A Helmholtz' Theorem

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A Helmholtz’ Theorem

Because 1 ∇2 = −4πδ(R) (A.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 B), then any sufficiently well-behaved vector function F(r)= F(x, y, z) can be represented as F(r)= F(r)δ(r − r) d3r V 1 1 = − F(r)∇2 d3r 4π R V 1 F(r) = − ∇2 d3r, (A.2) 4π V R the integration extending over any region V that contains the point r. With the identity ∇×∇×= ∇∇ · −∇2, Eq. (A.2) may be written as 1 F(r) 1 F(r) F(r)= ∇×∇× d3r − ∇∇ · d3r. (A.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. (A.4) 4π V R 4π V R Moreover, the integrand appearing in this expression may be expressed as 1 1 F(r) ·∇ = −F(r) ·∇ R R F(r) 1 = −∇ · + ∇ · F(r), (A.5) R R where the superscript prime on the vector differential operator ∇ denotes differentiation with respect to the primed coordinates alone. Substitution of 422 A Helmholtz’ Theorem

Eq. (A.5) into Eq. (A.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 R 4π R 4π R V V V 1 1 1 ∇ · F(r) = − F(r) · nˆd2r + d3r 4π S R 4π V R = φ(r), (A.6) whichisthedesiredformofthescalarpotentialφ(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. (A.3) one has that 1 F(r) 1 1 ∇× d3r = − F(r) ×∇ d3r 4π R 4π R V V 1 1 = F(r) ×∇ d3r. (A.7) 4π V R Moreover, the integrand appearing in the final form of the integral in Eq. (A.7) may be expressed as 1 ∇ × F(r) F(r) F(r) ×∇ = −∇ × , (A.8) R R R so that 1 F(r) 1 ∇ × F(r) 1 F(r) ∇× d3r = d3r − ∇ × d3r 4π R 4π R 4π R V V V 1 ∇ × F(r) 1 1 = d3r + F(r) × nˆd2r 4π V R 4π S R = a(r), (A.9) whichisthedesiredformofthevectorpotential. The relations given in Eqs. (A.3), (A.6), and (A.9) then show that

F(r)=−∇φ(r)+∇×a(r), (A.10) where the scalar potential φ(r) is given by Eq. (A.6) and the vector potential a(r) is given by Eq. (A9). This expression may also be written as

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

F (r)=−∇φ(r) ∇ · − 1 ∇ F(r ) 3 1 ∇ F(r ) · 2 = d r + nˆd r (A.12) 4π V |r − r | 4π S |r − r | References 423

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

F (r)=∇×a(r) t 1 F(r) = ∇×∇× d3r 4π |r − r| V ∇ × 1 ∇× F(r ) 3 1 ∇× F(r ) × 2 = d r + nˆd r (A.13) 4π V |r − r | 4π S |r − r | 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)isregular at infinity, then the surface integrals appearing in the above expressions and Eqs. (A.12)–(A.13) become

F (r)=−∇φ(r) ∇ · − 1 ∇ F(r ) 3 = d r , (A.14) 4π V |r − r | F (r)=∇×a(r) t ∇ × 1 ∇× F(r ) 3 = d r . (A.15) 4π V |r − r | Taken together, the above results constitute what is known as Helmholtz’ theorem [1]. Theorem 12. Helmholtz’ Theorem. Let F(r) be any continuous vector ﬁeld with continuous ﬁrst partial derivatives. Then F(r) can be uniquely expressed in terms of the negative gradient of a scalar potential φ(r) and the curl of a vector potential a(r), as embodied in Eqs. (A.10) and (A.11).

References

1. H. B. Phillips, Vector Analysis. New York: John Wiley & Sons, 1933. B The Dirac Delta Function

B.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 , (B.1) ∞ δ(x) dx =1. (B.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 pointandtheintegralofthisfunctionoveritsentiredomainofdefinition 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 (B.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 (B.4) ε→∞ −∞ 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 their oscillations (if any) near the origin x = 0 are not too violent [2]. Each of the following functions forms a sequence with respect to the parameter ε that satisfies the required properties. 426 B The Dirac Delta Function

ε 2 2 δ(x, ε)=√ e−ε x , π

δ(x, ε) = rect1/ε(x), ε δ(x, ε)= sinc(εx), π δ(r, ε)=circ1/ε(r), ε δ(r, ε)= J1(2πεr), r where rect1/ε(x) ≡ ε/2when|x| < 1/ε and is zero otherwise, circ1/ε(r) ≡ ε2/π when r<1/ε and is zero otherwise, and sinc(x) ≡ sin (x)/x when x =0 and is equal to its limiting value of unity when x = 0, where the last two of the above set of functions are appropriate for polar coordinates. 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)byf(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). (B.5) −∞ This result is referred to as the sifting property of the delta function. That is, the process of multiplying a continuous function by δ(x − a) and integrating over all values of the variable x is equivalent to the process of evaluating the function at the point x = a. 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 in its interior, so that a+∆2 f(x)δ(x − a)dx = f(a), (B.6) a−∆1 where ∆1 > 0, ∆2 > 0. It is then seen that f(x) need only be continuous at the point x = a. The above results may be written symbolically as

