ ∞ = g(x)δ(ξ − x)dx = g(ξ) −∞ and ∞ g(η)δ(ξ − η)dη = g(ξ), −∞ then the expression in Eq. (A.12) is again obtained. Consider next what interpretation may be given to the derivatives of the delta function, accomplished through use of the function sequence δ(x,ε). Consider then ∞ the ordinary integral −∞ f(x)δ(x, ε)dx which may be evaluated by application of the method of integration by parts with u = f(x)and dv = δ (x, ε)dx, so that

∞ ∞

f(x)δ(x, ε)dx = f(∞)δ(∞,ε)− f(−∞)δ(−∞,ε)− f (x)δ(x, ε)dx. −∞ −∞

Upon proceeding to the limit as ε →∞, the ﬁrst two terms appearing on the right- hand side of this equation both vanish because

lim δ(±∞,ε)= 0, (A.13) ε→∞ with the result ∞

f(x)δ(x)dx =−f (0). (A.14) −∞ 646 A The Dirac Delta Function

Upon repeating this procedure n times for the nth-order derivative of the delta function, one obtains the general result

∞ f(x)δ(n)(x)dx = (−1)nf (n)(0). (A.15) −∞

As a special case of Eq. (A.14), let f(x)= x so that

∞ ∞

xδ (x)dx =−1 =− δ(x)dx, −∞ −∞ and one then has the equivalence

xδ (x) =−δ(x). (A.16)

Because δ(x) is an even function and x is an odd function, it then follows that δ (x) is an odd function of its argument; that is

δ (−x) =−δ (x). (A.17)

The generalization of Eq. (A.16) may be directly obtained from Eq. (A.15) by letting f(x)= xn. In that case, f (n)(x) = n! and this relation gives

∞ ∞

xnδ (x)dx = (−1)nn!=(−1)nn! δ(x)dx, −∞ −∞ and one then has the general equivalence

xnδ(n)(x) = (−1)nn!δ(x). (A.18)

The even-order derivatives of the delta function are even functions and the odd-order derivatives are odd functions of the argument. It is often convenient to express the Dirac delta function in terms of the Heaviside unit step function U(x) deﬁned by the relations U(x) = 0 when x<0, U(x) = 1 when x>0. Consider the behavior of the derivative of U(x). If, as before, a superscript prime denotes differentiation with respect to the argument, one obtains formally upon integration by parts (with the limits −x1 < 0 and x2 > 0), x x 2 2 f(x)U (x)dx = f(x)U(x) x2 − f (x)U(x)dx [ ]−x1 −x −x 1 1 x 2 = f(x2) − f (x)dx 0

= f(x2) −[f(x2) − f(0)]=f(0), A.1 The One-Dimensional Dirac Delta Function 647 where f(x)is any continuous function. Upon setting x = y − a and f(x)= f(y− a) = F(y) and then proceeding to the limits as −x1 →−∞and x2 →+∞,the above result becomes ∞

F(y)U (y − a)dy = F(a), −∞ and the derivative U (x) is seen to satisfy the sifting property given in Eq. (A.5). In particular, with F(y) = 1 and a = 0, this expression becomes

∞

U (y)dy = 1, −∞ and U (x) also satisfies the property given in Eq. (A.2) which serves to partially define the delta function. Moreover, U (x) = 0 for all x = 0 and property (A.1)is also satisfied. Hence, one may identify the derivative of the unit step function with the delta function, so that

dU(x) δ(x) = . (A.19) dx In addition, it is seen that [from Eqs. (A.6) and (A.19)] x U(x) = δ(ξ)dξ. (A.20) −∞

The Dirac delta function may also be introduced through the use of the Fourier integral theorem [3], which may be written as

∞ ∞ − f(a)= dν dx f(x)ei2πν(x a) (A.21) −∞ −∞ for any sufﬁciently well-behaved, continuous function f(x). Deﬁne the function sequence ε − sin (2π(x − a)ε) K(x − a,ε) ≡ ei2πν(x a)dν = (A.22) −ε π(x − a) with limit

K(x − a) ≡ lim K(x − a,ε). (A.23) ε→∞

Strictly speaking, this limit does not exist in the ordinary sense when x = a; however, the limit does exist and has the value zero when x = a if it is interpreted in 648 A The Dirac Delta Function the sense of a Cesáro limit [4]. Upon inversion of the order of integration, Eq. (A.21) may be formally rewritten as

∞ f(a)= f(x)K(x− a)dx, (A.24) −∞ which should be interpreted as meaning that

∞ f(a)= lim f(x)K(x− a,ε)dx. (A.25) ε→∞ −∞

Thus, the function K(x−a) satisfies the sifting property (A.5) of the delta function. ∞ If one sets f(x) = 1 and a = 0inEq.(A.24), there results −∞ K(x)dx = 1 and K(x) satisfies the property given in Eq. (A.2) which serves to partially define the delta function. Because K(x) = limε→∞ K(x,ε) = 0 when x = 0, so that the property given in Eq. (A.1) is also satisfied, one then obtains from Eq. (A.23)the relation ∞ δ(x) = ei2πνxdν. (A.26) −∞

That is, the Dirac delta function may be regarded as the Fourier transform of unity. The reciprocal relation follows from Eq. (A.25) upon setting f(x) = exp(i2πνx) and a = 0, so that

∞ − 1 = δ(x)e i2πνxdx, (A.27) −∞ which also follows directly from the sifting property given in Eq. (A.5). Notice that this relation by itself is not sufﬁcient to imply the validity of Eq. (A.26).

A.2 The Dirac Delta Function in Higher Dimensions

The deﬁnition of the Dirac delta function may easily be extended to higher- dimensional spaces. In particular, consider three-dimensional vector space in which case the deﬁning relations given in Eqs. (A.1) and (A.2) become

δ(r) = 0; r = 0, (A.28) ∞ δ(r)d3r = 1. (A.29) −∞ A.2 The Dirac Delta Function in Higher Dimensions 649

The function

δ(r) ≡ δ(x,y,z) ≡ δ(x)δ(y)δ(z), (A.30) ˆ ˆ ˆ where r = 1xx + 1yy + 1zz is the position vector with components (x,y,z) clearly satisﬁes Eqs. (A.28) and (A.29) and so deﬁnes a three-dimensional Dirac delta function. The sifting property given in Eq. (A.5) then becomes

∞ f(r)δ(r − a)d3r = f(a), (A.31) −∞ and the similarity relationship or scaling law given in Eq. (A.9) now states that

1 δ(ar) = δ(r), (A.32) |a|3 where a is a scalar constant. The Fourier transform pair relationship expressed in Eqs. (A.26) and (A.27) becomes

∞ 1 · δ(r) = eik rd3k, (A.33) (2π)3 −∞ ∞ − · 1 = δ(r)e ik rd3r, (A.34) −∞

ˆ ˆ ˆ ˆ ˆ ˆ where k = 1xkx + 1yky + 1zkz = 2π(1xνx + 1yνy + 1zνz). The generalization of the three-dimensional Dirac delta function to more general coordinate systems requires more careful attention. Suppose that a function Δ(r) is given in Cartesian coordinates as

Δ(r) = δ(x)δ(y)δ(z) (A.35) and it is desired to express Δ(r) in terms of the orthogonal curvilinear coordinates (u,v,w)deﬁned by

u = f1(x,y,z), v = f2(x,y,z), w = f3(x,y,z), (A.36) where f1,f2,f3 are continuous, single-valued functions of x,y,z with a unique = −1 = −1 = −1 inverse x f1 (u,v,w),y f2 (u,v,w),z f3 (u,v,w). That is, an expression for Δ(r) is desired in terms of the coordinate variables (u,v,w) that satisﬁes the relation ∞

Δ(r − r )ϕ(u, v, w)dV = ϕ(u ,v ,w), (A.37) −∞ 650 A The Dirac Delta Function where dV is the differential volume element in u, v, w-space and (u ,v ,w ) is the point corresponding to (x ,y ,z ) under the coordinate transformation given in Eq. (A.36). If the point r = (x,y,z) is varied from r to r + δr1 by changing the coordinate variable u to u + δu while keeping v and w ﬁxed, then

∂r δr = δu. 1 ∂u

Similarly, if the point r = (x,y,z) is varied from r to r + δr2 by changing the coordinate variable v to v + δv while keeping u and w ﬁxed, then

∂r δr = δv. 2 ∂v

The parallelogram with sides δr1 and δr2 then has area

∂r ∂r δA =|δA|=|δr1 × δr2|= × δuδv. (A.38) ∂u ∂v

If the point r = (x,y,z)is now varied from r to r + δr3 by changing the coordinate variable w to w + δw while keeping u and v ﬁxed, then

∂r δr = δw, 3 ∂w and the volume of the parallelepiped with edges δr1, δr2, and δr3 is then given by

∂r ∂r ∂r δV =|δr3 · (δr1 × δr2)|= · × δuδvδw. (A.39) ∂w ∂u ∂v

The quantity x,y,z ∂(x,y,z) ∂r ∂r ∂r J ≡ ≡ · × (A.40) u, v, w ∂(u,v,w) ∂w ∂u ∂v is recognized as the Jacobian of the coordinate transformation of x,y,z with respect to u, v, w. With this result for the differential element of volume, Eq. (A.37) becomes ∞ x,y,z Δ(r − r )ϕ(u, v, w) J dudvdw = ϕ(u ,v,w), (A.41) −∞ u, v, w from which it is immediately seen that

x,y,z δ(u)δ(v)δ(w) = J δ(x)δ(y)δ(z). (A.42) u, v, w A.2 The Dirac Delta Function in Higher Dimensions 651

Because this transformation is assumed to be single-valued, then

= δ(u)δ(v)δ(w) = u, v, w δ(x)δ(y)δ(z) J δ(u)δ(v)δ(w), (A.43) x,y,z x,y,z J u,v,w where J(u,v,w/x,y,z) is the Jacobian of the inverse transformation. Consider ﬁnally the description of a function (r) that vanishes everywhere in three-dimensional space except on a surface S and is such that

∞ (r)ϕ(r)d3r = ς(r)ϕ(r)d2r, (A.44) −∞ S where ς(r) is the value of (r) on the surface S, that is, when r ∈ S. Choose orthogonal curvilinear coordinates (u,v,w)such that w = w0 describes the surface S for some constant w0, in which case ∇w is parallel to the normal to the surface S, and is such that ∇u and ∇v are both perpendicular to the normal to the surface S. The differential element of area of the surface S is then given by Eq. (A.38). Furthermore, both ∂r/∂w and (∂r/∂u) × (∂r/∂v) are normal to S so that

x,y,z ∂r ∂r ∂r J = × . (A.45) u, v, w ∂w ∂u ∂v

With this result, Eq. (A.44) may be written as ∞ ∂r ∂r ∂r (r)ϕ(r) × dudvdw = ς(r)ϕ(r)d2r, −∞ ∂w ∂u ∂v S which, with Eq. (A.38), may be expressed as ∞ ∂r (r)ϕ(r) d2rdw = ς(r)ϕ(r)d2r. (A.46) −∞ ∂w S

From this result it then follows that

= ς(r) − (r) ∂r δ(w w0), (A.47) ∂w which is the solution of Eq. (A.44). This result can be simpliﬁed somewhat by noting that when the variable w is varied while u and v are held ﬁxed, then the changes in r and w are related by δw =∇w · δr, so that

∂r ·∇w = 1. ∂w 652 A The Dirac Delta Function

Moreover, because both ∂r/∂w and ∇w are normal to the surface S described by w = w0, then

∂r |∇w| = 1, ∂w and, as a result, Eq. (A.47) becomes

(r) = |∇w| ς(r)δ(w − w0) (A.48) as the solution to Eq. (A.44).

References

1. P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford: Oxford University Press, 1930. Sect. 15. 2. M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions. London, England: Cambridge University Press, 1970. 3. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 4. B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Integral. London: Cambridge University Press, 1950. pp. 100–104. Appendix B Helmholtz’ Theorem

Because 1 ∇2 =−4πδ(R) (B.1) R where R = r − r with magnitude R =|R| and where δ(R) = δ(r − r ) = δ(x − x )δ(y − y )δ(z − z ) is the three-dimensional Dirac delta function (see Appendix A), then any sufﬁciently well-behaved vector function F(r) = F(x,y,z) can be represented as 1 1 F(r) = F(r )δ(r − r )d3r =− F(r )∇2 d3r V 4π V R 1 F(r ) =− ∇2 d3r , (B.2) 4π V R the integration extending over any region V that contains the point r. With the identity ∇×∇×=∇∇·−∇2,Eq.(B.2) may be written as

1 F(r ) 1 F(r ) F(r) = ∇×∇× d3r − ∇∇ · d3r . (B.3) 4π V R 4π V R

Consider ﬁrst the divergence term appearing in this expression. Because the vector differential operator ∇ does not operate on the primed coordinates, then

1 F(r ) 1 1 ∇· d3r = F(r ) ·∇ d3r . (B.4) 4π V R 4π V R

© Springer Nature Switzerland AG 2019 653 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 654 B Helmholtz’ Theorem

Moreover, the integrand appearing in this expression may be expressed as 1 1 F(r ) ·∇ =−F(r ) ·∇ R R F(r ) 1 =−∇ · + ∇ · F(r ), (B.5) R R where the superscript prime on the vector differential operator ∇ denotes differentiation with respect to the primed coordinates alone. Substitution of Eq. (B.5)into Eq. (B.4) and application of the divergence theorem to the ﬁrst term then yields

1 F(r ) 1 F(r ) 1 ∇ · F(r ) ∇· d3r =− ∇ · d3r + d3r 4π V R 4π V R 4π V R 1 1 1 ∇ · F(r ) =− F(r ) · nˆd2r + d3r 4π S R 4π V R = φ(r), (B.6) which is the desired form of the scalar potential φ(r) for the vector ﬁeld F(r).Here S is the surface that encloses the regular region V and contains the point r. For the curl term appearing in Eq. (B.3) one has that

1 F(r ) 1 1 ∇× d3r =− F(r ) ×∇ d3r 4π V R 4π V R 1 1 = F(r ) ×∇ d3r . (B.7) 4π V R

Moreover, the integrand appearing in the ﬁnal form of the integral in Eq. (B.7)may be expressed as

1 ∇ × F(r ) F(r ) F(r ) ×∇ = −∇ × , (B.8) R R R so that 1 F(r ) 1 ∇ × F(r ) 1 F(r ) ∇× d3r = d3r − ∇ × d3r 4π V R 4π V R 4π V R 1 ∇ × F(r ) 1 1 = d3r + F(r ) × nˆd2r 4π V R 4π S R = A(r), (B.9) which is the desired form of the vector potential A(r). B Helmholtz’ Theorem 655

The relations given in Eqs. (B.3), (B.6), and (B.9) then show that

F(r) =−∇φ(r) +∇×A(r), (B.10) which may also be written as

F(r) = F(r) + Ft (r), (B.11) where

F(r) =−∇φ(r) ∇ · =− 1 ∇ F(r ) 3 + 1 ∇ F(r ) · ˆ 2 d r nd r (B.12) 4π V |r − r | 4π S |r − r |

is the longitudinal or irrotational part of the vector ﬁeld (where ∇×F(r ) = 0), and where

Ft (r) =∇×A(r) = 1 ∇×∇× F(r ) 3 d r 4π V |r − r | ∇ × = 1 ∇× F(r ) 3 + 1 ∇× F(r ) × ˆ 2 d r nd r 4π V |r − r | 4π S |r − r | (B.13)

is the transverse or solenoidal part of the vector field (where ∇·F(r ) = 0). If the surface S recedes to infinity and if the vector field F(r) is regular at infinity, then the surface integrals appearing in the above expressions vanish and Eqs. (B.12) and (B.13) become

F(r) =−∇φ(r) ∇ · =− 1 ∇ F(r ) 3 d r , (B.14) 4π V |r − r |

Ft (r) =∇×A(r) ∇ × = 1 ∇× F(r ) 3 d r . (B.15) 4π V |r − r |

Taken together, the above results constitute what is known as Helmholtz’ theorem [1]. Theorem B.1 (Helmholtz’ Theorem) Let F(r) be any continuous vector ﬁeld with continuous ﬁrst partial derivatives. Then F(r) can be uniquely expressed in terms 656 B Helmholtz’ Theorem of the negative gradient of a scalar potential φ(r) and the curl of a vector potential A(r), as embodied in Eqs. (B.10)–(B.13).

Reference

1. H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933. Appendix C The Effective Local Field

The average electric field intensity acting on a given molecule in a dielectric is called the local or effective field. In a linear isotropic dielectric, the spatially averaged induced molecular charge separation is directly proportional to, and in the same direction as, the local field at that molecular site so that the average induced molecular dipole moment is given by ˜ p˜(r,ω) = 0α(ω) Eeff (r,ω) (C.1) at the fixed angular frequency ω of the applied time-harmonic field [see Eq. (4.171)]. The molecular polarizability α(ω) characterizes the frequency-dependent linear response of the molecules comprising the dielectric to the applied electric field. The effective local field may be determined by removing the molecule under consideration while maintaining all of the remaining molecules in their time- averaged polarized states (time-averaging being used only to remove the effects of thermal fluctuations), the spatially averaged electric field intensity then being calculated in the cavity left vacant by that removed molecule [1–3]. Let Vm denote the volume of that single molecular cavity region. The effective local field at that molecular site is then given by the difference ˜ 1 3 1 3 Eeff (r,ω) = e˜(r ,ω)d r − e˜m(r ,ω)d r , (C.2) Vm Vm Vm Vm

where e˜(r ,ω)is the total microscopic electric field at the point r ∈ Vm, and where e˜m(r ,ω) is the electric field due to the charge distribution of the molecule under consideration evaluated at the point r ∈ Vm. If the dielectric material is locally homogeneous (i.e., its dielectric properties at any point in the material are essentially constant over a macroscopically small but microscopically large region), then the first integral appearing in Eq. (C.2)is essentially the macroscopic electric field defined in Eq. (4.6). In particular, if the

© Springer Nature Switzerland AG 2019 657 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 658 C The Effective Local Field

Fig. C.1 Spherical cavity region V of radius R with point charge q situated a distance r

weighting function w(r ) is taken to be given by 1/Vm when r ∈ Vm and 0 otherwise, then Eq. (4.1) gives 1 E˜ (r,ω) ≡ e˜(r,ω) = e˜(r − r ,ω)d3r , (C.3) Vm Vm

where r ∈ Vm. With the change of variable r = r − r , the first integral appearing in Eq. (C.2) is then obtained. For the second integral appearing in Eq. (C.2), consider determining the average electric field intensity inside a sphere of radius R containing a point charge q that is located a distance r from the center O of the sphere, as illustrated in Fig. C.1.The z-axis is chosen to be along the line from the center O of the sphere passing through the point charge q. With this choice, symmetry shows that the average field over the spherical volume must be along the z-axis. The average electric field in V is then given by the scalar quantity 1 3 e¯z = e˜zd r, (C.4) V V where V is the volume of the spherical region. It is convenient to separate this volume integral into two parts, one taken over the spherical shell V1 between the radii r and R, and the other taken over the inner sphere V2 of radius r , so that 1 3 1 3 e¯z = e˜zd r + e˜zd r. (C.5) V V1 V V2 C The Effective Local Field 659

Fig. C.2 Spherical polar coordinate system about the point charge q for the integral over the inner spherical region V2 with center at O and radius r

The integral over the spherical shell region V1 vanishes because of the equal and opposite contributions arising from the pair of volume elements dτ1 and dτ2 that are intercepted by the element of solid angle dΩ, as depicted in Fig. C.1. Because the magnitude of e˜z decreases with the square of the distance from the point charge q, whereas the volume element dτ = r2dΩ increases with the square of this distance, their product remains constant. For positive q, e˜z is positive at dτ1 and it is negative at dτ2, whereas for negative q, e˜z is negative at dτ1 and positive at dτ2. In either case, the two contributions to the integral over V1 cancel and that integral then vanishes. The integral of e˜z over the inner volume V2 is then equal to the same integral over the entire spherical region V . In order to evaluate this ﬁnal volume integral, choose spherical polar coordinates (r ,θ,ϕ)with origin at the point charge q, as illustrated in Fig. C.2. At any point P ∈ V2,

4π q ˜ = e˜ · 1ˆ = ez z 2 cos(θ), (C.6) 4π0 r so that 2π π −2r cos(θ) 3 4π e˜zd r = q dϕ sin(θ) cos(θ) dr dθ V 4π0 0 π/2 0 π 4π 4π =− qr cos2(θ) sin(θ)dθ =− qr . (C.7) 0 π/2 30

The average electric ﬁeld intensity inside the sphere due to the point charge q is then given by

¯ = 1 ˜ 3 =− 4π qr ez 3 ezd r 3 . (C.8) (4/3)πR V 4π0 R 660 C The Effective Local Field

The electric dipole moment of the point charge q with reference to the center O of ˆ the spherical region is given by p˜ ≡ qr 1z, so that Eq. (C.8) may be written in the general form

4π e¯ =− p˜ 3 . (C.9) 4π0R

The second integral appearing in Eq. (C.2) is then given by 1 4π e˜ r 3 =− p˜ r m( ,ω)d r 3 m( ,ω), (C.10) Vm Vm 4π0rm where p˜ m is the dipole moment of the molecule under consideration. Because N = 1/Vm is the local volume density of molecules, then with the assumption that all of the local molecules have parallel and equal polarization vectors, the macroscopic polarization is given by ˜ P(r,ω) = Np˜ m(r,ω) so that the spatially averaged self-ﬁeld of the molecule is given by

1 3 4π ˜ e˜m(r ,ω)d r =− P(r,ω). (C.11) Vm Vm 30

With these substitutions, Eq. (C.2)forthespatially averaged effective ﬁeld becomes

˜ ˜ 4π ˜ Eeff (r,ω) = E(r,ω)+ P(r,ω), (C.12) 30 and the local field is larger than the macroscopic electric field. This expression for the effective local field was first derived by Lorentz [1] who used a somewhat different definition of the local field as the field value at the center of the molecule rather than that averaged over the molecular volume.

