A Helmholtz’ Theorem€¦ · B The Dirac Delta Function B.1 The One-Dimensional Dirac Delta...

A Helmholtz’ Theorem

Because

∇2(

1R

)= −4πδ(R) (A.1)

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) =∫

V

F(r′)δ(r− r′) d3r′

= − 14π

∫V

F(r′)∇2(

1R

)d3r′

= − 14π∇2∫

V

F(r′)R

d3r′, (A.2)

the integration extending over any region V that contains the point r. Withthe identity ∇×∇× = ∇∇ · −∇2, Eq. (A.2) may be written as

F(r) =14π∇×∇×

∫V

F(r′)R

d3r′ − 14π∇∇ ·

∫V

F(r′)R

d3r′. (A.3)

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

14π∇ ·∫

V

F(r′)R

d3r′ =14π

∫V

F(r′) · ∇(

1R

)d3r′. (A.4)

Moreover, the integrand appearing in this expression may be expressed as

F(r′) · ∇(

1R

)= −F(r′) · ∇′

(1R

)= −∇′ ·

(F(r′)R

)+

1R∇′ · F(r′), (A.5)

where the superscript prime on the vector differential operator ∇′ denotesdifferentiation 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 thefirst term then yields

14π∇ ·∫

V

F(r′)R

d3r′ = − 14π

∫V

∇′ ·(

F(r′)R

)d3r′ +

14π

∫V

∇′ · F(r′)R

d3r′

= − 14π

∮S

1R

F(r′) · nd2r′ +14π

∫V

∇′ · F(r′)R

d3r′

= φ(r), (A.6)

which is the desired form of the scalar potential φ(r) for the vector fieldF(r). Here S is the surface that encloses the regular region V and containsthe point r.

For the curl term appearing in Eq. (A.3) one has that

14π∇×

∫V

F(r′)R

d3r′ = − 14π

∫V

F(r′)×∇(

1R

)d3r′

=14π

∫V

F(r′)×∇′(

1R

)d3r′. (A.7)

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

F(r′)×∇′(

1R

)=∇′ × F(r′)

R−∇′ ×

(F(r′)R

), (A.8)

so that

14π∇×

∫V

F(r′)R

d3r′ =14π

∫V

∇′ × F(r′)R

d3r′ − 14π

∫V

∇′ ×(

F(r′)R

)d3r′

=14π

∫V

∇′ × F(r′)R

d3r′ +14π

∮S

1R

F(r′)× nd2r′

= a(r), (A.9)

which is the desired form of the vector potential.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 potentiala(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)

= − 14π∇∫

V

∇′ · F(r′)|r− r′| d

3r′ +14π∇∮

S

F(r′)|r− r′| · nd

2r′ (A.12)

References 423

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

Ft(r) = ∇× a(r)

=14π∇×∇×

∫V

F(r′)|r− r′|d

3r′

=14π∇×

∫V

∇′ × F(r′)|r− r′| d3r′ +

14π∇×

∮S

F(r′)|r− r′| × nd2r′(A.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 andEqs. (A.12)–(A.13) become

F(r) = −∇φ(r)

= − 14π∇∫

V

∇′ · F(r′)|r− r′| d

3r′, (A.14)

Ft(r) = ∇× a(r)

=14π∇×

∫V

∇′ × F(r′)|r− r′| d3r′. (A.15)

Taken together, the above results constitute what is known as Helmholtz’theorem [1].

Theorem 12. Helmholtz’ Theorem. Let F(r) be any continuous vectorfield with continuous first partial derivatives. Then F(r) can be uniquely ex-pressed in terms of the negative gradient of a scalar potential φ(r) and thecurl 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 thepair 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 ordinarymathematical sense, because if a function is zero everywhere except at a singlepoint and the integral of this function over its entire domain of definitionexists, then the value of this integral is necessarily also equal to zero. Becauseof this, it is more appropriate to regard δ(x) as a functional quantity witha certain well-defined symbolic meaning. For example, one can consider asequence 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 originand 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 Diracdelta 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 valuesof the independent variable x. However, the limit

limε→∞

∫ ∞

−∞δ(x, ε) dx = 1 (B.4)

must exist. As a consequence, one may interpret any operation that involvesthe delta function δ(x) as implying that this operation is to be performedwith a function δ(x, ε) of a suitable sequence and that the limit as ε→∞ isto be taken at the conclusion of the calculation. The particular choice of thesequence 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 followingfunctions forms a sequence with respect to the parameter ε that satisfies therequired properties.

426 B The Dirac Delta Function

δ(x, ε) =ε√πe−ε2x2

,

δ(x, ε) = rect1/ε(x),

δ(x, ε) =ε

πsinc(εx),

δ(r, ε) = circ1/ε(r),

δ(r, ε) =ε

rJ1(2πεr),

where rect1/ε(x) ≡ ε/2 when |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 = 0and is equal to its limiting value of unity when x = 0, where the last two ofthe 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

ε→∞

∫ ∞

−∞f(x)δ(x− a, ε)dx.

When the parameter ε is large, the value of the integral appearing on theright-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 thatresults from the replacement of f(x) by f(a) may be made as small as desiredby taking ε sufficiently large. Hence

limε→∞

∫ ∞

−∞f(x)δ(x− a, ε)dx = f(a) lim

ε→∞

∫ ∞

−∞δ(x− a, ε)dx,

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 integratingover all values of the variable x is equivalent to the process of evaluating thefunction at the point x = a. Notice that, for this result to hold, the domain ofintegration need not be extended over all x ∈ (−∞,∞); it is only necessarythat the domain of integration contain the point x = a in its interior, so that∫ a+∆2

a−∆1

f(x)δ(x− a)dx = f(a), (B.6)

where ∆1 > 0, ∆2 > 0. It is then seen that f(x) need only be continuous atthe 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 sameresult when integrated over any domain containing the point x = a. For thespecial case when f(x) = xk with k > 0 and a = 0, Eq. (B.7) yields

xkδ(x) = 0, ∀k > 0. (B.8)

Theorem 13. Similarity Relationship (Scaling Law). For all a = 0

δ(ax) =1|a|δ(x). (B.9)

Proof. In order to prove this relationship one need only compare the integralsof f(x)δ(ax) and f(x)δ(x)/|a| for any sufficiently well-behaved continuousfunction f(x). For the first integral one has (for any a = 0)∫ ∞

−∞f(x)δ(ax)dx = ±1

a

∫ ∞

−∞f(y/a)δ(y)dy =

1|a|f(0),

where the upper or lower sign choice is taken accordingly as a > 0 or a < 0,respectively, and for the second integral one obtains∫ ∞

−∞f(x)

1|a|δ(x)dx =

1|a|f(0).

