1 Introduction
###### Abstract

Vector and scalar potential formulation is valid from quantum theory to classical electromagnetics. The rapid development in quantum optics calls for electromagnetic solutions that straddle quantum physics as well as classical physics. The vector potential formulation is a good candidate to bridge these two regimes. Hence, there is a need to generalize this formulation to inhomogeneous media. A generalized gauge is suggested for solving electromagnetic problems in inhomogenous media which can be extended to the anistropic case. The advantages of the resulting equations are their absence of low-frequency catastrophe. Hence, usual differential equation solvers can be used to solve them over multi-scale and broad bandwidth. It is shown that the interface boundary conditions from the resulting equations reduce to those of classical Maxwell’s equations. Also, classical Green’s theorem can be extended to such a formulation, resulting in similar extinction theorem, and surface integral equation formulation for surface scatterers. The integral equations also do not exhibit low-frequency catastrophe as well as frequency imbalance as observed in the classical formulation using - fields. The matrix representation of the integral equation for a PEC scatterer is given.

Vector Potential Electromagnetic Theory with Generalized Gauge for Inhomogeneous Anisotropic Media

W. C. Chew111U of Illinois, Urbana-Champaign; visiting professor, HKU.

July 17, 2019 at 21 : 32

## 1 Introduction

Electromagnetic theory has been guided by Maxwell’s equations for 150 years now [2]. The formulation of electromagnetic theory based on , , , and , simplified by Heaviside [3], offers physical insight that results in the development of myriads of electromagnetic-related technologies. However, there are certain situations where the - formulation is not ideal. This is in the realm of quantum mechanics where the - formulation is needed. In certain situations where - are zero, but is not zero, and yet, the effect of is felt in quantum mechanics. This is true of the Aharonov-Bohm effect [4, 5]. Moreover, the quantization of electromagnetic field can be done more expediently with the vector potential rather than and . More importantly, when the electromagnetic effect needs to be incorporated in Schrödinger equation, vector and scalar potentials are used. This will be important in many quantum optics studies [6, 7, 8, 9, 10, 11, 12].

Normally, electromagnetic equations formulated in terms of - have low-frequency breakdown or catastrophe. Hence, many numerical methods based on - formulation are unstable at low frequencies or long wavelength. Therefore, the - formulation is not truly multi-scale, as it has catastrophe when the dimension of objects are much smaller than the wavelength. Different formulations using tree-cotree, or loop-tree decomposition [13, 14, 15, 16, 17], have to be sought when the frequency is low or the wavelength is long. Due to the low-frequency catastrophe encountered by - formulation, the vector potential formulation has been very popular for solving low frequency problems [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

This work will arrive at a general theory of vector potential formulation for inhomogeneous anisotropic media, together with the pertinent integral equations. This vector potential formulation does not have apparent low-frequency catastrophe of the - formulation and it is truly multi-scale. It can be shown that with the proper gauge, which is the extension of the simple Lorentz gauge to inhomogeneous anisotropic media, the scalar potential equation is decoupled from the vector potential equation.

## 2 Pertinent Equations–Inhomogeneous Isotropic Case

The vector potential formulation for homogeneous medium has been described in most text books [29, 30, 31, 32]. We derive the pertinent equations for the inhomogeneous isotropic medium case first. To this end, we begin with the Maxwell’s equations:

 ∇×E =−∂tB, (1) ∇×H =∂tD+J, (2) ∇⋅B =0, (3) ∇⋅D =ρ. (4)

From the above we let

 B =∇×A, (5) E =−∂tA−∇Φ (6)

so that the first and third of four Maxwell’s equations are satisfied. Then, using for isotropic, non-dispersive, inhomogeneous media, we obtain that

 −∂t∇⋅εA−∇⋅ε∇Φ=ρ, (7)
 ∇×μ−1∇×A=−ε∂2tA−ε∂t∇Φ+J. (8)

For homogeneous medium, the above reduce to

 −∂t∇⋅A−∇2Φ=ρ/ε, (9)
 μ−1(∇∇⋅A−∇2A)=−ε∂2tA−ε∂t∇Φ+J. (10)

By using the simple Lorentz gauge

 ∇⋅A=−με∂tΦ (11)

we have the usual

 ∇2Φ−με∂2tΦ=−ρ/ε, (12)
 ∇2A−με∂2tA=−μJ (13)

Lorentz gauge is preferred because it treats space and time on the same footing as in special relativity [29].

