Maxwell's Equations and the Helmholtz Equation

This chapter establishes the electromagnetic framework that underlies the uniform theory of diffraction. We fix the time-harmonic convention, derive the frequency-domain Maxwell equations, and reduce them to the scalar Helmholtz equation that governs two-dimensional wedge diffraction.

Time-harmonic convention

Throughout this package and its documentation, all electromagnetic fields are represented as complex phasors with time dependence $\exp(+i\omega t)$. A physical, real-valued field $\mathcal{E}(\mathbf{r},t)$ is recovered from its phasor $\mathbf{E}(\mathbf{r})$ by

\[\mathcal{E}(\mathbf{r}, t) = \operatorname{Re}\!\bigl[\mathbf{E}(\mathbf{r})\,e^{+i\omega t}\bigr].\]

Under this convention, the time derivative of a phasor quantity $\mathbf{A}(\mathbf{r})\,e^{+i\omega t}$ produces a factor of $+i\omega$:

\[\frac{\partial}{\partial t}\bigl[\mathbf{A}\,e^{+i\omega t}\bigr] = +i\omega\,\mathbf{A}\,e^{+i\omega t}.\]

Medium parameters

Consider a homogeneous, isotropic, lossless background medium with permittivity $\epsilon$ and permeability $\mu$. The wavenumber and intrinsic impedance are

\[k \equiv \omega\sqrt{\mu\epsilon}, \qquad \eta \equiv \sqrt{\frac{\mu}{\epsilon}}.\]

For free space, $\epsilon = \epsilon_0 \approx 8.854 \times 10^{-12}\,\text{F/m}$ and $\mu = \mu_0 = 4\pi \times 10^{-7}\,\text{H/m}$, giving $k = \omega/c$ with $c = 1/\sqrt{\mu_0\epsilon_0}$.

Frequency-domain Maxwell's equations

Starting from the time-domain Maxwell curl equations (source-free region):

\[\nabla \times \mathcal{E} = -\frac{\partial \mathcal{B}}{\partial t}, \qquad \nabla \times \mathcal{H} = \frac{\partial \mathcal{D}}{\partial t},\]

and substituting the phasor representation with $\partial/\partial t \to +i\omega$, the constitutive relations $\mathcal{B} = \mu\mathcal{H}$ and $\mathcal{D} = \epsilon\mathcal{E}$, and cancelling the common $e^{+i\omega t}$ factor, we obtain:

Faraday's law:

\[\nabla \times \mathbf{E} = -i\omega\mu\,\mathbf{H}.\]

Derivation: $\nabla \times \mathbf{E}\,e^{+i\omega t} = -\partial(\mu\mathbf{H}\,e^{+i\omega t})/\partial t = -\mu\mathbf{H}\cdot(+i\omega)\,e^{+i\omega t}$. Dividing both sides by $e^{+i\omega t}$ gives the result.

Ampère's law:

\[\nabla \times \mathbf{H} = +i\omega\epsilon\,\mathbf{E}.\]

Derivation: $\nabla \times \mathbf{H}\,e^{+i\omega t} = \partial(\epsilon\mathbf{E}\,e^{+i\omega t})/\partial t = \epsilon\mathbf{E}\cdot(+i\omega)\,e^{+i\omega t}$. Dividing by $e^{+i\omega t}$ gives the result. Note the positive sign on the right-hand side, which is characteristic of the $\exp(+i\omega t)$ convention.

Vector Helmholtz equation

Eliminating $\mathbf{H}$ from the two curl equations yields the vector Helmholtz equation. Taking the curl of Faraday's law:

\[\nabla \times (\nabla \times \mathbf{E}) = -i\omega\mu\,(\nabla \times \mathbf{H}).\]

Substituting Ampère's law for $\nabla \times \mathbf{H}$:

\[\nabla \times (\nabla \times \mathbf{E}) = -i\omega\mu\,(+i\omega\epsilon\,\mathbf{E}) = -i^2\omega^2\mu\epsilon\,\mathbf{E} = +\omega^2\mu\epsilon\,\mathbf{E} = k^2\,\mathbf{E}.\]

Rearranging:

\[\nabla \times \nabla \times \mathbf{E} - k^2\,\mathbf{E} = \mathbf{0}, \qquad \mathbf{r} \in \mathbb{R}^3 \setminus \Gamma,\]

where $\Gamma$ denotes the scattering boundary (the wedge surface).

Scalar Helmholtz equation for 2D problems

For a wedge that is uniform along the $z$-axis and with fields that do not vary in $z$, the vector problem decouples into two independent scalar problems. In cylindrical coordinates $(\rho, \phi, z)$ about the wedge edge:

• Soft polarisation (also called TM$_z$ or E-polarisation): The electric field is $\mathbf{E} = E_z(\rho,\phi)\,\hat{z}$. The scalar field $u = E_z$ satisfies the Dirichlet boundary condition $u = 0$ on the PEC surface.