Comparison of these two results then shows that δ(ax) = δ(x)/|a|, as was tobe 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 contin-uous 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

δ(f(x)) =∑

i

1|f ′(xi)|

δ(x− xi). (B.11)

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 whichy = 0. Under the change of variable x = f−1(y) one has that∫ ∞

−∞g(x)δ(f(x))dx =

∫R

g(f−1(y)

)δ(y)

1|f ′ (f−1(y)) |dy

=∑xi

g(f−1(0)

) 1|f ′ (f−1(0)) |

=∑

i

g(xi)1

|f ′(xi|,


where R denotes the range of f(x). In addition,∫ ∞

−∞g(x)

∑i

1|f ′(xi)|

δ(x− xi)

dx =

∑i

1|f ′(xi)|

∫ ∞

−∞g(x)δ(x− xi)d

=∑

i

g(xi)1

|f ′(xi|.

Comparison of these two expressions then proves the theorem. !

As an example, consider the function f(x) = x2−a2 which has simple zeroesat x = ±a. Then |f ′(±a)| = 2|a| so that, for a = 0,

δ(x2 − a2) =1

2|a| (δ(x− a) + δ(x+ a)) .

An additional relationship of interest that employs the Dirac delta func-tion 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 functionitself. This equation then implies that if both sides are multiplied by a con-tinuous function of either ξ or η and the result integrated over all values ofeither 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)


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 sequenceδ(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 theright-hand side of this equation both vanish because

limε→∞ δ(±∞, ε) = 0, (B.13)

with the result ∫ ∞

−∞f(x)δ′(x)dx = −f ′(0). (B.14)

Upon repeating this procedure n times for the nth-order derivative of thedelta 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 followsthat 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,


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 deltafunction are even functions and the odd-order derivatives are odd functionsof the argument.

It is often convenient to express the Dirac delta function in terms of theHeaviside unit step function U(x) that is defined by the relations U(x) = 0when x < 0, U(x) = 1 when x > 0. Consider the behavior of the derivative ofU(x). If, as before, a superscript prime denotes differentiation with respectto the argument, one obtains formally upon integration by parts (with thelimits −x1 < 0 and x2 > 0),∫ x2

−x1

f(x)U ′(x)dx = [f(x)U(x)]x2−x1

−∫ x2

−x1

f ′(x)U(x)dx

= f(x2)−∫ x2

0f ′(x)dx

= f(x2)− [f(x2)− f(0)]= f(0),

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 → −∞ andx2 → +∞, 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) = 1 and a = 0, this expression becomes∫ ∞

−∞U ′(y)dy = 1,

and U ′(x) also satisfies the property given in Eq. (B.2) which serves to par-tially 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 stepfunction with the delta function, so that

δ(x) =dU(x)dx

. (B.19)

In addition, it is seen that

U(x) =∫ x

−∞δ(ξ)dξ, (B.20)

which follows from Eqs. (B.6) and (B.19).


The Dirac delta function may also be introduced through the use of theFourier 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 func-tion sequence

K(x− a, ε) ≡∫ ε

−ε

ei2πν(x−a)dν

=sin (2π(x− a)ε)

π(x− a) (B.22)

with limitK(x− a) ≡ lim

ε→∞K(x− a, ε). (B.23)

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 itis interpreted in the sense of a Cesaro limit [4]. Upon inversion of the orderof 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) = limε→∞

∫ ∞

−∞f(x)K(x− a, ε)dx. (B.25)

Thus, the function K(x − a) satisfies the sifting property (B.5) of the deltafunction. If one sets f(x) = 1 and a = 0 in Eq. (B.24), there results∫ ∞

−∞K(x)dx = 1,

and K(x) satisfies the property given in Eq. (B.2) which serves to partiallydefine the delta function. Because K(x) = limε→∞K(x, ε) = 0 when x = 0,so that the property given in Eq. (B.1) is also satisfied, one then obtains fromEq. (B.23) the relation

δ(x) =∫ ∞

−∞ei2πνxdν. (B.26)

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

1 =∫ ∞

−∞δ(x)e−i2πνxdx, (B.27)

which also follows directly from the sifting property given in Eq. (B.5). Noticethat this relation by itself is not sufficient to imply the validity of Eq. (B.26).


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 inwhich 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 = 1xx + 1yy + 1zz is the position vector with components (x, y, z)clearly satisfies Eqs. (B.28)–(B.29) and so defines a three-dimensional Diracdelta 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 statesthat

δ(ar) =1|a|3 δ(r), (B.32)

where a is a scalar constant. The Fourier transform pair relationship ex-pressed in Eqs. (B.26)–(B.27) becomes

δ(r) =1

(2π)3

∫ ∞

−∞eik·rd3k, (B.33)

1 =∫ ∞

−∞δ(r)e−ik·rd3r, (B.34)

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 afunction ∆(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 coor-dinates (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 uniqueinverse x = f−1

1 (u, v, w), y = f−12 (u, v, w), z = f−1

3 (u, v, w). That is, an ex-pression for ∆(r) is desired in terms of the coordinate variables (u, v, w) thatsatisfies 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 transformationgiven in Eq. (B.36). If the point r = (x, y, z) is varied from r to r + δr1 bychanging the coordinate variable u to u + δu while keeping v and w fixed,then

δr1 =∂r∂uδu.

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

δr2 =∂r∂vδv.