For inhomogeneous media, we can choose the generalized Lorentz gauge. This gauge has been suggested previously, for example in [25].

 ε−1∇⋅εA=−με∂tΦ. (14)

However, we can decouple (7) and (8) with an even more generalized gauge, namely

 ∇⋅εA=−χ∂tΦ (15)

Then we get from (7) and (8) that

 ∇⋅ε∇Φ−χ∂2tΦ=−ρ, (16)
 −∇×μ−1∇×A−ε∂2tA+ε∇χ−1∇⋅εA=−J. (17)

It is to be noted that (16) can be derived from (17) by taking the divergence of (17) and then using the generalized gauge and the charge continuity equation that . In gneral, we can choose

 χ=αε2μ (18)

where can be a function of position. When , it reduces to the generalized Lorentz gauge used in (14).

For homogeneous medium, (16) and (17) reduce to (12) and (13) when we choose in (18), which is the case of the simple Lorentz gauge. Unlike the vector wave equations for inhomogeneous electromagnetic fields, the above do not have apparent low-frequency breakdown when . Hence, the above equations can be used for electrodynamics as well as electrostatics when the wavelength tends to infinity.

We can rewrite the above as a sequence of three equations, namely,

 ∇⋅εA =−χ∂tΦ (19) ∇×A =μH (20) ∇×H+ε∂2tA+ε∇∂tΦ =J (21)

The last equation can be rewritten as

 ∇×H−ε∂t(−∂tA−∇Φ)=J (22)

which is the same as solving Ampere’s law. Hence, solving (17) is similar to solving Maxwell’s equations.

It is to be noted that (17) resembles the elastic wave equation in solids where both longitudinal and transverse waves can exist [36]. Furthermore, these two waves can have different velocities in a homogeneous medium if in (18). The longitudinal wave has the same velocity as the scalar potential, which is , while the transverse wave has the velocity of light, or . If we choose , or , we have the Coulomb gauge where the scalar potential has infinite velocity.

## 3 Boundary Conditions for the Potentials

Many problems can be modeled with piecewise homogeneous medium. In case, the solutions can be sought in each of the piecewise homogenous region, and then sewn together using boundary conditions. The above equations, (16) and (17), are the governing equations for the scalar and vector potentials and for inhomogeneous media. The boundary conditions at the interface of two homogeneous media are also embedded in these equations. By eyeballing equation (17), we see that must be finite at an interface. This induces the boundary condition that

 ^n×A1=^n×A2 (23)

across an interface. Assuming that is finite, we also have

 ^n×1μ1∇×A1=^n×1μ2∇×A2. (24)

The above is equivalent to

 ^n×H1=^n×H2 (25)

at an interface. When a surface current sheet is present, we have to augment the above with the current sheet as is done in standard electromagnetic boundary conditions. Furthermore, due to the finiteness of at an interface, it is necessary that

 ^n⋅ε1A1=^n⋅ε2A2. (26)

It can be shown that if a surface dipole layer exists at an interface, we will have to augment the above with the correct discontinuity or jump condition. A surface current with a normal component to the surface will constitute a surface dipole layer.

By the same token, we can eyeball the scalar potential equation (16), and notice that

 ^n⋅ε1∇Φ1=^n⋅ε2∇Φ2. (27)

The above will be augmented with the necessary jump or discontinuity condition if a surface charge layer exists at an interface. Equations (26) and (27) together mean that

 ^n⋅ε1E1=^n⋅ε2E2. (28)

where we have noted that from (6). This is the usual boundary condition for the normal component of the electric field.

Equation (16) also implies that

 Φ1=Φ2, (29)

or

 ^n×∇Φ1=^n×∇Φ2. (30)

Equations (23) and (30) imply that

 ^n×E1=^n×E2. (31)

This is the normal boundary condition for the tangential component of the electric field.