f(x)δ(x − a)=f(a)δ(x − a), (B.7) B.1 The One-Dimensional Dirac Delta Function 427 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>0anda = 0, Eq. (B.7) yields xkδ(x)=0, ∀k>0. (B.8) Theorem 13. Similarity Relationship (Scaling Law). For all a =0 1 δ(ax)= δ(x). (B.9) |a| Proof. In order to prove this relationship one need only compare the integrals of f(x)δ(ax)andf(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|,aswasto be proved. ! For the special case a = −1, Eq. (B.9) yields δ(−x)=δ(x), (B.10) so that the delta function is an even function of its argument. Theorem 14. Composite Function Theorem. If y = f(x) is any continuous function of x with simple zeroes at the points xi [i.e., y =0at x = xi and f (xi) =0 ] and no other zeroes, then 1 δ(f(x)) = δ(x − x ). (B.11) |f (x )| i i i Proof. In order to prove this theorem, 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 1 g(x)δ(f(x))dx = g f (y) δ(y) −1 dy −∞ R |f (f (y)) | 1 = g f −1(0) |f (f −1(0)) | xi 1 = g(x ) , i |f (x | i i 428 B The Dirac Delta Function where R denotes the range of f(x). In addition, ∞ ∞ 1 − 1 − g(x) δ(x xi) dx = g(x)δ(x xi)d −∞ |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), (B.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), −∞ 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. (B.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) −∞ B.1 The One-Dimensional Dirac Delta Function 429 and ∞ g(η)δ(ξ − η)dη = g(ξ), −∞ then the expression in Eq. (B.12) is again obtained. Consider next what interpretation may be given to the derivatives of the delta function. This is 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,sothat

∞ ∞ 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, (B.13) ε→∞ with the result ∞ f(x)δ(x)dx = −f (0). (B.14) −∞ 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). (B.15) −∞

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

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

xδ(x)=−δ(x). (B.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). (B.17)

The generalization of Eq. (B.16) may be directly obtained from Eq. (B.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, −∞ −∞ 430 B The Dirac Delta Function and one then has the general equivalence

xnδ(n)(x)=(−1)nn!δ(x). (B.18)

This final relationship shows that 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)thatisdefinedbytherelationsU(x)=0 when x<0, U(x)=1whenx>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 < 0andx2 > 0), x2 x2 x2 − f(x)U (x)dx =[f(x)U(x)]−x1 f (x)U(x)dx −x1 −x1 x2 = f(x2) − f (x)dx 0 = f(x2) − [f(x2) − f(0)] = f(0), where f(x) is any continuous function. Upon setting x = y − a,andf(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. (B.5). In particular, with F (y)=1anda = 0, this expression becomes ∞ U (y)dy =1, −∞ and U (x) also satisfies the property given in Eq. (B.2) which serves to partially define the delta function. Moreover, U (x) = 0 for all x = 0 and property (B.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)= . (B.19) dx In addition, it is seen that x U(x)= δ(ξ)dξ, (B.20) −∞ which follows from Eqs. (B.6) and (B.19). B.1 The One-Dimensional Dirac Delta Function 431

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) (B.21) −∞ −∞ for any sufficiently well-behaved, continuous function f(x). Define the function sequence ε K(x − a, ε) ≡ ei2πν(x−a)dν −ε sin (2π(x − a)ε) = (B.22) π(x − a) with limit K(x − a) ≡ lim K(x − a, ε). (B.23) ε→∞ Strictly speaking, this limit does not exist in the ordinary sense when x−a = 0; however, the limit does exist and has the value zero when x − a = 0 if it is interpreted in the sense of a Cesáro limit [4]. Upon inversion of the order of integration, Eq. (B.21) may be formally rewritten as ∞ f(a)= f(x)K(x − a)dx, (B.24) −∞ which should be interpreted as meaning that ∞ f(a) = lim f(x)K(x − a, ε)dx. (B.25) ε→∞ −∞ Thus, the function K(x − a) satisfies the sifting property (B.5) of the delta function. If one sets f(x)=1anda =0inEq.(B.24),thereresults ∞ K(x)dx =1, −∞ and K(x) satisfies the property given in Eq. (B.2) which serves to partially define the delta function. Because K(x) = limε→∞ K(x, ε)=0whenx =0, so that the property given in Eq. (B.1) is also satisfied, one then obtains from Eq. (B.23) the relation ∞ δ(x)= ei2πνxdν. (B.26) −∞ That is, the Dirac delta function may be regarded as the Fourier transform of unity. The reciprocal relation follows from Eq. (B.25) upon setting f(x)= exp(i2πνx)anda =0,sothat ∞ 1= δ(x)e−i2πνxdx, (B.27) −∞ which also follows directly from the sifting property given in Eq. (B.5). Notice that this relation by itself is not sufficient to imply the validity of Eq. (B.26). 432 B The Dirac Delta Function B.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. (B.1)–(B.2) become

δ(r)=0; r = 0, (B.28) ∞ δ(r)d3r =1. (B.29) −∞ The function

δ(r) ≡ δ(x, y, z) ≡ δ(x)δ(y)δ(z), (B.30) where r = 1ˆxx + 1ˆyy + 1ˆzz is the position vector with components (x, y, z) clearly satisﬁes Eqs. (B.28)–(B.29) and so deﬁnes a three-dimensional Dirac delta function. The sifting property given in Eq. (B.5) then becomes ∞ f(r)δ(r − a)d3r = f(a), (B.31) −∞ and the similarity relationship or scaling law given in Eq. (B.9) now states that 1 δ(ar)= δ(r), (B.32) |a|3 where a is a scalar constant. The Fourier transform pair relationship expressed in Eqs. (B.26)–(B.27) becomes ∞ 1 ik·r 3 δ(r)= 3 e d k, (B.33) (2π) −∞ ∞ 1= δ(r)e−ik·rd3r, (B.34) −∞ where k = 1ˆxkx + 1ˆyky + 1ˆzkz =2π(1ˆxνx + 1ˆyνy + 1ˆzνz). The generalization of the three-dimensional Dirac delta function to more general coordinate systems requires more careful attention. Suppose that a function ∆(r) is given in Cartesian coordinates as

