Beyond Green’s Functions: Inverse Helmholtz and “Om” $\mathrm{ॐ}$-Potential Methods for Macroscopic Electromagnetism in Isotropy-Broken Media

Abstract

The applicability ranges of macroscopic and microscopic electromagnetism are contrasting. While microscopic electromagnetism deals with point sources, singular fields, and discrete atomistic materials, macroscopic electromagnetism concerns smooth average distributions of sources, fields, and homogenized effective metamaterials. Green’s function method (GFM) involves finding fields of point sources and applying the superposition principle to find fields of distributed sources. When utilized to solve microscopic problems, GFM is well within the applicability range. Extension of GFM to simple macroscopic problems is convenient, but not fully logically sound, since point sources and singular fields are technically not a subject of macroscopic electromagnetism. This explains the difficulty of both finding the Green’s functions and applying the superposition principle in complex isotropy-broken media, which are very different from microscopic environments. In this manuscript, we lay out a path to the solution of macroscopic Maxwell’s equations for distributed sources, bypassing GFM by introducing an inverse approach and a method based on “Om” $\mathrm{ॐ}$-potential, which we describe here. To the researchers of electromagnetism, this provides access to powerful analytical tools and a broad new space of solutions for Maxwell’s equations.

1. Introduction

The main problem of electromagnetism is predicting the interaction between arbitrary charge distributions placed in arbitrary environments [1]. The path to solving this problem is most typically understood as finding fields of point sources. The fields of complex sources can then be obtained via the superposition principle. This approach, known as the Green’s function method, is not only mathematically natural for singular microscopic fields but is grounded in the physics of elementary particles, which do not have dimensions according to relativistic considerations [2]. The ubiquitous position of Green’s functions in microscopic electromagnetism is best expressed in the essay by Julian Schwinger titled “The Greening of Quantum Field Theory: George and I” [3]. The “Greening” of macroscopic electromagnetism is less obvious since photonics researchers do not deal with elementary particles or singular microscopic fields and do not claim the applicability of macroscopic photonics to elementary particles. Not unexpectedly, the extension of the microscopic Green’s function approach to macroscopic electromagnetism faces difficulties due to fundamental differences between the corresponding sets of Maxwell’s equations.

A lot of effort is invested in extending Green’s function method to various macroscopic electromagnetic media; this cannot be considered very successful, however, at the rugged frontier of isotropy-broken media, due to the complexity and inherent non-locality of these media [4]. Although an integral representation of Green’s function can be obtained for the most general case of isotropy-broken media [5], closed-form expressions of dyadic Green’s functions are only available for a limited set of relatively simple isotropy-broken media [6,7]. Another complication arises from utilizing the superposition principle in the Green’s function method, as finding fields created by non-point sources involves untenable integration, even in the depolarization dyadics approximation [7]. Finding fields of non-point sources in isotropic media, in many cases, relies on symmetries of those sources [1]. This can be extended to isotropy-broken media by means of spectral eigenfunction representations; however, this results in infinite series, requiring truncation [7,8].

In this manuscript, we introduce two approaches to directly obtain fields created by a very broad class of sources immersed in generic isotropy-broken media. First, we apply the inverse approach to the inhomogeneous Helmholtz equation for the vector potential to obtain sources that create desired vector potentials. In the second approach, we draw inspiration from the teachings of Hindu philosophy about the primordial fore-sound of the universe encompassing all creation. We introduce the “Om” $\mathbf{ॐ}$-potential that underlies both sources and fields and provides for the direct method of evaluation of the solutions of macroscopic Maxwell’s equations in isotropy-broken media. Please note that the introduction of auxiliary vector fields to aid the solution or analysis of Maxwell equations is not unprecedented in the history of science, as exemplified by the scalar potential, vector potential, Hertz potential [9], and Beltrami fields [10]. The power of our methods is demonstrated by the mappings we uncover between different sources that create identical potential across all materials and between field-source pairs that come from the same “Om” $\mathbf{ॐ}$ in materials as they transition between symmetries, topology classes, and so on.

It is important to clarify that the Inverse Helmholtz and Om-potential methods proposed here are fully analytical in nature. Their primary purpose is to reveal structural and symmetry-based relationships between macroscopic sources and fields, not to replace numerical solvers such as FEM or FDTD or fast methods for band field calculations [11,12]. Accordingly, no computational speed or convergence comparisons are made or implied in this work.

