Maxwell's Equations and the Helmholtz Equation

From Maxwell's Equations to the Wave Equation

Every electromagnetic scattering problem begins with Maxwell's equations. In this section we derive the vector and scalar wave equations that govern field propagation in a source-free, inhomogeneous medium. The key result is that the total field satisfies a wave equation with a source term proportional to the contrast function χ(r)\chi(\mathbf{r}) — the object we wish to image.

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Definition:

Time-Harmonic Maxwell's Equations

For monochromatic fields with e−jωte^{-j\omega t} dependence in a linear, isotropic medium with permittivity ε(r)\varepsilon(\mathbf{r}) and permeability μ0\mu_0 (non-magnetic), Maxwell's equations reduce to:

∇×E=jωμ0H,\nabla \times \mathbf{E} = j\omega\mu_0 \mathbf{H},

∇×H=−jωε(r)E+J,\nabla \times \mathbf{H} = -j\omega\varepsilon(\mathbf{r})\mathbf{E} + \mathbf{J},

∇⋅(ε(r)E)=ρ,\nabla \cdot (\varepsilon(\mathbf{r})\mathbf{E}) = \rho,

∇⋅(μ0H)=0,\nabla \cdot (\mu_0 \mathbf{H}) = 0,

where J\mathbf{J} is an impressed current source and ρ\rho is the charge density. In source-free regions (J=0\mathbf{J} = 0, ρ=0\rho = 0), the divergence equations follow from the curl equations.

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Time-harmonic field

A field whose time dependence is purely sinusoidal at a single frequency ω\omega, represented by the phasor convention E(r,t)=Re{E(r)e−jωt}\mathbf{E}(\mathbf{r}, t) = \text{Re}\{\mathbf{E}(\mathbf{r})e^{-j\omega t}\}.

Related: Wavenumber

Theorem: The Vector Wave Equation

Taking the curl of Faraday's law and substituting Ampere's law, the electric field in a source-free inhomogeneous medium satisfies the vector wave equation:

∇2E+κ2(r)E=∇(∇⋅E),\nabla^2 \mathbf{E} + \kappa^{2}(\mathbf{r})\mathbf{E} = \nabla(\nabla \cdot \mathbf{E}),

where κ2(r)=ω2μ0ε(r)\kappa^{2}(\mathbf{r}) = \omega^2\mu_0\varepsilon(\mathbf{r}) is the spatially varying wavenumber squared. The right-hand side vanishes in homogeneous media where ∇⋅E=0\nabla \cdot \mathbf{E} = 0, but is nonzero at dielectric interfaces.

Proof
Curl of Faraday's law

Taking the curl of ∇×E=jωμ0H\nabla \times \mathbf{E} = j\omega\mu_0 \mathbf{H}:

∇×∇×E=jωμ0∇×H=jωμ0(−jωε(r)E)=ω2μ0ε(r)E.\nabla \times \nabla \times \mathbf{E} = j\omega\mu_0 \nabla \times \mathbf{H} = j\omega\mu_0(-j\omega\varepsilon(\mathbf{r})\mathbf{E}) = \omega^2\mu_0\varepsilon(\mathbf{r})\mathbf{E}.

Vector identity

Using ∇×∇×E=∇(∇⋅E)−∇2E\nabla \times \nabla \times \mathbf{E} = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}, we obtain

∇2E+κ2(r)E=∇(∇⋅E).■\nabla^2 \mathbf{E} + \kappa^{2}(\mathbf{r})\mathbf{E} = \nabla(\nabla \cdot \mathbf{E}). \quad \blacksquare

Definition:

Object Contrast Function

We decompose the permittivity into background and perturbation:

ε(r)=ε0(1+χ(r)),\varepsilon(\mathbf{r}) = \varepsilon_0(1 + \chi(\mathbf{r})),

where the contrast function χ(r)\chi(\mathbf{r}) vanishes outside the object domain DD. The wavenumber becomes

κ2(r)=κ02(1+χ(r)),\kappa^{2}(\mathbf{r}) = \kappa_{0}^{2}(1 + \chi(\mathbf{r})),

where κ0=ωμ0ε0=2π/λ\kappa_{0} = \omega\sqrt{\mu_0\varepsilon_0} = 2\pi/\lambda is the free-space wavenumber. The contrast function χ(r)\chi(\mathbf{r}) is the quantity we wish to reconstruct from measured scattered fields.

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Contrast function

The function χ(r)=εr(r)−1\chi(\mathbf{r}) = \varepsilon_r(\mathbf{r}) - 1 that describes the deviation of the medium's relative permittivity from the background. It vanishes outside the scatterer and is the primary unknown in inverse scattering.

Related: Wavenumber

Theorem: The Scalar Helmholtz Equation

For a scalar field u(r)u(\mathbf{r}) (valid for 2D TM polarization or scalar approximations), the scattered field satisfies the inhomogeneous Helmholtz equation:

∇2usca+κ02usca=−κ02χ(r) u(r),\nabla^2 u^{\text{sca}} + \kappa_{0}^{2} u^{\text{sca}} = -\kappa_{0}^{2} \chi(\mathbf{r})\,u(\mathbf{r}),

where u=uinc+uscau = u^{\text{inc}} + u^{\text{sca}} is the total field. The right-hand side −κ02χ(r)u(r)-\kappa_{0}^{2}\chi(\mathbf{r})u(\mathbf{r}) acts as an equivalent source — the contrast χ\chi illuminated by the total field uu radiates the scattered field.