References

1. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Ch. IV. 2. C. Kittel, Introduction to Solid State Physics. New York: John Wiley & Sons, fourth ed., 1971. Ch. 13. 3. J. D. Jackson, Classical Electrodynamics. New York: John Wiley & Sons, third ed., 1999. Appendix D Magnetic Field Contribution to the Lorentz Model of Resonance Polarization

With the complete Lorentz force relation as the driving force, the equation of motion of a harmonically bound electron is given by 2 d rj drj q 1 drj + 2δ + ω2r =− e E (r,t)+ × B (r,t) , (D.1) dt2 j dt j j m eff c dt eff

where Eeff(r,t) is the effective local electric field intensity and Beff(r,t) is the effective local magnetic induction field, and where rj = rj (r,t) describes the displacement of the electron from its equilibrium position. Here qe denotes the magnitude of the charge and m the mass of the harmonically bound electron with undamped resonance frequency ωj and phenomenological damping constant δj . The temporal Fourier integral representation of the electric and magnetic field vectors of the effective local plane electromagnetic wave is given by

∞ ˜ = iωt Eeff(r,ω) Eeff(r,t)e dt, (D.2) −∞ ∞ ˜ = iωt = c × ˜ Beff(r,ω) Beff(r,t)e dt k(ω) Eeff(r,ω), (D.3) −∞ ω where k(ω) is the wave vector of the plane wave field with magnitude given by the wavenumber k(ω) = ω/c because the effective local field is essentially a microscopic field. With the temporal Fourier integral representation

∞ iωt r˜j (ω) = rj (t)e dt, (D.4) −∞

© Springer Nature Switzerland AG 2019 661 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 662 D Magnetic Field Contribution to the Lorentz Model the dynamical equation of motion (D.1) becomes q ω2 + 2iδ ω − ω2 r˜ = e E˜ − ir˜ × k × E˜ j j j m eff j eff q = e 1 + ir˜ · k E˜ − i r˜ · E˜ k , (D.5) m j eff j eff with formal solution + ˜ · ˜ · ˜ q 1 irj k rj Eeff r˜ = e E˜ − i k . (D.6) j 2 − 2 + eff 2 − 2 + m ω ωj 2iδj ω ω ωj 2iδj ω

The electron displacement vector may then be expressed as a linear combination of ˜ the orthogonal pair of vectors k and Eeff

˜ = ˜ + rj aj Eeff bj k, (D.7)

· ˜ = where, because of the transversality relation k Eeff 0,

˜ · ˜ 2 rj Eeff c aj = ,bj = r˜j · k. (D.8) ˜ 2 ω2 Eeff

The pair of scalar products appearing in the above expression may be evaluated from Eq. (D.6)as

q r˜ · E˜ r˜ · k =−i e j eff k2, j 2 − 2 + m ω ωj 2iδj ω q 1 + ir˜ · k r˜ · E˜ = e j E˜ 2 . j eff 2 − 2 + eff m ω ωj 2iδj ω

Substitution of the second relation into the ﬁrst then yields

2 ˜ 2 2 (qe/mc) Eeffω r˜j · k =−i , (D.9) 2 − 2 + 2 − 2 ˜ 2 2 (ω ωj 2iδj ω) (qe/mc) Eeffω and substitution of this result into the second relation gives

q /m r˜ · E˜ = e j eff 2 − 2 + ω ωj 2iδj ω (q /mc)2E˜ 2 ω2 × 1 + e eff E˜ 2 . (D.10) 2 − 2 + 2 − 2 ˜ 2 2 eff (ω ωj 2iδj ω) (qe/mc) Eeffω D Magnetic Field Contribution to the Lorentz Model 663

The coefﬁcients aj and bj appearing in Eq. (D.7) are then given by

q /m a = e j 2 − 2 + ω ωj 2iδj ω (q /mc)2E˜ 2 ω2 × 1 + e eff , (D.11) 2 − 2 + 2 − 2 ˜ 2 2 (ω ωj 2iδj ω) (qe/mc) Eeffω 2 ˜ 2 (qe/m) Eeff bj =−i , (D.12) 2 − 2 + 2 − 2 ˜ 2 2 (ω ωj 2iδj ω) (qe/mc) Eeffω respectively. The local (or microscopic) induced dipole moment p˜ j ≡−qer˜j for the jth Lorentz oscillator type is then given by [compare with Eq. (4.209)] ˜ =− ˜ + pj (r,ω) qe aj Eeff(r,ω) bj k . (D.13)

If there are Nj Lorentz oscillators per unit volume of the jth type, then the macroscopic polarization induced in the medium is given by the summation over all oscillator types of the spatially averaged locally induced dipole moments as ˜ ˜ = ˜ P(r,ω,Eeff) Nj pj (r,ω) j = ˜ ˜ + ˜ Eeff(r,ω) Nj αj⊥(ω, Eeff) k Nj αj (ω, Eeff). j j (D.14)

Here

˜ ≡ (0) + (2) ˜ αj⊥(ω, Eeff) αj⊥(ω) αj⊥(ω, Eeff) − 2 = qe /m 2 − 2 + ω ωj 2iδj ω (q /mc)2E˜ 2 ω2 × 1 + e eff (D.15) 2 − 2 + 2 − 2 ˜ 2 2 (ω ωj 2iδj ω) (qe/mc) Eeffω is deﬁned here as the perpendicular component of the atomic polarizability, (0) ≡ = with αj⊥(ω) αj⊥(ω, 0) αj (ω), where αj (ω) is the classical expression 664 D Magnetic Field Contribution to the Lorentz Model

(4.211) for the atomic polarizability when magnetic ﬁeld effects are neglected, and

(q3/m2)E˜ 2 ˜ ≡ e eff αj (ω, Eeff) i (D.16) 2 − 2 + 2 − 2 ˜ 2 2 (ω ωj 2iδj ω) (qe/mc) Eeffω is the parallel component of the atomic polarizability, where αj (ω, 0) = 0. The atomic polarizability is then seen to be nonlinear in the local electric field strength when magnetic field effects are included. However, numerical calculations [1] show that these nonlinear terms are entirely negligible for effective field strengths that are typically less than ∼ 1012 V/m and that they begin to have a significant contribution for field strengths that are typically greater than ∼1015 V/m for a highly absorptive material. The physical origin of the nonlinear term considered here is due to the diamagnetic effect that appears in the analysis of the interaction of an electromagnetic field with a charged particle in the quantum theory of electrodynamics [2, 3]. The Hamiltonian for this coupled system is given by [see Eqs. (XIII.71)–(XIII.72) of Messiah [2]]

2 Z qe qe 2 2 H = H − H · L + H r ⊥, 0 2mc 8mc2 j j=1 in Gaussian units, where H0 is the Hamiltonian of the center of mass system of the isolated atom with Z spinless electrons, H(r) is the magnetic field intensity vector with magnitude H ≡|H|, r⊥ is the projection of the position vector r on the plane = Z × perpendicular to H(r), and L j=1(rj pj ) is the total angular momentum of the Z atomic electrons. The third term in the above expression for the Hamiltonian is the main factor in atomic diamagnetism. The order of magnitude of this factor ∼ 2 2 2 2 2 ∼ × −16 2 is given by (Zqe /12mc )H r , where r 1 10 cm for a bound electron. The ratio of this quantity to the level distance μB H, where μB ≡ qeh/¯ 2mc is the Bohr magneton, is found [2]tobe∼ 10−9 ZH gauss. For a single electron atom (Z = 1), the diamagnetic effect will become significant when H ≥ 109gauss, which corresponds to an electric field strength E ≥ 109 esu, or equivalently E ≥ 3 × 1013 V/m, in agreement with the preceding classical result that the nonlinear effects in the Lorentz model become significant for an applied field strength between 1012 V/m and 1015 V/m. In addition, nonlinear optical effects are found to dominate the linear response when the local field strength becomes comparable to the Coulomb field of the atomic nucleus [4]. As an estimate of this field strength, if the distance between the nucleus ≡ 2 2 ≈ and the bound electron is taken to be given by the Bohr radius a0 h¯ /mqe 5.29 × 10−9 cm, where h¯ ≡ h/2π and h is Planck’s constant, the electric field strength is E ≈ 5.13×1011 V/m, in general agreement with the preceding estimates. References 665

References

1. K. E. Oughstun and R. A. Albanese, “Magnetic ﬁeld contribution to the Lorentz model,” J. Opt. Soc. Am. A, vol. 23, no. 7, pp. 1751–1756, 2006. 2. A. Messiah, Quantum Mechanics, vol. II. Amsterdam: North-Holland, 1962. 3. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics. New York: John Wiley & Sons, 1989. Section III.D. 4. R. W. Boyd, Nonlinear Optics. San Diego: Academic, 1992. Ch. 1. Appendix E The Fourier–Laplace Transform

The complex temporal frequency spectrum of a vector function f(r,t) of both position r and time t that vanishes for t<0 is of central importance to the solution of problems in time-domain electromagnetics and optics and is considered here in some detail following, in part, the treatment by Stratton [1]. The Laplace transform of f(r,t)with respect to the time variable t is deﬁned here as

∞ L{f(r,t)}≡ f(r,t)eiωtdt, (E.1) 0 which is simply a Fourier transform with complex angular frequency ω that is taken over only the positive time interval. Let f (r,t) be another vector function of both position and time such that

f (r,t)= f(r,t); t>0, (E.2) but which may not vanish for t ≤ 0. The Laplace transform given in Eq. (E.1)may then be written as ∞

L{f(r,t)}= U(t)f (r,t)eiωtdt, (E.3) −∞ where U(t) = 0fort<0 and U(t) = 1fort>0 is the Heaviside unit step function. For real ω, the Laplace transform of f(r,t)is then seen to be equal to the Fourier transform of U(t)f (r,t),viz.

L{f(r,t)}=Fω{U(t)f (r,t)}; for real ω, (E.4)

© Springer Nature Switzerland AG 2019 667 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 668 E The Fourier–Laplace Transform where the subscript ω indicates that it is the Fourier transform variable. The inverse Fourier transform of this equation then gives − U(t)f (r,t)= F 1 L{f(r,t)} (E.5) for real ω. For complex ω,letω = ω +iω where ω ≡{ω} and ω ≡{ω}. The Laplace transform given in Eq. (E.3) then becomes

∞ − L{f(r,t)}= U(t)f (r,t)e ω t eiω t dt −∞ −ω t = Fω U(t)f (r,t)e . (E.6)

The inverse Fourier transform of this expression then yields −ω t = F−1 L{ } U(t)f (r,t)e ω f(r,t) ∞ 1 − = L{f(r,t)}e iω t dω , (E.7) 2π −∞ which may be rewritten as

∞ 1 − + U(t)f (r,t) = L{f(r,t)}e i(ω iω )t dω 2π −∞ ∞+iω 1 − = L{f(r,t)}e iωtdω 2π −∞+ iω 1 − = L{f(r,t)}e iωtdω. (E.8) 2π C

Here C denotes the straight line contour ω = ω + iω with ω ﬁxed and ω varying over the real domain from −∞ to +∞. Because f(r,t) = U(t)f (r,t),Eqs.(E.1) and (E.8) then deﬁne the Laplace transform pair relationship

∞ f˜(r,ω) ≡ L {f(r,t)} = f(r,t)eiωtdt, (E.9) 0 − 1 − f(r,t) ≡ L 1 f˜(r,ω) = f˜(r,ω)e iωtdω, (E.10) 2π C where f˜(r,ω)is the complex temporal frequency spectrum of f(r,t)with ω = ω + iω . Notice that ω ={ω} plays a passive role in the Laplace transform operation because it remains constant in both the forward and inverse transformations.

Nevertheless, its presence can be important because the factor e−ω t appearing in E The Fourier–Laplace Transform 669 the integrand of the transformation (E.9) may serve as a convergence factor when ω > 0. In particular,

∞ − f˜(r,ω)= f(r,t)e ω t eiω t dt (E.11) 0

−ω t is just the Fourier transform Fω {f(r,t)e }. The Fourier transform of f(r,t)alone is ∞ iω t Fω {f(r,t)}= f(r,t)e dt, 0 which exists provided that f(r,t)is absolutely integrable; viz., T lim |f(r,t)| dt < ∞. T →∞ 0

If f(r,t)does not vanish properly at inﬁnity, then the above integral fails to converge and the existence of the Fourier transform Fω {f(r,t)} is not guaranteed. However, if there exists a real number γ such that T − lim f(r,t)e γt dt < ∞, (E.12) T →∞ 0 then f(r,t)is transformable for all ω ≥ γ and its temporal frequency spectrum is given by the Laplace transform (E.9). The lower bound γa of all of the values of γ for which the inequality appearing in Eq. (E.12) is satisﬁed is called the abscissa of absolute convergence for the function f(r,t). The Laplace transform of the time derivative ∂f(r,t)/∂t can be related to the Laplace transform of f(r,t)through integration by parts as ∂f(r,t) ∞ ∂f(r,t) L = eiωtdt ∂t 0 ∂t ∞ ∞ = f(r,t)eiωt − iω f(r,t)eiωtdt 0 0 =−f(r, 0) − iωL {f(r,t)} , (E.13)

where the fact that |f(r,t)eiωt|=|f(r,t)|e−ω t must vanish as t →∞for all

ω ≥ γa has been used in obtaining the ﬁnal form of Eq. (E.13). For the appropriate form of the convolution theorem for the Laplace transform, consider determining the function whose Laplace transform is equal to the product 670 E The Fourier–Laplace Transform

˜ ˜ f1(ω)f2(ω) = L{f1(t)}L{f2(t)} so that −1 ˜ ˜ 1 ˜ ˜ −iωt L f1(ω)f2(ω) = f1(ω)f2(ω)e dω 2π C ∞ 1 ˜ iωτ −iωt = f1(ω) f2(τ)e dτ e dω 2π C 0 ∞ 1 ˜ −iω(t−τ) = dτ · f2(τ) f1(ω)e dω , 0 2π C and consequently

∞ −1 ˜ ˜ L f1(ω)f2(ω) = f1(t − τ)f2(τ)U(t − τ)dτ, 0 where the unit step function U(t − τ) is explicitly included in this expression to emphasize the fact that f1(t) vanishes for t<0. Because U(t − τ) vanishes for negative values of its argument, the upper limit of integration in τ must be t and the above equation becomes t −1 ˜ ˜ L f1(ω)f2(ω) = f1(t − τ)f2(τ)dτ, (E.14) 0 which may be rewritten as

∞ L f1(t − τ)f2(τ)dτ = L {f1(t)} L {f2(t)} , (E.15) 0 which is the convolution theorem for the Laplace transform. The spatiotemporal Fourier–Laplace transform of a vector function F(r,t) of both position r and time t that vanishes for t<0 is deﬁned here by the pair of relations ∞ ∞ − · − F(k,ω) ≡ FL{F(r,t)}= d3r dt · F(r,t)e i(k r ωt), (E.16) −∞ 0 ∞ 1 F r ≡ F−1L−1{F k }= 3 · F k i(k·r−ωt) ( ,t) ( ,ω) 4 d k dω ( ,ω)e , (2π) −∞ C (E.17) ˆ ˆ ˆ ˆ ˆ ˆ where k = 1xkx + 1yky + 1zkz and r = 1xx + 1yy + 1zz. Because,

∞ ∂F(r,t) 1 · − = d3k dω · ik F(k,ω)ei(k r ωt), 4 j ∂xj (2π) −∞ C Reference 671 from Eq. (E.17), then the transforms of the ﬁrst spatial derivatives of F(r,t) are given by ∂F(r,t) FL = ikj F(k,ω), (E.18) ∂xj where x1 = x, x2 = y, x3 = z and k1 = kx, k2 = ky, k3 = kz. With this result, the spatiotemporal transform of the divergence of the vector ﬁeld F(r,t)is found to be FL {∇·F(r,t)} = ik · F(k,ω), (E.19) and the spatiotemporal transform of the curl of F(r,t)is given by FL {∇×F(r,t)} = ik × F(k,ω). (E.20)

The spatiotemporal transforms of higher-order spatial derivatives of F(r,t) may then be obtained through repeated application of the above relations.

Reference

1. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. Appendix F Reversible and Irreversible, Recoverable and Irrecoverable Electrodynamic Processes in Dispersive Dielectrics

A critical re-examination of the Barash and Ginzburg result [1], presented in Sect. 5.2.2, for the electromagnetic energy density and evolved heat in a dispersive dissipative medium is presented here based upon the analysis by Glasgow et al. [2, 3] for a simple dispersive dielectric [μ(ω) = μ0, σ(ω) = 0]. In that case the total electromagnetic energy density in the coupled ﬁeld–medium system is given by [cf. Eqs. (5.186) and (5.187)] t U = 1 1 2 + 2 + · ∂P(r,t ) (r,t) 0E (r,t) μ0H (r,t) E(r,t ) dt . 4π 2 −∞ ∂t (F.1)

The first two terms on the right-hand side of this equation describe the electromagnetic field energy density in the absence of the dispersive medium 1 1 2 2 Uem(r,t)≡ 0E (r,t)+ μ0H (r,t) , (F.2) 4π 2 and the last term describes the electromagnetic energy density in the coupled field– medium system t U ≡ · ∂P(r,t ) int(r,t) E(r,t ) dt , (F.3) −∞ ∂t which is defined by Glasgow et al. [2]astheinteraction energy. By comparison, Barash and Ginzburg [1] separated the interaction energy into two parts: one part which represents the electromagnetic energy that is reactively stored in the dispersive medium, and another part, Q(r,t), which represents the dissipation of

© Springer Nature Switzerland AG 2019 673 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 674 F Electrodynamic Processes in Dispersive Dielectrics electromagnetic energy in the medium. Notice that

U (r,t) U(r,t) = + Q(r,t) ∂t ∂t U (r,t) U (r,t) = em + int , (F.4) ∂t ∂t where the first form of this separation is due to Barash and Ginzburg [1][cf. Eq. (5.193)], and the second form is due to Glasgow et al. [2], where U(r,t) = Uem(r,t)+ Urev(r,t)and Uint(r,t)= Urev(r,t)+ Uirrev(r,t). The Helmholtz free energy (also known as the work function) ψ ≡ U − TS is defined [4] as the difference between the internal energy U of the system and the heat energy TS, where T is the absolute temperature and S the entropy. Von Helmholtz (1882) called this ψ function the free energy of a system, because [4] “its change in a reversible isothermal process equals the energy that can be ‘freed’ in the process and converted to mechanical work.” The decrease in ψ is then equal to the maximum work done on the system in a reversible isothermal process. Glasgow et al. [2] generalize this Helmholtz free energy to the irreversible, dissipative, nonequilibrium case by defining the dynamical free energy as “the work the system can do on, or return to, an external agent,” which naturally reduces to the Helmholtz free energy in the reversible case. With regard to Eq. (5.271), this dynamical free energy U(r,t) is given by the sum of the field energy term Uem(r,t) and the reactive (or reversible) energy term Urev(r,t). The evolved heat energy Q(r,t)then corresponds to the irreversible (or latent) part of the interaction energy. The interaction energy defined in Eq. (5.270) can be expressed in terms of the instantaneous (or causal) spectrum of the electric field vector,1 defined as [5] t ˜ 1 iωt Et (r,ω)≡ E(r,t )e dt , (F.5) 2π −∞ in the following manner due to Peatross, Ware, and Glasgow [6]. From Eq. (4.95), the macroscopic polarization density of a temporally dispersive HILL medium can be expressed as

∞

P(r,t)= E(r,t )G(t − t )dt (F.6) −∞

1Notice that this instantaneous spectrum is equal to the Fourier transform of E(r,t )U(t − t ), where U(t) is the Heaviside unit step-function. F Electrodynamic Processes in Dispersive Dielectrics 675 with Green’s function ∞ 0 −iωt G(t) ≡ χe(ω)e dω. (F.7) 2π −∞