2. Helmholtz Equation and Green’s Functions in Isotropy-Broken Media

Macroscopic fields satisfy Maxwell’s equations

$$\nabla \times \mathit{H}-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \mathit{D}}{\partial t}}={\displaystyle \frac{4\pi }{c}}\mathit{j},\nabla \times \mathit{E}+{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \mathit{B}}{\partial t}}=0$$

(1)

The constitutive relations are generally expressed as

$$\left(\begin{array}{c}\mathit{D}\\ \mathit{B}\end{array}\right)=\widehat{M}\left(\begin{array}{c}\mathit{E}\\ \mathit{H}\end{array}\right)=\left(\begin{array}{cc}\widehat{ϵ}& \widehat{X}\\ \widehat{Y}& \widehat{\mu }\end{array}\right)\left(\begin{array}{c}\mathit{E}\\ \mathit{H}\end{array}\right)\mathrm{or} \left(\begin{array}{c}\mathit{D}\\ \mathit{H}\end{array}\right)=\left(\begin{array}{cc}{\widehat{C}}_{DE}& {\widehat{C}}_{DB}\\ {\widehat{C}}_{HE}& {\widehat{C}}_{HB}\end{array}\right)\left(\begin{array}{c}\mathit{E}\\ \mathit{B}\end{array}\right)=\left(\begin{array}{cc}\left(\widehat{ϵ}-\widehat{X}{\widehat{\mu }}^{-1}\widehat{Y}\right)& \widehat{X}{\widehat{\mu }}^{-1}\\ -{\widehat{\mu }}^{-1}\widehat{Y}& {\widehat{\mu }}^{-1}\end{array}\right)\left(\begin{array}{c}\mathit{E}\\ \mathit{B}\end{array}\right)$$

(2)

Under the Weyl gauge, the relationship between fields and the vector potential reduces to $\mathit{B}=\nabla \times \mathit{A}$ and $\mathit{E}=-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \mathit{A}}{\partial t}}$. Combining Maxwell’s equations and constitutive relations, we obtain the wave equation for the vector potential in isotropy-broken media

$$\left({\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}},-\nabla \times \widehat{I}\right)\left(\begin{array}{cc}{\widehat{C}}_{DE}& {\widehat{C}}_{DB}\\ {\widehat{C}}_{HE}& {\widehat{C}}_{HB}\end{array}\right)\left(\begin{array}{c}-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}}\\ \nabla \times \widehat{I}\end{array}\right)\mathit{A}=\widehat{L}\left(\nabla ,{\displaystyle \frac{1}{\mathrm{c}}}{\displaystyle \frac{\partial }{\partial t}}\right)\mathit{A}=-{\displaystyle \frac{4\pi }{c}}\mathit{j}$$

(3)

or

$$-{\widehat{C}}_{DE}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2\mathit{A}}{\partial t^2}}+{\widehat{C}}_{DB}\nabla \times {\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \mathit{A}}{\partial t}}+\nabla \times {\widehat{C}}_{HE}{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \mathit{A}}{\partial t}}-\nabla \times {\widehat{C}}_{HB}\nabla \times \mathit{A}=-{\displaystyle \frac{4\pi }{c}}\mathit{j}$$

(4)

More conventionally it is expressed as

$$-\left(\nabla \times \widehat{I}-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}}\widehat{X}\right){\widehat{\mu }}^{-1}\left(\nabla \times \widehat{I}+\widehat{Y}{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial }{\partial t}}\right)\mathit{A}-\widehat{ϵ}{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2\mathit{A}}{\partial t^2}}=-{\displaystyle \frac{4\pi }{c}}\mathit{j}$$

(5)

