Show That Current Continuity is Embedded Within Maxwell s Equations

In this chapter we introduce Maxwell's equations in the time and frequency domains, examine the representation of fields by the Lorenz and Debye potentials, and look at the boundary conditions that fields need to satisfy across material interfaces and at infinity. The Lorenz and Coulomb gauge conditions are introduced and it is shown that the Coulomb gauge condition also leads to retarded fields despite the intermediate quantities becoming non-causal. A basic existence theorem for fields related to Maxwell's equations is discussed. Several theorems involving radiation conditions and multipole expansion of fields by spherical harmonics are presented. The pertinent theorems we discuss in this chapter with regards to the latter are the Wilcox theorems, the Whittaker theorem, the theorem of Bouwkamp and Casimir, and spherical harmonics expansion theorems. All of these theorems have a bearing on the field representations carried out external to the sphere enclosing all sources.

Maxwell's equations for linear and simple media ( ${\boldsymbol{D}}=\epsilon {\boldsymbol{E}}$ , ${\boldsymbol{B}}=\mu {\boldsymbol{H}}$ , and and μ are scalars independent of the field¹ ) can be written as

Equation (1.1)

Equation (1.2)

where ( ${\boldsymbol{E}},\,{\boldsymbol{H}}$ ) are the real-valued, time-instantaneous electric (V m⁻¹) and magnetic (A m⁻¹) field strengths, $({\boldsymbol{D}},\,{\boldsymbol{B}})$ are the real-valued, time-instantaneous electric flux (C m⁻²) and magnetic flux (Wb m⁻²) densities, $(\epsilon ,\,\mu )$ are the permittivity (F m⁻¹) and permeability (H m⁻¹) of the medium, $({\boldsymbol{J}},{\boldsymbol{M}})$ are the real-valued, time-instantaneous electric (A m⁻²) and magnetic (V m⁻²) volume current densities, and $({q}_{\mathrm{ev}},{q}_{\mathrm{mv}})$ are the real-valued, time-instantaneous electric (C m⁻³) and magnetic (V sm⁻³) volume charge densities, respectively. The electric current density is assumed to comprise both the conduction current density and the source current density. The equations of continuity relate the current and charge densities:

Equation (1.3)

and are implied by Maxwell's equations. For field computations it is very advantageous to introduce intermediate field quantities called the scalar and vector potentials. The four relevant quantities are the electric and magnetic scalar potentials ${\psi }_{{\rm{e}}}$ (V) and ${\psi }_{{\rm{m}}}$ (A), respectively, and the electric and magnetic vector potentials ${\boldsymbol{F}}$ (V m⁻¹) and ${\boldsymbol{A}}$ (Wb m⁻¹), respectively. For example, in the absence of magnetic current density, the fields can be related to the magnetic vector potential ${\boldsymbol{A}}$ and the scalar electric potential ${\psi }_{{\rm{e}}}$ via

Equation (1.4)

In a similar fashion the fields in the absence of electric currents may be related to the electric vector potential ${\boldsymbol{F}}$ and magnetic scalar potential ${\psi }_{{\rm{m}}}$ via

Equation (1.5)

1.1.1. The Lorenz gauge, Coulomb gauge, and causality

We explore equation (1.4) further in the determination of fields from potentials. Writing ${\boldsymbol{D}}={\epsilon }_{0}{\boldsymbol{E}}+{{\boldsymbol{P}}}_{{\rm{e}}}$ and ${\boldsymbol{B}}={\mu }_{0}({\boldsymbol{H}}+{{\boldsymbol{P}}}_{{\rm{m}}})$ , where ${{\boldsymbol{P}}}_{{\rm{e}}}$ (C m⁻²) and ${{\boldsymbol{P}}}_{{\rm{m}}}$ (A m⁻¹) represent electric and magnetic polarization vectors, respectively, it is easy to see from Maxwell's equations that the potentials satisfy the equations

Equation (1.6)

Equation (1.7)

where $c=1/\sqrt{{\mu }_{0}{\epsilon }_{0}}$ is the speed of the wave in a vacuum. Additional constraints must be placed on the potentials ${\boldsymbol{A}}$ and/or ${\psi }_{{\rm{e}}}$ through any one of the gauge conditions to decouple and determine them uniquely.

In the Lorenz gauge

Equation (1.8)

and the potentials satisfy the non-homogeneous wave equations

Equation (1.9)

Equation (1.10)

where ${\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times {\boldsymbol{A}}={\rm{\nabla }}{\boldsymbol{\nabla }}\cdot {\boldsymbol{A}}-{{\boldsymbol{\nabla }}}^{2}{\boldsymbol{A}}$ and ${\square }^{2}=\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}-{{\rm{\nabla }}}^{2}$ is the d'Alembertian operator. The sources for the potentials are impressed and the polarization charges and currents, and the d'Alembertian operator carries the signal speed c at which the source fluctuations are transmitted. Since the solution of a wave equation is causal² , the potentials ${\psi }_{{\rm{e}}}$ and ${\boldsymbol{A}}$ (as will be shown in example 1.1.1.1 below) are both retarded functions³ of the sources, which, in turn, generate retarded electric and magnetic fields. Under the Lorenz gauge condition, the complete electromagnetic field in any region, including the source region, is completely determined in terms of three scalar functions out of the four functions $({\boldsymbol{A}}\,{\psi }_{{\rm{e}}})$ .

In the Coulomb gauge

Equation (1.11)

and the potentials satisfy

Equation (1.12)

Equation (1.13)

The scalar potential satisfies Poisson's equation, wherein it responds instantaneously to the charge density fluctuations at any distance as evidenced by the presence of the purely spatial Laplacian operator in equation (1.12). The presence of the time-instantaneous term proportional to $\frac{\partial {\psi }_{{\rm{e}}}}{\partial t}$ on the right-hand side (rhs) of equation (1.13) raises the question of whether the magnetic vector potential is also non-causal. If that were true then the entire electromagnetic field would be non-causal. If the total current density (impressed plus polarized) is decomposed⁴ into longitudinal and solenoidal parts as

Equation (1.14)

such that ${\boldsymbol{\nabla }}\times {{\boldsymbol{J}}}_{\ell }=0$ and ${\boldsymbol{\nabla }}\cdot {{\boldsymbol{J}}}_{\mathrm{so}}=0$ , then the equation of continuity together with equation (1.12) yields

Equation (1.15)

where the latter is due to the definition of ${{\boldsymbol{J}}}_{\ell }$ and the identity ${\boldsymbol{\nabla }}\times {\rm{\nabla }}{\psi }_{{\rm{e}}}=0$ . It is seen that both the divergence and the curl of the quantity within parenthesis is zero, suggesting that the quantity is a constant vector. Since the only constant function that vanishes at infinity is the constant function zero we have

Equation (1.16)

which reduces equation (1.13) to the non-homogeneous wave equation

Equation (1.17)

Thus the Coulomb gauge once again results in a retarded solution for the quantities ${\boldsymbol{A}},\,{\boldsymbol{B}},\,{\boldsymbol{H}}$ in terms of the solenoidal current ${{\boldsymbol{J}}}_{\mathrm{so}}$ and polarization current ${\boldsymbol{\nabla }}\times {{\boldsymbol{P}}}_{{\rm{m}}}$ . Since the electric field is proportional to the gradient of ${\psi }_{{\rm{e}}}$ , but not ${\psi }_{{\rm{e}}}$ itself, we will demonstrate in the example below that the electric field also remains retarded relative to the sources.

Example 1.1.1.1Causality in the Coulomb gauge.

Show that the electric field generated under the Coulomb gauge in an unbounded medium remains causal and that it coincides with the field produced under the Lorenz gauge [2].

The starting point is the expression for the causal Green's function, G _a, in spacetime for the d'Alembert's equation

Equation (1.18)

It is easy to derive this expression by solving the d'Alembert differential equation either in the spacetime [3, p 838] or in wavenumber-time [4, p 267] domains. For example, the steps in the spacetime approach are as follows:

(i)

Use the simpler representation $\delta ({\bf{r}}-{\bf{r}}^{\prime} )=\delta (R)/2\pi {R}^{2}$ from equation (18.29) for the spatial delta function. Note that ${\int }_{0}^{\infty }\delta (R)\,{\rm{d}}R=1/2$ .
(ii)

Write ${G}_{{\rm{a}}}=g/R$ and use spherical coordinates $(R,\theta ,\phi )$ to reduce the d'Alembert equation to

where $u=t-\tau$ . This is now a simple 1D wave equation in spacetime .
(iii)

Employ the change of variables $\xi =u-R/c$ , $\eta =u+R/c$ and the corresponding Jacobian relation ${\rm{d}}{t}{\rm{d}}{R}={\rm{d}}{u}{\rm{d}}{R}=(c/2){\rm{d}}\xi {\rm{d}}\eta$ .
(iv)

Using equation (18.118), the effect of distribution $\delta (R)\delta (u)/2\pi R$ (whose mapping into the $\xi \eta$ domain is simply denoted by $\{\cdot \}$ in the ensuing material) on a testing function $\varphi (R,u)$ is

However, since the singularity of the distribution is all concentrated at the origin

Therefore

which implies that the distribution $\delta (R)\delta (u)/2\pi R$ is transformed as
(v)

The operator in equation (1.19) is transformed to

resulting in the equation
(vi)

Finally, integrating this equation and retaining only the non-zero terms yields the desired Green's function

where ${\rm{\Theta }}(\cdot )$ is the unit step function.