This Green’s function may be expressed as the sum of two terms, the ﬁrst associated ≡{ } with the real part χe(ω) χ(ω) and the second with the imaginary part ≡{ } χe (ω) χ(ω) of the electric susceptibility, as

G(t) = G (t) + G (t), (F.8) where ∞ ≡ 0 −iωt G (t) χe(ω)e dω, (F.9) 2π −∞ ∞ ≡ 0 −iωt G (t) χe (ω)e dω. (F.10) 2π −∞

Causality is introduced through the fact that the real and imaginary parts of the electric susceptibility satisfy the Plemelj formulae given in Eqs. (4.155) and (4.156), viz. ∞ = 1 P χe(ω ) χe(ω) dω , (F.11) π −∞ ω − ω ∞ =−1 P χe(ω ) χe (ω) dω . (F.12) π −∞ ω − ω

Substitution of Eq. (5.278) into (5.276) then yields ∞ ∞ −iωt = 0 P e G (t) dω χe (ω) dω. (F.13) 2π 2 −∞ −∞ ω − ω

Because ∞ −iωt −iω t P e = iπe ,t>0 dω −iω t , (F.14) −∞ ω − ω −iπe ,t<0 then Eq. (5.280) shows that G (t) = G (t) for t>0 and G (t) =−G (t) for t<0, so that 2G (t), t > 0 G(t) = . (F.15) 0,t<0 676 F Electrodynamic Processes in Dispersive Dielectrics

Substitution of this result in Eq. (5.273) then gives t

P(r,t) = 2 E(r,t )G (t − t )dt −∞ ∞ t = 0 −iωt iωt i dωχe (ω)e dt E(r,t )e , (F.16) π −∞ −∞ where causality is now explicitly expressed in the upper limit of integration, in agreement with the expression given in Eq. (4.95). The time derivative of this expression is then given by ∞ t ∞ ∂P(r,t) = 0 −iωt iωt + dωωχe (ω)e dt E(r,t )e iE(r,t) χe (ω)dω . ∂t π −∞ −∞ −∞ (F.17)

Because χe (ω) is an odd function of ω, the ﬁnal integral in the above expression is equal to zero and consequently ∞ t ∂P(r,t) = 0 −iωt iωt dωωχe (ω)e dt E(r,t )e ∂t π −∞ −∞ ∞ = ˜ −iωt 20 ωχe (ω)Et (r,ω)e dω. (F.18) −∞

Substitution of this result into Eq. (5.270) then yields ∞ t − U = · ˜ iωt int(r,t) 20 dωωχe (ω) E(r,t ) Et (r,ω)e dt . −∞ −∞

The time derivative of the complex conjugate of Eq. (5.275) results in the identi- ˜ ∗ = −iωt ﬁcation ∂Et (r,ω)/∂t (1/2π)E(r,t)e , so that the above expression for the interaction energy becomes ∞ t ˜ ∗ ∂E (r,ω) U = ˜ · t int(r,t) 4π0 dωωχe (ω) Et (r,ω) dt . −∞ −∞ ∂t

Because Uint(r,t)is a real-valued quantity, the above expression may be written as

∞ U = int(r,t) 2π0 dωωχe (ω) −∞ t ˜ ∗ ˜ ∂E (r,ω) ∗ ∂Et (r,ω) × ˜ · t + ˜ · Et (r,ω) Et (r,ω) dt . −∞ ∂t ∂t F.1 Reversible and Irreversible Electrodynamic Processes 677

˜ 2 ∞ t ∂ Et (r,ω) = 2π0 dωωχe (ω) dt −∞ −∞ ∂t ∞ 2 = ˜ 2π0 ωχe (ω) Et (r,ω) dω. (F.19) −∞

It is then seen that

Uint(r,t)≥ 0 (F.20) for all time t, where Uint(r, −∞) = 0, in agreement with the Landau–Lifshitz result [7] for the asymptotic heat that Uint(r, +∞) ≥ 0. This generalization [6]of the Landau–Lifshitz result shows that the interaction energy can never run a deficit; that is, at any instant of time t the work that the electromagnetic field does on the medium always exceeds the work that the medium does against the field.

F.1 Reversible and Irreversible Electrodynamic Processes

The energy dissipated and consequently lost to a dielectric medium from some physically realizable electromagnetic field since time t =−∞maybeexpressedas 3 3 Uint(r, ∞)d r = Uint(r, ∞) − Uint(r, −∞) d r, D D because Uint(r, −∞) = 0, where D is some fixed region contained within the 3 dielectric body. The quantity D(Uint(r, ∞) − Uint(r, t))d r then represents the electromagnetic energy lost to the medium in the region D over the future time interval t >tgiven the present state of the medium at time t that has been established by the electromagnetic field over the past time interval t 0 D may then be interpreted as representing a reversible process of energy that is “borrowed” from the electromagnetic field and returned to it at a later time as 3 t →∞. The value of this energy difference D(Uint(r,t)−Uint(r, ∞))d r will be different for different future fields, the physically proper value then being given by an appropriate extremum principle. To that end, the interaction energy density given in Eq. (5.286) is denoted as

∞ U [ ] ≡ | ˜ |2 int E (r,t) 2π0 ωχe (ω) Et (r,ω) dω, (F.21) −∞ 678 F Electrodynamic Processes in Dispersive Dielectrics where t ˜ 1 iωt Et (r,ω)≡ E(r,t )e dt , (F.22) 2π −∞ from Eq. (5.272). The reversible and irreversible energy densities of the electromagnetic field in a simple temporally dispersive dielectric [μ = μ0 and σ(ω) = 0] are then given by the following definitions due to Broadbent, Hovhannisyan, Clayton, Peatross, and Glasgow [2] Definition F.1 (Reversible Electromagnetic Energy Density) The reversible energy density Urev(r,t) = Urev[E](r,t) of the electromagnetic field at time t in a simple temporally dispersive dielectric is given by the supremum (least upper bound) of values that the quantity Uint(r,t)−Uint(r, ∞) can attain over all possible alternative future fields; that is,

Urev[E](r,t)≡ sup Uint[E](r,t)− Uint[EU(t − t ) + Ef ](r, ∞) , (F.23) Ef where U(t) denotes the Heaviside unit step function, and where Ef denotes the possible future electric ﬁelds with Ef (r,t ) = 0 for all t

Notice that the quantity Uint[EU(t − t ) + Ef ](r,t) = Uint[E](r,t)in the above expression, where t denotes the “past” time variable appearing in the integrand of Eq. (5.289). Definition F.2 (Irreversible Electromagnetic Energy Density) The irreversible energy density Uirrev(r,t) = Uirrev[E](r,t) of the electromagnetic field at time t in a simple temporally dispersive dielectric is given by the complement of the reversible energy density as the infimum (greatest lower bound) of values that the quantity Uint(r, ∞) can attain over all possible alternate future fields; that is,

Uirrev[E](r,t)≡ inf Uint[EU(t − t ) + Ef ](r, ∞) . (F.24) Ef

Notice that

Uint[E](r,t)= Urev[E](r,t)+ Uirrev[E](r,t) (F.25) as required by conservation of energy. This formulation of the irreversible energy density is in a form that is now appropriate for the calculus of variations. Because any future ﬁeld solution Ef (r,t ) vanishes for all t

F E(r,t )U(t − t ) + Ef (r,t ) t ∞ 1 iωt 1 iωt = E(r,t )e dt + Ef (r,t )e dt 2π −∞ 2π t ˜ ˜ = Et (r,ω)+ Ef (r,ω). (F.26)

The variational derivative of the irreversible electromagnetic energy density deﬁned in Eq. (5.291) with respect to future possible ﬁelds then gives, with Eq. (5.286), U [ − + ] ∞ δEf int EU(t t ) Ef (r, ) ∞ 2 = ˜ + ˜ = δE˜ ωχe (ω) Et (r,ω) Ef (r,ω) dω 0. f −∞ (F.27)

Glasgow et al. [2] then deﬁne the new electric ﬁeld vector functions

E+(r,t ) ≡ Ef (r,t + t), (F.28)

E−(r,t ) ≡ E(r,t + t)U(−t ), (F.29) whose temporal Fourier transforms

∞ ˜ 1 iωt E+(r,ω; t) = Ef (r,t + t)e dt 2π −∞ ∞ −iωt 1 iωτ −iωt ˜ = e Ef (r,τ)e dτ = e Ef (r,ω), (F.30) 2π −∞ ∞ ˜ 1 iωt E−(r,ω; t) = E(r,t + t)U(−t )e dt 2π −∞ t −iωt iωτ −iωt ˜ = e E(r,τ)e dτ = e Et (r,ω), (F.31) −∞ are analytic in the upper and lower half-planes, respectively. Substitution of these ˜ ﬁeld quantities in Eq. (5.294) with the variation now taken with respect to E+ then results in the expression

∞ ˜ ; + ˜ ; ˜ ∗ ; = ωχe (ω) E−(r,ω t) E+(r,ω t) δE+(r,ω t)dω 0. (F.32) −∞

˜ Because E+(r,ω; t) is analytic in the upper half of the complex ω-plane, then so ˜ ˜ ∗ also is its variation δE+(r,ω; t). The variation δE+(r,ω; t) is then analytic in the 680 F Electrodynamic Processes in Dispersive Dielectrics

Fig. F.1 Completion of the contour of integration in the lower half of the complex ω-plane with a semicircular arc of radius |ω|=Ω →∞ lower half plane. The relation given in Eq. (5.299) cannot then be satisﬁed unless the remainder of the integrand is also analytic in the lower half plane. If the integrand in Eq. (5.299) is analytic in the lower half-plane and has magnitude that goes to zero sufﬁciently fast as |ω|→∞in the lower half-plane (Fig. F.1), then by completion of the contour with a semicircle in the lower half-plane centered at the origin with radius Ω →∞, as illustrated in Fig. 5.7, application of Cauchy’s theorem results in Eq. (5.299). This result then provides a general variational approach for the determination of the irreversible electromagnetic energy density in a simple dispersive dielectric [2]. However, this result does not contradict the Barash and Ginzburg result [1] presented in Sect. 5.2.2 because complete knowledge of the temporal frequency dependence of the imaginary part of the electric susceptibility is required, such as that provided by an analytic model. As an illustration, consider a single-resonance Lorentz model dielectric with resonance angular frequency ω0, phenomenological damping constant δ, and plasma = 2 frequency b ( 4π /0)Nqe /m. In their analysis, Glasgow et al. [2] take the electric susceptibility for this causal model medium to be given by the expression

b2 χe(ω) =− 4π , (F.33) (ω − (ω0 − iδ)) (ω + (ω0 + iδ)) F.2 Recoverable and Irrecoverable Electrodynamic Processes 681

2 + − 2 with a denominator that differs from the usual expression (ω 2iδω ω0) by the quantity δ2 [cf. Eq. (4.211)]. The reversible and irreversible energy densities are then found as 2 1 0 b ˜ 2 U [ ] r = − E− r − ; rev E ( ,t) 2 (ω0 iδ) ( ,ω0 iδ t) 4π 2 ω0 2 + 2 + 2 ˜ − ; ω0 δ E−(r,ω0 iδ t) , (F.34) 2 t 1 δb ˜ 2 Uirrev[E](r,t) = 20 (ω0 − iδ)E−(r,ω0 − iδ; t ) dt . 2 −∞ 4π ω0 (F.35) ˜ ˜ From the constitutive relation P(r,ω) = 0χe(ω)E(r,ω) relating the macroscopic polarization and electric ﬁeld spectra in a temporally dispersive HILL medium, it is found that [2] ˜ ω0 ˙ (ω − iδ)E−(r,ω − iδ; t) = P(r,t), (F.36) 0 0 b2 ˜ ω0 E−(r,ω − iδ; t) =− P(r,t). (F.37) 0 b2 With substitution of these two expressions, Eqs. (5.301) and (5.302) yield 2 1 0 1 ˙ 2 ω0 2 Urev[E](r,t) = P (r,t)+ P (r,t) , (F.38) 4π 2 2b2 2b2 t 1 δ ˙ 2 Uirrev[E](r,t) = 0 P (r,t )dt , (F.39) 4π b2 −∞ where the δ2 term in Eq. (5.305) has been accordingly neglected. Equation (5.305) shows that the reversible energy density for a single Lorentz oscillator type is given by the sum of its kinetic and potential energy densities.

F.2 Recoverable and Irrecoverable Electrodynamic Processes

Based upon the expression given in Eq. (5.286) for the electromagnetic interaction energy density Uint(r,t), it is seen that as t →∞, any given point in the (ﬁnitely extended) dielectric medium will have interacted with the entire electromagnetic 3 pulse (provided that it has compact temporal support) so that Uint(r, ∞)d r ≡ 3 limt→∞ Uint(r,t)d r speciﬁes the total amount of electromagnetic energy that was dissipated into the medium at that point due the entire pulse evolution at that point. 682 F Electrodynamic Processes in Dispersive Dielectrics

However, as the electromagnetic pulse propagates through the dielectric medium, a continuous interchange of electromagnetic energy takes place between the field and the atoms and molecules comprising the medium. This real-time energy exchange, for example, forms the physical basis for understanding both “slow” and “fast” light phenomena [8]. In particular, the phenomenon of “slow” light occurs when energy from the leading edge of the pulse is reactively stored in the medium and subsequently returned to the trailing edge of the pulse [9] as occurs in self-induced transparency [10]. The phenomenon of “fast” light, on the other hand, involves the distortion of the pulse envelope shape due to the unrecoverable dissipation of the leading edge energy such that the peak in the pulse envelope is shifted forward in time [11]. This forward temporal shift of the peak amplitude point in a pulse may then result in an apparent superluminal pulse envelope velocity; however, because the peak amplitude points in the dynamical pulse evolution are not causally related, this apparent superluminal velocity does not violate the Special Theory of Relativity. The analysis of this real-time energy dissipation requires a modified interpretation to that given in the preceding subsection, as reflected in the following definitions due to Glasgow, Meilstrup, Peatross, and Ware [3]: Definition F.3 (Recoverable Electromagnetic Energy Density) The recoverable energy density Urev[E](t) of the electromagnetic field interaction at time t is the supremum of the energy density that the dielectric body can subsequently return to the field under the influence of a well-chosen future field; that is, t t Urec[E](t) ≡ sup Uint[E](t) − Uint[E− + E+](+∞) t E+ t t = Uint[E](t) − inf Uint[E− + E+](+∞) , (F.40) t E+

t t where E− denotes the fixed past field evolution with E−(t ) = E(t)U(t − t ) so t t t that E−(t ) = E(t) for t ≤ t and E−(t ) = 0fort >t, and where E+ denotes a t variable future electric field with E+(t ) = 0 for all t

Both the recoverable and irrecoverable electromagnetic energy densities are naturally causal because [3] “the ﬁeld evolution after time t does not affect the energy F.2 Recoverable and Irrecoverable Electrodynamic Processes 683 and loss at that time”, where

t t t Urec[E](t) = Urec[E− + E+](t) = Urec[E−](t), t t t Uirrec[E](t) = Uirrec[E− + E+](t) = Uirrec[E−](t).

In addition, these two deﬁnitions satisfy the conservation of energy expression

Uint[E](t) = Urec[E](t) + Uirrec[E](t) (F.42) at each instant of time t. As stated in their paper [3], the supremum in Eq. (5.307) and the infimum in Eq. (5.308) are accomplished by treating the past field as fixed and allowing the future field to vary as necessary to accomplish the extrema. Thus Urec[E](t) gives the largest portion of Uint(t) that could possibly be converted back to field energy after t, given a fixed past field. Conversely, Uirrec[E](t) represents the smallest amount of eventual loss Uint(+∞) that is required for fields that share a common history prior to t.. . . both extrema are accomplished by the same future field. The difference between the recoverable and reversible energy densities, as well as that between the irrecoverable and irreversible energy densities, may be explained in the following manner [12]: Let U− be the “creation energy”, defined as the minimum electromagnetic energy required to create a state of a dispersive dissipative dielectric. This is the energy that is required to bring the dielectric from its quiescent state in the distant past at t =−∞to its present state at time t. The recoverable energy defined in Def. 5.3 is the maximum energy that can be extracted from the dispersive dissipative dielectric, starting from the present time t and ending up in its original quiescent state in the distant future at t =+∞. Because the dielectric body is dissipative, it then requires a non-negative influx of electromagnetic energy in order for the dielectric to be taken from its quiescent state at t =−∞toanewstateattimet and then returned to its quiescent state at t =+∞. Expressed mathematically,

U− − Urec ≥ 0 ⇒ U− ≥ Urec, (F.43) stating that “it takes no less energy to create a state than is extractable from it.”2 The irrecoverable energy in a dispersive dissipative dielectric is deﬁned in Def. 5.4 as the work that has been done on the dielectric from its quiescent state in the distant past at t =−∞to the present time t in order to establish the dielectric state at time t minus the energy that can be recovered, so that

Uirrec = W − Urec, (F.44)

2Scott Glasgow, personal correspondence (August 25, 2016). 684 F Electrodynamic Processes in Dispersive Dielectrics where W is the work that has been expended (up to the time t) in order to establish that dielectric state. Notice that W depends upon the path taken to establish the present dielectric state, and hence, is not a state variable. In contrast, the irreversible energy in a dispersive dissipative dielectric is given by the work W expended to establish the dielectric state at time t minus the optimal creation energy U− that could have been done in establishing this state, so that

Uirrev = W − U−. (F.45)

Because of the inequality in Eq. (5.310), one then has that

Uirrev ≤ Uirrec, (F.46) where both Uirrev and Uirrec are non-negative because the creation energy U− is the minimum over all possible values of W; that is, U− is the minimum energy that could have been used to establish the dielectric state at time t. In summary, as stated by Glasgow (see footnote 15): the irreversible energy is a measure of the remorse you feel in having created the given dielectricstateinefﬁciently...Theirrecoverableenergyontheotherhandisa measureof the work you won’t get back from the dielectric in the future despite even best efforts to create the state. Neither of these notions of loss is a state variable, but very much path dependent.

The minimum possible value of the irrecoverable energy Uirrec, given by min W − Urec = U− − Urec ≥ 0, (F.47) is a state variable and is a measure of the intrinsic “lossiness” of any given dielectric state. In a nonconducting dielectric medium, the differential form of Poynting’s theorem is given by Eq. (5.176) as ∂U (r,t)/∂t =−∇·S(r,t). Integration of this expression over the region of space D occupied by the dielectric body followed by application of the divergence theorem to the right-hand side then yields ∂ U (r,t)d3r =− S(r,t)· nˆd2r, ∂t D S where S is the surface enclosing D. If the integration domain is extended to all of space, then the surface integral of the normal component of the Poynting vector vanishes because the electromagnetic ﬁeld vectors go to zero faster than r−1 as r =|r| recedes to inﬁnity, so that

∞ ∞ ∂ U (r,t)d3r = 0 ⇒ U (r,t)d3r = W, (F.48) ∂t −∞ −∞ F.2 Recoverable and Irrecoverable Electrodynamic Processes 685 where W is a constant independent of the time t. From Eq. (5.271), one has that

U (r,t)= Uem(r,t)+ Uint(r,t), and hence

∞ ∞ ∂ 3 ∂ 3 Uem(r,t)d r + Uint[E](r,t)d r = 0. (F.49) ∂t −∞ ∂t −∞

Deﬁne the total electromagnetic ﬁeld energy as [3]

∞ 3 E[E](t) ≡ Uem(r,t)d r, (F.50) −∞ so that, with substitution from Eq. (5.315)

∞ ∞ 3 3 E[E](t) = U (r,t)d r − Uint(r,t)d r −∞ −∞ ∞ 3 = W − Uint(r,t)d r. (F.51) −∞

The net change in the total electromagnetic field energy after the so-called “distinguished” time t separating the past (fixed) field evolution from the future (possible) field evolution is then defined as

Δt E[E](t) ≡ E[E](∞) − E[E](t), (F.52) so that ∞ 3 Δt E[E](t) = Uint[E](t) − Uint[E](∞) d r −∞ ∞ ∞ 3 3 = Urec[E](t)d r − Uint[E](∞) − Uirrec[E](t) d r. −∞ −∞ (F.53)

Because Uint[E](∞) ≥ Uirrec[E](t), one ﬁnally obtains the inequality set by the global recoverable energy [3]

∞ 3 Δt E[E](t) ≤ Urec[E](t)d r, (F.54) −∞ stating that the net change in the total field energy in all of space after the distinguished instant of time t is always either equal to or bounded above by the recoverable energy in all of space at that instant of time t. However, the implications of this result are tempered by the simplifying assumption used in obtaining Eq. (5.315) that tacitly requires the dielectric material to fill all of space. One may, nevertheless, think of the result as applying to each point in the infinitely 686 F Electrodynamic Processes in Dispersive Dielectrics extended medium provided that the electromagnetic field behavior is the same at each point in the medium; that is, provided that it is an ideal time-harmonic uniform plane wave. For a multiple-resonance Lorentz model dielectric with susceptibility [see Eq. (4.219)]

2 fj ωp χe(ω) = χj (ω) = , (F.55) ω2 − 2iδ ω − ω2 j j j j

= 2 where fj is the oscillator strength of the dipole line and ωp 4π Nqe /m is the plasma frequency, the electromagnetic energy density in the coupled ﬁeld–medium system, or interaction energy Uint(t), deﬁned in Eq. (5.270), may be separated into reversible and irreversible parts as [cf. Eqs. (5.305) and (5.306)] ω2 1 0 1 ˙ 2 j 2 Urev[E](r,t) = P (r,t)+ P (r,t) , (F.56) 4π 2 f ω2 j f ω2 j j j p j p δ t U [ ] = 1 j ˙ 2 irrev E (r,t) 20 Pj (r,t )dt . (F.57) 4π f ω2 −∞ j j p

Glasgow et al. [3] refer to these two parts of the interaction energy as the collected energy Ue[E](r,t) and loss U[E](r,t) terms, respectively, stating that they are “model-dependent”, as described in the following: Barash and Ginzburg point out that the parametrization of a given χ is generally not unique. Furthermore, two distinct parametrizations for the same χ generally result in different energy allocations between U and Ue for a given field. Thus, in the Barash and Ginzburg approach one can change the fraction of energy that is allocated as “lost” at a given time by changing the parametrization of χ, even though the choice of parametrization has no effect on physically measurable quantities. Glasgow et al. [3] go on to state that “this is unsatisfactory at the physical level,” the purpose of their approach being to “develop concepts of recoverable energy and loss that depend only on χ and the electric field, and not on the parametrization of χ.” It is important to note here that the same can be said for the mathematical description of any physical system, from mechanics to thermodynamics to electromagnetism to circuit theory. As stated by Maxwell [13] in the introductory chapter on the “Nature of Physical Science” in his text on Matter and Motion, “Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events.” The physical sciences rely upon the interplay between the development of physical models to describe natural phenomena and experimental observation and measurement of that phenomena. If different physical models (based upon fundamental physical laws) used to describe a physical system predict different results, then the model that most closely agrees with experiment is References 687 likely the best one, and there may even be a better one. Such is the case with the Lorentz, Drude, Debye, and Rocard–Powles–Debye models described in Chap. 4. The difference between the irrecoverable energy of Definition 5.4 and the irreversible loss given in Eq. (F.57) for a multiple-resonance Lorentz model dielectric is found to be given by [3]

t t Uirrec[E](r,t)− Uirrev[E](r,t)= U[E− + E+](r,t), (F.58) where δ t U [ t + t ] ≡ 1 j ˙ 2[ t + t ] E− E+ (r,t) 20 inf Pj E− E+ (r,t )dt 4π Et f ω2 −∞ + j j p (F.59) is the inevitable future loss that will occur using the Lorentz model to describe the dispersive properties of the dielectric. That this model “determines how loss is allocated between past and future” [3] is of no real consequence because its applicability is restricted to an ideal time-harmonic uniform plane wave.