If is a perfect electric conductor (PEC), . From (17), it implies that , if or . Then (23) for a PEC surface becomes

 ^n×A1=0. (32)

Also, by eyeballing (16), we see that for a PEC, . This together with (29), (30), and (32) imply that on a PEC surface. For a perfect magnetic conductor (PMC), , from (24) and (25)

 ^n×H1=0. (33)

When or , does not contribute to . But from (27), when , we deduce that , implying that for for arbitrary . Hence, from (29) and (30), on a PEC surface even when .

## 4 General Anisotropic Media Case

For inhomogeneous, dispersionless, anisotropic media, the generalized gauge becomes

 ∇⋅¯¯¯ε⋅A=−χ∂tΦ (34)

In the above, is arbitrary, but we can choose

 χ=α|¯¯¯ε⋅¯¯¯μ⋅¯¯¯ε| (35)

where the vertical bar means determinant. When the medium is inhomogeneous and isotropic, the above gauge reduces to the generalized gauge previously discussed. When , the above reduces to the generalized Lorentz gauge for inhomogeneous isotropic medium. In general, (16) and (17) become

 ∇⋅¯¯¯ε⋅∇Φ−χ∂2tΦ=−ρ (36)
 ∇×¯¯¯μ−1∇×A+¯¯¯ε⋅∂2tA−¯¯¯ε⋅∇χ−1∇⋅¯¯¯ε⋅A=J (37)

The above can be rewritten in the manner of (19) to (22), showing that solving the above is the same as solving the original Maxwell’s equations. The boundary condition (23) remains the same. Boundary condition (24) becomes

 ^n×¯¯¯μ−11⋅∇×A1=^n×¯¯¯μ−12⋅∇×A2 (38)

and boundary condition (25) remains the same. Similarly, boundary condition (26) becomes

 ^n⋅¯¯¯ε1⋅A1=^n⋅¯¯¯ε2⋅A2. (39)

The boundary condition (27) becomes

 ^n⋅¯¯¯ε1⋅∇Φ1=^n⋅¯¯¯ε2⋅∇Φ2 (40)

while the other boundary conditions, similar to the isotropic case, can be similarly derived.

## 5 Green’s Theorem—Time Harmonic Case

As mentioned previously, for inhomogeneous media consisting of piecewise homogeneous regions, it is best to seek the solution in each region first, and then the solutions sewn together by boundary conditions. Consequently, surface (boundary) integral equations can be derived to solve such problems where unknowns only need to be assigned to the interfaces or boundaries between regions. In this manner, a 3D problem is reduced to a problem on a 2D manifold, beating the tyranny of dimensionality. Moreover, in recent years, fast algorithms have been developed to solve these surface integral equations rapidly [33, 34, 35], greatly underscoring their importance.

To this end, we need to derive the equivalence of the Green’s theorem for vector potential formulation. In the following, we assume a simple Lorentz gauge so that the equations for homogeneous region greatly simplify. In other words, we need to derive Green’s theorem’s equivalence for

 (∇2+k2)A(r)=−μJ(r). (41)

where and the time dependence is . It is more expedient to write the above as222If , the ensuing equation is of the form (42) But the dyadic Green’s function of such an equation can still be found using methods outlined in [36].

 ∇×∇×A(r)−∇∇⋅A(r)−k2A(r)=μJ(r). (43)

We can define a dyadic Green’s function that satisfies

 ∇×∇×¯¯¯G(r,r′)−∇∇⋅¯¯¯G(r,r′)−k2¯¯¯G(r,r′)=¯¯¯Iδ(r−r′). (44)

The solution to the above is simply

 ¯¯¯G(r,r′)=¯¯¯Ieik|r−r′|4π|r−r′|=¯¯¯I g(r,r′). (45)

From the above, using methods outlined in [37, Chapter 8], as well as in Appendix A, we have for region 1,

 r∈V1,A1(r)r∈V2,0 }=Ainc(r)+∫SdS′{μ1¯¯¯G1(r,r′)⋅^n′×H1(r′)−[∇′×¯¯¯G1(r,r′)]⋅^n′×A1(r′)}+∫SdS′ ^n′⋅{¯¯¯G1(r,r′)∇′⋅A1(r′)−A1(r′)∇′⋅¯¯¯G1(r,r′)}. (46)

We can rewrite the above using scalar Green’s function as

 r∈V1,A1(r)r∈V2,0 }=Ainc(r)+∫SdS′{μ1g1(r,r′)^n′×H1(r′)−∇′g1(r,r′)×^n′×A1(r′)}+∫SdS′{−^n′g1(r,r′)∇′⋅A1(r′)+(^n′⋅A1(r′))∇′g1(r,r′)}. (47)

A similar equation can be derived for region 2. These equations can be used to formulate surface integral equations for scattering. The lower parts of the above equations are known as the extinction theorem [37, 38].

