Uniqueness Theorem for Time-Harmonic EM Fields

• The uniqueness theorem for time-harmonic electromagnetic fields establishes conditions under which Maxwell's equations yield a single, well-posed solution for given material parameters and boundary conditions.
• It underpins both direct scattering and inverse problems, enabling accurate determination of media properties and geometric structures in complex electromagnetic environments.
• Key analytical tools include energy identities, unique continuation, and analytic continuation, with extensions addressing nonlocal, anisotropic, and periodic media.

The uniqueness theorem for time-harmonic electromagnetic fields governs the circumstances under which Maxwell’s equations—under time-harmonic (frequency-domain) excitation—admit at most one solution given material parameters, boundary conditions, and prescribed sources. This property is fundamental for both well-posedness (direct scattering) and the identifiability of media (inverse problems) in complex electromagnetic environments. The mathematical and physical insight afforded by these theorems underpins the theory of electromagnetic scattering, the development of direct and inverse algorithms, and the analysis of engineered materials, including nonlocal and anisotropic responses.

1. Classical Uniqueness Theorems: Local and Global Formulations

For a bounded, connected Lipschitz domain $\Omega\subset\mathbb{R}^3$, with electromagnetic parameters $\varepsilon(x)$, $\mu(x)$, and source $\mathbf{J}$, the time-harmonic Maxwell system at angular frequency $\omega$ is: $$\begin{aligned} &\nabla\times\mathbf{E} = i\omega\mu\mathbf{H}, \ &\nabla\times\mathbf{H} = \mathbf{J} - i\omega\varepsilon\mathbf{E}. \end{aligned}$$ Boundary value formulations (PEC: $\mathbf{E}\times\mathbf{n}=0$; PMC: $(\mu^{-1}\nabla\times\mathbf{E})\times\mathbf{n}=0$; impedance: $$\mathbf{n}\times[(\mu^{-1}\nabla\times\mathbf{E})\times\mathbf{n}]+Z_{\mathrm{imp}}(\mathbf{E}\times\mathbf{n})=g_{\mathrm{imp}}$$) manifest uniqueness when the associated variational form is coercive or Fredholm with trivial nullspace (Jr. et al., 13 Jul 2024).

The critical analytic mechanism is the construction of an energy (sesquilinear) form: $$a(u,u) = \int_\Omega \left[ (\mu^{-1}\nabla\times u)\cdot(\nabla\times u)^* - \omega^2(\varepsilon u)\cdot u^* \right] dx$$ with boundary terms for Robin/impedance data.

Sufficient conditions for uniqueness include:

• Strong (Gårding) ellipticity: existence of angle $\theta$ such that $$\operatorname{Re}(e^{i\theta}a(u,u)) \geq C \|u\|_V^2$$ for the problem’s Hilbert space $V$ (e.g., $H_0(\operatorname{curl},\Omega)$).
• Nontrapping (dissipative) medium: For lossy media, $\operatorname{Im}\varepsilon>0$, $\operatorname{Im}\mu>0$ ensure strict decay of field energy (Jr. et al., 13 Jul 2024).

In the anisotropic or piecewise regular setting—with piecewise $W^{1,\infty}$ or even $L^\infty$ coefficients and finite or countably many $C^0$ subdomains—the unique continuation principle for Maxwell’s equations (Nguyen–Wang, Carleman estimates) propagates vanishing of solutions from subdomains to the entire region, ensuring uniqueness (Ball et al., 2012).

2. Uniqueness Results for Inverse Problems: Determination of Material and Geometric Parameters

Uniqueness theorems for inverse boundary value problems establish the identifiability of constitutive parameters ($\varepsilon,\mu,\sigma$) or geometric structures from boundary or far-field measurements. Central results include:

• Global uniqueness from full Cauchy data: For Lipschitz domains and $C^1$ material parameters, knowledge of the boundary impedance map at a single frequency suffices to recover $\mu(x)$, $\varepsilon(x)$, and conductivity $\sigma(x)$ everywhere in $\Omega$ (Caro et al., 2012).
• Partial data uniqueness: If the coefficients coincide near the domain boundary, equality of Cauchy data on an open subset $\Gamma\subset\partial\Omega$ determines the coefficients globally, assuming $C^{1,1}$ boundary and coefficient regularity (Brown et al., 2014).
• Uniqueness for scatterer geometry: For a bounded obstacle $D\subset\mathbb{R}^3$, the knowledge of the entire far-field pattern $E^\infty(\beta;\alpha_0, k_0)$ for all observation directions $\beta$ due to a single incident plane wave uniquely determines both the domain $D$ and, in the impedance case, the function $\lambda(x)$ (Liu, 2016).
• Minimal-data uniqueness: For locally perturbed rough surfaces, a single dipole or incident wave measurement suffices to uniquely reconstruct polyhedral perturbations, using analytic continuation, mixed reciprocity, and reflection principles (Zhao et al., 2018).

3. Extensions: Nonlocal Media, Anisotropic and Periodic Structures

Nonlocal electromagnetic models, typified by the hydrodynamic Drude model (HDM) and generalized nonlocal optical response (GNOR), introduce spatial derivatives of higher order. In these settings, uniqueness is maintained if and only if:

• The material is passive ($\operatorname{Im}\varepsilon(\omega)>0$, $\operatorname{Im}\mu(\omega)>0$).
• One additional boundary condition (ABC) per material interface is imposed, such as the vanishing of normal free-electron current (Sauter ABC) or continuity of weighted normal divergence of $\mathbf{E}$ (Forstmann–Stenschke ABC) (Mystilidis et al., 2023).