References

1. Y. S. Barash and V. L. Ginzburg, “Expressions for the energy density and evolved heat in the electrodynamics of a dispersive and absorptive medium,” Usp. Fiz. Nauk., vol. 118, pp. 523– 530, 1976. [English translation: Sov. Phys.-Usp. vol. 19, 163–270 (1976)]. 2. C. Broadbent, G. Hovhannisyan, M. Clayton, J. Peatross, and S. A. Glasgow, “Reversible and irreversible processes in dispersive/dissipative optical media: Electro-magnetic free energy and heat production,” in Ultra-Wideband, Short-Pulse Electromagnetics 6 (E. L. Mokole, M. Kragalott, and K. R. Gerlach, eds.), pp. 131–142, New York: Kluwer Academic, 2003. 3. S. Glasgow, M. Meilstrup, J. Peatross, and M. Ware, “Real-time recoverable and irrecoverable energy in dispersive-dissipative dielectrics,” Phys.Rev.E, vol. 75, pp. 16616–1–16616–12, 2007. 4. F. W. Sears, An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics. Reading, MA: Addison-Wesley, second ed., 1953. 5. J. H. Eberly and K. Wódkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am., vol. 67, no. 9, pp. 1252–1261, 1977. 6. J. Peatross, M. Ware, and S. A. Glasgow, “Role of the instantaneous spectrum on pulse propagation in causal linear dielectrics,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1719–1725, 2001. 7. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Ch. IX. 8. R. W. Boyd and D. J. Gauthier, “Slow and Fast Light,” in Progress in Optics (E. Wolf, ed.), vol. 43, pp. 497–530, Amsterdam: Elsevier, 2002. 9. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A, vol. 1, pp. 305–313, 1970. 10. L. Allen and J. H. Eberly, Optical Resonance & Two-Level Atoms. Wiley, 1975. 11. R. Y. Chiao and A. M. Steinberg, “Tunneling times and superluminality,” in Progress in Optics (E. Wolf, ed.), vol. 37, pp. 345–405, Amsterdam: Elsevier, 1997. 688 F Electrodynamic Processes in Dispersive Dielectrics

12. S. Glasgow, J. Corson, and C. Verhaaren, “Dispersive dielectrics and time reversal: Free energies, orthogonal spectra, and parity in dissipative media,” Phys. Rev. E, vol. 82, p. 011115, 2010. 13. J. C. Maxwell, Matter and Motion. London: Sheldon Press, 1925. Appendix G Stationary Phase Approximations of the Angular Spectrum Representation

The temporal frequency spectrum U˜ (r,ω) of a pulsed wave ﬁeld U(r,t) may be expressed in the positive half-space z>0 of a nonabsorptive (and hence, nondispersive) medium by the real direction cosine form of the angular spectrum of plane waves representation as (see Sect. 8.3.2) ˜ ˜ ˜ U(r,ω)= UH (r,ω)+ UE(r,ω) (G.1) with [see Eqs. (8.195) and (8.196)] ˜ ˜ ik(px+qy+mz) UJ (r,ω)= U(p,q,ω)e dpdq (G.2) DJ for z>0 with J = H,E.Herek = ω/c is real-valued and 1/2 2 2 2 2 m =+ 1 − p − q ,(p,q)∈ DH = (p, q)| 0 ≤ p + q ≤ 1 , (G.3) 1/2 2 2 2 2 m =+i p + q − 1 ,(p,q)∈ DE = (p, q)| p + q > 1 . (G.4)

˜ For a sufﬁciently well behaved spatiotemporal frequency spectrum U˜ (p,q,ω),the spectral wave ﬁeld U˜ (r,ω) is found [1, 2] to satisfy the homogeneous Helmholtz equation ∇2 + k2 U˜ (r,ω)= 0, (G.5)

© Springer Nature Switzerland AG 2019 689 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 690 G Stationary Phase Approximations and the Sommerfeld radiation condition [3, page 250, Eq. (1d)] ! " ∂U˜ (r,ω) lim r − ikU˜ (r,ω) = 0 (G.6) r→∞ ∂r in the half-space z>0, which guarantees that there are only outgoing waves at ˜ infinity. The spectral wave field component UH (r,ω) is expressed in Eq. (8.195) as a superposition of homogeneous plane waves that are propagating in directions specified by the real-valued direction cosines (p,q,m), illustrated in Fig. 8.9, ˜ whereas the spectral wave field component UE(r,ω) is expressed in Eq. (8.196) as a superposition of evanescent plane waves that are propagating in directions perpendicular to the z-axis as specified by the real-valued direction cosines(p, q, 0) and exponentially decaying at different rates in the z-direction. Because the behavior of the evanescent plane wave components is so vastly different from that for the homogeneous plane wave components, the two component spec- ˜ ˜ tral wave fields UH (r,ω) and UE(r,ω) are usually treated separately, as done here. In view of the fact that the integral appearing in the angular spectrum representation in Eq. (G.2) cannot be evaluated analytically (except in special cases), meaningful approximations are then required. Typically, the most appropriate approximations in the majority of applications are those that are valid as the distance # 2 2 2 R = (x − x0) + (y − y0) + (z − z0) (G.7) from the fixed point (x0,y0,z0) becomes large in comparison with the wavelength λ = 2π/k = 2πc/ω of the spectral amplitude wave field. This approximation is ˜ ˜ ˜ provided by the asymptotic expansion of U(r,ω), UH (r,ω), and UE(r,ω)as kR → ∞ with fixed k and fixed observation direction specified by the (fixed) direction cosines (ξ1,ξ2,ξ3) defined by x − x y − y z − z ξ ≡ 0 ,ξ≡ 0 ,ξ≡ 0 . (G.8) 1 R 2 R 3 R The analysis presented here follows that given by Sherman, Stamnes, and Lalor [4] in 1975 who consider the asymptotic behavior of U˜ (r,ω)as the point of observation (x,y,z)recedes to infinity in a fixed direction (ξ1,ξ2,ξ3) with positive z-component ξ3 > 0 through the point (x0,y0,z0), as well as when the point of observation recedes to infinity in a fixed direction (ξ1,ξ2, 0) perpendicular to the z-axis through a fixed point (x0,y0,z0) in the positive half-space z>0, all obtained through an extension of the method of stationary phase. G.1 The Method of Stationary Phase 691

G.1 The Method of Stationary Phase

The method of stationary phase was originally developed by G. G. Stokes [5]in 1857 and Lord Kelvin [6] in 1887 for the asymptotic approximation of Fourier transform type integrals of the form b f(λ)= g(t)eiλh(t)dt (G.9) a as λ →∞, where a, b, g(t), h(t), λ and t are all real-valued. The functions h(ζ) and g(ζ) are assumed to be analytic functions of the complex variable ζ = t + iτ in some domain containing the closed interval [a,b] along the real axis [7]. The exponential kernel eiλh(t) appearing in the integrand is then purely oscillatory so that as λ becomes large, its oscillations become very dense and destructive interference occurs almost everywhere. The exceptions occur at any stationary phase point tj deﬁned by

h (tj ) ≡ 0, (G.10) because the phase term λh(t) is nearly constant in a neighborhood of each such point, as well as from the lower and upper end points t = a and t = b of the integration domain because the effects of destructive interference will be incomplete there. Assume that there is a single interior stationary phase point at t = t0 where aTaylor series expansion

1 h(t) = h(t ) + h (t )(t − t )2 +··· (G.11) 0 2 0 0 1 2 so that h(t) − h(t0) = O (t − t0) , whereas in a neighborhood of any other point τ ∈[a,b], h(t) − h(τ) = O {(t − τ)}. The expansion given in Eq. (G.11) then suggests the change of variable

2 h(t) − h(t0) ≡±s , (G.12)

2 2 where +s is used when h (t0)>0 while −s is used when h (t0)<0. For values of t in a neighborhood of t0,Eq.(G.12) may be inverted with use of the Taylor series

1The order symbol O is deﬁned as follows. Let f(z) and g(z) be two functions of the complex variable z that possess limits as z → z0 in some domain D.Thenf(z) = O(g(z)) as z → z0 iff there exist positive constants K and δ such that |f(z)|≤K|g(z)| whenever 0 < |z − z0| <δ. 692 G Stationary Phase Approximations expansion given in Eq. (G.11)as

1/2 − = 2 + O 2 t t0 s s , (G.13) |h (t0)|

valid when either h (t0)>0orh (t0)<0. With the Taylor series expansion

g(t) = g(t0) + g (t0)(t − t0) +··· , (G.14) the integral in Eq. (G.9) becomes b f(λ) = g(t)eiλh(t)dt a 1/2 s 2 2 ± 2 ∼ iλh(t0) iλs + O { } | | g(t0)e e [1 s ] ds h (t0) −s1 1/2 ∞ ∞ 2 ± 2 ± 2 ∼ iλh(t0) iλs + O 2 iλs g(t0)e e ds s e ds |h (t0)| −∞ −∞ as λ →∞, so that b 1/2 2π ± − iλh(t) ∼ iλh(t0) iπ/4 + O 1 g(t)e dt g(t0)e e λ (G.15) a λ |h (t0)|

as λ →∞, where +π/4 corresponds to the case when h (t0)>0 while −π/4 corresponds to h (t0)<0. The end point contributions to the asymptotic behavior of this integral are obtained through a straightforward integration by parts as b b b iλh(t) = g(t) − d g(t) iλh(t) g(t)e dt e dt a iλh (t) a a dt iλh (t) 1 g(b) g(a) − ∼ eiλh(b) − eiλh(a) + O λ 2 , (G.16) iλ h (b) h (a) as λ →∞. Combination of Eqs. (G.15) and (G.16) then results in the asymptotic approximation b 1/2 2π ± iλh(t) ∼ iλh(t0) iπ/4 g(t)e dt g(t0)e e a λ |h (t0)| 1 g(b) g(a) − + eiλh(b) − eiλh(a) + O λ 2 iλ h (b) h (a) (G.17) G.2 Generalization to Angular Spectrum Integrals 693 as λ →∞. As an example, consider the asymptotic behavior of the Fourier integral of a continuous function f(t) that identically vanishes outside of the ﬁnite domain [α, β]; i.e. has compact support. Since h(t) = t does not possess any stationary phase points, the asymptotic behavior is due solely to the endpoints. Repeated integration by parts N times then results in the expression β N−1 n+1 − −i − − f(t)e iωtdt = f (n)(α)e iωα − f (n)(β)e iωβ + R (ω), ω N α n=0 where − N β i (N) −iωt RN (ω) ≡ f (t)e dt. ω α

By the Riemann–Lebesgue lemma [8], if f (N)(t) is continuous in the interval [α, β], then the integral appearing in the remainder term RN (ω) tends to zero in the limit as −N ω →∞, so that RN (ω) = O{ω }. The above result is then the asymptotic series as ω →∞of the Fourier transform of the N th-order continuous function f(t)with compact support [α, β].

G.2 Generalization to Angular Spectrum Integrals

The generalized asymptotic approximation presented here for the angular spectrum integral appearing in Eq. (G.2) is based upon an earlier extension of the method of stationary phase to multiple integrals of the form I(kR) = g(p,q)eikRf(p,q)dpdq, (G.18) D as kR →∞. In this extended method of stationary phase, originally due to Focke [9], Braun [10], Jones and Kline [11], and Chako [12], the functions f(p,q) and g(p,q) are taken to be independent of kR and to be sufficiently smooth in the domain D and the quantity kRf(p,q) is required to be real-valued for all (p, q) ∈ D. Heuristically, as kR becomes large, small variations in (p, q) result in rapid oscillations of the exponential factor appearing in the integrand of Eq. (G.18) and this, in turn, results in destructive interference in the integration about that point in the integration domain D so that most of the domain D provides only a negligible contribution to the integral I(kR). Significant contributions to I(kR) come from the respective neighborhoods of specific types of critical points that are ordered according to their relative asymptotic order as follows: 694 G Stationary Phase Approximations

Critical Point of the First Kind: An interior point (ps,qs) ∈ D where

∂f (p, q) ∂f (p, q) = = 0, (G.19) ∂p ∂q (ps ,qs ) (ps ,qs )

at which point the phase function f(p,q)is stationary, called an interior stationary phase point because the exponential factor in the integrand of Eq. (G.18) does not oscillate at that point, is a critical point of the first kind because it results in an asymptotic contribution to the integral appearing in Eq. (G.18) that is of order O (kR)−1 as kR →∞. Critical Point of the Second Kind: A boundary point (pb,qb) ∈ ∂D on the curve D | = that forms the boundary of the integration domain at which ∂f /∂ s (pb,qb) 0, where ds describes a differential element of arc length along the boundary curve, is a critical point of the second kind because it yields a contribution of order O (kR)−3/2 as kR →∞. Critical Point of the Third Kind: A corner point (pc,qc) ∈ ∂D on the curve that forms the boundary of the integration domain D is a critical point of the third kind because the slope of the boundary curve is discontinuous, resulting in a contribution of order O (kR)−2 as kR →∞. Additional critical points can arise at any singularities of the function g(p,q) appearing in the integrand of Eq. (G.18). Unfortunately, this heuristic description does not lead to a rigorous asymptotic description of integrals of the form given in Eq. (G.18). Although these rigorous extensions of the stationary phase method have little resemblance to the heuristic stationary phase argument presented above, they do provide justification of that heuristic argument provided that certain restrictions (as specified above) are placed on both f(p,q) and g(p,q). Unfortunately, these restrictions are not satisfied by the angular spectrum integrals given in Eq. (G.2). In particular, the restriction that the functions f(p,q) and g(p,q) must possess a finite number of continuous partial derivatives in D except possibly at a finite number of isolated integrable singularities of g(p,q) is not satisfied by the angular spectrum integral given in Eq. (G.2) on the unit circle 2 2 p + q = 1 that forms the boundary between DH and DE. In addition, the requirement that kRf(p,q) is real-valued in D cannot be applied to either ˜ ˜ U(r,ω) or UE(r,ω) when ξ3 > 0 since m is imaginary in DE. Nevertheless, ˜ if the evanescent contribution UE(r,ω) is indeed negligible in comparison to ˜ the homogeneous contribution UH (r,ω) then, at the very least, the leading term in the asymptotic expansion of U˜ (r,ω) can be obtained from the asymptotic ˜ expansion of UH (r,ω). This is not always the case, however, as shown by the following example due to Sherman et al. [4]. A time-harmonic spherical wave

2 2 2 1/2 eikr eik(x +y +z ) U˜ (r,ω)= = s r (x2 + y2 + z2)1/2 G.2 Generalization to Angular Spectrum Integrals 695 radiating away from the origin has spectral amplitude [13]

˜ ik ik Us(p,q,ω)= = . 2πm 2π(1 − p2 − q2)1/2

When the point of observation lies along the z-axis, p = q = 0 and the integrals in Eq. (G.2) may be directly evaluated to obtain

eikz 1 U˜ (r,ω) = − , sH z z 1 U˜ (r,ω) = . sE z

It is then seen that the homogeneous and evanescent wave contributions are of the same order in 1/z, so that, in this special case, the evanescent contribution to the wave field cannot be neglected in comparison to the homogeneous contribution. This ˜ result is shown to be a direct consequence of the singular behavior of Us(p,q,ω)= ik/2πm on the unit circle p2 + q2 = 1. In the generalized asymptotic method due to Sherman, Stamnes and Lalor [4] that is considered here, specific restrictions are placed on the spectral amplitude function ˜ ˜ U(p,q,ω). In particular, U(p,q,ω) ∈ TN , where TN is defined for positive, even ˜ integer N as the set of all spectral functions U˜ (p,q,ω)that are independent of the spatial coordinates x,y,z and that satisfy the following conditions for all ω: ˜ 1. U˜ (p,q,ω) can be expressed in the form (with the dependence on the angular frequency ω suppressed)

˜ V(p,q,m) U˜ (p,q,ω)= , (G.20) m

where m is as defined in Eqs. (G.3) and (G.4), and where V(p,q,m) ≡ |V(p,q,m)| is bounded for all p, q. 2. V(p,q,s) is a complex, continuous vector function of the three independent variables p, q, s that are defined (a) for all real p and q, (b) for real s ∈[0, 1], and (c) for pure imaginary s = iσ with σ>0. 3. V(p,q,s) possesses continuous, bounded partial derivatives up to the order N with respect to p, q, s for all p, q, s within its domain of definition. Notice that condition 1 ensures the convergence of the angular spectrum integrals appearing in Eq. (G.2) for all z>0, while conditions 2 and 3 are required for the derivation of the asymptotic approximations of these integrals. 696 G Stationary Phase Approximations

G.3 Approximations Valid Over a Hemisphere

The asymptotic approximation of U˜ (r,ω) = U˜ (x,y,z,ω)that is valid as the point of observation r = (x,y,z) recedes to infinity in a fixed direction with z>0 through the point r0 = (x0,y0,z0) is considered first. This is done in two steps. First, the method of stationary phase is extended using the technique due to Focke ˜ [9] and Chako [12] in order to show that when U(p,q,ω) ∈ TN has an interior stationary phase point at (ps,qs) = (ξ1,ξ2), only an arbitrarily small neighborhood of that stationary phase point contributes to the asymptotic behavior as R →∞ up to terms of order O{(kR)−N }. A standard application of the method of stationary phase is then used to obtain the asymptotic contribution due to this interior stationary phase point.