Clearly, in the case of d'Alembert's equation, an impulse excited at the point ${\bf{r}}^{\prime}$ at epoch $t=\tau$ is only felt at a later time $t=\tau +R/c\gt \tau$ at an ${\bf{r}}$ that is located at a distance R from ${\bf{r}}^{\prime}$ . The solution is thus causal and is a retarded function. By linearity the solution of equation (1.17) with an arbitrary rhs leads to the solution

Equation (1.21)

where the prime on the curl indicates that it is w.r.t. the primed coordinates and the subscript 'c' on ${\boldsymbol{A}}$ qualifies that the expression is for the Coulomb gauge. It is seen that vector potential remains causal under the Coulomb gauge condition.

When specialized to $c\to \infty$ , the expression for Green's function in equation (1.18) also gives the Green's function, G _ℓ, in spacetime for the Laplace equation involved in equation (1.12):

Equation (1.22)

In this case an impulse excited at $({\bf{r}}^{\prime} ,\tau )$ is felt instantly at any ${\bf{r}}$ no matter how far its distance from ${\bf{r}}^{\prime}$ is. By linearity this leads to the non-causal solution for the scalar potential in equation (1.12):

Equation (1.23)

where the subscript 'c' on ${\psi }_{{\rm{e}}}$ once again qualifies that the expression is for the Coulomb gauge. For ease we use the notation $[\cdot ]$ such that the quantity within the square brackets is evaluated at a retarded time $\tau =t-R/c$ . According to this notation, $[{{\boldsymbol{J}}}_{\mathrm{so}}]={{\boldsymbol{J}}}_{\mathrm{so}}({\bf{r}}^{\prime} ;\tau =t-R/c)$ . Note also that the time derivatives $\partial /\partial t=\partial /\partial \tau$ . Under this notation, equation (1.21) may be rewritten as

Equation (1.24)

where the definition of ${{\boldsymbol{J}}}_{\mathrm{so}}$ given in equation (1.14) and the expression for ${{\boldsymbol{J}}}_{\ell }$ given in equation (1.16) were used in obtaining the rhs above. The time derivative of this expression is

Equation (1.25)

Since the electric field ${\boldsymbol{E}}$ is equal to $-{\rm{\nabla }}{\psi }_{\mathrm{ec}}-\partial {{\boldsymbol{A}}}_{{\rm{c}}}/\partial t$ from equation (1.4), let us explore the behavior of the gradient of the scalar potential term. By adding a term $-(1/{c}^{2}){\partial }^{2}{\psi }_{\mathrm{ec}}/\partial {t}^{2}$ to the left-hand side (lhs) of equation (1.12) and then taking its negative gradient we obtain the non-homogeneous d'Alembert's equation,

Equation (1.26)

for ${\rm{\nabla }}{\psi }_{\mathrm{ec}}$ . Solving this formally using the Green's function (1.18) leads to the integral equation for the unknown ${\rm{\nabla }}{\psi }_{\mathrm{ec}}$

Equation (1.27)

The last integrals on the rhs of equations (1.25) and (1.27) are equal and opposite and will cancel with each other when added. Hence the expression for electric field under Coulomb gauge is

Equation (1.28)

clearly indicating that the electric field is a retarded function of the sources.

To show that this is identical to the electric field derived under the Lorenz gauge condition, note that the first integral on the rhs of equation (1.28) is identical to what would be obtained from the vector potential equation (1.10). The explicit solution for the vector potential equation (1.10) under the Lorenz gauge condition is

Equation (1.29)

where the subscript 'l' is a qualifier for the Lorenz gauge. The solution of the scalar potential under the Lorenz gauge condition in equation (1.9) is

with a gradient

Equation (1.30)

assuming that the charge density and electric polarization have finite support to make the surface integral at infinity vanish. The conversion from the volume to surface integral of the last integral on the rhs above is due to the gradient theorem (B.7). The final expression in equation (1.30) is identical to the last integral on the rhs in equation (1.28), both of which are causal in nature. Clearly, the electromagnetic fields derived under the Coulomb gauge coincide with those obtained under the Lorenz gauge despite the fact that the scalar potential under the former fails to be causal. Thus use of the Coulomb gauge does not violate finite speed of signal propagation for the electromagnetic fields. ■■

1.1.2. Existence theorem for fields

At this point it is appropriate to state without proof an existence theorem for fields from given charge conservation.

Theorem 1.1.1.Existence theorem for fields [5].

Given localized sources ${q}_{{\rm{e}}}({\bf{r}};t)$ and ${\boldsymbol{J}}({\bf{r}};t)$ which satisfy the equation of continuity

Equation (1.31)

there exist retarded fields ${\boldsymbol{F}}({\bf{r}};t)$ and ${\boldsymbol{G}}({\bf{r}}:t)$ defined by

Equation (1.32)

Equation (1.33)

that satisfy the field equations

Equation (1.34)

Equation (1.35)

where the constants $\alpha ,\,\beta ,\,\gamma ,$ and c (not necessarily the speed of light) are related by $\alpha =\beta \gamma {c}^{2}$ , $\hat{{\bf{R}}}={\bf{R}}/R=({\bf{r}}-{\bf{r}}^{\prime} )/\left|{\bf{r}}-{\bf{r}}^{\prime} \right|$ , and the square brackets $[\cdot ]$ indicate that the quantity enclosed is to be evaluated at retarded time $t^{\prime} =t-R/c$ .

The proof is straightforward but lengthy and the reader is referred to [5] for details. Identifying ${\boldsymbol{J}}$ with electric current density, q _e with electric charge density, c with the speed of light in vacuum, ${\boldsymbol{F}}$ with electric field, ${\boldsymbol{G}}$ with magnetic flux density, γ with 1, α with ${\epsilon }_{0}^{-1}$ , β with ${\mu }_{0}$ , we arrive at Maxwell's equations in free-space given the equation of continuity. In other words, Maxwell's equations can be obtained by postulating charge conservation at the outset. Thus the equation of continuity assumes a fundamental status as a precursor to Maxwell's equations, rather than as an induced result from Maxwell's equations.

Assuming an ${{\rm{e}}}^{j\omega t}$ convention in the time variable t, where ω is the radian frequency, Maxwell's equations for simple media can be written as

Equation (1.36)

Equation (1.37)

where ( ${\bf{E}},\,{\bf{H}}$ ) are the complex-valued, phasor electric, and magnetic field strengths, $({\bf{D}},\,{\bf{B}})$ are the complex-valued, phasor electric, and magnetic flux densities, $(\epsilon (\omega ),\,\mu (\omega ))$ are the permittivity and permeability of the medium, respectively, $({\bf{J}},\,{\bf{M}})$ are the complex-valued, phasor electric, and magnetic current densities, $({q}_{\mathrm{ev}},\,{q}_{\mathrm{mv}})$ are the complex-valued, phasor electric, and magnetic charge densities and $j=\sqrt{-1}$ . The relation between a phasor field quantity and the original time-dependent field quantity, illustrated below for the electric field, is

Equation (1.38)

where ${\mathfrak{R}}$ stands for 'real part of'.

The media constants can change from point to point in space in general. To accommodate lossy media, the permittivity is assumed to be complex of the form $\epsilon =\epsilon ^{\prime} -j\epsilon ^{\prime\prime}$ with $\epsilon ^{\prime} ,\epsilon ^{\prime\prime} \gt 0$ . Similarly, the permeability can be made complex by assuming $\mu =\mu ^{\prime} -j\mu ^{\prime\prime}$ with $\mu ^{\prime} ,\mu ^{\prime\prime} \gt 0$ . For simple media, Maxwell's equations also imply the equations of continuity:

Equation (1.39)

By taking ${\boldsymbol{\nabla }}\times$ of a Maxwell curl equation in (1.36) or (1.37) and substituting from the other, we obtain the non-homogeneous, vector Helmholtz equations

Equation (1.40)

Equation (1.41)

satisfied by the electric and magnetic fields, where $k=\omega \sqrt{\mu \epsilon }={k}_{r}-{{jk}}_{i},{k}_{r}\gt 0,{k}_{i}\gt 0$ is the wavenumber at the frequency ω in the medium. By observing the rhs of equations (1.40) and (1.41) it is clear that ${\bf{J}}$ and ${\boldsymbol{\nabla }}\times {\bf{M}}/j\omega \mu$ generate the same electric field in the sense that in an expression relating ${\bf{E}}$ to ${\bf{J}}$ one can replace ${\bf{J}}$ with ${\boldsymbol{\nabla }}\times {\bf{M}}/j\omega \mu$ and obtain the expression for the electric field generated by the magnetic current ${\bf{M}}$ . Hence ${\bf{J}}$ and ${\boldsymbol{\nabla }}\times {\bf{M}}/j\omega \mu$ are equivalent. Likewise the current densities $-{\bf{M}}$ and ${\boldsymbol{\nabla }}\times {\bf{J}}/j\omega \epsilon$ are equivalent. For instance, an azimuthal electric current density of the form ${\bf{J}}=j\omega \epsilon \hat{{\boldsymbol{\phi }}}{J}_{\phi }(\rho )$ in cylindrical coordinates is equivalent to a vertical magnetic current density ${\bf{M}}=-{\boldsymbol{\nabla }}\times (\hat{{\boldsymbol{\phi }}}{J}_{\phi })=\hat{{\bf{z}}}\left(\frac{\partial {J}_{\phi }}{\partial \rho }+\frac{1}{\rho }{J}_{\phi }\right)$ .