As a side note, one can use the scalar Green’s theorem directly on (41) and obtain

 r∈V1,A1(r)r∈V2,0 }=Ainc(r)−∫SdS′{g1(r,r′)^n′⋅∇′A1(r′)−^n′⋅∇′g1(r% ,r′)A1(r′)}. (48)

After some lengthy manipulations, (47) becomes (48) as shown in Appendix A. In the above derivation, there is a surface integral at infinity that can be shown to vanish as in [37, Chapter 8] when radiation condition is invoked.

## 6 PEC Scatterer Case

For a PEC scatterer, we have proved that . Since and that on a PEC surface. Hence, for surface sources that satisfy the PEC scattering solution, the above becomes

 r∈V1,A1(r)r∈V2,0 }=Ainc(r)+∫SdS′{μ1g1(r,r′)^n′×H1(r′)+^n′⋅A1(r′)∇′g1(r,r′)}. (49)

The first term in the integral comes from the induced surface current flowing on the PEC surface. It will be interesting to ponder the meaning of the second term. It is to be noted that the surface charge on the PEC surface is given by

 ^n⋅ε1E1=^n⋅ε(iωA1−∇Φ1) (50)

The scalar potential can be obtained from the vector potential using Lorentz gauge, namely, . Hence, one can view that as the contribution to the surface charge from the vector potential . In fact, using the Lorentz gauge, and that , one can recover from the above that the electric field in region 1 outside the PEC is given by (see Appendix B)

 E1=Einc+∫SdS′{iωμ1g1(r,r′)J1(r′)−∇g1(r,r′)σ1(r′)ε1(r′)} (51)

where and . The above is just the traditional relationship between the field in region 1 and the sources on the PEC surface.

We can rewrite (49) in terms of two integral equations

 A1(r)=Ainc(r)+∫SdS′{μ1g1(r,r′)J1(r′)+Σ1(r′)∇′g(r,r′)} (52)
 Σ1(r)=Σinc(r)+∫SdS′{μ1g1(r,r′) ^n⋅J1(r′)+Σ1(r′) ^n⋅∇′g(r,r′)} (53)

where the second equation is obtained by the first equation. Also, the boundary condition is such that on . The above can be solved by the subspace projection method such as the Galerkin’s [39] or moments methods [40, 41]. The unknowns are and while and are known. We expand the unknowns in terms of basis functions and that span the subspaces of and , respectively. Namely,

 J1(r′)=N∑n=1jnJn(r′) (54)
 Σ1(r′)=M∑m=1smσm(r′) (55)

We choose to be divergence conforming tangential current so that the vector potential that it produces is also divergence conforming [42]. In the above, can be chosen to approximate a surface charge well. After expanding the unknowns, we project the field that they produce onto the subspace spanned by the same unknown set as in the process of testing in the Galerkin’s method. Consequently, (52) and (53) become

 0=⟨Jn′(r),Ainc(r)⟩ + μ1N∑n=1⟨Jn′(r),g1(r,r′),Jn(r)⟩jn (56) + M∑m=1sm⟨Jn′(r),∇′g1(r,r′),σm(r′)⟩ M∑m=1sm⟨σm′(r),σm(r)⟩ = ⟨σm′(r),Σinc(r)⟩ (57) + μ1N∑n=1⟨σm′(r),^ng1(r,r′),Jn(r′)⟩jn + M∑m=1⟨σm′(r),^n⋅∇′g1(r,r)′,σm(r′)⟩sm

The above is a matrix system of the form

 0=ainc+¯¯¯Γ1,J,J⋅j+¯¯¯Γ1,J,σ⋅s (58)
 ¯¯¯B⋅s=\boldmathΣinc+¯¯¯Γ1,σ,J⋅%j+¯¯¯Γ1,σ,σ⋅s (59)

where and are unknowns, while and are known. In detail, elements of the above matrices and vectors are given by