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

u = f1(x, y, z),

v = f2(x, y, z), (B.36)

w = f3(x, y, z), B.2 The Dirac Delta Function in Higher Dimensions 433 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 ∆(r) is desired in terms of the coordinate variables (u, v, w)that satisfies the relation ∞ ∆(r − r)ϕ(u, v, w)dV = ϕ(u,v,w), (B.37) −∞ 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 giveninEq.(B.36).Ifthepointr =(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 δr1 = δu. ∂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 δr2 = δv. ∂v

The parallelogram with sides δr1 and δr2 then has area

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

If the point r =(x, y, z)isnowvariedfromr to r + δr3 by changing the coordinate variable w to w + δw while keeping u and v ﬁxed, then ∂r δr3 = δw, ∂w and the volume of the parallelepiped with edges δr1, δr2,andδr3 is then given by

δV = |δr3 · (δr1 × δr2)| ∂r ∂r ∂r = · × δuδvδw. (B.39) ∂w ∂u ∂v The quantity x, y, z ∂(x, y, z) J ≡ u, v, w ∂(u, v, w) ∂r ∂r ∂r ≡ · × (B.40) ∂w ∂u ∂v 434 B The Dirac Delta Function 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. (B.37) becomes ∞ x, y, z ∆(r − r )ϕ(u, v, w) J dudvdw = ϕ(u ,v ,w ), (B.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). (B.42) u, v, w Because this transformation is assumed to be single-valued, then

δ(u)δ(v)δ(w) δ(x)δ(y)δ(z)= J x,y,z u,v,w u, v, w = J δ(u)δ(v)δ(w), (B.43) x, y, z where J(u, v, w/x, y, z) is the Jacobian of the inverse transformation. Consider ﬁnally the description of a function (r)thatvanishesevery- where in three-dimensional space except on a surface S andissuchthat ∞ (r)ϕ(r)d3r = ς(r)ϕ(r)d2r, (B.44) −∞ S where ς(r)isthevalueof(r) on the surface S,thatis,whenr ∈ S. Choose orthogonal curvilinear coordinates (u, v, w)suchthatw = w0 describes the surface S for some constant w0, in which case ∇w is parallel to the normal to the surface S,andissuchthat∇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. (B.38). Furthermore, both ∂r/∂w and (∂r/∂u) × (∂r/∂v) are normal to S so that x, y, z ∂r ∂r ∂r J = × . (B.45) u, v, w ∂w ∂u ∂v

With this result, Eq. (B.44) may be written as ∞ ∂r ∂r ∂r 2 (r)ϕ(r) × dudvdw = ς(r)ϕ(r)d r, −∞ ∂w ∂u ∂v S which, with Eq. (B.38), may be expressed as ∞ ∂r 2 2 (r)ϕ(r) d rdw = ς(r)ϕ(r)d r. (B.46) −∞ ∂w S From this result it then follows that References 435 ς(r) − 0 (r)= ∂r δ(w w ), (B.47) ∂w which is the solution of Eq. (B.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,sothat ∂r ·∇w =1. ∂w 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. (B.47) becomes

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

References

1. P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford: Oxford Univer- sity Press, 1930. §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. C 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, (C.1) 0 which is simply a Fourier transform with complex angular frequaency 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, (C.2) but which may not vanish for t ≤ 0. The Laplace transform given in Eq. (C.1) may then be written as ∞ L{f(r,t)} = U(t)f (r,t)eiωtdt, (C.3) −∞ where U(t)=0fort<0andU(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, (C.4) where the subscript ω indicates that it is the Fourier transform variable. The inverse Fourier transform of this equation then gives 7 8 U(t)f (r,t)=F−1 L{f(r,t)} (C.5) for real ω. For complex ω,letω = ω + iω where ω ≡{ω} and ω ≡{ω}.The Laplace transform given in Eq. (C.1) then becomes ∞ L{f(r,t)} = U(t)f (r,t)e−ω t eiω tdt −∞# $ −ωt = Fω U(t)f (r,t)e . (C.6) 438 C The Fourier–Laplace Transform

The inverse Fourier transform of this expression then yields 7 8 −ωt F−1 L{ } U(t)f (r,t)e = ω f(r,t) ∞ 1 = L{f(r,t)}e−iω tdω, (C.7) 2π −∞ which may be rewritten as ∞ 1 U(t)f (r,t)= L{f(r,t)}e−i(ω +iω )tdω 2π −∞ 1 ∞+ω = L{f(r,t)}e−iωtdω 2π −∞+iω 1 = L{f(r,t)}e−iωtdω. (C.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. (C.1) and (C.8) then deﬁne the Laplace transform pair relationship ∞ ˜f(r,ω) ≡ L {f(r,t)} = f(r,t)eiωtdt, (C.9) 0 # $ 1 f(r,t) ≡ L−1 ˜f(r,ω) = ˜f(r,ω)e−iωtdω, (C.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 the integrand of the transformation (C.9) may serve as a convergence factor when ω > 0. In particular, ∞ ˜f(r,ω)= f(r,t)e−ω teiω tdt (C.11) 0