As with time-domain equations, magnetic vector potential ${\bf{A}}$ and electric vector potential ${\bf{F}}$ (the so-called Lorenz potentials) are introduced such that ${\bf{B}}={\boldsymbol{\nabla }}\times {\bf{A}},{\bf{D}}=-{\boldsymbol{\nabla }}\times {\bf{F}}$ . The general relationship between the field quantities and the Lorenz potentials are

Equation (1.42)

Equation (1.43)

There is complete duality between the electric and magnetic quantities, which will be elaborated upon in the duality principle (see section 6.1.1).

In an infinite homogeneous medium carrying electric and magnetic currents $({\bf{J}},\,{\bf{M}})$ , the magnetic vector potential ${\bf{A}}$ is generally associated with the electric current density and the electric vector potential ${\bf{F}}$ is associated with the magnetic current density. Consequently, the complete field can be expressed in terms of three scalar functions (the three components of ${\bf{A}}$ or the three components of ${\bf{F}}$ ) when only one of these two sources is present. However, the freedom provided by a particular choice of gauge can potentially reduce the required scalar functions to two. For instance, under Coulomb gauge ${\boldsymbol{\nabla }}\cdot {\bf{A}}=0$ and one component of ${\bf{A}}$ can be determined from the other two. Additional gauge conditions under which the complete electromagnetic field can be expressed in terms of only two scalar functions is discussed in section 3.1.2. Under the Lorenz gauge condition ${\boldsymbol{\nabla }}\cdot {\bf{A}}=-j\omega \mu \epsilon {\psi }_{{\rm{e}}},\,{\boldsymbol{\nabla }}\cdot {\bf{F}}=-j\omega \mu \epsilon {\psi }_{{\rm{m}}}$ , and the vector potential satisfies ${{\rm{\nabla }}}^{2}{\bf{A}}+{k}^{2}{\bf{A}}=-\mu {\bf{J}}$ , where ${\psi }_{{\rm{e}}}$ and ${\psi }_{{\rm{m}}}$ are time-harmonic electric and magnetic scalar potentials. One consequently obtains

Equation (1.44)

Equation (1.45)

These expressions are valid at all points in space including the source region. On adapting the solution (1.18) for each component of ${\bf{A}}$ in the time-harmonic case it is easy to see that the solution of the vector Helmholtz equation

Equation (1.46)

with the dipole source, where the unit vector $\hat{{\bf{q}}}$ is an arbitrary constant vector representing the dipole orientation and the quantity p is the dipole moment (units (A m)), is

Equation (1.47)

where $R=\left|{\bf{r}}-{\bf{r}}^{\prime} \right|$ is the distance between the source and field points. Using equation (1.29), an expression for ${\bf{A}}$ for an arbitrary electric current distribution radiating in a homogeneous medium with parameters $(\epsilon ,\,\mu )$ can be written as

Equation (1.48)

Thus causality in the time-domain with a delay ${t}_{{\rm{d}}}=R/c$ is manifested in the frequency domain by the appearance of the phase lag factor ${{\rm{e}}}^{-{jkR}}$ .

In the far-zone where the approximations $R\approx r-{\bf{r}}^{\prime} \cdot \hat{{\bf{r}}}$ in the phase term and $R\approx r$ in the denominator of the integrand hold, the expression for the vector potential reduces to

Equation (1.49)

where $\widetilde{{\bf{J}}}({\boldsymbol{\kappa }})$ is the 3D vector Fourier transform of the current distribution as defined in equation (1.149). Thus the vector potential in the far-zone is proportional to the Fourier transform of the current distribution and the uncertainty relations associated with Fourier transforms hold in the space and wavenumber domains. For the θ and ϕ components of the electric field in the far-zone, only the first term on the rhs of equation (1.44) remains. The dominant electric field components in the far-zone are then

Equation (1.50)

Thus the transverse components of $\widetilde{{\bf{J}}}({\boldsymbol{\kappa }})$ on the circle $\left|{\boldsymbol{\kappa }}\right|=k$ are completely determined by the far-zone electric field components and vice versa.

Example 1.2.0.1.Decomposition of current distribution by the Helmholtz theorem.

In this example we apply the Helmholtz theorem (see theorem B.2.6) and show how to decompose a compact-support current density into a solenoidal part and an irrotational part. We do this for electric current density by assuming that there are no magnetic sources (i.e. the magnetic current and charge densities are both zero, ${\bf{M}}=0,\,{q}_{\mathrm{mv}}=0$ ). The goal is to see how the decomposition of current by the Helmholtz theorem translates to the properties of the electromagnetic field [6]. Furthermore, as already seen in section 1.1.1, the proof that the Coulomb gauge gives rise to retarded fields relies on this decomposition.

By the Helmholtz theorem, the electric current density ${\bf{J}}$ can be expressed as

Equation (1.51)

where ${{\bf{J}}}_{{\mathscr{l}}}={\rm{\nabla }}{\boldsymbol{\theta }}$ is the longitudinal part and ${{\bf{J}}}_{{\rm{t}}}={\boldsymbol{\nabla }}\times {\bf{w}}$ is the transverse part. Here ${\bf{J}}$ consists of both the impressed currents ${{\bf{J}}}_{{\rm{i}}}$ (charge density ${q}_{\mathrm{ev}}=-{\boldsymbol{\nabla }}\cdot {{\bf{J}}}_{{\rm{i}}}/j\omega$ ) and polarization currents ${{\bf{J}}}_{{\rm{p}}}=j\omega {\bf{P}}$ , ${\bf{P}}$ being electric polarization, so that the divergence equation becomes

Equation (1.52)

by virtue of the equation of continuity. Hence it is possible to express the electric field in a material medium as

Equation (1.53)

and

Equation (1.54)

where ${{\bf{A}}}_{{\rm{c}}}$ is the magnetic vector potential and ${\bf{B}}={\boldsymbol{\nabla }}\times {{\bf{A}}}_{{\rm{c}}}$ is the magnetic flux density. Since ${\boldsymbol{\nabla }}\cdot {{\bf{E}}}_{{\rm{t}}}=0$ , the Coulomb gauge condition ${\boldsymbol{\nabla }}\cdot {{\bf{A}}}_{{\rm{c}}}=0$ is implied in this decomposition. The longitudinal current generates an electric field, ${{\bf{E}}}_{{\mathscr{l}}}$ , whose curl vanishes, thereby resembling an electrostatic field.

Expressing ${\bf{B}}={\mu }_{0}({\bf{H}}+{{\bf{P}}}_{{\rm{m}}})$ , where ${{\bf{P}}}_{{\rm{m}}}$ is the magnetization vector, and using Maxwell's equation ${\boldsymbol{\nabla }}\times {\bf{H}}={\mu }_{0}^{-1}{\boldsymbol{\nabla }}\times {\bf{B}}-{\boldsymbol{\nabla }}\times {{\bf{P}}}_{{\rm{m}}}$ = $j\omega ({\epsilon }_{0}{\bf{E}}+{\bf{P}})+{{\bf{J}}}_{{\rm{i}}}$ = $j\omega {\epsilon }_{0}{\bf{E}}+{\bf{J}}$ we obtain

Equation (1.55)

Thus the transverse current together with the magnetization vector generates the vector potential ${{\bf{A}}}_{{\rm{c}}}$ and, consequently, the magnetic field. Clearly, the magnetic vector potential still satisfies the Helmholtz equation under the Coulomb gauge condition and, thereby, generates a retarded solution as also shown in section 1.1.1. The magnetic vector potential under the Coulomb gauge condition contributes to the transverse electric field and is responsible for the radiated field. If the transverse current is zero, both ${{\bf{A}}}_{{\rm{c}}}$ and the radiated field vanish. ■■

1.2.1. Classification of media

It is most convenient to classify media in the frequency domain because the relations involving them turn out to be algebraic rather than of convolutional type. The most general linear medium is described in terms of four matrices or dyadics, $\bar{\bar{\epsilon }},\bar{\bar{\xi }},\bar{\bar{\zeta }}$ , and $\bar{\bar{\mu }}$ . They relate the flux densities to the field quantities [7]

Equation (1.56)

Equation (1.57)

Whether one regards them as matrices or dyadics, note that the medium parameters have to be placed anterior to the field variables. The most general linear medium above is termed bianisotropic or magnetoelectric. If the medium parameters are all scalars, then the medium is called bi-isotropic. Special cases of bi-isotropic media are the isotropic chiral medium with $\xi =-\zeta$ and the Tellegen medium $\xi =\zeta$ . A chiral medium can be produced by inserting suitable base material particles with specific handedness, i.e. particles whose mirror image cannot be brought into coincidence with the original particles; for example, helical particles (for a detailed discussion see [8, 9]). A Tellegen medium can be produced by combining permanent electric and magnetic dipoles in similar parallel pairs and making a mixture with such particles. If $\bar{\bar{\xi }}=0=\bar{\bar{\zeta }}$ and $\bar{\bar{\epsilon }}$ and $\bar{\bar{\mu }}$ are scalars, then the medium is simply isotropic. If the medium dyadics are functions of frequency, the medium is said to be time-dispersive or simply dispersive.