The parallelogram with sides δr1 and δr2 then has area

δA = |δA|= |δr1 × δr2|

=∣∣∣∣ ∂r∂u × ∂r

∂v

∣∣∣∣ δuδv. (B.38)

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

δr3 =∂r∂w

δw,

and the volume of the parallelepiped with edges δr1, δr2, and δr3 is thengiven by

δV = |δr3 · (δr1 × δr2)|

=∣∣∣∣ ∂r∂w ·

(∂r∂u

× ∂r∂v

)∣∣∣∣ δuδvδw. (B.39)

The quantity

J

(x, y, z

u, v, w

)≡ ∂(x, y, z)∂(u, v, w)

≡ ∂r∂w

·(∂r∂u

× ∂r∂v

)(B.40)


is recognized as the Jacobian of the coordinate transformation of x, y, z withrespect to u, v, w. With this result for the differential element of volume, Eq.(B.37) becomes∫ ∞

−∞∆(r− r′)ϕ(u, v, w)

∣∣∣∣J ( x, y, zu, v, w

)∣∣∣∣ dudvdw = ϕ(u′, v′, w′), (B.41)

from which it is immediately seen that

δ(u)δ(v)δ(w) =∣∣∣∣J ( x, y, zu, v, w

)∣∣∣∣ δ(x)δ(y)δ(z). (B.42)

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

δ(x)δ(y)δ(z) =δ(u)δ(v)δ(w)∣∣∣J ( x,y,z

u,v,w

)∣∣∣=∣∣∣∣J (u, v, wx, y, z

)∣∣∣∣ δ(u)δ(v)δ(w), (B.43)

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 every-

where in three-dimensional space except on a surface S and is such that∫ ∞

−∞(r)ϕ(r)d3r =

∫S

ς(r)ϕ(r)d2r, (B.44)

where ς(r) is the value of (r) on the surface S, that is, when r ∈ S. Chooseorthogonal curvilinear coordinates (u, v, w) such that w = w0 describes thesurface S for some constant w0, in which case ∇w is parallel to the normalto the surface S, and is such that ∇u and ∇v are both perpendicular to thenormal to the surface S. The differential element of area of the surface S isthen given by Eq. (B.38). Furthermore, both ∂r/∂w and (∂r/∂u)× (∂r/∂v)are normal to S so that

J

(x, y, z

u, v, w

)=∣∣∣∣ ∂r∂w

∣∣∣∣ ∣∣∣∣ ∂r∂u × ∂r∂v

∣∣∣∣ . (B.45)

With this result, Eq. (B.44) may be written as∫ ∞

−∞(r)ϕ(r)

∣∣∣∣ ∂r∂w∣∣∣∣ ∣∣∣∣ ∂r∂u × ∂r

∂v

∣∣∣∣ dudvdw =∫

S

ς(r)ϕ(r)d2r,

which, with Eq. (B.38), may be expressed as∫ ∞

−∞(r)ϕ(r)

∣∣∣∣ ∂r∂w∣∣∣∣ d2rdw =

∫S

ς(r)ϕ(r)d2r. (B.46)

From this result it then follows that

References 435

(r) =ς(r)∣∣ ∂r∂w

∣∣δ(w − w0), (B.47)

which is the solution of Eq. (B.44). This result can be simplified somewhatby 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

· ∇w = 1.

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

∣∣∣∣ |∇w| = 1,

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-SidedLaplace 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) ofboth position r and time t that vanishes for t < 0 is of central importance tothe solution of problems in time-domain electromagnetics and optics and isconsidered 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 isdefined here as

Lf(r, t) ≡∫ ∞

0f(r, t)eiωtdt, (C.1)

which is simply a Fourier transform with complex angular frequaency ω thatis taken over only the positive time interval. Let f ′(r, t) be another vectorfunction 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

Lf(r, t) =∫ ∞

−∞U(t)f ′(r, t)eiωtdt, (C.3)

where U(t) = 0 for t < 0 and U(t) = 1 for t > 0 is the Heaviside unit stepfunction. For real ω, the Laplace transform of f(r, t) is then seen to be equalto the Fourier transform of U(t)f ′(r, t), viz.

Lf(r, t) = FωU(t)f ′(r, t); for real ω, (C.4)

where the subscript ω indicates that it is the Fourier transform variable. Theinverse Fourier transform of this equation then gives

U(t)f ′(r, t) = F−1Lf(r, t) (C.5)

for real ω.For complex ω, let ω = ω′ + iω′′ where ω′ ≡ ω and ω′′ ≡ ω. The

Laplace transform given in Eq. (C.1) then becomes

Lf(r, t) =∫ ∞

−∞

[U(t)f ′(r, t)e−ω′′t

]eiω′tdt

= Fω′U(t)f ′(r, t)e−ω′′t

. (C.6)

438 C The Fourier–Laplace Transform

The inverse Fourier transform of this expression then yields

U(t)f ′(r, t)e−ω′′t = F−1ω′Lf(r, t)

=

12π

∫ ∞

−∞Lf(r, t)e−iω′tdω′, (C.7)

which may be rewritten as

U(t)f ′(r, t) =12π

∫ ∞

−∞Lf(r, t)e−i(ω′+iω′′)tdω′

=12π

∫ ∞+ω′

−∞+iω′Lf(r, t)e−iωtdω

=12π

∫C

Lf(r, t)e−iωtdω. (C.8)

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) =∫ ∞

0f(r, t)eiωtdt, (C.9)

f(r, t) ≡ L−1f(r, ω)

=

12π

∫C

f(r, ω)e−iωtdω, (C.10)

where f(r, ω) is the complex temporal frequency spectrum of f(r, t) withω = ω′ + iω′′. Notice that ω′′ = ω plays a passive role in the Laplacetransform operation because it remains constant in both the forward andinverse transformations. Nevertheless, its presence can be important becausethe factor e−ω′′t appearing in the integrand of the transformation (C.9) mayserve as a convergence factor when ω′′ > 0. In particular,

f(r, ω) =∫ ∞

0f(r, t)e−ω′′teiω′tdt (C.11)

is just the Fourier transform Fω′f(r, t)e−ω′′t. The Fourier transform off(r, t) alone is

Fω′f(r, t) =∫ ∞

0f(r, t)eiω′tdt,

which exists provided that f(r, t) is absolutely integrable; viz.,

limT→∞

∫ T

0|f(r, t)| dt <∞.