−ωt is just the Fourier transform Fω {f(r,t)e }. The Fourier transform of f(r,t) alone is ∞ iωt 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 C The Fourier–Laplace Transform 439 T lim f(r,t)e−γt dt < ∞, (C.12) T →∞ 0 then f(r,t) is transformable for all ω ≥ γ and its temporal frequency spectrum is given by the Laplace transform (C.9). The lower bound γa of all of thevaluesofγ for which the inequality appearing in Eq. (C.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 ∂t 0 ∞ iωt ∞ − iωt = f(r,t)e 0 iω f(r,t)e dt 0 = −f(r, 0) − iωL {f(r,t)} , (C.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. (C.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 f˜1(ω)f˜2(ω)=L{f1(t)}L{f2(t)} so that # $ −1 1 −iωt L f˜1(ω)f˜2(ω) = f˜1(ω)f˜2(ω)e dω 2π C ∞ 1 iωτ −iωt = f˜1(ω) f2(τ)e dτ e dω 2π C 0 ∞ 1 −iω(t−τ) = dτ · f2(τ) f˜1(ω)e dω , 0 2π C and consequently

# $ ∞ −1 L f˜1(ω)f˜2(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 f˜1(ω)f˜2(ω) = f1(t − τ)f2(τ)dτ, (C.14) 0 which may be rewritten as

∞ L f1(t − τ)f2(τ)dτ = L {f1(t)} L {f2(t)} , (C.15) 0 440 C The Fourier–Laplace Transform 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), (C.16) −∞ 0 1 ∞ ≡ F−1L−1{ } 3 · i(k·r−ωt) F(r,t) F(k,ω) = 4 d k dω F(k,ω)e , (2π) −∞ C (C.17) where k = 1ˆxkx + 1ˆyky + 1ˆzkz and r = 1ˆxx + 1ˆyy + 1ˆzz. Because, ∂F(r,t) 1 ∞ 3 · i(k·r−ωt) = 4 d k dω ikjF(k,ω)e , ∂xj (2π) −∞ C from Eq. (C.17), then the transforms of the first spatial derivatives of F(r,t) are given by ∂F(r,t) FL = ikjF(k,ω), (C.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), (C.19) and the spatiotemporal transform of the curl of F(r,t)isgivenby FL {∇ × F(r,t)} = ik × F(k,ω). (C.20)

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

References

1. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. D 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)E˜eff (r,ω) (D.1) at the fixed angular frequency ω of the applied time-harmonic field [see Eq. (4.164)]. 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 E˜eff (r,ω) = ˜e(r ,ω)d r − ˜em(r ,ω)d r , (D.2) Vm Vm Vm Vm where ˜e(r ,ω) is the total microscopic electric field at the point r ∈ Vm, and where ˜em(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 macroscopi- cally small but microscopically large region), then the first integral appearing in Eq. (D.2) is essentially just the macroscopic electric field as defined in Eq. (4.6). In particular, if the weighting function w(r) is taken to be given by 1/Vm if r ∈ Vm and 0 otherwise, then Eq. (4.1) gives 1 E˜(r,ω) ≡˜e(r,ω) = ˜e(r − r,ω)d3r, (D.3) Vm Vm where r ∈ Vm. With the change of variable r = r − r , the first integral appearing in Eq. (D.2) is then obtained. 442 D The Effective Local Field

Τ d 1

q Τ r' d 2 O

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

For the second integral appearing in Eq. (D.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 Figure D.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 then 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, (D.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. (D.5) V V1 V V2

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 Figure D.1. Because the magnitude ofe ˜z decreases with the square of the D The Eﬀective Local Field 443 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,˜ez is positive at dτ1 and it is negative at dτ2, whereas for negative q,˜ez 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.

Θ q

r'' O

P V2

Fig. D.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 ofe ˜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)withoriginat the point charge q, as illustrated in Figure D.2. At any point P ∈ V2, · ˆ 4π q e˜z = e˜ 1z = 2 cos(θ), (D.6) 4π 0 r so that 2π π −2r cos(θ) 3 4π e˜zd r = q dϕ sin(θ)cos(θ) dr dθ V 4π 0 0 π/2 0 4π π = − qr cos2(θ) sin(θ)dθ 0 π/2 4π = − qr. (D.7) 3 0 The average electric field intensity inside the sphere due to the point charge q is then given by 444 D The Effective Local Field 1 3 − 4π qr e¯z = 3 e˜zd r = 3 . (D.8) (4/3)πR V 4π 0 R Theelectricdipolemomentofthepointchargeq with reference to the center O of the spherical region is given by p˜ ≡ qr 1ˆz,sothatEq.(D.8)maybe written in the general form − 4π e¯ = 3 p˜. (D.9) 4π 0R

The second integral appearing in Eq. (D.2) is then given by 1 3 − 4π ˜em(r ,ω)d r = 3 p˜m(r,ω), (D.10) Vm Vm 4π 0rm 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 ˜ ˜em(r ,ω)d r = − P(r,ω). (D.11) Vm Vm 3 0