G.3.1 Extension of the Method of Stationary Phase

The method of approach used by Focke [9] and Chako [12] employs a neutralizer function ν(p,q) to isolate the interior stationary point. Notice that since the point of observation r = (x,y,z)recedes to infinity in a fixed direction with z>0 through the point r0 = (x0,y0,z0), then the direction cosine values ξ1,ξ2,ξ3 are constant with ξ3 > 0, and consequently, the stationary phase point (ps,qs) = (ξ1,ξ2) is fixed within the interior of the region DH . Define Ω1 and Ω2 to be two arbitrarily small neighborhoods of the point (ps,qs), both of which are completely contained within the interior of DH , where Ω1 ⊂ Ω2 is a proper subset of Ω2.Letν(p,q) be a real, continuous function of the independent variables p and q with continuous partial derivatives of all orders for all real values of p and q that satisfies the property

0 ≤ ν(p,q) ≤ 1, (G.21) with 1, when (p, q) ∈ Ω ν(p,q) = 1 . (G.22) 0, when (p, q)∈ / Ω1

The existence of such a continuous function for arbitrary regions Ω1 and Ω2 with Ω1 ⊂ Ω2 has been given by Bremmerman [14]. Additional details about the explicit form of this neutralizer function ν(p,q) are unnecessary. The principal result of this section is expressed in the following theorem due to Sherman, Stamnes and Lalor [4]. ˜ Theorem G.1 Let U(p,q,ω) ∈ TN for some positive even integer N. Then for z>0, the spectral wave field U˜ (r,ω)= U˜ (x,y,z,ω)given in Eqs. (G.1) and (G.2) G.3 Approximations Valid Over a Hemisphere 697 with k a positive real constant satisfies ˜ ˜ U(x,y,z,ω)= U0(x,y,z)+ R(x,y,z), (G.23) where2 ˜ ˜ ik(px+qy+mz) U0(x,y,z)= ν(p,q)U(p,q,ω)e dpdq, (G.24) DH R = O −N →∞ and where (x,y,z) (kR) #as kR uniformly with respect to ξ1 and − 2 − 2 ≤ ξ2 for all real ξ1,ξ2 such that δ< 1 ξ1 ξ2 1 for any positive constant δ<1. Notice that the dependence on the angular frequency ω has been suppressed in the above expressions. In addition, the statement that R(x,y,z) = O (kR)−N as kR →∞uniformly with respect to ξ1 and ξ2 means that R(x,y,z)is bounded as |R(x,y,z)|≤M(kR)−N as kR →∞with M a positive constant independent of ξ1 and ξ2. This theorem then directly leads to the result that the asymptotic behavior of the spectral wave field U˜ (r,ω) of order lower than (kR)−N as kR →∞is completely determined by an arbitrarily small neighborhood of the point (ps,qs). ˜ In particular, if U(p,q,ω) ∈ TN when N is arbitrarily large, then the asymptotic ˜ ˜ expansion of U(r,ω)is equal to the asymptotic expansion of U0(x,y,z). The proof of this theorem [4] is based upon a modification of the proof of Theorem 1 by Chako [12] or Theorem 3 by Focke [9].

Proof Construct three neutralizer functions νj (p, q), j = 1, 2, 3, each of which is a real continuous function of p and q with continuous partial derivatives of all orders for all p, q and which satisfy

0 ≤ νj (p, q) ≤ 1 with

ν1(p, q) + ν2(p, q) + ν3(p, q) = 1.

Choose positive constants C1,C2,C3,C4 such that # 2 + 2 ξ1 ξ2 δ>0, the constants C1,C2,C3,C4 clearly exist.

2Notice the typographical error in Eq. (2.5) of Ref. [4] where the spectral amplitude function ˜ U˜ (p,q,ω)was inadvertently omitted from that equation. 698 G Stationary Phase Approximations

D D 1 D3 p C1 C2 1 C3 C4

Fig. G.1 Graphical illustration of the notation used in the proof of Theorem 1. The shaded area indicates the region where m = 1 − p2 − q2 is real-valued, and the unshaded area indicates the region where m = i p2 + q2 − 1 is pure imaginary

With these four constants speciﬁed, the three neutralizer functions νj (p, q), j = 1, 2, 3 are now required to satisfy the conditions 1, for p2 + q2 ≤ C ν (p, q) = 1 , 1 2 2 0, for p + q ≥ C2 1, for C ≤ p2 + q2 ≤ C ν (p, q) = 2 3 , 2 2 2 2 2 0, for p + q ≤ C1 & p + q ≥ C4 1, for p2 + q2 ≥ C ν (p, q) = 4 . 3 2 2 0, for p + q ≤ C3 G.3 Approximations Valid Over a Hemisphere 699

Finally, three overlapping regions D1,D2,D3 in the p, q-plane are deﬁned as # 2 2 D1 ≡ (p, q) p + q ≤ C2 + ε , # 2 2 D2 ≡ (p, q) C1 − ε ≤ p + q ≤ C4 + ε , # 2 2 D3 ≡ (p, q) C3 − ε ≤ p + q , where ε>0 is a positive constant that satisﬁes the three inequalities

ε<1 − C2,

Each of these three regions is illustrated in Fig. G.1. Features of note that are important for this proof are the following:

• each of the neutralizer functions νj (p, q), j = 1, 2, 3, vanishes both for points (p, q)∈ / Dj and for points (p, q) in some neighborhood of the boundary ∂Dj of Dj , • the stationary phase point (ps,qs) lies within the interior of D1 and is exterior to both regions D2 and D3, • the direction cosine m =+ 1 − p2 − q2 is real-valued for all points (p, q) ∈ D1, 2 2 • and m =+i p + q − 1 is pure imaginary for all points (p, q) ∈ D3. Separate the angular spectrum integral for U˜ (r,ω)that is given in Eqs. (G.1) and (G.2) into three parts as ˜ ˜ ˜ ˜ U(r,ω)= U1(r,ω)+ U2(r,ω)+ U3(r,ω), where ˜ ˜ ik(px+qy+mz) Uj (r,ω)≡ νj (p, q)U(p,q,ω)e dpdq Dj for j = 1, 2, 3. Each of these integrals is now separately treated. Consider first the asymptotic approximation of the integral representation of ˜ U1(r,ω)which is in a form that is appropriate for the method of stationary phase. Since the stationary phase point (ps,qs) is the only critical point of significance in the domain D1, and since the critical points of the phase function k · r = k(px + qy + mz) on the boundary ∂D1 of D1 do not contribute to the integral 700 G Stationary Phase Approximations because of the neutralizer function ν1(p, q), and since the region Ω2 which contains the neighborhood Ω1 of the stationary phase point (ps,qs) lies entirely within D1, then it follows from the proof of Theorem 1 in Chako [12] that ˜ ˜ U1(r,ω)= U0(x,y,z)+ R(x,y,z), R = O −N →∞ where (x,y,z) (kR) as kR with fixed# k>0, uniformly with − 2 − 2 ≤ respect to ξ1,ξ2 for all real ξ1 and ξ2 such that δ< 1 ξ1 ξ2 1 for any positive constant δ with 0 <δ<1. Consider next the asymptotic approximation of the integral representation of ˜ 2 2 U3(r,ω). In this case, the quantity m =+i p + q − 1 is pure imaginary with 2 im ≤− (C3 − ε) − 1 < 0 for all (p, q) ∈ D3. For a sufficiently large value 2 2 2 of R = (x − x0) + (y − y0) + (z − z0) , a positive constant a then exists such that for z>a, ˜ ˜ ikmz U3(r,ω) ≤ U(p,q,ω) e dpdq D3 √ − − − 2− ˜ ≤ e k(z a) (C3 ε) 1 U˜ (p,q,ω) eikmadpdq. D3

˜ Since U(p,q,ω) ∈ TN , it then follows from the first condition [see Eq. (1)] in the definition of TN that the final integral appearing in the above inequality converges to a finite, nonnegative constant M3 that is independent of the direction cosines ξ1,ξ2,ξ3. Since z = z0 + ξ3R [see Eq. (G.8)] and since ξ3 > 0, then the above inequality becomes √ 2 ˜ −k[(z0−a)+Rδ] (C3−ε) −1 U3(r,ω) ≤ M3e , ˜ −N independent of ξ1,ξ2. Hence, U3(r,ω) = O (kR) uniformly with respect to ξ1,ξ2 as kR →∞for all positive even integer values N. Consider finally the asymptotic approximation of the integral representation of ˜ U2(r,ω), which turns out to be much more involved than the previous two cases. The analysis begins with the change of variables

p = sin α cos β, q = sin α sin β, with Jacobian J (p, q/α, β) = sin α cos β. In addition, 1/2 m = 1 − p2 − q2 = cos α, G.3 Approximations Valid Over a Hemisphere 701 where α is, in general, complex as the region D2 extends from inside the inner region where m is real-valued into the outer region where m is pure imaginary, as illustrated in Fig. G.1. In addition, let the angles ϑ, ϕ describe the intersection point on the unit sphere with center at the origin r = 0 in coordinate space with the line through the origin that is parallel to the direction of observation r = (x,y,z)as it recedes to inﬁnity through the ﬁxed point r0 = (x0,y0,z0) with z>0, where

ξ1 = sin ϑ cos ϕ,

ξ2 = sin ϑ sin ϕ,

ξ3 = cos ϑ, with 0 ≤ ϑ<π/2 and 0 ≤ ϕ<2π. Under these two changes of variables, the ˜ integral for U2(r,ω)becomes 2π ˜ ikR[sin ϑ sin α cos (β−ϕ)+cos ϑ cos α] U2(r,ω)= A(α, β)e dαdβ, 0 C where

ik(px0+qy0+mz0) A(α, β) ≡ ν2(p, q)V (p, q, m)e sin α, with V(p,q,m)deﬁned in Eq. (1). The√ contour of integration C in the complex α- = − plane, which extends from α1 arcsin C1 √ε along the real α -axis to π/2 and −1 then to the endpoint at α4 = π/2 − i cosh C4 + ε , depicted in Fig. G.2, then results in a complete, single covering of the original integration domain D2. Notice that, in contrast with the original form of the integral, the phase

Φ ≡ sin ϑ sin α cos (β − ϕ) + cos ϑ cos α ˜ appearing in the above transformed integral for U2(r,ω) is analytic and the amplitude function A(α, β) is continuous with continuous partial derivatives with respect to α and β up to order N over the entire integration domain D2. However, the contour of integration C is now complex, as illustrated in Fig. G.2, and this results in the phase Φ being complex-valued when α = α + iα varies from π/2toα4 along C. As α varies over the contour C with the angle β held ﬁxed, the phase function Φ varies over a simple curve C(β) of ﬁnite length in the complex α-plane. Since the partial derivative ∂Φ = sin ϑ cos α cos (β − ϕ) − cos ϑ sin α ∂α β 702 G Stationary Phase Approximations

Fig. G.2 Contour of '' integration C for the ˜ α-integral in U2(r,ω)

is an entire function of complex α for all β, then −1 ∂α = ∂Φ ∂Φ β ∂α β is an analytic function of complex α for all α, β provided that ∂Φ = 0. Since ∂α β cos ϑ = ξ >δ, then ∂Φ = 0 when cos α = 0. It then follows that the zeros of 3 ∂α β ∂Φ ∂α β occur for those values of α that satisfy the relation

tan α = tan ϑ cos (β − ϕ).

On the portion of the contour C over which α is real, α1 ≤ α ≤ π/2, and since cos ϑ = ξ3 ≤ C1 − ε (see Fig. G.1), then ϑ ≤ α ≤ π/2 and one obtains the inequality tan α>tan ϑ cos (β − ϕ). On the portion of the contour C over which α is complex, {α}=π/2 and tan α =−i tanh ({α}) is pure imaginary whereas tan ϑ cos (β − ϕ) is real. As a consequence, the quantity ∂Φ has no zeros for ∂α β all α on the contour C with 0 ≤ β<2π, and this in turn implies that ∂α = ∂Φ β −1 ∂Φ ∈ ≤ ∂α β is an analytic function of complex α C when 0 β<2π.The G.3 Approximations Valid Over a Hemisphere 703

˜ change of variable of integration in the α-integral of U2(r,ω)to the variable Φ can then be made with the result 2π ˜ ∂α ikRΦ U2(r,ω)= A α(Φ),β e dΦdβ. 0 C(β) ∂Φ β

As a consequence of the prescribed properties for both the function V(p,q,m)= ˜ mU(p,q,ω)[cf. Eq. (1)] and the neutralizer function ν2(p, q), the function A(α, β) possesses N continuous partial derivatives with respect to the variable α ∈ C for all fixed β ∈[0, 2π).3 Because the quantity ∂α is an analytic function of α ∈ C,the ∂Φ β ∂α product A(α, β) ∂Φ β also possesses N continuous partial derivatives taken with respect to α along the contour C with fixed β ∈[0, 2π). Notice that differentiation with respect to Φ along the contour C(β) with fixed β is equivalent to differentiation with respect to α along C with fixed β followed by multiplication by the quantity ∂α ∂α ∂α ∂Φ β . Because ∂Φ β is analytic with respect to α, the quantity A(α, β) ∂Φ β has N continuous partial derivatives with respect to Φ ∈ C(β) for all fixed β ∈[0, 2π). The Φ-integral ∂α I(kR,β) = A α(Φ),β eikRΦdΦ C(β) ∂Φ β

˜ appearing in the integral expression for U2(r,ω) is now integrated by parts N times by integrating the exponential factor eikRΦ each time and differentiating the remaining factor, with the result [15]

I(kR,β) = LN (Φ4) − LN (Φ1) + RN (kR, β), where − N1 n ikRΦj = n−1 ∂ ∂α e LN (Φj ) i n A α(Φ),β n+1 , ∂Φ ∂Φ = (kR) n=0 β β Φ Φj with Φj = sin ϑ sin αj cos (β − ϕ) + cos ϑ cos αj for j = 1, 4, where α1 and α4 denote the endpoints of the contour C (see Fig. G.2), and where N R = − −N ∂ ∂α ikRΦ N (kR, β) ( ikR) N A α(Φ),β e dΦ. C(β) ∂Φ ∂Φ β β

3Notice that the derivatives with respect to α must be taken along the contour C since both V(p,q,m) and ν2(p, q) are deﬁned only for real-valued p and q. That is, variation of α along the contour C corresponds to p, q varying along the real axis. 704 G Stationary Phase Approximations

As a consequence of the prescribed properties for the neutralizer function ν2(p, q), ∂nA(α,β) = − the function A(α, β) and all N of its partial derivatives ∂αn , n 0, 1,...,N 1, taken along the contour C, vanish at the endpoints Φ1 and Φ4 of the contour C. ∂nA(α,β) Each of the N partial derivatives ∂Φn taken along the contour C(β) then also vanish at the endpoints Φ1 and Φ4, so that

LN (Φ1) = LN (Φ4) = 0.

Furthermore, because the integrand in the above expression for the remainder term RN (kR, β) is continuous along the contour C(β) and as this contour is of ﬁnite length for all ϑ, ϕ, and β, then that integral is bounded by some positive constant M2 independent of ϑ, ϕ, and β, so that

−N |RN (kR, β)| ≤ M2(kR) .

Consequently,

˜ −N U2(r,ω) ≤ 2πM2(kR) , ˜ −N so that U2(r,ω) = O (kR) uniformly with respect to ξ1,ξ2 as kR →∞with ﬁxed k>0. In summary, following the method of proof given by Sherman, Stamnes, and ˜ Lalor [4], it has been established that the three terms Uj (r,ω), j = 1, 2, 3, whose sum gives U˜ (r,ω), satisfy the order relations ˜ ˜ −N U1(r,ω) = U0(x,y,z)+ O (kR) , ˜ −N U2(r,ω) = O (kR) , ˜ −N U3(r,ω) = O (kR) , uniformly with respect to ξ1,ξ2 as kR →∞with ﬁxed k>0. This then completes the proof of the theorem.

G.3.2 Asymptotic Approximation of U˜ (r,ω)

Theorem G.1 establishes that any terms of order lower than (kR)−N in the asymptotic behavior of U˜ (r,ω)as kR →∞with ﬁxed k>0 must be contributed G.3 Approximations Valid Over a Hemisphere 705 by the term [see Eq. (G.24)] ˜ ˜ ik(px+qy+mz) U0(x,y,z)= ν(p,q)U(p,q,ω)e dpdq. (G.25) DH

This integral representation can be expressed in the form of the integral I(kR) in Eq. (G.18) with phase function

f(p,q)= ξ1p + ξ2q + ξ3m (G.26) and amplitude function (now a vector)

˜ + + g(p, q) = ν(p,q)U˜ (p,q,ω)eik(px0 qy0 mz0). (G.27)

Because ν(p,q) = 0 when (p, q)∈ / Ω2, the domain of integration DH appearing in Eq. (G.25) can be replaced by a domain D that is taken as any region within the interior of the region DH that contains the subregion Ω2 within its interior (see Fig. G.3). With that choice, the phase function f(p,q) defined in Eq. (G.26) is real-valued and infinitely differentiable in D, and the amplitude function g(p,q) has continuous, bounded partial derivatives up to order N in D. The integral in Eq. (G.25) then satisfies all of the conditions required for the direct application of the stationary phase method due to Braun [10]. Because there is only the single critical point [the interior stationary phase point (p, q) = (ξ1,ξ2)]inD, and because the integrand and its derivatives all vanish on the boundary of the domain D, application of the method of stationary phase due to Braun [10] then gives4

N −1 ikR 2 ˜ e Bn(ϑ, ϕ) − N U(x,y,z)= + O (kR) 2 , (G.28) kR (kR)n n=0 as kR →∞with fixed k>0, positive z, and fixed direction cosines ξ1,ξ2, and with ˜ U(p,q,ω) ∈ TN , where the coefficients Bn(ϑ, ϕ) are independent of R. Notice that, as required by the definition of the set TN of spectral functions introduced just prior to Eq. (1), N is a positive, even integer. Finally, notice that the order of the remainder term in Eq. (G.28) is a refinement of Braun’s estimate as reported by Sherman, Stamnes, and Lalor [4]. Application of the result given in Sect. 5.1 of the paper by Jones and Kline [11] yields the zeroth-order term

ik(ξ1x0+ξ2y0+ξ3z0) B0(ϑ, ϕ) =−2πiV(ξ1,ξ2,ξ3)e , (G.29)

4Although the stationary phase result given by Braun [10] is for a scalar field, the result may be directly generalized to a vector field by separately applying it to each of the scalar components of that vector field. 706 G Stationary Phase Approximations

H1 E1

p 2 + q 2-1

H2 E2

Fig. G.3 Illustration of the integration regions DJ 1 and DJ 2, J = H,E, appearing in Eqs. (G.2)– (G.4)

˜ where [cf. Eq. (1)] V(p,q,m) ≡ mU˜ (p,q,ω) with m =+ 1 − p2 − q2, and where [see Eqs. (G.7) and (G.8)] ξ1 = (x−x0)/R, ξ2 = (y−y0)/R, ξ3 = (z−z0)/R 2 2 2 with R = (x − x0) + (y − y0) + (z − z0) are related to the angles ϑ and ϕ through the relations

ξ1 = sin ϑ cos ϕ,

ξ2 = sin ϑ sin ϕ, (G.30)

ξ3 = cos ϑ, with 0 ≤ ϑ<π/2 and 0 ≤ ϕ<2π. By making use of the fact that the spectral wave field U˜ (r,ω) satisfies the Helmholtz equation (G.5)forz>0, Sherman [16] has shown that if a solution U˜ (r,ω) of the Helmholtz equation has an asymptotic expansion of the form given in Eq. (G.28) for arbitrarily large (even) values of N, and if the asymptotic expansion of each of the partial derivatives of U˜ (r,ω)with respect to the spherical coordinate variables R,ϑ,ϕ up to order 2 can be obtained by term by term differentiation of the asymptotic expansion given in Eq. (G.28), then the coefficients Bn(ϑ, ϕ) satisfy the recursion formula (see Problem 8.9) i 2 (n + 1)B + (ϑ, ϕ) = L − n(n + 1) B (ϑ, ϕ), (G.31) n 1 2 n where L2 is the differential operator defined by 1 ∂ ∂ 1 ∂2 L2 ≡− sin ϑ − . (G.32) sin ϑ ∂ϑ ∂ϑ sin2 ϑ ∂ϕ2

Since the partial derivatives of the spectral wave field U˜ (r,ω)given in Eqs. (8.195) and (8.196) satisfy these requirements, then the coefficients Bn+1(ϑ, ϕ) appearing in the asymptotic expansion (G.28) of this wave field satisfy the recursion formula ˜ given in Eq. (G.31) provided that U(p,q,ω) ∈ T∞. Since the functional depen- ˜ dence of the coefficients Bn+1(ϑ, ϕ) on the spatiotemporal spectrum U(p,q,ω)of G.4 Approximations Valid on the Plane z = z0 707

˜ the wave field is independent of N, then Eq. (G.31) also holds for U(p,q,ω)∈ TN with N finite. Sherman’s recursion formula (G.31) then provides a straightforward method for obtaining higher-order terms in the asymptotic approximation of the spectral wave ˜ field U(r,ω)from the first term in the expansion (G.28). For example, B1(ϑ, ϕ) = 2 ˜ (i/2)L B0(ϑ, ϕ), so that if U(p,q,ω)∈ TN with N ≥ 6, ikR e i + + U˜ (r,ω) =−2πi 1 + L2 V(ξ ,ξ ,ξ )eik(ξ1x0 ξ2y0 ξ3z0) kR 2kR 1 2 3 − +O (kR) 3 (G.33) as kR →∞with fixed k>0 and fixed ξ1,ξ2 with z>0.