• Hard polarisation (also called TE$_z$ or H-polarisation): The magnetic field is $\mathbf{H} = H_z(\rho,\phi)\,\hat{z}$. The scalar field $u = H_z$ satisfies the Neumann boundary condition $\partial u/\partial n = 0$ on the PEC surface.

Why soft = Dirichlet and hard = Neumann

These boundary conditions follow directly from the PEC condition $\hat{\mathbf{n}} \times \mathbf{E} = \mathbf{0}$ (tangential electric field vanishes on a perfect conductor). The two cases arise because the $z$-directed field component has a different electromagnetic role in each polarisation.

Soft (TM$_z$): $E_z = 0$ on the surface (Dirichlet). When the electric field is purely $z$-directed, $\mathbf{E} = E_z\,\hat{z}$, the component $E_z$ is tangential to any surface whose normal $\hat{\mathbf{n}}$ lies in the $(\rho, \phi)$ plane (which is the case for the wedge faces). The PEC boundary condition requires this tangential component to vanish:

\[\hat{\mathbf{n}} \times \mathbf{E} = \mathbf{0} \quad\Longrightarrow\quad E_z\,(\hat{\mathbf{n}} \times \hat{z}) = \mathbf{0} \quad\Longrightarrow\quad E_z = 0 \;\text{ on the surface}.\]

Since $\hat{\mathbf{n}} \times \hat{z} \neq \mathbf{0}$ (the surface normal is perpendicular to $\hat{z}$), the only way to satisfy this is $E_z = 0$. This is a Dirichlet condition on the scalar field $u = E_z$.

The name "soft" comes from acoustics: a pressure-release (soft) boundary forces the pressure (the scalar field) to zero, exactly like the Dirichlet condition here.

Hard (TE$_z$): $\partial H_z / \partial n = 0$ on the surface (Neumann). When the magnetic field is purely $z$-directed, $\mathbf{H} = H_z\,\hat{z}$, the PEC condition $\hat{\mathbf{n}} \times \mathbf{E} = \mathbf{0}$ does not directly constrain $H_z$, because $H_z$ is a magnetic field component. Instead, we must use Faraday's law to relate the electric field to $H_z$.

From $\nabla \times \mathbf{E} = -i\omega\mu\,\mathbf{H} = -i\omega\mu\,H_z\,\hat{z}$, the $z$-component of the curl gives a constraint on the transverse electric field components $E_\rho$ and $E_\phi$. More directly, the transverse electric field components are obtained from $\mathbf{H} = H_z\,\hat{z}$ via Ampère's law. In cylindrical coordinates, the relevant component is:

\[E_\phi = \frac{i\omega\mu}{k^2}\,\frac{\partial H_z}{\partial\rho}, \qquad E_\rho = -\frac{i\omega\mu}{k^2}\,\frac{1}{\rho}\frac{\partial H_z}{\partial\phi}.\]

On the wedge face (say $\phi = 0$), the outward normal is $\hat{\mathbf{n}} = \hat{\phi}$ (or $-\hat{\phi}$). The tangential electric field component in the $z$-direction is zero (there is no $E_z$ in this polarisation), but the tangential component along $\hat{\rho}$ must also vanish:

\[\hat{\mathbf{n}} \times \mathbf{E}\big|_{\text{surface}} = \mathbf{0} \quad\Longrightarrow\quad E_\rho\big|_{\phi=0} = 0 \quad\Longrightarrow\quad \frac{\partial H_z}{\partial\phi}\bigg|_{\phi=0} = 0.\]

Since $\partial/\partial\phi$ is proportional to $\partial/\partial n$ (the normal derivative) on the wedge face, this is a Neumann condition on the scalar field $u = H_z$.

The name "hard" comes from acoustics: a rigid (hard) boundary forces the normal velocity (proportional to the normal derivative of pressure) to zero, matching the Neumann condition.

In both cases, the scalar field $u(\rho,\phi)$ satisfies the scalar Helmholtz equation:

\[\bigl(\nabla^2 + k^2\bigr)\,u = 0,\]

where $\nabla^2$ is the two-dimensional Laplacian in cylindrical coordinates:

\[\nabla^2 u = \frac{1}{\rho}\frac{\partial}{\partial\rho}\!\left(\rho\frac{\partial u}{\partial\rho}\right) + \frac{1}{\rho^2}\frac{\partial^2 u}{\partial\phi^2}.\]

Radiation condition

The scattered (diffracted) field must satisfy the Sommerfeld radiation condition to ensure that energy propagates outward. For the $\exp(+i\omega t)$ convention, outgoing cylindrical waves have the spatial dependence $e^{-ik\rho}/\sqrt{\rho}$ as $\rho \to \infty$. Formally:

\[\lim_{\rho\to\infty} \sqrt{\rho}\left(\frac{\partial u^{\text{sc}}}{\partial\rho} + ik\,u^{\text{sc}}\right) = 0.\]

This selects outgoing waves and excludes incoming waves.

Summary of sign conventions

These conventions are locked in by PhasorConvention(+1) (aliased as EXP_IWT) and are assumed throughout the library.