With these substitutions, Eq. (D.2) for the spatially averaged effective field becomes 4π ˜ E˜eff (r,ω) = E˜(r,ω)+ P(r,ω), (D.12) 3 0 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. E Magnetic Field Contribution to the Classical 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 2 qe 1 drj +2δ + ω r = − Eeff(r,t)+ × Beff(r,t) , (E.1) dt2 j dt j j m c dt where Eeff(r,t) is the effective local electric field intensity and Beff(r,t)is the effective local magnetic induction field at the space–time point (r,t), 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 E˜ eff(r,ω)= Eeff(r,t)e dt, (E.2) −∞ ∞ iωt B˜ eff(r,ω)= Beff(r,t)e dt −∞ c = k(ω) × E˜ eff(r,ω), (E.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 ˜rj(ω)= rj(t)e dt, (E.4) −∞ the dynamical equation of motion (E.1) becomes 2 − 2 qe ˜ − × × ˜ ω +2iδjω ωj ˜rj = Eeff i˜rj k Eeff m qe = (1 + i˜r · k) E˜ eff − i ˜r · E˜ eff k , (E.5) m j j with formal solution 446 E Magnetic Field Contribution to the Lorentz Model ˜ qe 1+i˜rj · k ˜rj · Eeff ˜r = E˜ eff − i k . (E.6) j 2 − 2 2 − 2 m ω ωj +2iδjω ω ωj +2iδjω

The electron displacement vector may then be expressed as a linear combi- nation of the orthogonal pair of vectors k and E˜ eff ˜ ˜rj = ajEeff + bjk, (E.7)

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

2 ˜r · E˜ eff c a = j ,b= ˜r · k. (E.8) j ˜2 j 2 j Eeff ω The pair of scalar products appearing in the above expression may be evaluated from Eq. (E.6) as

q ˜r · E˜ eff ˜r · k = −i e j k2, j 2 − 2 m ω ωj +2iδjω qe 1+i˜rj · k 2 ˜r · E˜ eff = E˜ . j 2 − 2 eff m ω ωj +2iδjω Substitution of the second relation into the ﬁrst then yields

(q /mc)2E˜2 ω2 ˜r · k = −i e eff , (E.9) j 2 − 2 2 − 2 ˜2 2 (ω ωj +2iδjω) (qe/mc) Eeffω and substitution of this result into the second relation gives

qe/m ˜r · E˜ eff = j ω2 − ω2 +2iδ ω j j (q /mc)2E˜2 ω2 × 1+ e eff E˜2 . (E.10) 2 − 2 2 − 2 ˜2 2 eff (ω ωj +2iδjω) (qe/mc) Eeffω

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

q /m a = e j ω2 − ω2 +2iδ ω j j (q /mc)2E˜2 ω2 × 1+ e eff , (E.11) 2 − 2 2 − 2 ˜2 2 (ω ωj +2iδjω) (qe/mc) Eeffω (q /m)2E˜2 b = −i e eff , (E.12) j 2 − 2 2 − 2 ˜2 2 (ω ωj +2iδjω) (qe/mc) Eeffω respectively. E Magnetic Field Contribution to the Lorentz Model 447

The local (or microscopic) induced dipole moment p˜j ≡−qe˜rj for the jth Lorentz oscillator type is then given by [compare with Eq. (4.202)] ˜ p˜j(r,ω)=−qe ajEeff(r,ω)+bjk . (E.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 summa- tion over all oscillator types of the spatially averaged locally induced dipole moments as ˜ ˜ P(r,ω,Eeff)= Njp˜j(r,ω) j ,, -- ˜ ˜ ˜ = Eeff(r,ω) Njαj⊥(ω, Eeff)+k Njαj(ω, Eeff). j j (E.14)

Here ˜ ≡ (0) (2) ˜ αj⊥(ω, Eeff) αj⊥(ω)+αj⊥(ω, Eeff) −q2/m = e ω2 − ω2 +2iδ ω j j (q /mc)2E˜2 ω2 × 1+ e eff (E.15) 2 − 2 2 − 2 ˜2 2 (ω ωj +2iδjω) (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 (4.204) for the atomic polarizability when magnetic field effects are neglected, and 3 2 ˜2 (qe /m )Eeff α (ω, E˜eff) ≡ i (E.16) j 2 − 2 2 − 2 ˜2 2 (ω ωj +2iδjω) (qe/mc) Eeffω is defined here as 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]} 448 E Magnetic Field Contribution to the Lorentz Model

2 Z qe qe 2 2 H = H0 − H · L + H r , 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|,wherer⊥ is the projection of the position vector r on the plane perpendicular to H(r), and where L = Z × 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 is given by ∼ 2 2 2 2 2∼ × −16 2 (Zqe /12mc )H r ,where r 1 10 cm for a bound electron. The ratio of this quantity to the level distance µBH,whereµB ≡ qe¯h/2mc is the Bohr magneton, is found [2] to be ∼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 preceeding 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 and the bound electron is taken to be given ≡ 2 2 ≈ × −9 ≡ by the Bohr radius a0 ¯h /mqe 5.29 10 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

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. Index

abscissa of absolute convergence, 282, macroscopic (r,t), 170 439 macroscopic bound b(r,t), 170 acceleration field, 130 macroscopic free f (r,t), 170 advanced potentials, 118 macroscopic, conductive, 290 Ampère’s law, 63 microscopic ρ(r,t), 48 analytic signals, 402 microscopic bound ρb(r,t), 167 angular spectrum representation microscopic free ρf (r,t), 167 freely propagating wave field, 332, surface, 264 366 Clausius–Mossotti relation, 197 geometric form, 341 Cole–Cole extended Rocard–Powles– of radiation field, 297 Debye model asymptotic heat, 253 triply distilled water, 206 atomic polarizability Cole–Cole plot, 199 parallel component, 447 complex dielectric permittivity, 188 perpendicular component, 447 complex index of refraction n(ω), 210, attenuation factor, 7 224, 297 auxiliary field vectors, 57 complex permittivity, 224 complex velocity, 280 Barash, Y. S. and Ginzburg, V. L., 242 complex wavenumber, 7 baryon, 79 conservation laws Baños, A., 324 electromagnetic energy, 91, 236 Boltzmann’s constant KB , 197 electromagnetic momentum Born, M. and Wolf, E., 13, 346 angular, 97 boundary condition linear, 93 normal, 264 energy tangential, 265, 266 local form, 89 Brillouin precursor, 12 relativistic momentum, 76 Brillouin, L., 1 conservation of charge microscopic, 50 Cauchy principal value, 183 constitutive relations, 178 causality and linearity, 179 dispersion relations, 183 general form, 179 Kramers–Kronig relations, 186 induction fields, 178 Plemelj formulae, 183 primitive fields, 178 primitive, 179, 182 convective derivative, 72 relativistic, 182, 260 convolution theorem, Laplace trans- charge density form, 440 macroscoopic, nonconductive, 290 cophasal surfaces, 160 450 Index