Transforming into the Fourier domain as $\widehat{L}\left(\nabla ,{\displaystyle \frac{1}{\mathrm{c}}}{\displaystyle \frac{\partial }{\partial t}}\right)\to \widehat{L}\left(i\mathit{k},-i{k}_0\right)$ results in the Helmholtz operator for isotropy-broken media

$$\widehat{L}\left(i\mathit{k},-i{k}_0\right)=\left(k_0,\mathit{k}\times \widehat{I}\right)\left(\begin{array}{cc}{\widehat{C}}_{DE}& {\widehat{C}}_{DB}\\ {\widehat{C}}_{HE}& {\widehat{C}}_{HB}\end{array}\right)\left(\begin{array}{c}{k}_0\\ \mathit{k}\times \widehat{I}\end{array}\right)=\left(\mathit{k}\times \widehat{I}+k_0\widehat{X}\right){\widehat{\mu }}^{-1}\left(\mathit{k}\times \widehat{I}-k_0\widehat{Y}\right)+k_0^2\widehat{ϵ}$$

(6)

The vector potential can be now expressed as $\mathit{A}(\mathit{k},k_0)=-{\displaystyle \frac{4\pi }{c}}{\displaystyle \frac{\mathrm{a}\mathrm{d}\mathrm{j} \widehat{L}}{\left|\widehat{L}\right|}}\mathit{j}(\mathit{k},k_0)$ and

$$\mathit{A}\left(\mathit{r},k_0\right)=-{\displaystyle \frac{4\pi }{c}}\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\displaystyle \frac{\mathrm{a}\mathrm{d}\mathrm{j} \widehat{L}}{\left|\widehat{L}\right|}}\mathit{j}\left(\mathit{k},k_0\right)\exp \left(i\mathit{k}\mathit{r}\right),$$

(7)

where the $\left|\widehat{L}\right|={\displaystyle \frac{1}{k_0^2}}{\sum }_{i+j+l+m=4}[{\alpha }_{ijlm}{k}_x^i{k}_y^j{k}_z^l{k}_0^m]$ is the determinant of the operator Equation (4), with ${\alpha }_{ijlm}$ being the Tamm–Rubilar tensor [13,14], and the corresponding adjoint operator is $\mathrm{a}\mathrm{d}\mathrm{j}\widehat{L}={\displaystyle \frac{1}{k_0^4}}{\sum }_{i+j+l+m=4}[{\widehat{\beta }}_{ijlm}{k}_x^i{k}_y^j{k}_z^l{k}_0^m]$, where ${\widehat{\beta }}_{ijlm}$ are 3 × 3 matrices.

The usual approach is to use the superposition principle to express Equation (5) in terms of dyadic Green’s function $\widehat{G}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)$

$$\mathit{A}\left(\mathit{r}\right)=-{\displaystyle \frac{4\pi }{c}}\int {\mathit{d}\mathit{r}}^{\mathit{\prime }} \widehat{G}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right) \mathit{j}\left({\mathit{r}}^{\mathit{\prime }}\right)$$

(8)

Correspondingly, from Equations (7) and (8), the dyadic Green’s function can be expressed as [5,7]

$$\widehat{G}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)=\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\displaystyle \frac{\mathrm{a}\mathrm{d}\mathrm{j} \widehat{L}}{\left|\widehat{L}\right|}}\exp \left(i\mathit{k}(\mathit{r}-{\mathit{r}}^{\mathit{\prime }})\right),$$

(9)

The problem of finding fields created by arbitrary sources is thus reduced to finding the Green’s function, which is a response to a delta-functional point source and always has a singular part [7].

$$\widehat{L}\left(\nabla ,-i{k}_0\right) \widehat{G}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)=\widehat{I} \delta \left(\mathit{r}-{\mathit{r}}^{\mathit{\prime }}\right)$$

(10)

As described in the introduction, in general, both finding the Green’s function from Equations (9) and (10) and utilizing the superposition principle, Equation (8), are challenges in macroscopic electromagnetism and are unnatural due to the limited validity of point sources and singular fields in macroscopic environments.