 [ainc]n′=⟨Jn′(r),Ainc(r)⟩ (60)
 [¯¯¯Γ1,J,J]n′,n=μ⟨Jn′(r),g1(r,r′),Jn(r′)⟩ (61)
 [¯¯¯Γ1,J,σ]n′,m=⟨Jn′(r),∇′g1(r,r′),σm(r′)⟩ (62)
 [¯¯¯B]m′,m=⟨σm′(r),σm(r)⟩ (63)
 [\boldmathΣinc]m′=⟨σm′(r),Σinc(r)⟩ (64)
 [¯¯¯Γ1,σ,J]m′,n=μ1⟨σm′(r),^n g1(r,r′),Jn(r′)⟩ (65)
 [¯¯¯Γ1,σ,σ]m′,m=⟨σm′(r),^n⋅∇′g1(r,r′),σm(r′)⟩ (66)
 [j]n=jn,[s]m=sm (67)

Furthermore, in the above,

 ⟨f(r),h(r)⟩=∫SdSf(r)⋅h(r) (68)
 ⟨f(r),γ(r,r′),h(r)⟩=∫SdSf(r)⋅∫SdS′γ(r,r′)h(r′) (69)

where and can be replaced by scalar functions, and can be replaced by a vector function with the appropriate inner products between them.

The matrices above are different matrix representations of the scalar Green’s function and its derivative. It is to be noted that all the matrices above do not have low-frequency catastrophe as in the matrix representation of the dyadic Green’s function. Hence, the above behaves like the augmented electric field integral equation (A-EFIE) [43].

## 7 Vector Potential Plane Wave

A time-harmonic vector potential plane wave is the solution to the equation

 (∇2+k2)A=0 (70)

But it seems odd that , , and are decoupled from each other. To dispel this notion, we should think of as the solution to

 (∇2+k2)A=−μJ (71)

The vector potential above satisfies the Lorentz gauge via the charge continuity equation. By taking the divergence of the above, we have

 (∇2+k2)∇⋅A=−μ∇⋅J=−iωμρ (72)

where .

If is due to a Hertzian dipole source

 J(r)=Iℓ^ℓδ(r) (73)

the corresponding vector potential is

 A(r)=μIℓ^ℓeikr4πr (74)

We can produce a locally plane wave by letting where . Then the above spherically wave can be approximated by a locally plane wave:

 A(r)≈μIℓ^ℓeikr04πr0eik0⋅s=aeik0⋅s (75)

where and is a unit vector that points in the direction of . It is seen that the components of generated this way satisfies the gauge condition and are not independent of each other. We have to keep this notion in mind when we generate a vector potential plane wave.

Hence, for a plane wave incident,

 Ainc(r)=(a⊥+aki)eiki⋅r (76)

where , and . Therefore

 ∇⋅Ainc=iki⋅akieiki⋅r=ia0kieiki⋅r (77)
 Binc=∇×Ainc=iki×Ainc=iki×a⊥eiki⋅r (78)

and

 Einc =∇×Binc−iωμε=iki×(ki×a⊥)eiki⋅rωμε (79) =i[k2ia⊥−ki(ki⋅a⊥)]eiki⋅rωμε =iω[¯¯I−^ki^ki]⋅a⊥eiki⋅r (80)

It is to be noted that if has only the longitudinal component, then both and are zero even though is not zero. This could happen to leading order along the axial direction of a Hertzian dipole.