Courant condition, 18 static zone, 149 current density wave zone, 152 conduction, 172 Dirac delta function δ(ξ), 124, 425 convective (microscopic) j(r,t), 50 composite function theorem, 427 effective, 272 derivatives of, 429 irrotational or longitudinal j(r,t), extension to higher dimensions, 432 114 sifting property, 426 macroscopic J(r,t), 174 similarity relationship (scaling law), macroscopic bound Jb(r,t), 175 427 macroscopic free Jf (r,t), 175 direction cosines, plane wave microscopic “bound” jb(r,t), 172 complex, 334 microscopic free jf (r,t), 172 real, 361 solenoidal or transverse jt(r,t), 114 dispersion surface, 266 anomalous, 212 current source J0(r,t), 277 normal, 212 dispersion relations Debye model dielectric permittivity, 186 and rotational Brownian motion, electric conductivity, 188 197, 201 electric susceptibility, 185 causality, 198 magnetic permeability, 192 effective relaxation time, 198 magnetic susceptibility, 192 of orientational (dipolar) polariza- dispersive interface, 16 tion, 197 dissipation, 240 permittivity, 198 Drude model relaxation equation, 197 causality, 217 Rocard–Powles extension, 201 damping constant γ, 217 susceptibility, 198 dielectric permittivity, 217 Debye-type dielectric, 4 electric conductivity σ(ω), 232 Devaney, A. J., 299 static conductivity σ0, 232 diamagnetic, 191 dynamical free energy, 250 diamagnetic effect, 447 dielectric permittivity effective electric field, 196, 445 complex c(ω), 188 effective field, 441 composite model, 214 spatially averaged, 444 free space 0,52 effective magnetic field, 445 of sea-water, 232 eikonal equation of triply distilled water, 193 generalized, 160 temporally dispersive (ω), 184 geometrical optics, 161 dipolar relaxation time τm, 197 Einstein, A., 1 dipole mass–energy relation, 78 Hertzian, elemental, 125 special theory of relativity, 67 linear electric, 124 electric 2n-pole, 144 moment, electric, 148 electric conductivity σ(ω), 188 dipole oscillator electric dipole approximation, 144 electric dipole contribution, 144 electric displacement vector electric monopole contribution, 143 macroscopic D(r,t), 176 magnetic dipole contribution, 144 microscopic d(r,t), 58 dipole radiation electric energy density intermediate zone, 150 time-harmonic, 244 Index 451 electric field vector free-field, 387 macroscopic E(r,t), 167 Laplace–Fourier integral representa- microscopic e(r,t), 52 tion, 390 electric moment, 120 plane wave expansion, 391–405 electric susceptibility χe(ω), 184 freely propagating wave field, 329 electromagnetic angular momentum Fröhlich distribution function, 205 microscopic density lem,97 microscopic total lem,97 Gabor cells, 20 electromagnetic beam field, 367 Galilean invariance, 62, 64 separable, 381 Galilean transformation electromagnetic energy electric field intensity, 62 irreversible, 254 magnetic field intensity, 65 macroscopic gauge density, total U (r,t), 235 Coulomb, 113 microscopic function, 111, 286 density u(r,t), 90 invariance, 111, 286 total U(t), 89 Lorenz, 113, 287 reversible, 253 radiation, 113 electromagnetic energy density, 89, 235 transformation, 111, 122, 286 electromagnetic field, 53 transformation, restricted, 113, 287 electromagnetic linear momentum transverse, 113 microscopic density pem(r,t), 93 Gauss’ law microscopic total pem,93 microscopic electromagnetic wave electric field, 66 freely propagating, 329 magnetic field, 66 freely propagating boundary Gaussian pulse dynamics conditions, 329 Balictsis and Oughstun, 23 plane, time-harmonic, 227 Garrett and McCumber, 23 source-free, 329 Tanaka, Fujiwara, and Ikegami, 23 source-free initial conditions, 387 Gaussian units electrostatic field, 53 electrostatic unit (esu) of electric energy density field intensity, 58 electromagnetic field Uem, 258 gauss, 58 reversible, 258 maxwell, 60 energy transport velocity, 21, 261 statampere, 58 energy velocity description statcoulomb, 58 Sherman and Oughstun, 24 statvolt, 58 equation of continuity Goos–Hänchen effect, 16 conduction current, 222 Green’s function macroscopic, 177 free-space, 345 microscopic, 51 group method evolved heat (dissipation) Q(r,t), 240 Havelock,T.H.,11 extensive variable, 90 group velocity, 4, 7 Hamilton, Sir W. R., 4 Faraday’s law Rayleigh, Lord, 5 microscopic, 60 Stokes, G. G., 5 forerunner, 1 group velocity approximation, 11 Fourier integral theorem, 431 Eckart, C., 13 Fourier transform Fω {f(r,t)}, 438 Lighthill, M. J., 13 452 Index