3. The “Om” $\mathbf{ॐ}-\mathbf{P}\mathbf{o}\mathbf{t}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}$

To bypass the complications related to GFM we introduce differential operators based on Fourier space operators $\left|\widehat{L}\right|$ and $\mathrm{a}\mathrm{d}\mathrm{j} \widehat{L}$

$$D\left({\partial }_x,{\partial }_y,{\partial }_z\right)={\displaystyle \frac{1}{k_0^2}}\sum _{i+j+l+m=4}{\left(-i\right)}^{i+j+l}[{\alpha }_{ijlm}{\partial }_x^i{\partial }_y^j{\partial }_z^l{k}_0^m]$$

(11)

$$\widehat{U}\left({\partial }_x,{\partial }_y,{\partial }_z\right)={\displaystyle \frac{1}{k_0^4}}\sum _{i+j+l+m=4}{\left(-i\right)}^{i+j+l}[{\widehat{\beta }}_{ijlm}{\partial }_x^i{\partial }_y^j{\partial }_z^l{k}_0^m]$$

(12)

Note that differential operators $D$ and $\widehat{U}$ have constant coefficients in homogeneous media and, therefore, commute. We recast Equation (7) as

$$D\left({\partial }_x,{\partial }_y,{\partial }_z\right)\mathit{A}\left(\mathit{r}\right)=-{\displaystyle \frac{4\pi }{c}}\widehat{U}\left({\partial }_x,{\partial }_y,{\partial }_z\right)\mathit{j}\left(\mathit{r}\right)$$

(13)

Instead of representing the source $\mathit{j}\left(\mathit{r}\right)$ as a superposition of point charges, as is done in the Green’s function method, we express the source via the underlying “Om” $\mathbf{ॐ}-\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$ vector field

$$\mathit{j}\left(\mathit{r}\right)=D\left({\partial }_x,{\partial }_y,{\partial }_z\right)\mathbf{ॐ}\left(\mathit{r}\right) ,$$

(14)

From Equation (13) an expression for the vector potential corresponding to source current Equation (14) can be obtained as

$$\mathit{A}\left(\mathit{r}\right)=-{\displaystyle \frac{4\pi }{c}}\widehat{U}\left({\partial }_x,{\partial }_y,{\partial }_z\right)\mathbf{ॐ}\left(\mathit{r}\right)$$

(15)

where the Devanagari script “Om” $\mathbf{ॐ}\left(\mathit{r}\right)$ is a vector field, which underlies both the source $\mathit{j}$ in Equation (14) and the vector potential $\mathit{A}$ in Equation (15) in a unified paradigm of Equations (13)–(15).

The Om-potential we introduce in this work can be interpreted as a vector field that underlies both the sources and the fields [see Equations (14)–(15)], which are observables in macroscopic electromagnetism. This distantly mirrors the role of the wavefunction in quantum mechanics, which generates all observables via operator action, or the role of partition functionals in statistical mechanics and quantum field theory, from which correlation functions are derived. Like the Hertz vector potential, the Om-potential is not directly observable but simplifies the analytical framework and reveals invariant structures that may otherwise remain hidden. The Om-potential formalism, therefore, acts as a unifying substrate across varying isotropy-broken electromagnetic media, organizing both the geometric and material complexity of Maxwell’s theory into a single functional framework.

Note that for an arbitrary source, the underlying “Om” vector field $\mathbf{ॐ}\left(\mathit{r}\right)$ can be found as

$$\mathbf{ॐ}\left(\mathit{r}\right)=\int {dr}^{\prime } g_{ॐ}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right) \mathit{j}\left({\mathit{r}}^{\mathit{\prime }}\right),$$

where the scalar “Om” Green’s function is $g_{ॐ}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)$

$$g_{ॐ}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)=\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\left|\widehat{L}\right|}}\exp \left(i\mathit{k}\left(\mathit{r}-{\mathit{r}}^{\mathit{\prime }}\right)\right)$$

It is important to clarify that using $g_{ॐ}\left(\mathit{r},{\mathit{r}}^{\mathit{\prime }}\right)$ is challenging for complex media, as described in the introduction, and we do not use it in any of the methods we propose here.

The summary of relationships between the “Om” $\mathbf{ॐ}$-potential, vector potential $\mathit{A}$, and sources $\mathit{j}$ is shown in Figure 1.

To improve readability, we list major quantities discussed in this paper into

Table 1. Symbols and notation used in the manuscript.