## Appendix A Derivations of (46), (47), and (48)

We will ignore the source term in order to derive some identities similar to Green’s theorem. We begin with the following equations:

 ∇×∇×A−∇∇⋅A−k2A=0 (A.1)
 ∇×∇×¯¯¯¯¯G−∇∇⋅¯¯¯¯¯G−k2¯¯¯¯¯G=¯Iδ(r−r′) (A.2)

In the above, and , but we will suppress these dependent variables for the time being in the following. First, we dot-multiply (A.1) from the right by where is an arbitrary vector, and then dot-multiply (A.2) from the left by and the right by . We take their difference, and ignoring the term for the time being, to get

 ∇×∇×A⋅¯G⋅a−A⋅∇×∇×¯¯¯¯¯G⋅a=∇⋅(∇×A×¯¯¯¯¯G⋅a+A×∇×¯¯¯¯¯G⋅a) (A.3)

Integrating right-hand side of the above over , we have

 I1⋅a=∫S^n⋅(∇×A×¯¯¯¯¯G⋅a+A×∇×¯¯¯¯¯G⋅a)dS       =∫S[^n×(∇×A)⋅¯¯¯¯¯G⋅a+(^n×A)⋅∇×¯G⋅a]dS (A.4)

Including now the term gives

 −∇∇⋅A⋅¯¯¯¯¯G⋅a+A⋅∇∇⋅¯¯¯¯¯G⋅a=∇⋅(−∇⋅A ¯¯¯¯¯G⋅a+A⋅∇⋅¯¯¯¯¯G⋅a) (A.5)

Integrating the right-hand side of the above over , we have

 I2⋅a=∫S^n⋅(−∇⋅A ¯¯¯¯¯G⋅a+A⋅∇⋅¯¯¯¯¯G⋅a)dS (A.6)

Letting , where , the scalar Green’s function, the above becomes

 (A.7)

or

 I1=∫S[^n×(∇×A)g+(^n×A)×∇g]dS (A.8)

Similarly, we have

 I2=∫S[−^n(∇⋅A)g+^n⋅A ∇g]dS (A.9)

Using the above, we get (47).

To get (48), more manipulations are needed. Using ,

 I1=∫S[−(^n⋅∇A)g+(∇A)⋅^ng−^n(A⋅∇g)+(^n⋅∇g)A]dS (A.10)

First, we look at

 I3 = ∫S^n[−∇⋅¯A g−¯A⋅∇g]dS (A.11) = ∫VdV ∇(−∇⋅¯A g−¯A⋅∇g) = ∫VdV [−∇∇⋅¯A g−¯A⋅∇∇g−∇¯A⋅∇g−∇⋅¯A ∇g] I4 = ∫SdS [g ∇A⋅^n+^n⋅A ∇g] (A.12) = ∫VdV ∇⋅[(g∇A)t+A ∇g]

Furthermore, with the knowledge that

 ∇⋅[(g∇A)t−A∇g] = ∂i[g∂kAi+Ai∂jg] (A.13) = (∂ig)∂kAi+g∂i∂kAi+∂iAi∂jg+Ai∂i∂jg (A.14) = ∇A⋅∇g+g ∇∇⋅A+∇⋅A ∇g+A⋅∇∇g (A.15)

it is seen that . Using this fact, we can show (48), or that

 I1+I2=∫SdS [(^n⋅∇g)A−g ^n⋅∇A] (A.16)

## Appendix B Derivation of (51)

 A1=Ainc+∫SdS′{μ1g1J1+(^n′⋅A1)∇′g1} (B.1)
 ∇⋅A1=∇⋅Ainc+∫SdS′{μ1g1∇′⋅J1−(^n′⋅A1)∇2g1} (B.2)

Since , the above becomes

 iωμ1ε1Φ1=iωμ1ε1Φinc+∫SdS′{μ1g1iωσ1+(^n′⋅A1)k21g1} (B.3)

or

 Φ1=Φinc+∫SdS′{g1σ1ε1−iω(^n′⋅A1)g1} (B.4)

Since , using (B.1) and (B.4), we have

 E1=Einc+∫SdS′{iωμ1g1J1−∇g1σ1ε1+iω(^n′⋅A1)∇′g1+iω(^n′⋅A1)∇g1} (B.5)

The last two terms cancel each other.

### Acknowledgements

This work was supported in part by the USA NSF CCF Award 1218552, SRC Award 2012-IN-2347, at the University of Illinois at Urbana-Champaign, by the Research Grants Council of Hong Kong (GRF 711609, 711508, and 711511), and by the University Grants Council of Hong Kong (Contract No. AoE/P-04/08) at HKU. The author is grateful to M. Wei, H. Gan, C. Ryu, T. Xia, Y. Li, Q. Liu, and L. Meng for helping to typeset the manuscript.

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