For a non-dispersive Tellegen medium, writing $\zeta =\xi =\chi /c$ , $\mu ={\mu }_{0}{\mu }_{{\rm{r}}},\,\epsilon ={\epsilon }_{0}{\epsilon }_{{\rm{r}}}$ , and ${\alpha }_{{\rm{T}}}=({\mu }_{{\rm{r}}}{\epsilon }_{{\rm{r}}}-{\chi }^{2})$ , where c is the speed of light in a vacuum, it can be shown that with the modified Lorenz condition

Equation (1.58)

the Lorenz potentials satisfy the decoupled d'Alembertian equations

Equation (1.59)

Equation (1.60)

in the presence of electric sources $({\boldsymbol{J}},\,{q}_{\mathrm{ev}})$ .

1.2.1.1. The plasmonic medium and Drude model

Metals behave as good conductors at radio frequencies, but have other interesting properties at optical frequencies in that the real part of permittivity turns negative at sufficiently high frequencies. A simple approach to describe the effect of the free electrons in a metal is to approximate the metal as a homogeneous domain with a complex dielectric permittivity described by the Drude model [10]. The equation of motion of a bound electron under the action of an applied electromagnetic field is

Equation (1.61)

where ${{\bf{E}}}_{\mathrm{eff}}$ is the local electric field intensity that acts on the electron of charge q _e and effective mass m, γ is the damping constant, and ${\omega }_{0}$ is the undamped resonance frequency of the transverse vibrational mode of the ionic lattice structure. The Drude model is derived based on setting the oscillating force term $m{\omega }_{0}^{2}{\bf{r}}$ to zero for time-harmonic excitation:

Equation (1.62)

where ${\omega }_{{\rm{p}}}=\sqrt{{{Nq}}_{{\rm{e}}}^{2}/m{\epsilon }_{0}}$ is known as the plasma frequency, N is the number of free electrons per unit volume, and ${\epsilon }_{\infty }$ is the relative permittivity at infinite frequency. Commonly used values are ${\epsilon }_{\infty }=5,\,{\omega }_{{\rm{p}}}=1.443\,3\times {10}^{16}\,$ rad s⁻¹, and $\gamma ={10}^{14}$ s⁻¹. Figure 1.1 shows a plot of the real and imaginary parts of the complex relative permittivity of the metal as a function of free-space wavelength ${\lambda }_{0}$ . Note that the real part ${\epsilon }_{\mathrm{rs}}$ becomes negative for wavelengths above 300 nm. The upper horizontal axis is the normalized wavelength ${\lambda }_{0}/{\lambda }_{{\rm{p}}}$ , where ${\lambda }_{{\rm{p}}}$ is the wavelength corresponding to the plasma frequency.

**Figure 1.1.** Complex permittivity of a typical metal at optical wavelengths.

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1.2.2. Boundary conditions

Maxwell's equations must be supplemented by boundary conditions that must be satisfied by the electric and magnetic fields at material interfaces or at infinity for open domains.

1.2.2.1. Material interface conditions

If ( ${\epsilon }_{1},\,{\mu }_{1})$ and ( ${\epsilon }_{2},\,{\mu }_{2})$ are the material parameters on two sides of an interface, and $\hat{{\bf{n}}}$ is a unit normal from medium 1 to medium 2, figure 1.2, the interface conditions are⁵ :

Equation (1.63)

Equation (1.64)

Equation (1.65)

where the subscripts '1' and '2' denote the interface fields in regions 1 and 2, respectively, and the subscript 's' on the current and charge densities denotes surface values. The boundary condition (1.65) follows from applying the Gauss divergence theorem to the equation of continuity (1.31). If medium 1 is a perfect electric/magnetic conductor (PEC/PMC), then the boundary conditions reduce to

Equation (1.66)

Equation (1.67)

**Figure 1.2.** Material boundary conditions at an interface.

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Another useful interface boundary condition is an impedance boundary condition in which the tangential electric and magnetic fields in medium 2 are related through a surface impedance Z _s (complex-valued):

Equation (1.68)

Definition 1.2.1. Scalar wave function. A complex-valued scalar function u is called a scalar wave function for a domain ${ \mathcal D }$ if it is defined and of class C ² in ${ \mathcal D }$ and satisfies the scalar homogeneous Helmholtz equation ${{\rm{\nabla }}}^{2}u+{k}^{2}u=0$ at every point in ${ \mathcal D }$ .

Definition 1.2.2. Vector wave function. A complex-valued vector function ${\bf{A}}({\bf{r}})$ is called a vector wave function for a domain ${ \mathcal D }$ if it is defined and of class C ² in ${ \mathcal D }$ and satisfies the vector homogeneous Helmholtz equation ${\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times {\bf{A}}-{k}^{2}{\bf{A}}=0$ .

The electric and magnetic fields ${\bf{E}}$ and ${\bf{H}}$ in source-free regions in the frequency domain given by equations (1.40) and (1.41) are examples of vector wave functions.

1.2.2.2. Radiation, absorbing boundary conditions, and related theorems

Radiation conditions are needed to yield a unique solution to Maxwell's equations when finite extent sources radiate in an infinite medium. Radiation conditions to time-harmonic waves are what causality conditions are for time-instantaneous waves. If r denotes the radial distance from a point within the sources and if $\eta =\sqrt{\mu /\epsilon }$ is the intrinsic impedance of the infinite medium, Silver–Müller radiation conditions state that⁶ for $r\to \infty$

Equation (1.69)

where $\hat{{\boldsymbol{r}}}$ is the unit vector in the radial direction.

For a scalar function ψ satisfying the Helmholtz equation, the radiation conditions are referred to as the Sommerfeld's radiation conditions:

Equation (1.70)

If ψ satisfies the non-homogeneous scalar Helmholtz equation

Equation (1.71)

with the source function $s({\bf{r}})$ , then the only solution that satisfies the Sommerfeld's radiation condition is

Equation (1.72)

where $R=\left|{\bf{r}}-{\bf{r}}^{\prime} \right|$ is the distance between the observation and source points. Wilcox [12] has shown that the radiation condition can be expressed in a variety of equivalent statements, which we quote here without proof. For the scalar case the relevant theorem is the following.

Theorem 1.2.1.Wilcox theorem (equivalent forms of radiation condition) [12].

Let $\psi ({\bf{p}})$ satisfy the Helmholtz equation in an exterior domain ${ \mathcal D }$ . Let ${ \mathcal R }({\bf{p}},r)$ denote a sphere with a center at ${\bf{p}}$ and radius r. Let ${r}_{0}({\bf{p}})$ be the least radius r for which ${ \mathcal R }({\bf{p}},r)$ contains the sources and other material boundaries in the problem. Then the following radiation conditions (Sommerfeld radiation conditions) are equivalent for ψ; i.e. each implies all others:

RC 1: $\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }({\bf{p}},r)}\left[\frac{\partial (r\psi )}{\partial r}+{jkr}\psi \right]\,{\rm{d}}s=0$ for all p and $r\gt {r}_{0}({\bf{p}})$ .
RC 2: ${\mathrm{lim}}_{r\to \infty }\frac{1}{r}\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }({\bf{p}},r)}\left(\frac{\partial \psi }{\partial r}+{jk}\psi \right)\,{\rm{d}}s=0$ for all p.
RC 3: ${\mathrm{lim}}_{r\to \infty }\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }({\bf{p}},r)}{\left|\frac{\partial \psi }{\partial r}+{jk}\psi \right|}^{2}\,{\rm{d}}s=0$ for all p.
RC 4: ${\mathrm{lim}}_{r\to \infty }r\left(\frac{\partial \psi }{\partial r}+{jk}\psi \right)=0$ for all p, uniformly in $\hat{{\bf{r}}}$ .
RC 5: ${\mathrm{lim}}_{r\to \infty }\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }(0,r)}{\left|\frac{\partial \psi }{\partial r}+{jkr}\right|}^{2}\,{\rm{d}}s=0$ for a single center 0.

Definition 1.2.3. Scalar radiation function. Let ${\boldsymbol{D}}$ be an exterior domain. A scalar function u in the domain ${ \mathcal D }$ is called a scalar radiation function if it is a scalar wave function for ${\boldsymbol{D}}$ and, in addition, satisfies the Sommerfeld radiation condition.

Theorem 1.2.2.Whittaker theorem [13, 14].

Every scalar radiation function $\psi ({\bf{r}})$ can be represented in a plane wave expansion of the form

Equation (1.73)

where the unit vector $\hat{{\bf{s}}}=\hat{{\bf{x}}}\,\sin \,\alpha \,\cos \,\beta +\hat{{\bf{y}}}\,\sin \,\alpha \,\sin \,\beta +\hat{{\bf{z}}}\,\cos \,\alpha$ , with $\alpha \in (0,\pi )$ , $\beta \in (0,2\pi )$ , and $\hat{\psi }({\bf{s}})$ is the spectrum of the representation.