Whitham, G. B., 13 Jordan’s lemma, 408 group velocity method, 11 Kramers–Kronig relations, 186 half-space, positive and negative, 296 Kronecker-delta function δij ,94 Hall effect, 222 Heaviside unit step function U(ξ), 430 Lagrange’s theorem, 140 Heaviside–Poynting theorem Lalor, E., 344 differential form, 235 Landau, L. D. and Lifshitz, E. M., 240, integral form, 235 253 time-harmonic form, 244 Laplace transform L{f(r,t)}, 437 Helmholtz equation, 7, 225, 367 Liénard–Wiechert potentials, 128 Helmholtz free energy (work function), Liénard-Wiechert potentials 250 and special relativity, 135 Helmholtz’ theorem, 423 local energy theorem, 90 Hertz local field, see effective field potential, 121 Lorentz covariant, 68 vector, 121 Lorentz force relation, 53 Hertz, H., 119, 149 Lorentz invariant, 68 Hilbert transform, 183, 185 Lorentz model homogeneous, isotropic, locally linear atomic polarizability, 208 (HILL) temporally dispersive Lorentz–Lorenz modified, 210 media magnetic field contribution, 207 constitutive relations, 221 of resonance polarization, 207 Huygen’s principle, 345 oscillator strength, 213 phenomenological damping constant idemfactor, 70, 94 δj , 207 impulse response function resonance frequency, undamped ωj , spatial, 342 207 incomplete Lipschitz–Hankel integrals sum rule, 213 Dvorak, S. and Dudley, D., 25 Lorentz theory, 48 indeterminacy principle, 19 Lorentz, H. A., 1 inertial reference frames, 67 Lorentz–Lorenz formula, 197 information diagram, 19 Lorentz-type dielectric, 4 initial field values Lorenz condition, 112, 286 in a closed convex surface, 410 generalized, 290 in a sphere, 406 Loudon, R., 21, 260 instantaneous (causal) spectrum, 250 luminiferous ether, 67 intensive variable, 90 luxon, 78 interaction energy, 250 intrinsic impedance macroscopic polarization density complex, 228 P(r,t), 171 free space, 228 macroscopic quadrupole moment good conductor, 231 density tensor Q(r,t), 171 near-ideal dielectric, 230 magnetic 2n−1-pole, 144 intrinsic impedance, complex, 228 magnetic energy density inverse problems, 21 time-harmonic, 244 isoplanatic, see space-invariant magnetic field vector macroscopic B(r,t), 167 Jackson, J. D., 14 microscopic b(r,t), 52 Index 453 magnetic flux, 60 spatiotemporal frequency domain magnetic intensity vector form, 389 macroscopic H(r,t), 176 Maxwell, J. C., 1 microscopic h(r,t), 58 mean polarizability αj (ω), 196 magnetic permeability mean square charge radius, 171 composite model, 215 metallic waveguides, 14 dispersive µ(ω), 190 microscopic Maxwell–Lorentz theory, 51 free space µ0,52 Minkowski formulation, 177 magnetic susceptibility χm(ω), 190 mksa units magnetization, macroscopic M(r,t), ampere, 59 175 coulomb, 59 magnetostatic field, 53 farad, 59 material relations, see cnstitutive volt, 59 relations178 weber, 59, 60 material response modern asymptotic description, 24 anisotropic, 181 molecular magnetic moment, 173 isotropic, 181 molecular multipole moments, 169 nondispersive, 180 molecular polarizability α(ω), 441 spatially dispersive, 180 monochromatic (time-harmonic) wave spatially homogeneous, 179 field, 159 spatially inhomogeneous, 179 temporally dispersive, 180 near-ideal dielectrics, 230 temporally homogeneous, 179 Newton’s second law of motion temporally inhomogeneous, 179 relativistic form, 75 material response tensors Nisbet, A. and Wolf, E., 345, 356 dielectric permittivity ˆ (r,t, r,t), nonlinear optical effects, 448 179 normalized velocity, 127 electric conductivityσ ˆ(r ,t , r,t), 179 number density Nj , 208 magnetic permeabilityµ ˆ(r,t, r,t), numerical techniques 179 discrete Fourier transform method, mature dispersion regime, 2, 24 17 Maxwell stress tensor fast Fourier transform method, 18 microscopic, 94 finite-difference time-domain method, Maxwell’s displacement current, 63 18 Hosono’s Laplace transform method, Maxwell’s equations 18 macroscopic temporal frequency-domain form, ohmic power loss, 273 224 Olver’s saddle point method, 24 time-domain differential form, 176, optical wave field, 158 223 order symbol O,76 microscopic, 52 Ott’s integral representation, 324 time-domain differential form, 58 time-domain integral form, 66 Panofsky, W. and Phillips, M., 129 phasor form, 226 paramagnetic, 191 source-free, 387 penetration depth dp, 231, 269 spatial average, 167 phase velocity, 5 spatiotemporal form, 280 good conductor, 231 454 Index