G.4 Approximations Valid on the Plane z = z0

The asymptotic approximation of U˜ (r,ω) = U˜ (x,y,z,ω)that is valid as the point of observation r = (x,y,z) recedes to inﬁnity in a ﬁxed direction perpendicular to the z-axis with z>0 through the point r0 = (x0,y0,z0) is now considered. This case is excluded in Sect. G.3 because the method of analysis used there cannot be extended to include the special case when ξ3 = 0. As in Theorem G.1,itis ˜ again assumed that U(p,q,ω) ∈ TN for some positive even integer N and that z = z0 > 0. Upon setting z = z0 in Eq. (G.2) an integral of the form given in Eq. (G.18)is obtained, viz. I(kR) = g(p,q)eikRf(p,q)dpdq, (G.34) D with

f(p,q) = ξ1p + ξ2q, (G.35) ˜ + + g(p,q) = U˜ (p,q,ω)eik(px0 qy0 mz0), (G.36) and D = DJ , J = H,E. The phase function f(p,q) is then real and analytic in both regions DH and DE. However, the amplitude function g(p,q) need not have any of its partial derivatives with respect to p and q exist on the unit circle p2 + q2 = 1. In order to circumvent this problem, Sherman, Stamnes, and Lalor [4] isolate that unit circle using a neutralizer function as follows. Let ν0(p, q) be a real, continuous function with continuous partial derivatives of all orders for all p, q that 708 G Stationary Phase Approximations satisﬁes the inequality

0 ≤ ν0(p, q) ≤ 1, (G.37) with 1for 1− ε ≤ p2 + q2 ≤ 1 + ε, ν (p, q) = (G.38) 0 0 for either p2 + q2 ≤ 1 − 2ε or p2 + q2 ≥ 1 + 2ε, where ε>0 is a positive constant with ε<1/3. The integral in Eq. (G.2) can then be written in the form ˜ ˜ ˜ UJ (x,y,z0,ω)= UJ 1(x,y,z0,ω)+ UJ 2(x,y,z0,ω) (G.39) with J = H,E, where ˜ ˜ ˜ ik(px+qy+mz0) UJ 1(x,y,z0,ω) = ν0(p, q)U(p,q,ω)e dpdq, (G.40) D J 1 ˜ ˜ ˜ ik(px+qy+mz0) UJ 2(x,y,z0,ω) = [1 − ν0(p, q)]U(p,q,ω)e dpdq. (G.41) DJ 2

The various regions of integration in the p, q-plane appearing in these integral expressions are deﬁned as 2 2 DH1 ≡ (p, q) | 1 − 3ε ≤ p + q ≤ 1 , 2 2 DH2 ≡ (p, q) | 0 ≤ p + q ≤ 1 − ε/2 , 2 2 DE1 ≡ (p, q) | 1 ≤ p + q ≤ 1 + 3ε , 2 2 DE2 ≡ (p, q) | p + q ≤ 1 + ε/2 , and are depicted in Fig. G.3. The asymptotic approximation of each of the integrals appearing in Eqs. (G.40) and (G.41) is now separately considered.

G.4.1 The Region DH 2

˜ The integrand appearing in the expression (9.225) for UH2(x,y,z0,ω)is found [4] to satisfy all of the conditions required in Theorem 1 of Chako [12]. Because the region DH2 does not contain any critical points of that integrand, and because the ˜ amplitude function [1 − ν0(p, q)]U(p,q,ω)and all N of its partial derivatives with G.4 Approximations Valid on the Plane z = z0 709 respect to p and q vanish on the boundary of DH2, it then follows that ˜ −N UH2(x,y,z0,ω)= O (kR) (G.42) as kR →∞with ﬁxed k>0.

G.4.2 The Region DE2

˜ The integrand appearing in the expression (G.41)forUE2(x,y,z0,ω)is also found [4] to satisfy the conditions in Theorem 1 of Chako [12], but the integration domain DE2 appearing in Eq. (G.41) extends to infinity whereas Chako’s proof is given only for a finite domain of integration. Its extension to the case of an infinite domain has been given by Sherman, Stamnes, and Lalor [4] who consider the same integral but with the region of integration now given by 1 + ε/2 ≤ p2 + q2 ≤ K, where K>1 + ε/2 is an arbitrary constant that is allowed to go to infinity. This change then introduces two complications into Chako’s proof. For the first, the amplitude ˜ function U˜ (p,q,ω) and its partial derivatives with respect to p and q do not, in general, vanish on the boundary p2 + q2 = K for finite K, but they do vanish in →∞ the limit as√K because each of these boundary terms contains the exponential −z K−1 5 factor e 0 with z0 > 0. The second complication occurs in the remainder integral after integration by parts N times, which is now taken over an infinite domain. However, all that is required in the proof is that this integral is√ convergent − 2+ 2− and this is guaranteed by the presence of the exponential factor e z0 p q 1 in the integrand. Hence, the result of the theorem due to Chako applies in this case, so that ˜ −N UE2(x,y,z0,ω)= O (kR) (G.43) as kR →∞with fixed k>0.

G.4.3 The Region DH 1

˜ Attention is now turned to obtaining the asymptotic approximation of UH1(x,y,z0, ω) as kR →∞with ﬁxed k>0. Under the change of variable p = sin α cos β,

5 Notice that the special case when z0 = 0 can be treated only if additional restrictions are placed on ˜ the behavior of the spectral amplitude function U˜ (p,q,ω)and its partial derivatives with respect to p and q in the limit as p2 + q2 →∞. 710 G Stationary Phase Approximations q = sin α sin β the integral representation appearing in Eq. (G.40) becomes β0+2π π/2 ˜ ikR sin α cos (β−ϕ) UH1(x,y,z0,ω)= A (α, β)e dαdβ, (G.44)

β0 α1 with ﬁxed direction cosines ξ1 = sin ϑ cos ϕ, ξ2 = sin ϑ sin ϕ [see Eq. (G.8)] with ≤ ≤ ≡ 0 ϑ<π/√ 2 and 0 √ϕ<2π.Hereβ0 is an arbitrary real constant, α1 arcsin 1 − 3ε = arccos 3ε, and

ik(px0+qy0+mz0) A (α, β) ≡ ν0(p, q)V (p, q, m)e sin α, (G.45) with m = (1 − p2 − q2)1/2 = cos α and V(p,q,m)is as deﬁned in Eq. (1). Since β0 is an arbitrary constant, it is chosen for later convenience to be given by β0 = ϕ − π/4. The integral appearing in Eq. (G.44) is now in the form of the integral given in Eq. (G.34) with phase and amplitude functions

f(α,β) = sin α cos (β − ϕ), (G.46)

g(α,β) = A (α, β), (G.47) respectively. Because both of these functions satisfy all of the requirements necessary for application of the method of stationary phase, this method may now be directly applied. The critical points of the integral in Eq. (G.44) occur at the stationary phase points that are deﬁned by the condition [cf. Eq. (G.19)]

∂f ∂f = = 0, (G.48) ∂α ∂β (αs ,βs ) (αs ,βs ) so that both cos α cos (β − ϕ) = 0 ⇒ α = π/2 ∨ β = ϕ + π/2,ϕ + 3π/2 and sin α sin (β − ϕ) = 0 ⇒ β = ϕ,ϕ + π. The stationary phase points are then

Point a: (αs,βs) = (π/2,ϕ),

Point b: (αs,βs) = (π/2,ϕ+ π), both of which occur on the boundary of the integration region DH1, as illustrated in Fig. G.4. Additional critical points occur at the two corners (π/2,β0) and (π/2,β0+ 2π). Because of the above choice that β0 = ϕ − π/4, neither of the stationary phase points coincide with a boundary corner of the integration domain, as depicted in Fig. G.4. Consider ﬁrst the contribution from the two corner critical points which are just an artifact of the change of variables of integration. Because their location along the line α = π/2 is completely determined by the choice of the constant β0,itis expected that, taken together, they do not contribute to the asymptotic behavior of G.4 Approximations Valid on the Plane z = z0 711

f/ ''

f/ b

'' f/

f a / '' '' ' I I I

Fig. G.4 Illustration of the integration region DH 1 appearing in Eq. (G.44) and the location of the critical points appearing in that integral

˜ UH1(x,y,z0,ω) as kR →∞. The veriﬁcation of this expected result is given by the following argument [4]. Construct a periodic neutralizer function ν (β) with period 2π that is a real, continuous function of β with continuous derivatives of all orders, and is such that

ν (β) = 1, when ϕ − π/8 ≤ β ≤ ϕ + 9π/8,

and ν (β) = 0 in some neighborhood of β0, as depicted on the left side of ˜ Fig. G.4. The integral representation (G.44) of the ﬁeld UH1(x,y,z0,ω)may then 712 G Stationary Phase Approximations be expressed as β0+2π π/2 ˜ ikR sin α cos (β−ϕ) UH1(x,y,z0,ω) = ν (β)A (α, β)e dαdβ

β0 α1 β0+2π π/2 − + 1 − ν (β) A (α, β)eikR sin α cos (β ϕ)dαdβ.

β0 α1

Because the integrand vanishes (due to the neutralizer function) at the corner points, the only critical points in the first integral are the stationary phase points labeled a and b in Fig. G.4. Moreover, because ν (β) = 1 in a separate neighborhood of each of the stationary phase points, the asymptotic behavior of this first integral is identical to the stationary phase point contributions to the asymptotic behavior of ˜ UH1(x,y,z0,ω)as kR →∞. On the other hand, because the quantity (1 − ν (β)) vanishes at both of the stationary phase points, the only critical points of importance in the second integral are the corner points. Because the integrand in this second integral is a periodic function of β with period 2π, the integrand can be made to vanish at the corner points simply by changing the region of integration so that it extends from ϕ − 9π/8toϕ + 7π/8. It then follows from Theorem 1 of the paper [12] by Chako that the second integral is O (kR)−N as kR →∞. ˜ The field UH1(x,y,z0,ω)may then be expressed as ˜ = ˜ (a) + ˜ (b) + O −N UH1(x,y,z0,ω) UH1(x,y,z0,ω) UH1(x,y,z0,ω) (kR) , (G.49)

˜ (a) ˜ (b) where UH1(x,y,z0,ω) and UH1(x,y,z0,ω) denote the separate contributions to ˜ the asymptotic behavior of UH1(x,y,z0,ω)from the stationary phase points a and b, respectively. In order to obtain the separate asymptotic approximations of both ˜ (a) ˜ (b) UH1(x,y,z0,ω)and UH1(x,y,z0,ω), the asymptotic expansion due to a boundary stationary phase point of the hyperbolic type must be used [4]. This has been shown to be of the same form as that for a boundary stationary phase point of the elliptic type, which has been given in Sects. 3.3–3.4 of Bremmerman [14]. Thus

˜ (j) = ˜ (j1) + ˜ (j2) UH1(x,y,z0,ω) UH1 (x,y,z0,ω) UH1 (x,y,z0,ω), (G.50) with

N ±ikR 2 (j1) e B (ϕ) − U˜ (j1)(x,y,z ,ω) = Hn + o (kr) N/4 , (G.51) H1 0 kR (kR)n n=0

N −1 ±ikR 2 (j2) e B (ϕ) − U˜ (j2)(x,y,z ,ω) = Hn + o (kr) N/4 , (G.52) H1 0 (kR)3/2 (kR)n n=0 G.4 Approximations Valid on the Plane z = z0 713

→∞ (j1) (j2) as kR with ﬁxed k>0, where BHn(ϕ) and BHn(ϕ) are both independent of R. The upper sign in the exponential appearing in Eqs. (G.51) and (G.52)isused when j = a and the lower sign is used when j = b. Explicit expressions for (j1) (j2) the coefﬁcients BHn(ϕ) and BHn(ϕ) are given in Appendix II of the paper [4]by Sherman, Stamnes, and Lalor.

G.4.4 The Region DE1

˜ Consider finally obtaining the asymptotic approximation of UE1(x,y,z0,ω) as kR →∞with fixed k>0. With the change of variable defined by

p = cosh μ cos λ, (G.53) q = cosh μ sin λ, (G.54) so that

m =+i sinh μ, (G.55) with Jacobian J(p,q/μ,λ) = sinh μ cosh μ, the integral appearing in Eq. (G.40) becomes β0+2π μ1 ˜ ikR cosh (μ) cos (λ−ϕ) UE1(x,y,z0,ω)= A (μ, λ)e dμdλ, (G.56) β0 0 √ √ −1 −1 where μ1 ≡ sinh 3 = cosh 1 + 3 and

+ + A (μ, λ) ≡−iν (p, q)V (p, q, m)eik(px0 qy0 mz0) cosh μ, (G.57) with p and q as given in Eqs. (G.53) and (G.54). The integral appearing in Eq. (G.56) is now in the form of the integral in Eq. (G.34) with amplitude and phase functions

g(μ,λ) = A (μ, λ), (G.58) f(μ,λ) = cosh (μ) cos (λ − ϕ), (G.59) respectively. As was found for the integral in Eq. (G.44) for the spectral field component ˜ UH1(x,y,z0,ω), all of the critical points lie on the boundary ∂DE1 of the integration domain. Just as in that case, the critical points on the boundary μ = μ1 and at the corners (μ, λ) = (0, 0) and (μ, λ) = (0, 2π) do not contribute to the asymptotic behavior of the integral in Eq. (G.56). The asymptotic behavior of the ˜ spectral field component UE1(x,y,z0,ω) is then completely determined by the 714 G Stationary Phase Approximations contributions from the stationary phase points (a) at (μs,λs) = (0,ϕ) and (b) at ˜ (μs,λs) = (0,ϕ+ π). As in Eq. (G.49), the field UE1(x,y,z0,ω)is expressed as ˜ = ˜ (a) + ˜ (b) + O −N UE1(x,y,z0,ω) UE1(x,y,z0,ω) UE1(x,y,z0,ω) (kR) , (G.60)

˜ (a) ˜ (b) where UE1(x,y,z0,ω) and UE1(x,y,z0,ω) denote the separate contributions to ˜ the asymptotic behavior of UH1(x,y,z0,ω) from the stationary phase points (a) and (b), respectively. An analysis similar to that following Eq. (G.49) then yields [4]

˜ (j) = ˜ (j1) + ˜ (j2) UE1(x,y,z0,ω) UE1 (x,y,z0,ω) UE1 (x,y,z0,ω), (G.61) with

N ±ikR 2 (j1) e B (ϕ) − U˜ (j1)(x,y,z ,ω) = En + o (kr) N/4 , (G.62) E1 0 kR (kR)n n=0

N −1 ±ikR 2 (j2) e B (ϕ) − U˜ (j2)(x,y,z ,ω) = En + o (kr) N/4 , (G.63) E1 0 (kR)3/2 (kR)n n=0

→∞ (j1) (j2) as kR with ﬁxed k>0, where BEn (ϕ) and BEn (ϕ) are both independent of R. The upper sign in the exponential appearing in Eqs. (G.62) and (G.63)isused when j = a and the lower sign is used when j = b.

(j ) (j ) 1 (ϕ) 2 (ϕ) G.4.5 Relationship Between the Coefﬁcients BHn , BHn (j ) (j ) 1 (ϕ) 2 (ϕ) and BEn , BEn

(j1) (j2) The various relationships between the coefficients BHn(ϕ) and BHn(ϕ) in the ˜ asymptotic expansion given in Eqs. (G.51) and (G.52)forUH1(x,y,z0,ω)and the (j1) (j2) coefficients BEn (ϕ) and BEn (ϕ) in the asymptotic expansion given in Eqs. (G.62) ˜ and (G.63)forUE1(x,y,z0,ω)are given by Jones and Kline [11], as modified by Sherman, Stamnes, and Lalor [4], as6

(a1) =+ (a1) BEn (ϕ) BHn (ϕ), (G.64) (a2) =− (a2) BEn (ϕ) BHn (ϕ), (G.65)

6Explicit expressions for all of these coefﬁcients are unnecessary for the ﬁnal asymptotic expansion and so are not given here. The interested reader should consult Appendices I–III in the 1976 paper [4] by Sherman, Stamnes, and Lalor. G.4 Approximations Valid on the Plane z = z0 715

(b1) =− (b1) BEn (ϕ) BHn (ϕ), (G.66) (b2) =− (b2) BEn (ϕ) BHn (ϕ). (G.67)

The pair of relations in Eqs. (G.66) and (G.67) then show that the contribution of ˜ the point b to UE1(x,y,z0,ω) is equal in magnitude but opposite in sign with ˜ the contribution of the point b to UH1(x,y,z0,ω). Consequently, the point b does ˜ not contribute to the asymptotic behavior of the field quantity UH1(x,y,z0,ω)+ ˜ −N/4 UE1(x,y,z0,ω)with order lower than O (kR) that arises from the sum of the second terms on the right hand side of Eqs. (G.49) and (G.60). In a similar manner, Eq. (G.65) implies that the contribution of the point a to the series involving inverse ˜ half powers of (kR) in the expansion of UE1(x,y,z0,ω)is equal in magnitude but ˜ opposite in sign to the same contribution of the point a to UH1(x,y,z0,ω). Taken together with the previous result, this implies that the asymptotic expansion of the ˜ ˜ field quantity UH1(x,y,z0,ω)+ UE1(x,y,z 0,ω)does not include terms with half powers of (kR) of order lower than O (kR)−N/4 . Finally, the relation given in ˜ ˜ Eq. (G.64) implies that the field components UH1(x,y,z0,ω)and UE1(x,y,z0,ω) contribute equally to the remaining terms in the asymptotic expansion of the total ˜ field U(x,y,z0,ω)involving only inverse integral powers of (kR). It then follows from this result together with Eqs. (G.1), (G.39), (G.43), and (G.51) that

N ikR 2 (a1) e B (ϕ) − U˜ (x,y,z ,ω)= 2 Hn + o (kr) N/4 , (G.68) 0 kR (kR)n n=0 as kR →∞with ﬁxed k>0 and z = z0. (a1) The coefﬁcients BHn (ϕ) appearing in the asymptotic expansion (G.68) are found [4] to be given by the limiting expression

(a1) 1 B (ϕ) = lim Bn(ϑ, ϕ) (G.69) Hn → π 2 θ 2 of the coefﬁcients Bn(ϑ, ϕ) appearing in the asymptotic expansion (G.28). It then follows that the results of this section can be combined with those of Sect. G.3.1 to yield the asymptotic expansion

N ikR 2 e B (ϑ, ϕ) − U˜ (x,y,z ,ω)= n + o (kr) N/4 , (G.70) 0 kR (kR)n n=0 as kR →∞with fixed k>0 and z0 > 0 that is uniformly valid with respect to the angles ϑ and ϕ in their respective domains 0 ≤ ϑ ≤ π/2 and 0 ≤ ϕ ≤ 2π. The zeroth-order coefficient B0(ϑ, ϕ) is given in Eq. (G.29) and the first-order 2 coefficient is given by B1(ϑ, ϕ) = (i/2)L B0(ϑ, ϕ), where the differential operator L2 is defined in Eq. (G.32). 716 G Stationary Phase Approximations

˜ ˜ G.5 Asymptotic Approximations of UH (r,ω)and UE(r,ω)

The respective dominant terms in the asymptotic expansion of the homogeneous ˜ ﬁeld component UH (r,ω)and in the asymptotic expansion of the evanescent ﬁeld ˜ component UE(r,ω)are now derived for each of the three direction cosine regions 0 <ξ3 < 1, ξ3 = 0, and ξ3 = 1.