For a proof see [13]. Equation (1.73) may also be regarded as a Fourier transform relation between the space function $\psi ({\bf{r}})$ and its spectrum $\hat{\psi }(\hat{{\bf{s}}})$ . The Fourier transform is performed on a unit sphere in the wavenumber domain. For the special case of a spherical harmonic ${\psi }_{n}^{m}({\bf{r}})={j}_{n}({kr}){Y}_{n}^{m}(\theta ,\phi )$ , where ${j}_{n}({kr})$ is the spherical Bessel function of order n and argument kr, and ${Y}_{n}^{m}(\theta ,\phi )$ is the surface spherical harmonic as defined in equation (1.103), we have in view of identity (D.127) that ${\hat{\psi }}_{n}^{m}(\hat{{\bf{s}}})={k}^{-1}{j}^{n}{Y}_{n}^{m}(\alpha ,\beta )$ so that

Equation (1.74)

Thus the Fourier transform of a spherical harmonic yields another surface spherical harmonic in the space domain whose strength is governed by the spherical Bessel function along the radial direction.

Example 1.2.2.1.Radial components of EM fields as scalar radiating functions.

Determine the equations satisfied by $u={\bf{r}}\cdot {\bf{E}}$ and $\mathit{\unicode[Book Antiqua]{x76}}={\bf{r}}\cdot {\bf{H}}$ in a homogeneous medium containing electric sources only, where ${\bf{r}}=\hat{{\bf{r}}}r$ , r being the radial coordinate in a spherical coordinate system and $\hat{{\bf{r}}}$ being the unit vector along r. Obtain a solution for ${\bf{r}}\cdot {\bf{E}}$ and ${\bf{r}}\cdot {\bf{H}}$ in terms of the given current density ${\bf{J}}$ .

Expressing the various quantities in rectangular coordinates, it is straightforward to see that for an arbitrary vector ${\bf{w}}$ belonging to class C ²,

Equation (1.75)

Applying the identity (1.75) to the divergenceless vector ${\bf{w}}={\bf{E}}+{\bf{J}}/j\omega \epsilon$ and noting that ${\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times {\bf{E}}-{k}^{2}{\bf{E}}=-j\omega \mu {\bf{J}}$ from equation (1.40), we arrive at the non-homogeneous, scalar Helmholtz equation

Equation (1.76)

for the radial component of the electric field. Using equation (1.72), the solution of this equation that satisfies the Sommerfeld's radiation condition is

Equation (1.77)

Similarly, on applying the identity (1.75) to the divergenceless vector ${\bf{w}}={\bf{H}}$ and noting that ${\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times {\bf{H}}-{k}^{2}{\bf{H}}={\boldsymbol{\nabla }}\times {\bf{J}}$ from equation (1.41), we obtain the non-homogeneous Helmholtz equation

Equation (1.78)

for the radial component of the magnetic field, whose only solution satisfying the Sommerfeld's radiation condition is

Equation (1.79)

Several remarks are in order at this point.

(i)

Equations (1.77) and (1.79) are valid everywhere in space including the source region.
(ii)

If all sources are contained within a sphere of radius r ₀ such that the current density and all its derivatives vanish outside the sphere, then it is clear from equations (1.76), (1.77), (1.78), and (1.79) that for $r\gt {r}_{0}$ the radial components ${\bf{r}}\cdot {\bf{E}}={{rE}}_{{\rm{r}}}$ and ${\bf{r}}\cdot {\bf{H}}={{rH}}_{{\rm{r}}}$ are scalar radiation functions. Since the spherical harmonics of the type ${h}_{n}^{(2)}({kr}){P}_{n}^{m}(\cos \,\theta ){{\rm{e}}}^{{jm}\phi }$ are also scalar radiation functions, it should be possible to expand the radial fields $({\bf{r}}\cdot {\bf{E}})$ and $({\bf{r}}\cdot {\bf{H}})$ in terms of these harmonics for $r\gt {r}_{0}$ .
(iii)

The current density ${\bf{J}}$ could represent either impressed sources or equivalent sources arising from material scatterers embedded inside S.
(iv)

Discontinuous currents can be accommodated in the current formulation by treating the integrands in equations (1.77) and (1.79) in the sense of distributions. In such a case additional terms will arise in equations (1.77) and (1.79) that are consistent with distribution theory as outlined in section 18.1 (see theorem 18.1.4).

■■

For the vector case the Wilcox theorem for radiation conditions may be stated as follows.

Theorem 1.2.3.Wilcox theorem (equivalent forms of Silver–Müller radiation condition) [12].

Let ${\bf{A}}({\bf{p}})$ satisfy the vector Helmholtz equation

Equation (1.80)

in an exterior domain ${ \mathcal D }$ (namely ${\bf{A}}$ is a vector wave function in ${ \mathcal D }$ ). Let ${ \mathcal R }({\bf{p}},r)$ denote a sphere with a center at ${\bf{p}}$ and radius r. Let ${r}_{0}({\bf{p}})$ be the least radius r for which ${ \mathcal R }({\bf{p}},r)$ contains the sources and other material boundaries in the problem. Then the following radiation conditions are equivalent; i.e. for A , each implies all others:

RC 1: $\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }({\bf{p}},r)}\left[\frac{\partial (r{\bf{A}})}{\partial r}+{jkr}{\bf{A}}\right]\,{\rm{d}}s=0$ for all p and $r\gt {r}_{0}({\bf{p}})$ .
RC 2: ${\mathrm{lim}}_{r\to \infty }\frac{1}{r}\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }({\bf{p}},r)}\left[\hat{{\bf{r}}}\times ({\boldsymbol{\nabla }}\times {\bf{A}})-{jk}{\bf{A}}\right]\,{\rm{d}}s=0$ for all p.
RC 3: ${\mathrm{lim}}_{r\to \infty }\mathop{\unicode{x0222f}}\limits_{{ \mathcal R }(0,r)}{\left|\hat{{\bf{r}}}\times ({\boldsymbol{\nabla }}\times {\bf{A}})-{jk}{\bf{A}}\right|}^{2}\,{\rm{d}}s=0$ for a single center 0.

Based on the radiation condition a vector radiation function is defined as follows.

Definition 1.2.4. Vector radiation function. Let ${ \mathcal D }$ be an exterior domain. A vector field ${\bf{A}}({\bf{r}})$ is a vector radiation function for ${ \mathcal D }$ provided it is vector wave function for ${ \mathcal D }$ and satisfies the Silver–Müller radiation condition

Equation (1.81)

Example 1.2.2.2Examples of vector radiation functions.

Show that the vector function ${\bf{A}}$ defined by ${\bf{A}}={\boldsymbol{\nabla }}\times ({\bf{r}}u)$ , where u is a scalar radiation function and ${\bf{r}}=\hat{{\bf{r}}}r$ , with r being the radial coordinate in the spherical coordinate system, is a vector radiation function.

Using the definition of the curl operator in spherical coordinates we have

Equation (1.82)

Equation (1.83)

Equation (1.84)

where D is the Beltrami operator given in equation (1.94). Therefore

Equation (1.85)

since u is a scalar radiation function. Furthermore,

Equation (1.86)

since u satisfies Sommerfeld's radiation condition. Therefore ${\bf{A}}={\boldsymbol{\nabla }}\times ({\bf{r}}u)$ is a vector radiation function. Similarly, the associated function ${\bf{B}}={\boldsymbol{\nabla }}\times {\bf{A}}$ is also a vector radiation function. ■■

Theorem 1.2.4Wilcox representation theorem [15].

Let ${\bf{A}}({\bf{r}})$ be a vector radiation function for an exterior domain ${ \mathcal D }$ bounded by a regular surface S (namely comprised of a finite number of smooth sections) and let ${\bf{A}}({\bf{r}})$ be of class C ² in the closure of ${ \mathcal D }$ . Then

Equation (1.87)

where $\psi ({\bf{r}},{\bf{r}}^{\prime} )=\frac{{{\rm{e}}}^{-{jkR}}}{4\pi R},R=\left|{\bf{r}}-{\bf{r}}^{\prime} \right|$ , ${\rm{\nabla }}^{\prime}$ is the gradient operator with respect to the point ${\bf{r}}^{\prime}$ , and $\hat{{\bf{n}}}$ is a unit outward normal on S.

The field representation here is a special case of the Stratton–Chu integral representation (see equations (10.12) and (10.13)) in a source-free region, although Wilcox proves the theorem without resorting to the stronger assumption of $\left|\left.{\bf{A}}({\bf{r}}\right)\right|={ \mathcal O }\left(\frac{1}{r}\right),\,r\to \infty$ . A direct consequence of the representation theorem is the expansion theorem.

Theorem 1.2.5.Expansion theorem [15].

Let ${\bf{A}}({\bf{r}})$ be a vector radiation function for a region $r\gt {r}_{0}$ where $(r,\theta ,\phi )$ are spherical coordinates. Then ${\bf{A}}({\bf{r}})$ has an expansion

Equation (1.88)

which is valid for $r\gt {r}_{0}$ and which converges absolutely and uniformly in the parameters $(r,\theta ,\phi )$ in any region $r\gt {r}_{0}$ . The series can be differentiated term by term with respect to r, $\,\theta$ , and ϕ any number of times and the resulting series all converge absolutely and uniformly.

Corollary 1.2.5.1Corollary 1 to expansion theorem.