Symbol	Description
$\mathit{A}\left(\mathit{r}\right)$	Vector potential
$\mathit{j}\left(\mathit{r}\right)$	Source current density
$\mathrm{ॐ}\left(\mathit{r}\right)$	$\mathrm{Om}\text{-}\mathrm{potential} (\mathrm{underlying} \mathrm{vector} \mathrm{field} \mathrm{generating} \mathrm{both} \mathit{A}\left(\mathit{r}\right)$$\mathrm{and} \mathit{j}\left(\mathit{r}\right)$)
$\widehat{L}$	$\mathrm{Helmholtz} \mathrm{operator} \mathrm{for} \mathrm{isotropy}-\mathrm{broken} \mathrm{media}: \widehat{L}\mathit{A}=-{\displaystyle \frac{4\mathit{\pi }}{\mathit{c}}}\mathit{j}$
$D$	$\mathrm{Differential} \mathrm{operator} \mathrm{used} \mathrm{to} \mathrm{generate} \mathit{j}\left(\mathit{r}\right)$$\mathrm{from} \mathrm{ॐ}\left(\mathit{r}\right)$$; \mathrm{its} \mathrm{Fourier} \mathrm{transform} \mathrm{is} \det \widehat{L}$
$\widehat{U}$	$\mathrm{Differential} \mathrm{operator} \mathrm{used} \mathrm{to} \mathrm{generate} \mathit{A}\left(\mathit{r}\right)$$\mathrm{from} \mathrm{ॐ}\left(\mathit{r}\right)$$; \mathrm{its} \mathrm{Fourier} \mathrm{transform} \mathrm{is} \mathrm{adj}\widehat{L}$

.

In vacuum, the operator $\widehat{L}$ has the following properties:

$$\widehat{L}\left(i\mathit{k},-i{k}_0\right)=\left(k_0,\mathit{k}\times \widehat{I}\right)\left(\begin{array}{c}{k}_0\\ \mathit{k}\times \widehat{I}\end{array}\right)=\left(\mathit{k}\times \widehat{I}\right)\left(\mathit{k}\times \widehat{I}\right)+k_0^2\widehat{I}=\mathit{k}\mathit{k}+\left(k_0^2-k^2\right)\widehat{I}$$

(16)

$$\left|\widehat{L}\right|=k_0^2{\left(k^2-k_0^2\right)}^2, \mathrm{a}\mathrm{d}\mathrm{j} \widehat{L}=\left(k^2-k_0^2\right)\left(\mathit{k}\mathit{k}+k_0^2\widehat{I}\right)$$

This means that for a vacuum, Equation (13) can be rewritten as

$${\left({\nabla }^2+k_0^2\right)}^2\mathit{A}\left(\mathit{r}\right)={\displaystyle \frac{4\pi }{c}}\left({\nabla }^2+k_0^2\right)\left(\widehat{I}-{\displaystyle \frac{1}{k_0^2}}\nabla \nabla \right)\mathit{j}\left(\mathit{r}\right)$$

$$D_{vac}\left({\partial }_x,{\partial }_y,{\partial }_z\right)=k_0^2{\left({\nabla }^2+k_0^2\right)}^2, {\widehat{U}}_{vac}\left({\partial }_x,{\partial }_y,{\partial }_z\right)=\left({\nabla }^2+k_0^2\right)\left(k_0^2\widehat{I}-\nabla \nabla \right)$$

For a point source at ${\mathit{r}}_0$ polarized in direction $\widehat{\mathit{e}}$ in vacuum $\mathit{j}=\widehat{\mathit{e}}\delta \left(\mathit{r}-{\mathit{r}}_0\right),$ the “Om” vector field $\mathbf{ॐ}\left(\mathit{r}\right)$ is a spherical wave propagating from the source location

$${\mathbf{ॐ}}_{vac-point}\left(\mathit{r}\right)={\left({\nabla }^2+k_0^2\right)}^{-1} \widehat{\mathit{e}}{\displaystyle \frac{e^{i{k}_0|\mathit{r}-{\mathit{r}}_0|}}{4\pi |\mathit{r}-{\mathit{r}}_0|}}=\widehat{\mathit{e}} e^{i{k}_0|\mathit{r}-{\mathit{r}}_0|}$$

4. Inverse Helmholtz Equation Method

The first method to find solutions of Equation (8) relies on inverse approach to the Helmholtz equation

$$\mathit{j}\left(\mathit{r}\right)=\widehat{L}\left(\nabla ,-i{k}_0\right) \mathit{A}\left(\mathit{r}\right),$$

(17)

where instead of looking for vector potential $\mathit{A}\left(\mathit{r}\right)$ for a given source $\mathit{j}\left(\mathit{r}\right)$, we set the vector potential $\mathit{A}\left(\mathit{r}\right)$ and obtain sources $\mathit{j}\left(\mathit{r}\right)$, which create the desired vector potential.