G.5.1 Approximations Valid Over the Hemisphere 0 <ξ3 < 1

In order to obtain the asymptotic approximation of the homogeneous wave contri- ˜ bution UH (r,ω) to the total spectral wave field, the notation and definitions used in Eqs. (G.37)–(G.41) are applied with z0 replaced by z. The only critical point of ˜ the integral for UH2(r,ω), where the integrand is nonzero, is the interior stationary ˜ phase point (ps,qs) = (ξ1,ξ2). The asymptotic expansion of UH2(r,ω) is then seen to be identical with the asymptotic expansion of U˜ (r,ω)given in Eqs. (G.28) and (G.33). It then follows from the above result together with Eq. (G.1) that the asymptotic ˜ expansion of the evanescent wave contribution UE(r,ω) to the total wave field is ˜ identical to that of −UH1(r,ω). The asymptotic expansion of the integral expression ˜ ˜ for UH1(r,ω) = UH1(x,y,z,ω) is treated in the same manner as was employed ˜ for UH1(x,y,z0,ω) in Sect. G.3.2. The change of integration variables given by p = sin α cos β, q = sin α sin β that was used in the proof of Theorem G.1 is first made, resulting in the integral [cf. Eq. (G.44)] β0+2π π/2 ˜ ikRf(α,β) UH1(x,y,z,ω)= A (α, β)e dαdβ, (G.71)

β0 α1 with phase function [cf. Eq. (G.46)]

f(α,β)= sin ϑ sin α cos (β − ϕ) + cos ϑ cos α. (G.72)

The location of the critical points for this phase function are the same as those depicted in Fig. G.4. As in that case, the points a and b are the only critical points that contribute to the asymptotic behavior of the integral under consideration. However, unlike that case, the saddle points here are not ordinary stationary phase points where both ∂f /∂ α and ∂f /∂β vanish, but rather they are boundary stationary phase points where the isotimic contours f(α,β) = constant are tangent to the boundary of the integration domain; in the present case they are both points on the boundary line α = π/2 where ∂f /∂β = 0. The resultant asymptotic expansion is then found ˜ ˜ G.5 Asymptotic Approximations of UH (r,ω)and UE(r,ω) 717 to be [4, 10, 11] √ 2π/sin ϑeiπ/4 ˜ ik(x0ξ +y0ξ ) ikR sin ϑ UH (x,y,z,ω) = V(ξ ,ξ , 0)e 1 2 e 1 (kR)3/2 cos ϑ 1 2 − + − − + − − ik(x0ξ1 y0ξ2) ikR sin ϑ + O 2 iV( ξ1, ξ2, 0)e e (kR) (G.73) as kR →∞with ﬁxed k>0 and N ≥ 8, where

= = ξ1 ξ1/ sin ϑ cos ϕ, (G.74) = = ξ2 ξ2/ sin ϑ sin ϕ. (G.75) ˜ ˜ The resultant asymptotic expansions for UH (r,ω)and UE(r,ω)are then given by (for N ≥ 8)

ikR e + + U˜ (x,y,z,ω) =−2πi V(ξ ,ξ ,ξ )eik(ξ1x0 ξ2y0 ξ3z0) H kR 1 2 3 ˜ −2 +UH1(x,y,z,ω)+ O (kR) , (G.76) ˜ ˜ −2 UE(x,y,z,ω) =−UH1(x,y,z,ω)+ O (kR) , (G.77) as kR →∞with fixed ξ1, ξ2, and k>0. ˜ These results then show that the evanescent wave contribution UE(r,ω) is of −1 ˜ higher order in (kR) than the homogeneous wave contribution UH (r,ω).This ˜ then provides a rigorous justification for neglecting UE(r,ω) in comparison to ˜ UH (r,ω) as kR →∞with fixed k>0 when 0 <ξ3 < 1. In addition, ˜ −1 the evanescent wave contribution UE(r,ω) is of even higher order in (kR) in ˜ comparison to the homogeneous wave contribution UH (r,ω) as kR →∞in the special case when V(p, q, 0) = 0.

G.5.2 Approximations Valid on the Plane z = z0

The asymptotic behavior of the spectral wave field on the plane z = z0 (or, equivalently, when ξ3 = 0) is directly obtained from the analysis presented in Sect. G.3.2, the present analysis focusing on obtaining explicit expressions for the dominant terms in the separate homogeneous and evanescent wave contributions. The asymptotic expansion of the homogeneous spectral wave contribution ˜ UH (x,y,z0,ω) is obtained from Eqs. (G.39), (G.42), and (G.49)–(G.52), taken j1 j2 together with the series expressions [4] for the coefficients BHn(ϕ) and BHn(ϕ), 718 G Stationary Phase Approximations with the result (for N ≥ 8) iπ + U˜ (x,y,z ,ω) =− V(ξ ,ξ , 0)eikReik(x0ξ1 y0ξ2) H 0 kR 1 2 −ikR −ik(x0ξ1+y0ξ2) −3/2 −V(−ξ1, −ξ2, 0)e e + O (kR) (G.78) as kR →∞with fixed ξ1, ξ2, and k>0. Similarly, the asymptotic expansion ˜ of the evanescent spectral wave contribution UE(x,y,z0,ω) is obtained from Eqs. (G.39), (G.43), (G.60), (G.64), and (G.66) with the result (for N ≥ 8) iπ + U˜ (x,y,z ,ω) =− V(ξ ,ξ , 0)eikReik(x0ξ1 y0ξ2) E 0 kR 1 2 −ikR −ik(x0ξ1+y0ξ2) −3/2 +V(−ξ1, −ξ2, 0)e e + O (kR) (G.79) as kR →∞with fixed ξ1, ξ2, and k>0. These two expressions then show that the homogeneous and evanescent wave contributions are of the same order in (kR)−1 as kR →∞. As a consequence, the evanescent wave contribution ˜ UE(x,y,z0,ω)cannot, in general, be neglected in comparison to the homogeneous ˜ wave contribution UH (x,y,z0,ω)on the plane z = z0.

G.6 Approximations Valid on the Line x = x0, y = y0

Along the line x = x0, y = y0, one has that ξ3 = 0 and the homogeneous spectral wave contribution can be written as 2π π/2 ˜ ikzcos α UH (x0,y0,z,ω)= dβ dα A(α, β)e , (G.80) 0 0 where

+ + A(α, β) = V(p,q,m)eik(px0 qy0 mz0) sin α, (G.81) with p = sin α cos β, q = sin α sin β, and m = 1 − p2 − q2 = cos α [see the proof of Theorem G.1]. Because the phase function in Eq. (G.80) does not involve the integration variable β, the asymptotic behavior of the integral can be obtained from the single integral π/2 ˜ ikR cos α UH (x0,y0,z,ω)= A(α)e dα, (G.82) 0 G.7 Summary 719 where R = z − z0, with 2π A(α) ≡ eikz0 cos α A(α, β)dβ. (G.83) 0

Application of the method of stationary phase for single integrals in Sect. G.1 shows that the only contributions to the asymptotic behavior of the integral in Eq. (G.82) arise from the endpoints of the integral at α = 0 and α = π/2 with the result [see Eq. (G.17)]

ikR e i − U˜ (x ,y ,z,ω)=−2πi V(0, 0, 1)eikz0 + A(π/2) + O (kR) 3/2 H 0 0 kR kR (G.84) as kR →∞with k>0 and for N ≥ 4. Because the ﬁrst term in Eq. (G.84) is the dominant term in the asymptotic ˜ approximation of U(x0,y0,z,ω),Eq.(G.1) then shows that the asymptotic behavior of the evanescent wave contribution is given by i − U˜ (x ,y ,z,ω)=− A(π/2) + O (kR) 3/2 (G.85) E 0 0 kR as kR →∞with k>0. The homogeneous and evanescent wave contributions are again seen to be of the same order in (kR)−1 as kR →∞so that, in general, ˜ ˜ UE(x0,y0,z,ω) cannot be neglected in comparison to UH (x0,y0,z,ω) for large kR →∞.

G.7 Summary

The stationary phase asymptotic expansions presented in this appendix are all valid as kR →∞with fixed wavenumber k>0 and fixed direction cosines ξ1 = (x − x0)/R, ξ2 = (y − y0)/R with N ≥ 12 in a nonabsorptive (and hence, strictly speaking, nondispersive) medium. Less restrictive conditions on N result in special cases. The asymptotic behavior of the spectral wave field U˜ (r,ω) is the same for all ξ3 = (z − z0)/R such that 0 ≤ ξ3 ≤ 1 and is given by [cf. Eq. (G.70)] ikR e i + + U˜ (r,ω) =−2πi 1 + L2 V(ξ ,ξ ,ξ )eik(ξ1x0 ξ2y0 ξ3z0) kR 2kR 1 2 3 − +O (kR) 3 (G.86) 720 G Stationary Phase Approximations as kR →∞with fixed k>0, where L2 is the differential operator defined in Eq. (G.32). The asymptotic behavior of the separate homogeneous and evanescent com- ˜ ponent wave fields UJ (r,ω), J = H,E, however, depends on the value of ξ3, separating into the three cases (a) 0 <ξ3 < 1, (b) ξ3 = 0, and (c) ξ3 = 1, as described in Sect. G.5. The results show that the evanescent wave contribution ˜ UE(r,ω) is negligible in comparison to the homogeneous wave contribution ˜ UH (r,ω)for large kR →∞in case (a), but not necessarily in cases (b) and (c). In some important applications of the angular spectrum representation in electromagnetic wave theory, integral representations of the form given in Eqs. (G.1) ˜ and (G.2) are obtained with U(p,q,ω)∈ / TN because of the presence of isolated singularities in the integrand. This occurs, for example, in the analysis of the reflection and refraction of a nonplanar wave field (e.g., an electromagnetic beam field) at a planar interface separating two different media [13, 17]. A neutralizer ˜ function can then be used to isolate each singularity. Because U(p,q,ω)∈ TN ,the asymptotic approximation of the resultant integral that does not contain any of the singularities can then be obtained using the two-dimensional stationary phase results of Sherman, Stamnes, and Lalor [4] presented here. Each of the remaining integrals ˜ contains one of the isolated singularities, U(p,q,ω)∈ / TN and so its asymptotic approximation must be obtained using some other technique. In some case, as in the reflection and refraction problem [17], a change of integration variable can result in a transformed integral that is amenable to the stationary phase method presented here. If that is not the case, uniform asymptotic methods [18–20] may then need to be employed.

References

1. C. J. Bouwkamp, “Diffraction theory,” Rept. Prog. Phys., vol. 17, pp. 35–100, 1954. 2. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am., vol. 58, pp. 1235–1237, 1968. 3. A. Sommerfeld, Optics,vol.IVofLectures in Theoretical Physics. New York: Academic, 1964. paperback edition. 4. G. C. Sherman, J. J. Stamnes, and É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys., vol. 17, no. 5, pp. 760–776, 1976. 5. G. G. Stokes, “On the discontinuity of arbitrary constants which appear in divergent develop- ments,” Trans. Camb. Phil. Soc., vol. X, pp. 106–128. 6. L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., vol. XLII, p. 80, 1887. 7. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 9.41. 9. J. Focke, “Asymptotische Entwicklungen mittels der Methode der stationären phase,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, vol. 101, no. 3, pp. 1–48, 1954. 10. G. Braun, “Zur Methode der stationären Phase,” Acta Phys. Austriaca, vol. 10, pp. 8–33, 1956. References 721

11. D. S. Jones and M. Kline, “Asymptotic expansion of multiple integrals and the method of stationary phase,” J. Math. Phys., vol. 37, pp. 1–28, 1958. 12. N. Chako, “Asymptotic expansions of double and multiple integrals occurring in diffraction theory,” J. Inst. Math. Appl., vol. 1, no. 4, pp. 372–422, 1965. 13. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon, 1966. Sect. 2.12. 14. H. Bremermann, Distributions, Complex Variables, and Fourier Transforms. Reading, MA: Addison-Wesley, 1965. Ch. 8. 15. A. Erdélyi, “Asymptotic representations of Fourier integrals and the method of stationary phase,” SIAM J. Appl. Math., vol. 3, pp. 17–27, 1955. 16. G. C. Sherman, “Recursion relations for coefficients in asymptotic expansions of wavefields,” Radio Science, vol. 8, pp. 811–812, 1973. 17. J. Gasper, “Reflection and refraction of an arbitrary wave incident on a planar interface,” M.S. dissertation, The Institute of Optics, University of Rochester, Rochester, NY, 1972. 18. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure and Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 19. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967. 20. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1975. Appendix H The Radon Transform

The Radon transform, named after the Czech mathematician Johann Radon (1887– 1956) has its origin in his 1919 paper [1] (which then led to the Riemann– Lebesgueým theorem). For the purpose of this brief development, it is best introduced in connection with the projection-slice theorem in Fourier analysis in the following manner. Consider a two-dimensional scalar (object) function O(r) = O(x,y), which may be expressed as O(R) = O(r)δ(R − r)d2r, (H.1) D where D is any region containing the point R, and where δ(r) is the two-dimensional Dirac delta function which has the Fourier integral representation

∞ 1 · − δ(R − r) = eik (R r)d2k. (H.2) 4π 2 −∞

Let nˆ be the unit vector in the direction of the vector k so that k = knˆ. With this substitution, Eq. (H.2) becomes

∞ 1 · − ·ˆ δ(R − r) = d2keik Re ikr n, 4π 2 −∞ and since ∞ − ·ˆ − e ikr n = dx δ(x − r · nˆ)e ikx, −∞

© Springer Nature Switzerland AG 2019 723 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 224, https://doi.org/10.1007/978-3-030-20835-6 724 H The Radon Transform one then obtains the expression

∞ ∞ 1 · − δ(R − r) = d2keik R dx δ(x − r · nˆ)e ikx. (H.3) 4π 2 −∞ −∞

Substitution of this result into Eq. (H.1) then gives

∞ ∞ 1 · − O(R) = d2keik R dx P(nˆ; x)e ikx, (H.4) 4π 2 −∞ −∞ where P(nˆ; x) ≡ O(r)δ(x − r · nˆ)d2r (H.5) D denotes the projection of the object function O(r) onto the direction deﬁned by the unit vector nˆ. The variable x appearing in this expression is now interpreted as the real variable deﬁning the location in the nˆ-direction at which the rectilinear line integral through the object function O(r) is taken. As an illustration, the vertical line integral through the object function depicted in Fig. H.1 is given by

O(x,y )dy = O(x ,y )δ(x − x )dx dy D = O(r)δ(x − r · nˆ)d2r, D which is precisely the expression deﬁned in Eq. (H.5).

Fig. H.1 Geometry of the vertical line integral through the object function O(r) H The Radon Transform 725

The two-dimensional Radon transform of a (sufﬁciently well-behaved) function O(r) = O(x,y) is then deﬁned as [2] R[O](nˆ; x) ≡ O(r)δ(x − r · nˆ)d2r. (H.6) D

With use of the notation (see Appendix E)

∞ −ikx Fk {f(x)} = f(x)e dx, (H.7) −∞ ∞ F−1 ˜ = 1 ˜ ikx x f(k) f(k)e dk, (H.8) 2π −∞ for the Fourier transform and its inverse, respectively, where x and k are referred to as Fourier conjugate variables, then as a special case of the Fourier integral theorem F−1 ◦ F = [3], x k 1, that is, the subsequent application of the forward and inverse Fourier transforms (on a sufﬁciently well-behaved class of functions) results in the identity operator. Eq. (H.4) may then be expressed as = F−1 ◦ F ˆ; O(R) k k P(n x) , (H.9) so that Fk {O(r)} = Fk P(nˆ; x) . (H.10)

This important result is known as the projection slice theorem (or central slice theorem) [4]. In its N-dimensional generalization, this theorem relates the (N − 1)-dimensional Fourier transforms of the projections of an object function onto hyperplanes to the N-dimensional Fourier transform of that original function. Simply stated, this theorem states that the (N − 1)-dimensional Fourier transform of a given projection of the function is equal to a “slice” through the N-dimensional Fourier transform of that function. The Fourier integral representation of the delta function given in Eq. (H.2) involves a vector k which may be expressed in polar coordinate form as being the vector with magnitude k ∈[0, ∞) and with direction speciﬁed by a unit vector nˆ which is at an angle θ ∈[0, 2π) with respect to some ﬁxed reference direction; one then has that ∞ 2π ∞ d2k = dθ kdk. −∞ 0 0

However, it is equally valid to regard the integration variable in Eq. (H.2) as being a vector κ with “magnitude” κ ∈ (−∞, ∞) and direction specified by the unit vector Nˆ which is at an angle φ ∈[0,π)with respect to the reference direction; in this case 726 H The Radon Transform one has that ∞ π ∞ d2k = dφ |κ|dκ, −∞ 0 −∞ where the absolute value of κ must be employed since the differential element of area d2k must always be nonnegative. Just as k = knˆ, one also has that κ = κNˆ . There is then a one-to-one correspondence between the points in k-space and the points in κ-space. In addition, Nˆ may be treated as a conventional unit vector with its range of directions (with respect to some fixed reference direction) appropriately restricted. The identity appearing in Eq. (H.3) may then be expressed in κ-space as π ∞ ∞ 1 · ˆ − δ(R − r) = dφ dκ dx |κ|ei(R N x)δ(x − R · Nˆ ). 2 (H.11) 4π 0 −∞ −∞ Substitution of this expression into Eq. (H.1) and using the definition of the projection of the object function onto the direction specified by the unit vector Nˆ given in Eq. (H.5) then results in the representation of the object function as π ∞ 1 · ˆ O(R) = dφ dκ |κ|eiκR NF P(Nˆ ; x) . 2 k (H.12) 4π 0 −∞ This result then yields the well-known filtered backprojection algorithm for object reconstruction [5]. It is said to be “filtered” because of the quantity |κ| appearing in the integrand of Eq. (H.12). The inverse of the two-dimensional Radon transformation given in Eq. (H.6)may be obtained in the following manner [2]. Since |k|=k sgn(k), where sgn(k) =+1 if k>0, sgn(k) = 0ifk = 0, and sgn(k) =−1ifk<0 is the signum function , then the representation of the delta function given in Eq. (H.11) may be rewritten as

∞ ∞ 1 π ˆ R−r = i(R·N−x) −R·Nˆ δ( ) 2 dφ dκ dx κ sgn(κ)e δ(x ). (H.13) 4π 0 −∞ −∞

Since [see Eq. (B.14)ofVol.1]

f(x)δ(x − a)dx =− f (x)δ(x − a)dx, where the prime denotes differentiation with respect to the variable x, then with ˆ ˆ f (x) = κ sgn(κ)ei(R·N−x) so that f(x) = i sgn(κ)ei(R·N−x), the ﬁnal integral in Eq. (H.13) becomes · ˆ − κ sgn(κ)ei(R N x)δ(x − R · Nˆ )dx · ˆ − =−i sgn(κ)δ (x − R · Nˆ )ei(R N x)dx, References 727 and hence π ∞ ∞ i · ˆ − δ(R−r) =− dφ dκ dx (κ)δ (x−R·Nˆ )ei(R N x). 2 sgn (H.14) 4π 0 −∞ −∞

With the identity [2] f(x) i ∞ P dx =− dx dk sgn(k)eikxf(x), (H.15) x 2 −∞ where P indicates that the Cauchy principal value of the integral is to be taken, Eq. (H.14) becomes 2 π ∞ δ (x − R · Nˆ ) δ(R − r) =− P dφ dx . (H.16) 2 ˆ π 0 −∞ R · N − x

Substitution of this expression in Eq. (H.1) then yields 1 π ∞ P (Nˆ ; x) O(r) = P dφ dx , (H.17) 2 ˆ 2π 0 −∞ R · N − x which is known as the inverse Radon transform relationship, where

P (Nˆ ; x) = O(r)δ (x − R · Nˆ )d2r (H.18) D is the derivative of the projection with respect to the variable x.