If f(r, t) does not vanish properly at infinity, then the above integral failsto converge and the existence of the Fourier transform Fω′f(r, t) is notguaranteed. However, if there exists a real number γ such that

C The Fourier–Laplace Transform 439

limT→∞

∫ T

0

∣∣f(r, t)e−γt∣∣ dt <∞, (C.12)

then f(r, t) is transformable for all ω′′ ≥ γ and its temporal frequency spec-trum is given by the Laplace transform (C.9). The lower bound γa of all ofthe values of γ for which the inequality appearing in Eq. (C.12) is satisfiedis 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 relatedto the Laplace transform of f(r, t) through integration by parts as

L

∂f(r, t)∂t

=∫ ∞

0

∂f(r, t)∂t

eiωtdt

=[f(r, t)eiωt

]∞0 − iω

∫ ∞

0f(r, t)eiωtdt

= −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 allω′′ ≥ γa has been used in obtaining the final form of Eq. (C.13).

For the appropriate form of the convolution theorem for the Laplace trans-form, consider determining the function whose Laplace transform is equal tothe product f1(ω)f2(ω) = Lf1(t)Lf2(t) so that

L−1f1(ω)f2(ω)

=

12π

∫C

f1(ω)f2(ω)e−iωtdω

=12π

∫C

f1(ω)[∫ ∞

0f2(τ)eiωτdτ

]e−iωtdω

=∫ ∞

0dτ · f2(τ)

[12π

∫C

f1(ω)e−iω(t−τ)dω

],

and consequently

L−1f1(ω)f2(ω)

=∫ ∞

0f1(t− τ)f2(τ)U(t− τ)dτ,

where the unit step function U(t− τ) is explicitly included in this expressionto emphasize the fact that f1(t) vanishes for t < 0. Because U(t−τ) vanishesfor negative values of its argument, the upper limit of integration in τ mustbe t and the above equation becomes

L−1f1(ω)f2(ω)

=∫ t

0f1(t− τ)f2(τ)dτ, (C.14)

which may be rewritten as

L

∫ ∞

0f1(t− τ)f2(τ)dτ

= L f1(t)L f2(t) , (C.15)

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 thepair of relations

˜F(k, ω) ≡ FLF(r, t) =∫ ∞

−∞d3r

∫ ∞

0dt · F(r, t)e−i(k·r−ωt), (C.16)

F(r, t) ≡ F−1L−1 ˜F(k, ω) =1

(2π)4

∫ ∞

−∞d3k

∫C

dω · ˜F(k, ω)ei(k·r−ωt),

(C.17)

where k = 1xkx + 1yky + 1zkz and r = 1xx+ 1yy + 1zz. Because,

∂F(r, t)∂xj

=1

(2π)4

∫ ∞

−∞d3k

∫C

dω · ikj˜F(k, ω)ei(k·r−ωt),

from Eq. (C.17), then the transforms of the first spatial derivatives of F(r, t)are given by

FL

∂F(r, t)∂xj

= ikj

˜F(k, ω), (C.18)

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) isfound to be

FL ∇ · F(r, t) = ik · ˜F(k, ω), (C.19)

and the spatiotemporal transform of the curl of F(r, t) is given by

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 iscalled the local or effective field. In a linear isotropic dielectric, the spatiallyaveraged induced molecular charge separation is directly proportional to,and in the same direction as, the local field at that molecular site so that theaverage induced molecular dipole moment is given by

〈〈p(r, ω)〉〉 = ε0α(ω)〈〈Eeff (r, ω)〉〉 (D.1)

at the fixed angular frequency ω of the applied time-harmonic field [seeEq. (4.164)]. The molecular polarizability α(ω) characterizes the frequency-dependent linear response of the molecules comprising the dielectric to theapplied electric field.

The effective local field may be determined by removing the moleculeunder consideration while maintaining all of the remaining molecules in theirtime-averaged polarized states (time-averaging being used only to remove theeffects of thermal fluctuations), the spatially averaged electric field intensitythen 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 effectivelocal field at that molecular site is then given by the difference

〈〈Eeff (r, ω)〉〉 =1Vm

∫Vm

e(r′, ω)d3r′ − 1Vm

∫Vm

em(r′, ω)d3r′, (D.2)

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 themolecule under consideration evaluated at the point r′ ∈ Vm.

If the dielectric material is locally homogeneous (i.e., its dielectric prop-erties at any point in the material are essentially constant over a macroscopi-cally small but microscopically large region), then the first integral appearingin 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 by1/Vm if r′ ∈ Vm and 0 otherwise, then Eq. (4.1) gives

E(r, ω) ≡ 〈〈e(r, ω)〉〉 =1Vm

∫Vm

e(r− r′, ω)d3r′, (D.3)

where r ∈ Vm. With the change of variable r′′ = r − r′, the first integralappearing in Eq. (D.2) is then obtained.

442 D The Effective Local Field

z

O

qr'

R

V

dΤ1

dΤ2

Fig. D.1. Spherical cavity region V of radius R with point charge q situated adistance r′ < R from the center O.

For the second integral appearing in Eq. (D.2), consider determining theaverage electric field intensity inside a sphere of radius R containing a pointcharge q that is located a distance r′ from the center O of the sphere, asillustrated in Figure D.1. The z-axis is chosen to be along the line from thecenter O of the sphere passing through the point charge q. With this choice,symmetry then shows that the average field over the spherical volume mustbe along the z-axis. The average electric field in V is then given by the scalarquantity

ez =1V

∫V

ezd3r, (D.4)

where V is the volume of the spherical region. It is convenient to separate thisvolume integral into two parts, one taken over the spherical shell V1 betweenthe radii r′ and R, and the other taken over the inner sphere V2 of radius r′,so that

ez =1V

∫V1

ezd3r +

1V

∫V2

ezd3r. (D.5)

The integral over the spherical shell region V1 vanishes because of the equaland opposite contributions arising from the pair of volume elements dτ1 anddτ2 that are intercepted by the element of solid angle dΩ, as depicted inFigure D.1. Because the magnitude of ez decreases with the square of the

D The Effective Local Field 443

distance from the point charge q, whereas the volume element dτ = r2dΩincreases 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 fornegative q, ez is negative at dτ1 and positive at dτ2. In either case, the twocontributions to the integral over V1 cancel and that integral then vanishes.

z

q

P

O

r'

r''