Every vector radiation function ${\bf{A}}({\bf{r}})$ has the asymptotic form

Equation (1.89)

where the vector field ${{\bf{A}}}_{0}(\theta ,\phi )$ , called the radiation pattern of ${\bf{A}}({\bf{r}})$ , is tangential to the sphere $r\,=$ constant, i.e. $\hat{{\bf{r}}}\cdot {{\bf{A}}}_{0}=0$ or ${{\bf{A}}}_{0}={A}_{0}^{\theta }\hat{{\boldsymbol{\theta }}}+{A}_{0}^{\phi }\hat{{\boldsymbol{\phi }}}$ .

Corollary 1.2.5.2.Corollary 2 to the expansion theorem.

The vector coefficients ${{\bf{A}}}_{n}(\theta ,\phi )={A}_{n}^{r}\hat{{\bf{r}}}+{A}_{n}^{\theta }\hat{{\boldsymbol{\theta }}}+{A}_{n}^{\phi }\hat{{\boldsymbol{\phi }}}$ , $n\gt 0$ of the expansion theorem are completely determined by the radiation pattern ( 1.88 ) through the recursion formulas

Equation (1.90)

Equation (1.91)

and

Equation (1.92)

Equation (1.93)

where

Equation (1.94)

is the Beltrami operator for the sphere, while D _θ and D _ϕ are first linear operators (taking vector fields ${\bf{F}}=\hat{{\bf{r}}}{F}^{r}+\hat{{\boldsymbol{\theta }}}{F}^{\theta }+\hat{{\boldsymbol{\phi }}}{F}^{\phi }$ into scalar functions) defined by

Equation (1.95)

Equation (1.96)

Example 1.2.2.3.Numerical absorbing boundary conditions.

As an example of the utility of the Wilcox expansion theorem, consider the development of absorbing boundary conditions on a sphere S _r of radius r circumscribing a scatterer. Such boundary conditions are useful in terminating computational domains of numerical algorithms and the scalar case is discussed in detail in [16]. We discuss here the vector case [17]. Let $\hat{{\bf{r}}}$ represent the unit outward normal on S _r. Exterior to S _r the scattered electromagnetic fields satisfy the vector Helmholtz equation. Let ${\bf{F}}$ denote a field vector, which could be the scattered electric field, or the scattered magnetic field, or the scattered vector potential, that takes the form given in equation (1.88) exterior to S _r. Then

Equation (1.97)

since $\hat{{\bf{r}}}\cdot {{\bf{F}}}_{0}=0$ . If the Silver–Müller type boundary condition ${B}_{{abc}1}:\ \hat{{\bf{r}}}\times {\boldsymbol{\nabla }}\times {\bf{F}}-{jk}{\bf{F}}=0$ is employed on S _r, then the residual error as seen from equation (1.97) will be of order ${ \mathcal O }({r}^{-2})$ . Judging from the form of equation (1.88) a better absorbing boundary condition is the one that removes the first term from the series expression. This results in the absorbing boundary condition ${B}_{{abc}2}:\ \hat{{\bf{r}}}\times {\boldsymbol{\nabla }}\times {\bf{F}}-{jk}{\bf{F}}+{jk}\hat{{\bf{r}}}{F}^{r}=0$ on S _r. From equation (1.97) this will result in a lower residual error⁷ of order ${ \mathcal O }({r}^{-3})$ . In fact a whole hierarchy of absorbing boundary conditions B _abcℓ may be constructed on S _r using the Wilcox expansion, which will yield successively lower residual errors of order ${ \mathcal O }({r}^{-(\ell +1)})$ , ${\ell }\geqslant 1$ . ■■

In many applications it is desirable to express the full vector fields in terms of a minimal set of scalar components. For example, in formulations involving Cartesian and cylindrical coordinates, the complete electromagnetic field in a source-free region can be expressed in terms of the z-components of the magnetic and electric vector potentials [18]. The fields produced separately by these z-components of potentials are labeled as transverse magnetic to z (TM_z) and transverse electric to z (TE_z), respectively, in the literature. In example 1.2.2.1, it was shown that the radial components of electric and magnetic fields are scalar radiation functions in an unbounded medium. These scalar functions are determined everywhere, including the source region, by the specification of electric current density. This raises the question whether the radial components of electric and magnetic fields are sufficient to describe the complete electromagnetic field in an unbounded medium. The answer to this is affirmative when one recognizes that the radial components of fields are related to the Debye potentials, as has been shown in [19]. The field generated separately by the radial component of the electric or magnetic field is labeled as transverse magnetic to r (TM_r) or transverse electric to r (TE_r), respectively. The methodology of TM_r and TE_r proves to be extremely useful in solving various problems in 3D in unbounded media. The radial components are expanded in terms of spherical harmonics and the coefficients of expansion are known as multipole strengths. The multipole strengths are completely determined by the current density specified in 3D space. In an alternative formulation involving the wavenumber space, the multipole strengths are determined by the transverse components of the spectral components of the current density. Important theorems in this regard are the theorem of Bouwkamp and Casimir [19] and the theorem of Devaney and Wolf [20].

1.3.1. Multipole expansion, Debye potentials, and related theorems

We begin by relating the radial components of fields to current density via spherical harmonics.

Example 1.3.1.1.Multipole expansion of radial electromagnetic fields.

Using the spherical harmonic expansion of the free-space Green's function, determine the multipole expansions of radial electromagnetic fields in a homogeneous medium outside the sphere containing all electric currents. (See also [19].)

Equations (1.77) and (1.79) of example 1.2.2.1 give the expressions for the radial electromagnetic fields in terms of the given electric current density ${\bf{J}}$ . Let all electric currents be contained within a sphere S of radius r ₀ so that ${\bf{J}}$ and all its derivatives vanish on S. We now employ the representation (15.21) for the free-space Green's function in the expression (1.77) for a radial electric field to obtain for $r\gt {r}_{0}$ that

Equation (1.98)

Applying the vector Green's second identity (B.15)

the radial electric field could also be expressed for $r\gt {r}_{0}$ as

Equation (1.99)

In a like manner, starting from equation (1.79) and the vector Green's first identity (B.14)

Equation (1.100)

it is easy to show that the radial magnetic field for $r\gt {r}_{0}$ can be expressed as

Equation (1.101)

Equation (1.102)

We now define the complex-valued surface spherical harmonic of order m and degree n as

Equation (1.103)

which implies on using equation (D.92) that

Equation (1.104)

where the overhead bar denotes complex conjugation. Using equation (D.116) we can then deduce the orthogonality relationship

Equation (1.105)

Defining multipole strengths ${I}_{n}^{m},{L}_{n}^{m}$ in terms of these surface harmonics as

Equation (1.106)

Equation (1.107)

Equation (1.108)

Equation (1.109)

Equation (1.110)

the radial fields in equations (1.99) and (1.102) can then be re-expressed as

Equation (1.111)

Equation (1.112)

The last form (1.108) for ${I}_{n}^{m}$ follows from the vector Green's first identity (B.14). Equations (1.111) and (1.112) provide a discrete representation of the radial electromagnetic fields outside S in terms of outgoing spherical harmonics whose strengths, ${I}_{n}^{m},{L}_{n}^{m}$ are governed by the currents flowing within the volume V. These currents could be a combination of impressed sources and induced sources on scattering objects residing in V. The radial fields ${{rE}}_{r}$ and ${{rH}}_{r}$ are both scalar radiation functions outside S. Since the surface spherical harmonics are complete for functions of class C ² (see theorem 1.3.2), and the Hankel functions are bounded for arguments different from zero, the above series converge uniformly and absolutely outside S and the terms can be differentiated any number of times.

Note that for ${\bf{P}}({\bf{r}})={\bf{r}}{j}_{n}({kr}){\bar{Y}}_{n}^{m}(\theta ,\phi )$ , we obtain

Equation (1.113)

Equation (1.114)

where ${\hat{J}}_{n}(z)={{zj}}_{n}(z)$ is the spherical Ricatti–Bessel function (see subsection C.3.8).

If the current density is all radial so that ${\bf{J}}({\bf{r}})=\hat{{\bf{r}}}{J}_{r}({\bf{r}})$ , then we obtain from equations (1.106) and (1.109) that

Equation (1.115)

For a radial electric dipole of moment ${I}_{0}{\ell }$ and located at $(b,{\theta }_{0},{\phi }_{0})$ , for instance, ${J}_{r}={I}_{0}{\ell }\delta (r-b)\delta (\theta -{\theta }_{0})\delta (\phi -{\phi }_{0})/{r}^{2}\,\sin \,\theta$ . In that case

Equation (1.116)

and the radial components of fields are

Equation (1.117)

and

Equation (1.118)

Obviously, the n = 0 terms will not influence the radial fields and may be dropped. As a check, for a radial dipole located at the origin and oriented along the z-axis, b = 0 and azimuthal symmetry implies m = 0. We then obtain upon using (i) ${P}_{n}^{0}(\cos \,\theta )={P}_{n}(\cos \,\theta )$ , (ii) ${P}_{1}(\cos \,\theta )=\cos \,\theta$ , (iii) the limit

Equation (1.119)

and (iv)

Equation (1.120)

that

Equation (1.121)

for the radial component of an electric field of a z-directed infinitesimal electric dipole located at the origin.