near-ideal dielectric, 230 precursor, 1 phase-space asymptotic description, 17 Brillouin, 12 phasor representation, 225 formation on transmission, 16 plane wave observation of attenuation factor α(ω), 228, 334 Aaviksoo, Lippmaa, and Kuhl, 29 evanescent, 339, 364 Choi and Osterberg,¨ 29 homogeneous, 298, 339, 364 Jeong, Dawes, and Gauthier, 29 inhomogeneous, 298, 339, 364 Pleshko and Palócz, 12 propagation factor β(ω), 228, 334 Sommerfeld, 12 plane wave expansion principle of superposition, 55 and mode expansions, 402 propagation factor, 7 and Poisson’s solution, 405 pulse centroid velocity, 29 Devaney, A. J., 399 pulse diffraction, 26 polar coordinate form, 401, 404 pulse distortion, 16 uniqueness, 399 quadratic dispersion relation, 16 plasma frequency bj , 208 Poincaré–Lorentz transformation radiated energy, 136 relations, 69–70 radiation coordinate transformation matrix, 70 electromagnetic, 119 electric field transformation, 84 reaction, 55 force transformation, 80 radiation field invariance of Maxwell’s equations, 87 Fourier–Laplace integral representa- magnetic field transformation, 84 tion, 281, 284 mass transformation, 80 scalar potential, 285 Poisson’s equation vector potential, 285 scalar potential, 113 ray techniques Poisson’s solution, 415 direct-ray method, 13 polarization Felsen, L. B., 14 left-handed, 350 Heyman, E., 14 right-handed, 351 Melamed, T., 14 state, 353 space–time ray theory, 13 polarization ellipse, 347, 353 Rayleigh–Sommerfeld diffraction polarized field integrals, 345 circular, 354 refractive index linear, 354 complex, 210 uniform, 353, 357 function, modified, 160 potential functions, electromagnetic relativity complex, 156 Newtonian, 67 Liénard–Wiechert, 125 special theory, 67 macroscopic, 285–291 relaxation times microscopic, 109–111 dipolar, 197 retarded, 116 distribution of, 204 Poynting vector Drude model, mean-free path τc, 217 complex S˜(r), 243 effective, 198, 202 macroscopic S(r,t), 235 retardation condition, 131 microscopic s(r,t), 89 retarded potentials, 118 Poynting’s theorem, 90 retarded time, 116 Poynting–Heaviside interpretation, 91 Riemann’s proof, 116 Index 455

Righi, A., 119 of a microscopic function, 166 Rocard–Powles–Debye model weighting function, 166 Cole–Cole extension, 205 spatial dispersion effects, 26 friction time τmf , 202 spatial transfer function, 343 of triply distilled water, 203 spatially locally linear, 179 permittivity, 202 spatiotemporal Fourier-Laplace susceptibility, 202 transform, 440 special theory of relativity sampling theorem, 20 dilation factor γ,72 scalar potential fundamental postulate, 67 macroscopic, 285 longitudinal mass, 78 microscopic, 110 Lorentz–Fitzgerald contraction, 73 semiconductor, 231 mass–energy, 78 Sherman expansion, see source-free postulate of the constancy of the wave field speed of light, 67 Sherman, G. C., 3, 31, 299, 342, 367 proper differential time interval, 72 SI (Système Internationale) units, see relativistic mass, 75 mksa units rest energy, 77 signal rest mass (proper mass), 75 arrival, 9, 12, 15 time dilation, 71 buildup, 15 transverse mass, 78 velocity, 2 speed of light in vacuum c,52 signal velocity, 2, 7 stationary phase method Baerwald, H., 11 Kelvin, L., 10 Brillouin, L., 7 steepest descent method Ehrenfest, P., 7 Debye, P., 8 Laue, A., 7 Olver, 24 Shiren, N. S., 12 Stone, J. M., 138 Sommerfeld, A., 7 Stratton, J. A., 12, 91 sound, 13 streamlines Voigt, W., 7 electric, 149 Weber and Trizna, 12 magnetic, 149 simple magnetizable medium, 190 substantial derivative, 72 simple polarizable dielectric, 184 superluminal pulse propagation, 28 simultaneity, 68 superluminal pulse velocities, 22 skin depth, 269 surface of constant phase, 227 slowly evolving wave approach, 27 symmetry property, 282 slowly varying envelope approximation, 13, 27 tachyon, 79 Sommerfeld precursor, 12 test particle, 56 Sommerfeld’s integral representation, time-average 322 of a periodic function, 243 Sommerfeld, A., 1 electromagnetic energy velocity, 257 source-free wave field, 366, 371 Poynting vector, 243, 249 separability, 382 Titchmarsh’s theorem, 183 Sherman expansion, 374 transient spatial series representation, 382 anterior, 15 space-invariant, 343 posterior, 15 spatial average transport equation, 160 456 Index transversality relation, 227 complex k˜+(ω), 331 complex k˜±(ω), 296 uniform asymptotic method wavelength λ, 147 Handelsman, R. and Bleistein, N., 12 wavenumber uniqueness theorem complex k˜(ω), 227, 297, 331 microscopic electromagnetic field, vacuum k0, 297 101 wavenumber k, 147 wavevector vacuum wavenumber k0, 224 vector potential complex part γ(ω), 294 macroscopic, 285 Weyl’s integral microscopic, 110 polar coordinate form, 320 velocity rectangular coordinate form, 319 complex, 280 Weyl’s proof, 317 energy transport, 21 Weyl’s proof , 310 front, 8 Weyl, H., 310 group, 4, 260 Weyl-type expansion, 299 phase, 5, 260 Whittaker-type expansion, 299 pulse centroid, 29 Wolf, E., 155 time-average energy transport, 260, world distance 261 lightlike, 70 velocity field, 131 spacelike, 70 virtual present radius vector, 135 timelike, 70 wave vector Yaghjian, A., 55