Θ

V2

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

The integral of ez over the inner volume V2 is then equal to the sameintegral over the entire spherical region V . In order to evaluate this finalvolume integral, choose spherical polar coordinates (r′′, θ, ϕ) with origin atthe point charge q, as illustrated in Figure D.2. At any point P ∈ V2,

ez = e · 1z =‖4π‖4πε0

q

r′′2 cos(θ), (D.6)

so that∫V

ezd3r =

‖4π‖4πε0

q

∫ 2π

0dϕ

∫ π

π/2sin(θ) cos(θ)

[∫ −2r′ cos(θ)

0dr′′]dθ

= −‖4π‖ε0

qr′∫ π

π/2cos2(θ) sin(θ)dθ

= −‖4π‖3ε0

qr′. (D.7)

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

444 D The Effective Local Field

ez =1

(4/3)πR3

∫V

ezd3r = −‖4π‖

4πε0qr′

R3 . (D.8)

The electric dipole moment of the point charge q with reference to the centerO of the spherical region is given by p ≡ qr′1z, so that Eq. (D.8) may bewritten in the general form

e = − ‖4π‖4πε0R3 p. (D.9)

The second integral appearing in Eq. (D.2) is then given by

1Vm

∫Vm

em(r′, ω)d3r′ = − ‖4π‖4πε0r3m

pm(r, ω), (D.10)

where pm is the dipole moment of the molecule under consideration. BecauseN = 1/Vm is the local volume density of molecules, then with the assumptionthat all of the local molecules have parallel and equal polarization vectors,the macroscopic polarization is given by

P(r, ω) = N pm(r, ω)

so that the spatially averaged self-field of the molecule is given by

1Vm

∫Vm

em(r′, ω)d3r′ = −‖4π‖3ε0

P(r, ω). (D.11)

With these substitutions, Eq. (D.2) for the spatially averaged effectivefield becomes

〈〈Eeff (r, ω)〉〉 = E(r, ω) +‖4π‖3ε0

P(r, ω), (D.12)

and the local field is larger than the macroscopic electric field. This expres-sion for the effective local field was first derived by Lorentz [1] who used asomewhat different definition of the local field as the field value at the centerof 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 ClassicalLorentz Model of Resonance Polarization

With the complete Lorentz force relation as the driving force, the equationof motion of a harmonically bound electron is given by

d2rj

dt2+ 2δj

drj

dt+ ω2

j rj = −qem

(Eeff(r, t) +

1‖c‖

drj

dt×Beff(r, t)

), (E.1)

where Eeff(r, t) is the effective local electric field intensity and Beff(r, t) isthe 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 itsequilibrium position. Here qe denotes the magnitude of the charge and m themass of the harmonically bound electron with undamped resonance frequencyωj and phenomenological damping constant δj . The temporal Fourier integralrepresentation of the electric and magnetic field vectors of the effective localplane electromagnetic wave is given by

Eeff(r, ω) =∫ ∞

−∞Eeff(r, t)eiωtdt, (E.2)

Beff(r, ω) =∫ ∞

−∞Beff(r, t)eiωtdt

=‖c‖ω

k(ω)× Eeff(r, ω), (E.3)

where k(ω) is the wave vector of the plane wave field with magnitude givenby the wavenumber k(ω) = ω/c because the effective local field is essentiallya microscopic field. With the temporal Fourier integral representation

rj(ω) =∫ ∞

−∞rj(t)eiωtdt, (E.4)

the dynamical equation of motion (E.1) becomes(ω2 + 2iδjω − ω2

j

)rj =

qem

(Eeff − irj ×

(k× Eeff

))=qem

[(1 + irj · k) Eeff − i

(rj · Eeff

)k], (E.5)

with formal solution

446 E Magnetic Field Contribution to the Lorentz Model

rj =qem

[1 + irj · k

ω2 − ω2j + 2iδjω

Eeff − irj · Eeff


k

]. (E.6)

The electron displacement vector may then be expressed as a linear combi-nation of the orthogonal pair of vectors k and Eeff

rj = ajEeff + bjk, (E.7)

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

aj =rj · Eeff

E2eff

, bj =c2

ω2 rj · k. (E.8)

The pair of scalar products appearing in the above expression may be eval-uated from Eq. (E.6) as

rj · k = −i qem

rj · Eeff


k2,

rj · Eeff =qem

1 + irj · kω2 − ω2

j + 2iδjωE2

eff.

Substitution of the second relation into the first then yields

rj · k = −i (qe/mc)2E2effω

2

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2, (E.9)

and substitution of this result into the second relation gives

rj · Eeff =qe/m


×[1 +

(qe/mc)2E2effω

2

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2

]E2

eff. (E.10)

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

aj =qe/m


×[1 +

(qe/mc)2E2effω

2

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2

], (E.11)

bj = −i (qe/m)2E2eff

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2, (E.12)

respectively.

E Magnetic Field Contribution to the Lorentz Model 447

The local (or microscopic) induced dipole moment pj ≡ −qerj for the jthLorentz oscillator type is then given by [compare with Eq. (4.202)]

pj(r, ω) = −qe(ajEeff(r, ω) + bjk

). (E.13)

If there are Nj Lorentz oscillators per unit volume of the jth type, thenthe macroscopic polarization induced in the medium is given by the summa-tion over all oscillator types of the spatially averaged locally induced dipolemoments as

P(r, ω, Eeff) =∑

j

Nj〈〈pj(r, ω)〉〉

=⟨⟨

Eeff(r, ω)⟩⟩∑

j

Njαj⊥(ω, Eeff) + k∑

j

Njαj‖(ω, Eeff).

(E.14)

Here

αj⊥(ω, Eeff) ≡ α(0)j⊥(ω) + α

(2)j⊥(ω, Eeff)

=−q2e/m


×[1 +

(qe/mc)2E2effω

2

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2

](E.15)

is defined here as the perpendicular component of the atomic polarizability,with α

(0)j⊥(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

αj‖(ω, Eeff) ≡ i(q3e/m

2)E2eff

(ω2 − ω2j + 2iδjω)2 − (qe/mc)2E2

effω2

(E.16)

is defined here as the parallel component of the atomic polarizability, whereαj‖(ω, 0) = 0.

The atomic polarizability is then seen to be nonlinear in the local electricfield strength when magnetic field effects are included. However, numericalcalculations [1] show that these nonlinear terms are entirely negligible foreffective field strengths that are typically less than ∼1012 V/m and that theybegin to have a significant contribution for field strengths that are typicallygreater than ∼1015 V/m for a highly absorptive material.