For a radial magnetic dipole of moment ${V}_{0}{\ell }$ located at $(b,{\theta }_{0},{\phi }_{0})$ , dual radial fields are generated, namely

Equation (1.122)

and

Equation (1.123)

■■

Theorem 1.3.1.Bouwkamp and Casimir theorem [19].

Let S be a sphere containing all sources in a homogeneous medium characterized by $(\epsilon (\omega ),\mu (\omega ))$ . The time-harmonic electromagnetic field at the frequency ω outside S is identically zero if the radial components of the electromagnetic field there are zero.

Proof.Let ${\bf{E}}=\hat{{\boldsymbol{\theta }}}{E}_{\theta }+\hat{{\boldsymbol{\phi }}}{E}_{\phi }$ and ${\bf{H}}=\hat{{\boldsymbol{\theta }}}{H}_{\theta }+\hat{{\boldsymbol{\phi }}}{H}_{\phi }$ outside S. We need to show that ${\bf{E}}\equiv 0$ and ${\bf{H}}\equiv 0$ . From source-free Maxwell's curl equations we have

Equation (1.124)

Equation (1.125)

Equation (1.126)

Introduce a change of variable such that $\sin \,\theta \partial /\partial \theta =\partial /\partial \xi$ , which implies that ${\rm{d}}\xi /{\rm{d}}\theta =1/\sin \,\theta$ or $\xi =\mathrm{ln}[(1-\cos \,\theta )/\sin \,\theta ]=\mathrm{ln}[\tan (\theta /2)]$ . Define $u(r,\xi ,\phi )=r\,\sin \,\theta {E}_{\theta }$ , $\mathit{\unicode[Book Antiqua]{x76}}(r,\xi ,\phi )=r\,\sin \,\theta {E}_{\phi }$ , $g(r,\xi ,\phi )=\eta r\,\sin \,\theta {H}_{\phi }$ , $h(r,\xi ,\phi )=-\eta r\,\sin \,\theta {H}_{\theta }$ . Then equations (1.124)–(1.126) can be rewritten as

Equation (1.127)

Equation (1.128)

Equation (1.129)

Upon writing $u={u}_{0}(\xi ,\phi ){{\rm{e}}}^{-{jkr}},\,v={v}_{0}(\xi ,\phi ){{\rm{e}}}^{-{jkr}},\,g={g}_{0}(\xi ,\phi ){{\rm{e}}}^{-{jkr}},\,h={h}_{0}(\xi ,\phi ){{\rm{e}}}^{-{jkr}}$ for waves outgoing in the exterior of S and substituting into equations (1.128)–(1.129), we obtain ${g}_{0}={u}_{0}$ and ${h}_{0}={\mathit{\unicode[Book Antiqua]{x76}}}_{0}$ . Letting ${w}_{0}={u}_{0}+{{jh}}_{0}={g}_{0}+{{jv}}_{0}$ and substituting into equation (1.127) yields

Equation (1.130)

which is the complex form of Cauchy–Riemann equations (A.4) in the complex variable $z=\phi +j\xi$ of the function ${w}_{0}(z)$ . The function w ₀ must be analytic in the whole complex plane of $z=\phi +j\xi ;$ it is bounded and periodic in ϕ with a period 2π. The fields are related to ${w}_{0}(z)$ through $({E}_{\theta }-j\eta {H}_{\theta })=j({E}_{\phi }-j\eta {H}_{\phi })={w}_{0}(\phi +j\xi ){{\rm{e}}}^{-{jkr}}/(r\,\sin \,\theta )$ . Finiteness of the fields at $\theta =0,\pi$ requires that $\left|{w}_{0}(\phi \pm j\infty )\right|\to 0$ . By Liouville's theorem [21, p 85], every bounded entire function such as ${w}_{0}(z)$ must be a constant. Since the constant has to vanish at infinity along the imaginary axis, it must be identically zero. Hence ${w}_{0}(z)\equiv 0\Longrightarrow {E}_{\phi }={E}_{\theta }={H}_{\phi }={H}_{\theta }\equiv 0$ and we arrive at the proof that ${\bf{E}}\equiv 0$ , ${\bf{H}}\equiv 0$ outside S if $\hat{{\bf{r}}}\cdot {\bf{E}}$ and $\hat{{\bf{r}}}\cdot {\bf{H}}$ are zero there.

We can draw the following conclusions from this theorem:

(i)

The theorem indicates that it is not possible to have a purely spherical TEM wave ( ${E}_{r}=0,{H}_{r}=0$ ) outside the circumscribing sphere of any radiator. In particular, it is not possible to design an antenna that generates zero radial fields. The quest to design antennas with minimal radial components forms the subject of electrically small, wideband antennas.
(ii)

From equations (1.107) and (1.110) it is clear that if the current density satisfies $\hat{{\bf{r}}}\cdot {\boldsymbol{\nabla }}\times {\boldsymbol{\nabla }}\times {\bf{J}}=0$ and $\hat{{\bf{r}}}\cdot {\boldsymbol{\nabla }}\times {\bf{J}}=0$ in the interior of S, then both ${I}_{n}^{m}$ and ${L}_{n}^{m}$ , and thereby, $({\bf{r}}\cdot {\bf{E}})$ and $({\bf{r}}\cdot {\bf{H}})$ are identically zero. Consequently, the field outside S is identically zero per this theorem. Currents such as these are referred to as non-radiating currents.
(iii)

Note that the theorem says nothing about the radial and other components of fields inside S.

Corollary 1.3.1.1Field determination by radial components [19].

Any electromagnetic field $({\bf{E}},{\bf{H}})$ satisfying a source-free Maxwell's equation exterior to a sphere S is completely determined by its radial components E _r and H _r . In particular, the field $({\bf{E}},{\bf{H}})$ due to currents contained inside S is fully characterized by the radial parts ${\bf{r}}\cdot {\bf{E}}$ and ${\bf{r}}\cdot {\bf{H}}$ . Equality of the radial parts for two fields exterior to S implies the equality of other components.

Example 1.3.1.2.Construction of total field from radial fields, Debye potentials.

Construct two independent radial potentials ${\bf{r}}{{\rm{\Pi }}}_{1}$ and ${\bf{r}}{{\rm{\Pi }}}_{2}$ , with ${{\rm{\Pi }}}_{1}$ and ${{\rm{\Pi }}}_{2}$ being scalar radiation functions, such that the electromagnetic field in the source-free region exterior to a sphere, S, enclosing all sources and defined by the expressions

Equation (1.131)

Equation (1.132)

recovers the radial field $({\bf{r}}\cdot {\bf{E}})$ and $({\bf{r}}\cdot {\bf{H}})$ specified in equations (1.111) and (1.112).

We shall construct the solution using linear superposition by treating ${{\rm{\Pi }}}_{1}$ and ${{\rm{\Pi }}}_{2}$ separately. We assume a spherical harmonic representation and let

Equation (1.133)

where the coefficients A _mn are to be determined from the equality of the radial parts. Substituting equation (1.133) into equations (1.131) and (1.132) and carrying out the curl operations in spherical coordinates using the identity (1.83) and noting ${{DY}}_{n}^{m}=-n(n+1)\,\sin \,\theta {Y}_{n}^{m}$ , we obtain

Equation (1.134)

Equation (1.135)

Equation (1.136)

Equation (1.137)

Equation (1.138)

Equation (1.139)

By comparing equation (1.134) with equation (1.111), we conclude that ${A}_{{mn}}=1/n(n+1),\,n\ne 0$ . Because the field generated by ${{\rm{\Pi }}}_{1}$ lacks H _r, it is appropriate to label this field as being transverse magnetic to r or simply TM_r. Note that the n = 0 term does not affect the electromagnetic field as determined by the above equations because ${Y}_{0}^{0}(\theta ,\phi )$ is a constant. By theorem 1.3.1, except for the addition of terms proportional to ${h}_{0}^{(2)}({kr})$ , the potential (1.133) is uniquely determined by the radial part $({\bf{r}}\cdot {\bf{E}})$ . With the choice ${A}_{{mn}}=1/n(n+1)$ , it generates the complete field (1.134)–(1.139) consistent with the specification of this $({\bf{r}}\cdot {\bf{E}})$ .

In a like manner letting

Equation (1.140)

where the coefficients B _mn are to be determined, we obtain upon substituting into equations (1.131) and (1.132)

Equation (1.141)

Equation (1.142)

Equation (1.143)

Equation (1.144)

Equation (1.145)

Equation (1.146)

By comparing equation (1.141) to equation (1.112), we conclude that ${B}_{{mn}}=1/n(n+1),n\ne 0$ . Because the field generated by ${{\rm{\Pi }}}_{2}$ lacks E _r, it is appropriate to label this field as being transverse electric to r or simply TE_r. As before, per theorem 1.3.1, except for the addition of terms proportional to ${h}_{0}^{(2)}({kr})$ , the potential (1.140) is uniquely determined by the radial part $({\bf{r}}\cdot {\bf{H}})$ . With the choice ${B}_{{mn}}=1/n(n+1)$ , it generates the complete field (1.141)–(1.146) consistent with the specification of this $({\bf{r}}\cdot {\bf{H}})$ . The potentials ${{\rm{\Pi }}}_{1}$ and ${{\rm{\Pi }}}_{2}$ are referred to in the literature as Debye potentials [22] and are given explicitly by

Equation (1.147)

Equation (1.148)

As expected, ${{\rm{\Pi }}}_{1}$ and ${{\rm{\Pi }}}_{2}$ are dual quantities (see section 6.1.1). ■■

Example 1.3.1.3Multipole strengths via spectral current components.