References

1. J. Radon, “Über lineare Funktionaltransformationen und Funktionalgleichungen,” Acad. Wiss. Wien, vol. 128, pp. 1083–1121, 1919. 2. H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics (E. Wolf, ed.), vol. XXI, pp. 217–286, Amsterdam: North-Holland, 1984. 3. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 4. R. M. Mersereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE, vol. 62, pp. 1319–1338, 1974. 5. A. C. Kak, “Computerized tomography with X-ray, emission, and ultrasound sources,” Proc. IEEE, vol. 67, pp. 1245–1272, 1979. Index

A B Aberration of light, 85 Backprojection, 602 Abscissa of absolute convergence, 434, 673 Barash, Y. S., 287 Absorption depth zd , 586 Baños, A., 480 Acceleration field, 140 Beer’s law, 21 Advanced potentials, 127 Bertilone, D. C., 555, 558 Ampère’s law, 66 Bianisotropic, 200 Analytic signals, 628 Bi-isotropic (BI), 201 Angular spectrum of plane waves Boltzmann’s constant KB , 218 representation, 484 Born, M., 16, 529 Angular spectrum representation Boundary condition freely propagating wave field, 513, 550 normal, 308 geometric form, 522 tangential, 309, 310 of radiation field, 452 Bradyon, 82 radiated wave field, 484 Brewster’s angle, 356, 383, 400 Anisotropic medium Brewster’s law, 357 principal axes, 277 Brillouin, L., 5 Anisotropy, 198 Brillouin precursor, 15 biaxial, 199 uniaxial, 199 C Antenna pattern Cauchy principal value, 202 IEEE definition, 502 Causality Area of a pulse, 20 dispersion relations, 202 m Associated Legendre polynomial P (u), 485 Kramers–Kronig relations, 205 Asymptotic expansions Plemelj formulae, 202 method of stationary phase, 695 primitive, 196, 201 Asymptotic heat, 681 relativistic, 201, 304 Atomic polarizability Centrovelocity, 30 parallel component, 668 Charge density perpendicular component, 668 macroscopic (r,t), 186 Attenuation factor, 9 macroscopic bound b(r,t), 187 Auxiliary field vectors, 60 macroscopic, conductive, 444 Axial vector, 278 macroscopic free f (r,t), 187

Charge density (cont.) macroscopic free Jf (r,t), 191 macroscopic, nonconductive, 444 macroscopic J(r,t), 191 microscopic ρ(r,t),51 microscopic “bound” jb(r,t), 188 microscopic bound ρb(r,t), 184 microscopic free jf (r,t), 188 microscopic free ρf (r,t), 184 solenoidal or transverse jt (r,t), 122 surface, 308 surface, 310 Chirality, 201 Current source J0(r,t), 427 Clausius–Mossotti relation, 217, 408 Cole–Cole extended Rocard–Powles–Debye D model De Broglie wavelength, 80 triply distilled water, 228 Debye model Cole–Cole plot, 220 causality, 219 Complex dielectric permittivity, 207 effective relaxation time, 218 Complex dielectric tensor ˜(k,ω), 272 of orientational (dipolar) polarization, 217 Complex index of refraction n(ω), 231, 251, permittivity, 218 451 relaxation equation, 217 Complex intrinsic impedance η(ω), 490 Rocard–Powles extension, 222 Complex permittivity, 251 and rotational Brownian motion, 217, 222 Complex permittivity tensor, 430 susceptibility, 218 Complex phase function φ(ω,θ),11 Devaney, A. J., 454, 480, 559 Complex phase velocity, 409 Devaney–Wolf representation, 483 Complex velocity, 432 Diamagnetic, 211 Complex wave number, 9 Diamagnetic effect, 668 Complex wave vector, 451 Dielectric permittivity Conjugate electromagnetic field, 105 complex c(ω), 207 Conservation laws composite model, 235, 239 electromagnetic energy, 97, 281 free space 0,54 electromagnetic momentum of sea-water, 260 angular, 103 temporally dispersive (ω), 203 linear, 99 of triply distilled water, 213 energy Dilation factor γ ,73 local form, 94 Dipolar relaxation time τm, 217 relativistic momentum, 78 Dipole Conservation of charge Hertzian, elemental, 134 microscopic, 53 linear electric, 133 Constitutive relations, 196 moment, electric, 158 general form, 196 Dipole oscillator induction fields, 196 electric dipole contribution, 153 and linearity, 197 electric monopole contribution, 153 primitive fields, 196 magnetic dipole contribution, 153 Convective derivative, 64, 75 Dipole radiation Convolution intermediate zone, 161 temporal, 603 static zone, 159 theorem, Laplace transform, 674 wave zone, 163 Cophasal surfaces, 170 Dirac delta function δ(ξ), 133, 645 Courant condition, 26 composite function theorem, 647 Critical angle Θc, 343, 357, 372, 393 derivatives of, 649 Current density extension to higher dimensions, 652 conduction, 188 sifting property, 646 convective (microscopic) j(r,t),52 similarity relationship (scaling law), 647 effective, 316 Direction cosines, plane wave irrotational or longitudinal j(r,t), 122 complex, 515 macroscopic bound Jb(r,t), 191 real, 545 Index 731

Dispersion Electromagnetic bullets, 597 anomalous, 234 Electromagnetic energy normal, 233 density, 95, 280 Dispersion relations irreversible, 682 dielectric permittivity, 205 macroscopic electric conductivity, 208 density, total U(r,t), 280 electric susceptibility, 205 microscopic magnetic permeability, 212 density u(r,t),95 magnetic susceptibility, 212 total U(t),95 Dispersive interface, 21 reversible, 682 Dissipation, 285 Electromagnetic field, 55 DNG medium, 263 Bateman’s definition, 104 Doppler effect, 86 conjugate, 105 Double negative (DNG) medium, 263 Electromagnetic linear momentum Drude model microscopic density pem(r,t),99 causality, 241 microscopic total pem,99 damping constant γ , 241 Electromagnetic momentum density dielectric permittivity, 241 macroscopic, 326 electric conductivity σ(ω), 259 Electromagnetic wave static conductivity σ0, 259 freely propagating, 509 Dynamical free energy, 678 freely propagating boundary conditions, 510 plane, time-harmonic, 254 E source-free, 509 Effective electric field, 216, 665 source-free initial conditions, 611 Effective field, 661 Electrostatic field, 55 spatially averaged, 664 Energy density Effective magnetic field, 665 electromagnetic field Uem, 302 Eikonal S, 269 reversible, 302 Eikonal equation, 269 Energy transport velocity, 29, 305 generalized, 171 Energy velocity description geometrical optics, 171 Sherman and Oughstun, 31 Einstein, A., 1 Equation of continuity mass–energy relation, 82 conduction current, 248 special theory of relativity, 70 macroscopic, 194 Einstein mass–energy relation, 79 microscopic, 54 Electric conductivity σ(ω), 207 Erdélyi, A., 486 Electric dipole approximation, 154 Evanescent field, 352 Electric displacement vector Evolved heat (dissipation) Q(r,t), 285 macroscopic D(r,t), 193 Ewald–Oseen extinction theorem, 408 microscopic d(r,t),60 Extensive variable, 96 Electric energy density Extraordinary refractive index, 277 time-harmonic, 289 Electric field vector macroscopic E(r,t), 183 F microscopic e(r,t),54 Faraday’s law Electric moment, 129 microscopic, 62 n Electric 2 -pole, 153 Filtered backprojection algorithm, 730 Electric susceptibility χe(ω), 204 Forerunner, 5 Electromagnetic angular momentum Fourier conjugate variables, 729 microscopic density lem, 103 Fourier integral microscopic total lem, 103 asymptotic approximation, 697 Electromagnetic beam field, 560 theorem, 651 separable, 575 Fourier transform, 729 732 Index

Fourier transform Fω {f(r,t)}, 673 Green’s function Fröhlich distribution function, 226 free-space, 526 Free-field, 611 Group method Laplace–Fourier integral representation, Havelock,T.H.,13 615 Group velocity, 7, 10 plane wave expansion, 616–631 Hamilton, Sir W. R., 7 Freely propagating wave field, 509 Rayleigh, Lord, 7 Fresnel approximation, 581 Stokes, G. G., 7 Fresnel drag coefficient, 196 Group velocity approximation, 14 Fresnel–Kirchhoff diffraction integral, 581 Eckart, C., 15 Fresnel reflection and transmission coefficients, Lighthill, M. J., 15 352, 368, 380 Whitham, G. B., 15 Group velocity method, 14 Gyrotropic media, 273 G Gabor cells, 26 Galilean invariance, 65, 67 H Galilean transformation Half-space, positive and negative, 450 electric field intensity, 65 Hall effect, 248 magnetic field intensity, 67 Hamilton–Jacobi ray theory, 15 Gauge Heaviside–Poynting theorem Coulomb, 121 differential form, 280 function, 119, 439 integral form, 280 invariance, 119, 439 time-harmonic form, 289 Lorenz, 121, 440 Heaviside unit step function U(ξ), 650 radiation, 121 Helmholtz equation, 9, 251, 560 transformation, 119, 131, 439 inhomogeneous, 266 transformation, restricted, 121, 439 Helmholtz free energy (work function), 678 transverse, 121 Helmholtz’ theorem, 659 Gaussian beam, 586 Hertz beam waist, 590 potential, 129 divergence angle, 592 vector, 129 Rayleigh range, 593 Hertz, H., 128, 159 spot size, 590 Hilbert transform, 202, 205 Gaussian pulse dynamics Homogeneous, isotropic, locally linear (HILL) Balictsis and Oughstun, 31 temporally dispersive media Garrett and McCumber, 30 constitutive relations, 248 Tanaka, Fujiwara, and Ikegami, 31 Huygens–Fresnel principle, 581 Gaussian units Huygen’s principle, 527 electrostatic unit (esu) of electric field intensity, 61 gauss, 61 I maxwell, 62 Idemfactor, 100, 112, 327 statampere, 61 Impulse response function statcoulomb, 61 spatial, 524, 553 statvolt, 61 Incomplete Lipschitz–Hankel integrals Gauss’ law Dvorak, S. and Dudley, D., 33 microscopic Indeterminacy principle, 26 electric field, 68 Inertial reference frames, 69 magnetic field, 68 Information diagram, 26 Generalized potential, 174 Initial field values Geometrical optics limit, 267, 579 in a closed convex surface, 636 Ginzburg, V. L., 287 in a sphere, 632 Goos–Hänchen effect, 21 Instantaneous (causal) spectrum, 678 Index 733

Intensive variable, 96 Lorentz, H. A., 1 Interaction energy, 677 Lorentz covariant, 71 Intrinsic impedance Lorentz–Fitzgerald contraction, 76 complex, 255 Lorentz force relation, 56 complex η(ω), 496 Lorentz invariant, 71 free space, 255 Lorentz–Lorenz formula, 217 of free space η0, 596 Lorentz–Lorenz relation, 408 good conductor, 259 Lorentz model near-ideal dielectric, 258 atomic polarizability, 229 Inverse problems, 28 Lorentz–Lorenz modified, 230 Inverse source problem magnetic field contribution, 228 nonuniqueness, 601 oscillator strength, 235 normal solution, 602 phenomenological damping constant δj , time-dependent, 595 228 Isoplanatic, see Space-invariant resonance frequency, undamped ωj , 228 Isotropy, 198 of resonance polarization, 228 sum rule, 235 Lorentz theory, 51 J Lorenz condition, 120, 439 Jackson, J. D., 16 generalized, 441, 443 Jordan’s lemma, 634 Loudon, R., 29, 304 Luminiferous ether, 69, 110 Luxon, 80, 82 K Kelvin functions, 320 Kramers–Kronig relations, 205 M Kronecker-delta function δij , 100 Macroscopic polarization density P(r,t), 187 Macroscopic quadrupole moment density tensor Q (r,t), 187 L Magnetic energy density Lagrange’s equations time-harmonic, 290 charged particle, 174 Magnetic field vector Lagrange’s theorem, 150 macroscopic B(r,t), 183 Lagrangian density L, 174 microscopic b(r,t),54 Lagrangian L, 174 Magnetic flux, 62 Lalor, É., 525, 559 Magnetic intensity vector Landau, L. D., 285, 681 macroscopic H(r,t), 193 Laplace transform L{f(r,t)}, 671 microscopic h(r,t),60 Law of reflection, 337, 342 Magnetic permeability complex media, 367 composite model, 241 Law of refraction dispersive μ(ω), 209 complex media, 367 free space μ0,54 n− perfect dielectrics, 343 Magnetic 2 1-pole, 153 Legendre polynomials P(u), 485 Magnetic susceptibility χm(ω), 210 Rodriques’ formula, 485 Magnetization, macroscopic M(r,t), 191 Liénard–Wiechert potentials, 138 Magnetohydrodynamic waves, 15 and special relativity, 144 Magneto-optical susceptibility, 273 Lifshitz, E. M., 285, 681 Magnetostatic field, 55 Light rays Masking function, 599 differential equations, 270 Material relations, see Constitutive relations Local energy theorem, 95 Material response Local field, see Effective field anisotropic, 198 Lommel functions, 555 isotropic, 198 Lord Kelvin, 695 nondispersive, 198 734 Index

Material response (cont.) Multipole expansion spatially dispersive, 198 electromagnetic wave field, 495 spatially homogeneous, 197 scalar wave field, 486 spatially inhomogeneous, 197 Multipole moments, 485, 486, 495 temporally dispersive, 198 temporally homogeneous, 197 temporally inhomogeneous, 197 N Material response tensors Near-ideal dielectrics, 257 dielectric permittivity ˆ(r,t, r,t), 196 Negative refraction, 393 electric conductivity σˆ (r,t, r,t), 196 Neutralizer function ν(p,q), 700 magnetic permeability μˆ (r,t, r,t), 196 Newton’s second law of motion Mature dispersion regime, 6, 31 relativistic form, 78 Maxwell, J. C., 1 Nisbet, A., 527, 539 Maxwell’s equations Nonlinear optical effects, 668 Bateman–Cunningham form, 596 Nonradiating sources, 600 frequency-domain form, 481 Normalized velocity, 137 homogeneous, 595 Number density Nj , 229 macroscopic Numerical techniques temporal frequency-domain form, 250 discrete Fourier transform method, 24 time-domain differential form, 193, 249 fast Fourier transform method, 24 microscopic, 54 finite-difference time-domain method, 25 time-domain differential form, 60 Hosono’s Laplace transform method, 24 time-domain integral form, 69 phasor form, 253 source-free, 612 O spatial average, 183 Ohmic power loss, 317 spatio-temporal form, 432 Olver’s saddle point method, 32 spatio-temporal frequency domain form, Onsager relation, 274 614 Optical wave field, 169 Maxwell stress tensor Orbital angular momentun operator Ls , 492 macroscopic, 327 Order symbol O,79 microscopic, 100 Ordinary refractive index, 277 Mean polarizability αj (ω), 216 Orthogonality relations Mean square charge radius, 187 spectral amplitude vectors, 491 Metallic waveguides, 16 Ott’s integral representation, 480 Metamaterial Veselago, 263 Microscopic Maxwell–Lorentz theory, 54 P Minkowski formulation, 195 Panofsky, W, 138 Minkowski stress tensor, 327 Parallel polarization, 336 Mksa units Paramagnetic, 211 ampere, 61 Paraxial approximation, 588 coulomb, 61 quadratic phase dispersion, 581 farad, 62 Penetration depth dp, 259, 313 volt, 61 Perpendicular polarization, 336 weber, 62 Phase-space asymptotic description, 23 Modern asymptotic description, 32 Phase velocity, 8, 267 Molecular magnetic moment, 190 good conductor, 259 Molecular multipole moments, 185 near-ideal dielectric, 258 Molecular polarizability α(ω), 661 Phasor representation, 252 Momentum density Phillips, M., 138 electromagnetic, 326 Photon, 80 Monochromatic (time-harmonic) wave field, Planck time tP , 237 170 Plane of incidence, 342 Index 735

Plane wave P polarization, 336 attenuation factor α(ω), 255, 515 Precursor, 5, 14 evanescent, 520, 548, 552 Brillouin, 15 homogeneous, 453, 520, 548, 552 formation on transmission, 21 inhomogeneous, 453, 520, 548 observation of, 36 propagation factor β(ω), 255, 515 Pleshko and Palócz, 15 Plane wave expansion Sommerfeld, 15 Devaney, A. J., 624 Principal dielectric permittivities, 277 and mode expansions, 627 Principal indices of refraction, 277 and Poisson’s solution, 630 Principle of superposition, 58 polar coordinate form, 627, 630 Project ELF, 263 uniqueness, 624 Projection slice theorem, 729 Plasma frequency bj , 229 Project Sanguine, 263 Plasma oscillations, 241 Propagation factor, 9 Poincaré-Lorentz transformation, 68 Propagation kernel, 580 Poincaré–Lorentz transformation relations, Propagation tensor, 431 73–111 Pseudo-Brewster angle Θpb, 384 coordinate transformation matrix, 112 Pseudoscalar, 278 electric field transformation, 90 Pseudovector, 278 force transformation, 84 Pulse area θ(z),20 invariance of Maxwell’s equations, 93 Pulse centroid velocity, 36 magnetic field transformation, 90 Pulse diffraction, 34 mass transformation, 84 Pulse distortion, 22 Poisson’s equation scalar potential, 121 Q Poisson’s solution, 641 Quadratic dispersion relation, 23 Polarization ellipse, 530, 536 left-handed, 533 R right-handed, 533 Radiated energy, 145 state, 536 Radiation Polarized field electromagnetic, 128 circular, 537 reaction, 57 linear, 537 Radiation field uniform, 536, 540 Fourier–Laplace integral representation, Polarizing angle, 356, 383, 384, 400 434, 436 Polar vector, 278 scalar potential, 438 Ponderable body, 50 vector potential, 437 Ponderomotive, 50 Radiation pattern, 489, 498 Potential functions, electromagnetic filtered, 602 complex, 166 frequency-domain, 600 Liénard–Wiechert, 135 IEEE definition, 502 macroscopic, 437, 444 scalar wave field, 489 microscopic, 117–119 time-domain, 598 retarded, 124 Radon transform, 599, 727, 729 Poynting–Heaviside interpretation, 97 inverse, 731 Poynting’s theorem, Poynting–Heaviside Rayleigh–Sommerfeld diffraction integrals, theorem, 96 527 Poynting vector Ray techniques complex, 550 direct-ray method, 16 complex S˜(r), 289 Felsen, L. B., 16 macroscopic S(r,t), 279 Heyman, E., 16 microscopic s(r,t),95 Melamed, T., 16 time-average, 551 space–time ray theory, 16 736 Index

Reflectivity R, 351 Signum function, 730 Refraction Simple magnetizable medium, 210 negative, 393 Simple polarizable dielectric, 204 normal, 367 Simultaneity, 71 Refractive index Skin depth, 313, 320 complex, 231 Skin effect, 319 function, modified, 171 Skin resistivity, 323 Relativity Slowly evolving wave approach, 35 Newtonian, 69 Slowly varying envelope approximation, 16, 34 special theory, 69 Snell–Descarte law, 353 Relaxation times Snellius, Willebrord, 353 dipolar, 217 Snell’s law, 343, 353 distribution of, 225 Sommerfeld, A., 5 Drude model, mean-free path τc, 241 Sommerfeld precursor, 15 effective, 218, 224 Sommerfeld radiation condition, 694 Retardation condition, 140 Sommerfeld’s integral representation, 478 Retarded potentials, 127 Source-free wave field, 559, 564 Retarded time, 124 Sherman expansion, 568 Riemann’s proof, 125 Source function, 599 Righi, A., 128 Space-invariant, 524 Rocard–Powles–Debye model Spatial average Cole–Cole extension, 227 of a microscopic function, 182 friction time τmf , 222 weighting function, 182 permittivity, 222 Spatial dispersion effects, 33 susceptibility, 222 Spatially inhomogeneous media, 266 of triply distilled water, 225 Spatially locally linear, 197 Spatial transfer function, 524 S Spatiotemporal Fourier–Laplace transform, Sampling theorem, 28 674 Scalar dipole field, 605 Special theory of relativity Scalar potential fundamental postulate, 71 macroscopic, 438 longitudinal mass, 81 microscopic, 118 Lorentz–Fitzgerald contraction, 75 Semiconductor, 258 mass–energy, 82 Sherman, G. C., 37, 454, 523, 559, 560 postulate of the constancy of the speed of Sherman’s expansion, 568, 578 light, 71 See also source-free wave field, 568 proper differential time interval, 75 Sherman’s recursion formula, 711 relativistic mass, 78 SI (Système Internationale) units, see Mksa rest energy, 79 units rest mass (proper mass) m0,78 Signal time dilation, 74 arrival, 11, 15, 19 transverse mass, 82 buildup, 20 Spectral amplitude vectors velocity, 6 orthogonality relations, 491 Signal velocity, 6, 10 Spectral distribution function, 552 Baerwald, H., 14 Speed of light in vacuum c,54 Brillouin, L., 10 Spherical Bessel function j(ζ ), 486 (+) Ehrenfest, P., 10 Spherical Hankel function h (ζ ), 486 m Laue, A., 10 Spherical harmonic functions Y (θ, ϕ), 484 Shiren, N. S., 14 S polarization, 336 Sommerfeld, A., 10 Stamnes, J. J., 559 sound, 15 Stationary phase method Voigt, W., 10 Kelvin, L., 13 Weber and Trizna, 15 Index 737

Stationary phase point, 695 V interior, 698 Vacuum wavenumber k0, 251 Steepest descent method Vector potential Debye, P., 10 macroscopic, 437 Olver, 32 microscopic, 118 m Stellar aberration, 85 Vector spherical harmonic functions Y (α, β), Stokes, G.G., 695 492 Stone, J. M., 148 Velocity Stratton, J. A., 14, 97 complex, 432 Streamlines energy transport, 29 electric, 159 field, 140 magnetic, 159 front, 10 Substantial derivative, 75 group, 7, 304 Superluminal pulse propagation, 35 phase, 8, 304 Superluminal pulse velocities, 30 pulse centroid, 36 Superluminal velocity of light, 340 time-average energy transport, 304, 305 Surface of constant phase, 254 Virtual present radius vector, 144 Surface resistivity Rs , 323 Symmetry property, 434 W Wave equation T reduced, 482 Tachyon, 82 Wave fronts Tardyon, 82 geometrical, 269 TE polarization, 336 Wavelength λ, 157 Test particle, 58 Wavenumber Time-average complex k(ω)˜ , 254, 451, 511 electromagnetic energy velocity, 300 k, 157 of a periodic function, 288 vacuum k0, 451 Poynting vector, 288, 295 Wave vector Time-reversal symmetry, 107 complex k˜ +(ω), 511 Time-reversal transformation, 107 complex k˜ ±(ω), 451 Time dilation, 75 complex part γ(ω), 449 Titchmarsh’s theorem, 202 Weyl’s integral TM polarization, 336 polar coordinate form, 476 Total internal reflection, 347, 350 rectangular coordinate form, 475 Transient Weyl’s proof, 466, 473 anterior, 20 Weyl, H., 465 posterior, 20 Weyl-type expansion, 454 Transmissivity T, 351 Whittaker, E. T., 480 Transport equation, 171 Whittaker-type expansion, 454, 480 Transversality relation, 254, 513 Whittaker’s multipole expansion, 480 Transverse electric TE, 335, 344 Whittaker’s representation, 483 Transverse electromagnetic TEM, 335 Wolf, E., 16, 165, 480, 527, 529, 539 Transverse magnetic TM, 335, 344 World distance light-like, 73 space-like, 73 U time-like, 73 Uniform asymptotic method Handelsman, R. and Bleistein, N., 15 Uniqueness theorem Y microscopic electromagnetic field, 109 Yaghjian, A., 57