To proceed, we utilize the Hermite functions ${\varphi }_{n_x{n}_y{n}_z}\left(\mathit{r}\right)={\psi }_{n_x}\left(x\right){\psi }_{n_y}\left(y\right){\psi }_{n_z}\left(z\right)$, which are eigenfunctions of the quantum harmonic oscillator ${\psi }_{n_x}\left(x\right)={\left(2^{n_x}{n}_x!\sqrt{\pi }{w}_x\right)}^{-{\displaystyle \frac{1}{2}}}{e}^{-{\displaystyle \frac{x^2}{2w_x^2}}}{H}_n\left(x/w_x\right)$.

The del operator applied to Hermite functions is

$$\nabla {\psi }_{n_x{n}_y{n}_z}\left(\mathit{r}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left\{\left({\widehat{a}}_x-{\widehat{a}}_x^{+}\right)/w_x,\left({\widehat{a}}_y-{\widehat{a}}_y^{+}\right)/w_y,\left({\widehat{a}}_z-{\widehat{a}}_z^{+}\right)/w_z\right\}{\psi }_{n_x{n}_y{n}_z}\left(\mathit{r}\right)$$

where the ladder operators ${\widehat{a}}_x{\psi }_n=\sqrt{n+1}{\psi }_{n+1},{\widehat{a}}_x^{+}{\psi }_n=\sqrt{n}{\psi }_{n-1}$.

For a vector potential in vacuum polarized in direction $\widehat{\mathit{x}}$ and given by $\mathit{A}\left(\mathit{r}\right)=\widehat{\mathit{x}}{\psi }_{000}\left(\mathit{r}\right),$ the source can be found as

$$\begin{gathered} \mathit{j}\left(\mathit{r}\right)=\widehat{L}\left(\nabla ,-i{k}_0\right)\mathit{A}\left(\mathit{r}\right)=\left\{\left(k_0^2+{\nabla }^2\right)\widehat{I}-\nabla \nabla \right\}\mathit{A}\left(\mathit{r}\right) \\ ={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{1}{w^2}}\left\{\sqrt{2}\left(k_0^2{w}^2-1\right){\psi }_{000}+{\psi }_{020}+{\psi }_{002},-{\psi }_{110},-{\psi }_{101}\right\} \end{gathered}$$

(18)

If the vector potential $\mathit{A}\left(\mathit{r}\right)$ is fixed, the only material-dependent factor in the RHS of Equation (17) is the operator $\widehat{L}\left(\nabla ,-i{k}_0\right)$. This allows us to create a cross-material mapping between sources $\mathit{j}\left(\mathit{r}\right),$ which create the same vector potential in different media. To demonstrate this, we consider a material with ${\widehat{M}}_{\kappa }=(1-\kappa )\widehat{1}+\kappa {\widehat{M}}_1$, where

$${\widehat{M}}_1=\left(\begin{array}{cccccc}0.63& 0.11& 0.59& 0.82i& -0.8i& -0.56i\\ 0.11& -0.078& -0.37& 0.81i& -0.65i& 0.87i\\ 0.59& -0.37& -0.021& 0.79i& 0.49i& 0.68i\\ -0.82i& -0.81i& -0.79i& 0.27& 0.046& -0.98\\ 0.8i& 0.65i& -0.49i& 0.046& 0.22& -0.73\\ 0.56i& -0.87i& -0.68i& -0.98& -0.73& -0.58\end{array}\right).$$

Note that the matrix ${\widehat{M}}_1$ is also color-coded in Figure 2e. As $\kappa$ changes from 0 to 1, the material undergoes several topological transitions (see Ref. [15]) from non-hyperbolic to mono-hyperbolic [Figure 2a], to bi-hyperbolic [Figure 2b], to tri-hyperbolic [Figure 2c], to tetra-hyperbolic [Figure 2d].