Using equation (1.74) express the multipole strengths ${I}_{n}^{m}$ and ${L}_{n}^{m}$ of the Debye potentials in terms of the vector Fourier transform of the current distribution:

Equation (1.149)

where ${\boldsymbol{\kappa }}=\hat{{\bf{x}}}{\kappa }_{x}+\hat{{\bf{y}}}{\kappa }_{y}+\hat{{\bf{z}}}{\kappa }_{z}$ is the variable conjugate to ${\bf{r}}$ , i.e. is the transformed variable.

Denoting the interior spherical harmonic as ${\psi }_{n}^{m}={j}_{n}({kr}){Y}_{n}^{m}(\theta ,\phi )$ , defining the gradient operator in the wavenumber domain with respect to a 'position' vector $\hat{{\bf{s}}}=(\hat{{\bf{x}}}{s}_{x},\hat{{\bf{y}}}{s}_{y},\hat{{\bf{z}}}{s}_{z})$ with three independent spectral coordinates $({s}_{x},{s}_{y},{s}_{z})$ as $\widetilde{{\rm{\nabla }}}=\hat{{\bf{x}}}\partial /\partial {s}_{x}+\hat{{\bf{y}}}\partial /\partial {s}_{y}+\hat{{\bf{z}}}\partial /\partial {s}_{z}$ , and a related transverse operator as ${\boldsymbol{L}}=-j\hat{{\bf{s}}}\times \tilde{{\rm{\nabla }}}$ , and employing the vector identity ${\boldsymbol{\nabla }}\times ({\bf{r}}{\psi }_{n}^{m})={\rm{\nabla }}{\psi }_{n}^{m}\times {\bf{r}}$ and the result (1.74) in the defining equation (1.109), the multipole strength ${L}_{n}^{m}$ can be obtained as

Equation (1.150)

where the Fourier transformed current is decomposed into a transverse part and a longitudinal part as $\widetilde{{\bf{J}}}={\widetilde{{\bf{J}}}}^{T}+\hat{{\bf{s}}}(\hat{{\bf{s}}}\cdot \widetilde{{\bf{J}}})$ , ${\widetilde{{\bf{J}}}}^{T}=(\hat{{\bf{s}}}\times \widetilde{{\bf{J}}})\times \hat{{\bf{s}}}$ being the transverse part of $\widetilde{{\bf{J}}}$ . The following simplification was used in arriving at the final step

Equation (1.151)

Equation (1.150) expresses the multipole strength ${L}_{n}^{m}$ of a TE_r mode in terms of (i) the transverse part of the Fourier transform of the current distribution evaluated at ${\boldsymbol{\kappa }}=k\hat{{\bf{s}}}$ , with $\hat{{\bf{s}}}=\hat{{\bf{x}}}\,\sin \,\alpha \,\cos \,\beta +\hat{{\bf{y}}}\,\sin \,\alpha \,\sin \,\beta +\hat{{\bf{z}}}\,\cos \,\alpha$ , and (ii) the vector ${\boldsymbol{L}}{\bar{Y}}_{n}^{m}$ obtained by operating the orbital angular momentum operator ${\boldsymbol{L}}=-j\hat{{\bf{s}}}\times \tilde{{\rm{\nabla }}}$ on the surface harmonic ${\bar{Y}}_{n}^{m}(\alpha ,\beta )$ in the ${\boldsymbol{\kappa }}$ -space. Its main advantage over the alternative spatial form in equation (1.109) arises from the fact that the far-zone radiated fields of an antenna and the transverse component of $\widetilde{{\bf{J}}}$ in the visible part of the spectrum (namely $\left|{\boldsymbol{\kappa }}\right|=k$ ) are proportional to each other. Note that the substitution $\cos \,\alpha ={s}_{z},\,\cos \,\beta ={s}_{x}/\sqrt{1-{s}_{z}^{2}},\,\sin \,\beta ={s}_{y}/\sqrt{1-{s}_{z}^{2}}$ may be used in ${\bar{Y}}_{n}^{m}$ before implementing the term ${\boldsymbol{L}}{\bar{Y}}_{n}^{m}$ in the wavenumber domain.

In an identical manner, defining ${\bf{M}}={\boldsymbol{\nabla }}\times {\bf{J}}$ , the multipole strength ${I}_{m}^{n}$ can be obtained as

Equation (1.152)

But the Fourier transform of ${\bf{M}}={\boldsymbol{\nabla }}\times {\bf{J}}$ is

Equation (1.153)

since the localized current density ${\bf{J}}$ vanishes on the bounding surface S. The conversion of the volume integral to a surface integral above follows from the curl theorem (B.5). Inserting this result into equation (1.152) gives

Equation (1.154)

for the multipole strength of a TM_r mode. Expressions for the multipole strengths in terms of the Fourier transform of the current have been given in [20], whose detailed derivation is completed in this example. Thus it is seen that the multipole strengths of both TM_r and TE_r modes depend solely on the transverse part of the current distribution evaluated on the circle $\left|{\boldsymbol{\kappa }}\right|=k$ . The latter is completely determined by the transverse components of the electric field in the far-zone (see equation (1.50)). Thus the transverse components of the far-zone electric field completely determine the electromagnetic field radiated in a homogeneous medium outside the sphere enclosing all electric sources. Furthermore, it is also clear from equations (1.150), (1.154), (1.131), (1.132), (1.133), and (1.140) that if the current is such that ${\widetilde{{\bf{J}}}}^{T}(k\hat{{\bf{s}}})\equiv 0$ , then the electromagnetic field outside the spherical surface enclosing the electric currents is identically zero. This is an alternative definition of a non-radiating current. ■■

1.3.2. Additional theorems related to spherical harmonics

A theorem related to the Wilcox theorem 1.2.4 is the expansion theorem of an arbitrary function in spherical surface harmonics.

Theorem 1.3.2.Spherical surface harmonics expansion theorem [23, p 513].

Let $g(\theta ,\phi )$ be an arbitrary function on the surface of a unit sphere, which together with all its first and second derivatives is continuous. Then $g(\theta ,\phi )$ may be expanded in an absolutely and uniformly convergent series of surface spherical harmonics

Equation (1.155)

whose coefficients are determined from

Equation (1.156)

Equation (1.157)

Equation (1.158)

An equivalent representation using the normalized surface harmonic ${Y}_{n}^{m}(\theta ,\phi )$ of equation (1.103) is

Equation (1.159)

where

Equation (1.160)

A direct consequence of the above theorem are the two following corollaries.

Corollary 1.3.2.1.Closure of surface spherical harmonics [24, p 62].

Let $g(\theta ,\phi )$ be an arbitrary function on the surface of a unit sphere, which together with all its first and second derivatives is continuous. If ${a}_{n0},\,{a}_{{nm}},\,{b}_{{nm}}$ , or equivalently, ${C}_{n}^{m}$ are all zero, then $g(\theta ,\phi )$ is identically zero.

Corollary 1.3.2.2.Completeness of surface spherical harmonics [24, p 62].

Let $g(\theta ,\phi )$ be an arbitrary function on the surface of a unit sphere Ω, which together with all its first and second derivatives is continuous. Then

Equation (1.161)

Theorem 1.3.3.Spherical harmonics representation interior to a sphere [24, p 83].

If the twice continuously differentiable function $U({\bf{r}})$ , $r\leqslant {r}_{0}$ , satisfies the scalar Helmholtz equation ${{\rm{\nabla }}}^{2}U+{k}^{2}U=0$ , then for $0\leqslant r\lt {r}_{0}$ , this function can be represented in a uniformly convergent series of the form

Equation (1.162)

where

and ${j}_{n}(z)$ is the spherical Bessel function of argument z and order n.

Theorem 1.3.4.Spherical harmonics representation exterior to a sphere [24, p 84].

If the twice continuously differentiable function $U({\bf{r}})$ , $r\geqslant {r}_{0}$ , satisfies the scalar Helmholtz equation ${{\rm{\nabla }}}^{2}U+{k}^{2}U=0$ , and the Sommerfeld's radiation condition

i.e. is a scalar radiation function, then for ${r}_{0}\lt r\lt \infty$ , this function can be represented in a uniformly convergent series of the form

Equation (1.163)

where

and ${h}_{n}^{(2)}(z)$ is the spherical Hankel function of the second kind of argument z and order n.

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Source: https://iopscience.iop.org/book/978-0-7503-1716-0/chapter/bk978-0-7503-1716-0ch1

Dunlop Maimad

Show That Current Continuity is Embedded Within Maxwell s Equations

1.1.1. The Lorenz gauge, Coulomb gauge, and causality

1.1.2. Existence theorem for fields

1.2.1. Classification of media

1.2.1.1. The plasmonic medium and Drude model

1.2.2. Boundary conditions

1.2.2.1. Material interface conditions

1.2.2.2. Radiation, absorbing boundary conditions, and related theorems

1.3.1. Multipole expansion, Debye potentials, and related theorems

1.3.2. Additional theorems related to spherical harmonics

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