The physical origin of the nonlinear term considered here is due to thediamagnetic effect that appears in the analysis of the interaction of an elec-tromagnetic field with a charged particle in the quantum theory of electrody-namics [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

H = H0 −qe

2mcH · L +

q2e8mc2

H2Z∑

j=1

r2j⊥,

in Gaussian units, where H0 is the Hamiltonian of the center of mass systemof the isolated atom with Z spinless electrons, H(r) is the magnetic fieldintensity vector with magnitude H ≡ |H|, where r⊥ is the projection ofthe 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 factorin atomic diamagnetism. The order of magnitude of this factor is given by∼ (Zq2e/12mc2)H2〈r2〉, where 〈r2〉 ∼ 1×10−16 cm2 for a bound electron. Theratio of this quantity to the level distance µBH, where µB ≡ qeh/2mc is theBohr 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 equivalentlyE ≥ 3× 1013 V/m, in agreement with the preceeding classical result that thenonlinear effects in the Lorentz model become significant for an applied fieldstrength between 1012 V/m and 1015 V/m.

In addition, nonlinear optical effects are found to dominate the linearresponse when the local field strength becomes comparable to the Coulombfield of the atomic nucleus [4]. As an estimate of this field strength, if thedistance between the nucleus and the bound electron is taken to be givenby the Bohr radius a0 ≡ h2/mq2e ≈ 5.29 × 10−9 cm, where h ≡ h/2π and his Planck’s constant, the electric field strength is E ≈ 5.13 × 1011 V/m, ingeneral agreement with the preceding estimates.

References

1. K. E. Oughstun and R. A. Albanese, “Magnetic field contribution to the Lorentzmodel,” 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, 282,439

acceleration field, 130advanced potentials, 118Ampere’s law, 63analytic signals, 402angular spectrum representation

freely propagating wave field, 332,366

geometric form, 341of radiation field, 297

asymptotic heat, 253atomic polarizability

parallel component, 447perpendicular component, 447

attenuation factor, 7auxiliary field vectors, 57

Barash, Y. S. and Ginzburg, V. L., 242baryon, 79Banos, A., 324Boltzmann’s constant KB , 197Born, M. and Wolf, E., 13, 346boundary condition

normal, 264tangential, 265, 266

Brillouin precursor, 12Brillouin, L., 1

Cauchy principal value, 183causality

dispersion relations, 183Kramers–Kronig relations, 186Plemelj formulae, 183primitive, 179, 182relativistic, 182, 260

charge densitymacroscoopic, nonconductive, 290

macroscopic (r, t), 170macroscopic bound b(r, t), 170macroscopic free f (r, t), 170macroscopic, conductive, 290microscopic ρ(r, t), 48microscopic bound ρb(r, t), 167microscopic free ρf (r, t), 167surface, 264

Clausius–Mossotti relation, 197Cole–Cole extended Rocard–Powles–

Debye modeltriply distilled water, 206

Cole–Cole plot, 199complex dielectric permittivity, 188complex index of refraction n(ω), 210,

224, 297complex permittivity, 224complex velocity, 280complex wavenumber, 7conservation laws

electromagnetic energy, 91, 236electromagnetic momentum

angular, 97linear, 93

energylocal form, 89

relativistic momentum, 76conservation of charge

microscopic, 50constitutive relations, 178

and linearity, 179general form, 179induction fields, 178primitive fields, 178

convective derivative, 72convolution theorem, Laplace trans-

form, 440cophasal surfaces, 160

450 Index

Courant condition, 18current density

conduction, 172convective (microscopic) j(r, t), 50effective, 272irrotational or longitudinal j(r, t),

114macroscopic J(r, t), 174macroscopic bound Jb(r, t), 175macroscopic free Jf (r, t), 175microscopic “bound” jb(r, t), 172microscopic free jf (r, t), 172solenoidal or transverse jt(r, t), 114surface, 266

current source J0(r, t), 277

Debye modeland rotational Brownian motion,

197, 201causality, 198effective relaxation time, 198of orientational (dipolar) polariza-

tion, 197permittivity, 198relaxation equation, 197Rocard–Powles extension, 201susceptibility, 198

Debye-type dielectric, 4Devaney, A. J., 299diamagnetic, 191diamagnetic effect, 447dielectric permittivity

complex εc(ω), 188composite model, 214free space ε0, 52of sea-water, 232of triply distilled water, 193temporally dispersive ε(ω), 184

dipolar relaxation time τm, 197dipole

Hertzian, elemental, 125linear electric, 124moment, electric, 148

dipole oscillatorelectric dipole contribution, 144electric monopole contribution, 143magnetic dipole contribution, 144

dipole radiationintermediate zone, 150

static zone, 149wave zone, 152

Dirac delta function δ(ξ), 124, 425composite function theorem, 427derivatives of, 429extension to higher dimensions, 432sifting property, 426similarity relationship (scaling law),

427direction cosines, plane wave

complex, 334real, 361

dispersionanomalous, 212normal, 212

dispersion relationsdielectric permittivity, 186electric conductivity, 188electric susceptibility, 185magnetic permeability, 192magnetic susceptibility, 192

dispersive interface, 16dissipation, 240Drude model

causality, 217damping constant γ, 217dielectric permittivity, 217electric conductivity σ(ω), 232static conductivity σ0, 232

dynamical free energy, 250

effective electric field, 196, 445effective field, 441

spatially averaged, 444effective magnetic field, 445eikonal equation

generalized, 160geometrical optics, 161

Einstein, A., 1mass–energy relation, 78special theory of relativity, 67

electric 2n-pole, 144electric conductivity σ(ω), 188electric dipole approximation, 144electric displacement vector

macroscopic D(r, t), 176microscopic d(r, t), 58

electric energy densitytime-harmonic, 244

Index 451

electric field vectormacroscopic E(r, t), 167microscopic e(r, t), 52

electric moment, 120electric susceptibility χe(ω), 184electromagnetic angular momentum

microscopic density lem, 97microscopic total lem, 97

electromagnetic beam field, 367separable, 381

electromagnetic energyirreversible, 254macroscopic

density, total U ′(r, t), 235microscopic

density u(r, t), 90total U(t), 89

reversible, 253electromagnetic energy density, 89, 235electromagnetic field, 53electromagnetic linear momentum