In Figure 3, we show how the source $\mathit{j}\left(\mathit{r}\right)$ required to create the vector potential $\mathit{A}\left(\mathit{r}\right)=\widehat{\mathit{x}}{\psi }_{000}\left(\mathit{r}\right)$ changes as $\kappa$ changes. The leftmost panels of Figure 3 correspond to vacuum $\kappa =0$ and follow Equation (18). One can see that the source distribution is deformed and rotated as the materials pass through the topological transitions shown in Figure 2.

5. The “Om” $\mathbf{ॐ}-\mathbf{P}\mathbf{o}\mathbf{t}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}$ Method

The second method to find solutions to Equation (8) is to use Equations (14) and (15). We select the “Om” $\mathbf{ॐ}\left(\mathit{r}\right)$-potential and find the corresponding source $\mathit{j}\left(\mathit{r}\right)$ and vector potential $\mathit{A}\left(\mathit{r}\right)$. If the “Om” $\mathbf{ॐ}\left(\mathit{r}\right)$-potential is fixed, the only material-dependent factors in the RHS of Equations (14) and (15) are operators $D\left({\partial }_x,{\partial }_y,{\partial }_z\right)$ and $\widehat{U}\left({\partial }_x,{\partial }_y,{\partial }_z\right)$. This creates cross-material mapping between the source–vector potential pairs $\mathit{j}\left(\mathit{r}\right)$ and $\mathit{A}\left(\mathit{r}\right)$, which corresponds to the same “Om” $\mathbf{ॐ}\left(\mathit{r}\right)$-potential as the material is modified.

In Figure 4, we show the x-y cross-sectional distributions of the sources $\mathit{j}\left(\mathit{r}\right)$ and vector potentials $\mathit{A}\left(\mathit{r}\right)$ corresponding to $\mathbf{ॐ}\left(\mathit{r}\right)=\widehat{\mathit{x}}{\psi }_{000}\left(\mathit{r}\right)$ for different materials ${\widehat{M}}_{\kappa }$. We see drastic modifications of both $\mathit{j}\left(\mathit{r}\right)$ and $\mathit{A}\left(\mathit{r}\right)$, which undergo both deformation and rotation. Interestingly, the rate of change in x-y cross-sections of $\mathit{j}\left(\mathit{r}\right)$ and $\mathit{A}\left(\mathit{r}\right)$ is not the same as $\kappa$ is changed. For $\kappa =0.63-0.69$, when the material is in the topological transition into the bi-hyperbolic phase, the source $\mathit{j}\left(\mathit{r}\right)$ is strongly modified, as can be seen from the leftmost panels in Figure 4b–d. At the same time, the vector potential $\mathit{A}\left(\mathit{r}\right)$ has minimal changes in the same range of $\kappa$.

6. Discussion

We would like to summarize the three methods discussed in this manuscript using

Table 2. The three approaches to solving macroscopic Maxwell’s equations.

Method	Uses Point Sources	Suitable for Isotropy-Broken Media	Structure	Notes
Green’s Function Method (GFM)	Yes	Limited	Obtain fields from sources via an integral with singularities/No closed form	Natural in microscopic EM; struggles in complex macroscopic media
Inverse Helmholtz Method	No	Yes	Obtain sources from fields via analytical differentiation/Closed form	Solves for the source from the desired vector potential. Applicable to complex macroscopic media
Om-Potential Method	No	Yes	Obtain source-field pairs from Om-potential via analytical differentiation/Closed form	Introduces unified Om-potential for source and potential. Applicable to complex macroscopic media

below.

While the operators $D$ and $\widehat{U}$ introduced in this paper contain high-order derivatives, no numerical discretization is performed in this work. If implemented numerically in future studies, we note that staggered-grid finite-difference schemes and spectral methods have been used successfully for similar operators, with proper stability constraints [16,17].

In conclusion, the Green’s function method with point sources and singular fields is inherent to microscopic electromagnetism and fundamentally stems from the properties of dimensionless elementary particles. Extension of GFM to macroscopic electromagnetism faces obvious and fundamental challenges. In this manuscript, we demonstrate that solutions to problems of macroscopic electromagnetism can be found without the use of Green’s functions by introducing two new approaches: the Inverse Helmholtz equation method and the “Om” ॐ-potential method. Although this work is restricted to the frequency domain, the analytical methods may be extended to time-dependent and dispersive media (see [18] as an example). These extensions are left for future investigation.