In periodic or locally perturbed periodic media, the Rayleigh expansion radiation condition is insufficient for uniqueness. Uniqueness is achieved by adopting equivalent open-waveguide conditions, separating the field into guided and radiative components, and applying integral identities to suppress nonphysical surface/guided modes (Hu et al., 17 Feb 2024).

For biperiodic dielectric structures on metallic substrates, uniqueness for all frequencies is established when the permittivity profile satisfies specific monotonicity and smallness assumptions, with a Rellich-type identity ruling out nontrivial waves (Lechleiter et al., 2012).

4. Special Geometric Configurations: Spherical Stratification, Corners, and Multipole Expansions

For spherically stratified dielectrics, the entire spectrum of interior transmission eigenvalues uniquely determines the radial refractive index profile. This is achieved by encoding the eigenvalue set as zeros of an entire function whose type is related to the path integral of $\sqrt{n(r)}$, and then applying Cartwright’s and Levin’s theorems to conclude spectral and hence material uniqueness (Chen, 2012). For the unique recovery of fields from their radial components on an external spherical surface, the orthogonality and completeness of vector spherical harmonics are crucial; the multipole coefficients are uniquely determined by the given data, and the result reduces to classical Poynting-based uniqueness inside the sphere (Talashila, 20 Dec 2025).

Sharp uniqueness results are also established for domains with corners: In truncated polyhedral cones, nontrivial sources must vanish at the apex. This leads to global uniqueness in source or medium support recovery from single far-field patterns if each geometric corner interrogates the support (no non-radiating corners) (Blåsten et al., 2019). For time-harmonic Maxwell fields, the Debye source representation yields an explicit isomorphism between boundary data and solution space, leading to strong uniqueness theorems in domains with smooth boundary components (Epstein et al., 2013).

5. Key Ingredients in Uniqueness Proofs

Uniqueness arguments feature several core analytic mechanisms:

• Energy identities and Rellich’s lemma: Integral identities (stepwise use of Green’s theorem, boundary terms, and the imaginary part of the sesquilinear form) imply that vanishing data at the boundary forces triviality of the solution in the interior, provided certain coercivity or dissipativity holds (Jr. et al., 13 Jul 2024, Lechleiter et al., 2012).
• Unique continuation principles: Strong unique continuation (through Carleman estimates) in various regularity classes ensures that such vanishing propagates and rules out nontrivial compactly supported fields (Ball et al., 2012).
• Complex geometrical optics (CGO) solutions: Inverse boundary value problem proofs construct oscillatory exponentially growing/decaying solutions to exploit integral identities for identifying parameters (Caro et al., 2012, Brown et al., 2014).
• Analytic continuation and reciprocity: For inverse scattering, extension of the far-field pattern and mixed reciprocity identities allow reduction from knowledge of one measurement to full boundary control (Liu, 2016, Zhao et al., 2018).
• Entire function theory and spectral asymptotics: Transmission eigenvalue problems for spherically symmetric media employ entire function and zero-density arguments to reach material uniqueness (Chen, 2012).

6. Representative Uniqueness Theorems: Context, Assumptions, and Implications

Problem Type	Data	Sufficient Conditions
Direct Maxwell problem	Tangential electric/magnetic field on ∂Ω	Elliptic, bounded, coercive parameters
Inverse boundary value	Cauchy data (tE, tH) on ∂Ω or open part Γ	$C^1$ parameters, boundary regularity
Inverse scattering	Far-field pattern for full/partial aperture	PEC/impedance boundaries, Lipschitz
Nonlocal media	Maxwell-HDM/ GNOR + ABCs at each interface	Loss (Im ε, Im μ), correct ABCs
Stratified/periodic media	Rayleigh/Open-waveguide radiation condition	Spectral assumptions, periodicity
Corner domains	Support, field vanishing at apex	Polyhedral geometry, regularity

Rigorous uniqueness guarantees imply that measured data—boundary, far field, or eigenvalue spectra—fundamentally determine the sought material or geometric variables, justifying the well-posedness of both direct and inverse problems in electromagnetic theory. These theorems form the mathematical foundation for computational and theoretical advances in inverse design, nonlocal photonics, and electromagnetic imaging.

7. Open Problems and Extensions

Several open directions emerge:

• Partial-spectrum uniqueness: For spherically stratified media, the sufficiency of real transmission eigenvalues (rather than angular wedge densities) remains unresolved (Chen, 2012).
• Nonlocal and dispersive boundaries: Correct specification of ABCs in complex material junctions remains an evolving subject, as do rigorous results for general nonlocal response models (Mystilidis et al., 2023).
• Lower regularity: Extension of unique continuation and well-posedness to less regular media (e.g., rough coefficients, fractal boundaries) is ongoing (Ball et al., 2012).
• Minimal data: Characterization of the minimal measurement data (in frequency, aperture, or polarization) required for uniqueness, particularly in inverse scattering from complex media or open structures.
• Uniqueness under severe geometrical degeneracies: For example, cloaking or the presence of corners which may in principle generate non-radiating configurations if field vanishing at singularities accidentally occurs (Blåsten et al., 2019).

These problems represent the interface between functional analysis, PDE theory, spectral theory, and mathematical physics, continuing to motivate advances in the understanding of electromagnetic field uniqueness in increasingly complex physical systems.