microscopic density pem(r, t), 93microscopic total pem, 93

electromagnetic wavefreely propagating, 329freely propagating boundary

conditions, 329plane, time-harmonic, 227source-free, 329source-free initial conditions, 387

electrostatic field, 53energy density

electromagnetic field Uem, 258reversible, 258

energy transport velocity, 21, 261energy velocity description

Sherman and Oughstun, 24equation of continuity

conduction current, 222macroscopic, 177microscopic, 51

evolved heat (dissipation) Q(r, t), 240extensive variable, 90

Faraday’s lawmicroscopic, 60

forerunner, 1Fourier integral theorem, 431Fourier transform Fω′f(r, t), 438

free-field, 387Laplace–Fourier integral representa-

tion, 390plane wave expansion, 391–405

freely propagating wave field, 329Frohlich distribution function, 205

Gabor cells, 20Galilean invariance, 62, 64Galilean transformation

electric field intensity, 62magnetic field intensity, 65

gaugeCoulomb, 113function, 111, 286invariance, 111, 286Lorenz, 113, 287radiation, 113transformation, 111, 122, 286transformation, restricted, 113, 287transverse, 113

Gauss’ lawmicroscopic

electric field, 66magnetic field, 66

Gaussian pulse dynamicsBalictsis and Oughstun, 23Garrett and McCumber, 23Tanaka, Fujiwara, and Ikegami, 23

Gaussian unitselectrostatic unit (esu) of electric

field intensity, 58gauss, 58maxwell, 60statampere, 58statcoulomb, 58statvolt, 58

Goos–Hanchen effect, 16Green’s function

free-space, 345group method

Havelock, T. H., 11group velocity, 4, 7

Hamilton, Sir W. R., 4Rayleigh, Lord, 5Stokes, G. G., 5

group velocity approximation, 11Eckart, C., 13Lighthill, M. J., 13

452 Index

Whitham, G. B., 13group velocity method, 11

half-space, positive and negative, 296Hall effect, 222Heaviside unit step function U(ξ), 430Heaviside–Poynting theorem

differential form, 235integral form, 235time-harmonic form, 244

Helmholtz equation, 7, 225, 367Helmholtz free energy (work function),

250Helmholtz’ theorem, 423Hertz

potential, 121vector, 121

Hertz, H., 119, 149Hilbert transform, 183, 185homogeneous, isotropic, locally linear

(HILL) temporally dispersivemedia

constitutive relations, 221Huygen’s principle, 345

idemfactor, 70, 94impulse response function

spatial, 342incomplete Lipschitz–Hankel integrals

Dvorak, S. and Dudley, D., 25indeterminacy principle, 19inertial reference frames, 67information diagram, 19initial field values

in a closed convex surface, 410in a sphere, 406

instantaneous (causal) spectrum, 250intensive variable, 90interaction energy, 250intrinsic impedance

complex, 228free space, 228good conductor, 231near-ideal dielectric, 230

intrinsic impedance, complex, 228inverse problems, 21isoplanatic, see space-invariant

Jackson, J. D., 14

Jordan’s lemma, 408

Kramers–Kronig relations, 186Kronecker-delta function δij , 94

Lagrange’s theorem, 140Lalor, E., 344Landau, L. D. and Lifshitz, E. M., 240,

253Laplace transform Lf(r, t), 437Lienard–Wiechert potentials, 128Lienard-Wiechert potentials

and special relativity, 135local energy theorem, 90local field, see effective fieldLorentz covariant, 68Lorentz force relation, 53Lorentz invariant, 68Lorentz model

atomic polarizability, 208Lorentz–Lorenz modified, 210magnetic field contribution, 207of resonance polarization, 207oscillator strength, 213phenomenological damping constant

δj , 207resonance frequency, undamped ωj ,

207sum rule, 213

Lorentz theory, 48Lorentz, H. A., 1Lorentz–Lorenz formula, 197Lorentz-type dielectric, 4Lorenz condition, 112, 286

generalized, 290Loudon, R., 21, 260luminiferous ether, 67luxon, 78

macroscopic polarization densityP(r, t), 171

macroscopic quadrupole momentdensity tensor Q′(r, t), 171

magnetic 2n−1-pole, 144magnetic energy density

time-harmonic, 244magnetic field vector

macroscopic B(r, t), 167microscopic b(r, t), 52

Index 453

magnetic flux, 60magnetic intensity vector

macroscopic H(r, t), 176microscopic h(r, t), 58

magnetic permeabilitycomposite model, 215dispersive µ(ω), 190free space µ0, 52

magnetic susceptibility χm(ω), 190magnetization, macroscopic M(r, t),

175magnetostatic field, 53material relations, see cnstitutive

relations178material response

anisotropic, 181isotropic, 181nondispersive, 180spatially dispersive, 180spatially homogeneous, 179spatially inhomogeneous, 179temporally dispersive, 180temporally homogeneous, 179temporally inhomogeneous, 179

material response tensorsdielectric permittivity ε(r′, t′, r, t),

179electric conductivity σ(r′, t′, r, t), 179magnetic permeability µ(r′, t′, r, t),

179mature dispersion regime, 2, 24Maxwell stress tensor

microscopic, 94Maxwell’s displacement current, 63Maxwell’s equations

macroscopictemporal frequency-domain form,

224time-domain differential form, 176,

223microscopic, 52

time-domain differential form, 58time-domain integral form, 66

phasor form, 226source-free, 387spatial average, 167spatiotemporal form, 280

spatiotemporal frequency domainform, 389

Maxwell, J. C., 1mean polarizability αj(ω), 196mean square charge radius, 171metallic waveguides, 14microscopic Maxwell–Lorentz theory,

51Minkowski formulation, 177mksa units

ampere, 59coulomb, 59farad, 59volt, 59weber, 59, 60

modern asymptotic description, 24molecular magnetic moment, 173molecular multipole moments, 169molecular polarizability α(ω), 441monochromatic (time-harmonic) wave

field, 159

near-ideal dielectrics, 230Newton’s second law of motion

relativistic form, 75Nisbet, A. and Wolf, E., 345, 356nonlinear optical effects, 448normalized velocity, 127number density Nj , 208numerical techniques

discrete Fourier transform method,17

fast Fourier transform method, 18finite-difference time-domain method,

18Hosono’s Laplace transform method,

18

ohmic power loss, 273Olver’s saddle point method, 24optical wave field, 158order symbol O, 76Ott’s integral representation, 324