Proof
Separation into incident and scattered parts

The total field satisfies ∇2u+κ02(1+χ)u=0\nabla^2 u + \kappa_{0}^{2}(1 + \chi)u = 0. The incident field satisfies ∇2uinc+κ02uinc=0\nabla^2 u^{\text{inc}} + \kappa_{0}^{2} u^{\text{inc}} = 0. Subtracting and using u=uinc+uscau = u^{\text{inc}} + u^{\text{sca}}:

∇2usca+κ02usca=−κ02χ(r) u(r).■\nabla^2 u^{\text{sca}} + \kappa_{0}^{2} u^{\text{sca}} = -\kappa_{0}^{2} \chi(\mathbf{r})\,u(\mathbf{r}). \quad \blacksquare

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Definition:

The Sommerfeld Radiation Condition

The Sommerfeld radiation condition ensures uniqueness of the exterior scattering problem by requiring that scattered fields propagate outward. In 3D:

lim⁡r→∞r(∂usca∂r−jκ0usca)=0,\lim_{r \to \infty} r\left(\frac{\partial u^{\text{sca}}}{\partial r} - j\kappa_{0} u^{\text{sca}}\right) = 0,

uniformly in all directions. In 2D the factor rr is replaced by r\sqrt{r}. Physically, this excludes incoming waves from infinity — the only source of radiation is the scatterer.

Theorem: Uniqueness of the Exterior Helmholtz Problem

The exterior Helmholtz equation ∇2usca+κ02usca=f\nabla^2 u^{\text{sca}} + \kappa_{0}^{2} u^{\text{sca}} = f in Rd∖D‾\mathbb{R}^d \setminus \overline{D} with the Sommerfeld radiation condition has a unique solution for all κ0>0\kappa_{0} > 0, provided the source ff has compact support.

Show Hint

Uniqueness fails at discrete frequencies for interior problems (resonances), but for the exterior scattering problem uniqueness holds for all positive wavenumbers.

Proof
Uniqueness via Rellich's lemma

Suppose u1,u2u_1, u_2 are two solutions. Their difference w=u1−u2w = u_1 - u_2 satisfies the homogeneous Helmholtz equation and the radiation condition. By Rellich's lemma, if ∮∣r∣=R∣w∣2 dS→0\oint_{|r|=R} |w|^2 \, dS \to 0 as R→∞R \to \infty, then w≡0w \equiv 0 outside DD. The radiation condition implies exactly this decay, establishing uniqueness. ■\blacksquare

TM and TE Polarizations in 2D

In 2D scattering problems, two independent polarizations decouple:

  • TM (Transverse Magnetic): E=Ezz^\mathbf{E} = E_z \hat{z}. The scalar field u=Ezu = E_z satisfies the Helmholtz equation directly.
  • TE (Transverse Electric): H=Hzz^\mathbf{H} = H_z \hat{z}. The scalar field u=Hzu = H_z satisfies a modified equation with 1/ε1/\varepsilon weighting at interfaces.

The TM case is simpler because the boundary conditions are continuous uu and continuous ∂u/∂n\partial u/\partial n. Most 2D imaging formulations use TM polarization for this reason.

Wavenumber

The spatial frequency of a monochromatic wave: κ=2π/λ=ω/c\kappa = 2\pi/\lambda = \omega/c. In an inhomogeneous medium the local wavenumber becomes κ(r)=κ01+χ(r)\kappa(\mathbf{r}) = \kappa_{0}\sqrt{1 + \chi(\mathbf{r})}.

Related: Contrast function

Historical Note: Hermann von Helmholtz and the Wave Equation

19th century

Hermann von Helmholtz (1821--1894) derived the equation bearing his name in the context of acoustics, not electromagnetism. The equation ∇2u+k2u=0\nabla^2 u + k^2 u = 0 governs all monochromatic wave phenomena — acoustic, electromagnetic, and quantum mechanical. Arnold Sommerfeld (1868--1951) later added the radiation condition that makes the exterior problem well-posed, enabling rigorous scattering theory.

Quick Check

In the inhomogeneous scalar Helmholtz equation for the scattered field, ∇2usca+κ02usca=−κ02χu\nabla^2 u^{\text{sca}} + \kappa_{0}^{2} u^{\text{sca}} = -\kappa_{0}^{2} \chi u, what role does the term −κ02χu-\kappa_{0}^{2} \chi u play?

It is the incident field that drives the scattering.

It acts as an equivalent source — the contrast illuminated by the total field radiates the scattered field.

It represents absorption losses in the medium.

It enforces the radiation condition at infinity.

Correction:
It acts as an equivalent source — the contrast illuminated by the total field radiates the scattered field.

The product χ(r)u(r)\chi(\mathbf{r}) u(\mathbf{r}) couples the object properties with the local field to create radiation.

Key Takeaway

Time-harmonic Maxwell's equations in an inhomogeneous medium yield the vector wave equation with spatially varying wavenumber κ2(r)=κ02(1+χ(r))\kappa^{2}(\mathbf{r}) = \kappa_{0}^{2}(1 + \chi(\mathbf{r})). The contrast function χ(r)\chi(\mathbf{r}) encodes the scattering object and is the unknown to be reconstructed. The scattered field satisfies a Helmholtz equation with source −κ02χutot-\kappa_{0}^{2}\chi u^{\text{tot}}, coupling contrast and total field. The Sommerfeld radiation condition ensures uniqueness of the exterior scattering problem.

Prerequisites & NotationGreen's Functions