Panofsky, W. and Phillips, M., 129paramagnetic, 191penetration depth dp, 231, 269phase velocity, 5

good conductor, 231

454 Index

near-ideal dielectric, 230phase-space asymptotic description, 17phasor representation, 225plane wave

attenuation factor α(ω), 228, 334evanescent, 339, 364homogeneous, 298, 339, 364inhomogeneous, 298, 339, 364propagation factor β(ω), 228, 334

plane wave expansionand mode expansions, 402and Poisson’s solution, 405Devaney, A. J., 399polar coordinate form, 401, 404uniqueness, 399

plasma frequency bj , 208Poincare–Lorentz transformation

relations, 69–70coordinate transformation matrix, 70electric field transformation, 84force transformation, 80invariance of Maxwell’s equations, 87magnetic field transformation, 84mass transformation, 80

Poisson’s equationscalar potential, 113

Poisson’s solution, 415polarization

left-handed, 350right-handed, 351state, 353

polarization ellipse, 347, 353polarized field

circular, 354linear, 354uniform, 353, 357

potential functions, electromagneticcomplex, 156Lienard–Wiechert, 125macroscopic, 285–291microscopic, 109–111retarded, 116

Poynting vectorcomplex S(r), 243macroscopic S(r, t), 235microscopic s(r, t), 89

Poynting’s theorem, 90Poynting–Heaviside interpretation, 91

precursor, 1Brillouin, 12formation on transmission, 16observation of

Aaviksoo, Lippmaa, and Kuhl, 29Choi and Osterberg, 29Jeong, Dawes, and Gauthier, 29Pleshko and Palocz, 12

Sommerfeld, 12principle of superposition, 55propagation factor, 7pulse centroid velocity, 29pulse diffraction, 26pulse distortion, 16

quadratic dispersion relation, 16

radiated energy, 136radiation

electromagnetic, 119reaction, 55

radiation fieldFourier–Laplace integral representa-

tion, 281, 284scalar potential, 285vector potential, 285

ray techniquesdirect-ray method, 13Felsen, L. B., 14Heyman, E., 14Melamed, T., 14space–time ray theory, 13

Rayleigh–Sommerfeld diffractionintegrals, 345

refractive indexcomplex, 210function, modified, 160

relativityNewtonian, 67special theory, 67

relaxation timesdipolar, 197distribution of, 204Drude model, mean-free path τc, 217effective, 198, 202

retardation condition, 131retarded potentials, 118retarded time, 116Riemann’s proof, 116

Index 455

Righi, A., 119Rocard–Powles–Debye model

Cole–Cole extension, 205friction time τmf , 202of triply distilled water, 203permittivity, 202susceptibility, 202

sampling theorem, 20scalar potential

macroscopic, 285microscopic, 110

semiconductor, 231Sherman expansion, see source-free

wave fieldSherman, G. C., 3, 31, 299, 342, 367SI (Systeme Internationale) units, see

mksa unitssignal

arrival, 9, 12, 15buildup, 15velocity, 2

signal velocity, 2, 7Baerwald, H., 11Brillouin, L., 7Ehrenfest, P., 7Laue, A., 7Shiren, N. S., 12Sommerfeld, A., 7sound, 13Voigt, W., 7Weber and Trizna, 12

simple magnetizable medium, 190simple polarizable dielectric, 184simultaneity, 68skin depth, 269slowly evolving wave approach, 27slowly varying envelope approximation,

13, 27Sommerfeld precursor, 12Sommerfeld’s integral representation,

322Sommerfeld, A., 1source-free wave field, 366, 371

separability, 382Sherman expansion, 374spatial series representation, 382

space-invariant, 343spatial average

of a microscopic function, 166weighting function, 166

spatial dispersion effects, 26spatial transfer function, 343spatially locally linear, 179spatiotemporal Fourier-Laplace

transform, 440special theory of relativity

dilation factor γ, 72fundamental postulate, 67longitudinal mass, 78Lorentz–Fitzgerald contraction, 73mass–energy, 78postulate of the constancy of the

speed of light, 67proper differential time interval, 72relativistic mass, 75rest energy, 77rest mass (proper mass), 75time dilation, 71transverse mass, 78

speed of light in vacuum c, 52stationary phase method

Kelvin, L., 10steepest descent method

Debye, P., 8Olver, 24

Stone, J. M., 138Stratton, J. A., 12, 91streamlines

electric, 149magnetic, 149

substantial derivative, 72superluminal pulse propagation, 28superluminal pulse velocities, 22surface of constant phase, 227symmetry property, 282

tachyon, 79test particle, 56time-average

of a periodic function, 243electromagnetic energy velocity, 257Poynting vector, 243, 249

Titchmarsh’s theorem, 183transient

anterior, 15posterior, 15

transport equation, 160

456 Index

transversality relation, 227

uniform asymptotic methodHandelsman, R. and Bleistein, N., 12

uniqueness theoremmicroscopic electromagnetic field,

101

vacuum wavenumber k0, 224vector potential

macroscopic, 285microscopic, 110

velocitycomplex, 280energy transport, 21front, 8group, 4, 260phase, 5, 260pulse centroid, 29time-average energy transport, 260,

261velocity field, 131virtual present radius vector, 135

wave vector

complex k+(ω), 331complex k±(ω), 296

wavelength λ, 147wavenumber

complex k(ω), 227, 297, 331vacuum k0, 297

wavenumber k, 147wavevector

complex part γ(ω), 294Weyl’s integral

polar coordinate form, 320rectangular coordinate form, 319

Weyl’s proof, 317Weyl’s proof , 310Weyl, H., 310Weyl-type expansion, 299Whittaker-type expansion, 299Wolf, E., 155world distance

lightlike, 70spacelike, 70timelike, 70

Yaghjian, A., 55

A Helmholtz’ Theorem€¦ · B The Dirac Delta Function B.1 The One-Dimensional Dirac Delta...

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