Optical Wave and Helmholtz Equations from Maxwell

Statement

In a linear, homogeneous, isotropic, source-free medium (permittivity \(\varepsilon\), permeability \(\mu\)), Maxwell's equations imply that the electric field obeys the vector wave equation \(\nabla^2\vec{E}-\mu\varepsilon\,\partial_t^2\vec{E}=\vec{0}\); each Cartesian component obeys the scalar wave equation with phase speed \(v=1/\sqrt{\mu\varepsilon}\); and for a monochromatic field \(\vec{E}(\vec{r},t)=\mathrm{Re}\!\left[\vec{E}(\vec{r})\,e^{-i\omega t}\right]\) the spatial part satisfies the Helmholtz equation \(\nabla^2\vec{E}(\vec{r})+k^2\vec{E}(\vec{r})=\vec{0}\) with \(k=\omega\sqrt{\mu\varepsilon}=n\omega/c\).

Why it matters

This is the moment optics becomes a branch of electromagnetism. The wave equation fixes that light propagates at \(v=1/\sqrt{\mu\varepsilon}\) from purely electric and magnetic constants, delivering the refractive index \(n=c/v=\sqrt{\mu\varepsilon/\mu_0\varepsilon_0}\) without any empirical input about "light."

The reduction to Helmholtz is the workhorse of physical optics. Once time is separated out, diffraction, Gaussian beams, waveguide modes, cavity resonances and scattering are all boundary-value problems for a single elliptic operator \(\nabla^2+k^2\), which is why the same mathematics recurs across the entire subject.

Assumptions

Derivation

Result

Limiting cases

• Vacuum: \(\varepsilon\to\varepsilon_0,\ \mu\to\mu_0\) gives \(v\to c=1/\sqrt{\mu_0\varepsilon_0}\) and \(k\to\omega/c\).
• Non-magnetic dielectric: \(\mu\approx\mu_0\) so \(n\approx\sqrt{\varepsilon/\varepsilon_0}=\sqrt{\varepsilon_r}\), the standard optical index.
• Static limit \(\omega\to0\): \(k\to0\), Helmholtz degenerates to Laplace's equation \(\nabla^2\psi=0\) — electrostatics/magnetostatics recovered.
• Plane wave \(\psi=\psi_0 e^{i\vec{k}\cdot\vec{r}}\): Helmholtz forces \(|\vec{k}|^2=k^2\), i.e. the dispersion relation \(\omega=v|\vec{k}|\).
• Paraxial limit: writing \(\psi=u\,e^{ikz}\) with slowly varying \(u\) reduces Helmholtz to the paraxial wave equation governing Gaussian beams.

Breaks when

Failure modes

• Dropping the double-curl term prematurely. Writing \(\nabla\times(\nabla\times\vec{E})=-\nabla^2\vec{E}\) before invoking \(\nabla\!\cdot\!\vec{E}=0\) — the \(\nabla(\nabla\!\cdot\!\vec{E})\) term must be shown to vanish, and it does not in inhomogeneous media.
• Applying the componentwise Laplacian in curvilinear coordinates. The vector Laplacian equals the scalar Laplacian per component only in Cartesian coordinates; in spherical/cylindrical coordinates \((\nabla^2\vec{E})_r\neq\nabla^2 E_r\).
• Sign of the time convention. Mixing \(e^{-i\omega t}\) and \(e^{+i\omega t}\) conventions flips the sign of the imaginary part of \(k\), turning decay into unphysical growth in lossy media.
• Confusing \(k\) with the free-space \(k_0\). Using \(k=\omega/c\) inside a medium instead of \(k=n\omega/c\) gives wrong wavelengths and phase.
• Treating \(\vec{E}(\vec{r})\) in Helmholtz as three independent scalars while ignoring \(\nabla\!\cdot\!\vec{E}=0\). The transversality constraint still links the components even after separation of time.

Discussion

The physical content is that Maxwell's coupled first-order equations for \(\vec{E}\) and \(\vec{B}\) conceal a decoupled second-order wave equation for each field alone. The price of decoupling is order: differentiating once (the curl in step 3) trades the mutual \(\vec{E}\)–\(\vec{B}\) coupling for a self-contained equation, at the cost of introducing spurious solutions that must be filtered by re-imposing the divergence conditions. That is why \(\nabla\!\cdot\!\vec{E}=0\) is not a byproduct here but an active constraint: it selects the physical, transverse solutions of the wave equation.

The Helmholtz reduction is the spectral decomposition of the wave operator. Because Maxwell's equations are linear, any field is a superposition of monochromatic components, and each component lives in the kernel of \(\nabla^2+k^2\). Optics then becomes the study of one self-adjoint operator on various domains: free space gives plane and spherical waves, a hollow pipe gives waveguide modes, a closed box gives cavity eigenfrequencies, and a lens aperture gives diffraction integrals via the Green's function of \(\nabla^2+k^2\).

At the deepest level the appearance of \(c=1/\sqrt{\mu_0\varepsilon_0}\) — a velocity built from static electric and magnetic measurements — is what forced the relativistic revision of space and time. The wave equation is not Galilean invariant; its invariance group is the Lorentz group, and the constancy of \(v\) in the medium's rest frame is the seed of special relativity. In covariant language the entire derivation collapses to \(\partial_\mu F^{\mu\nu}=0\) in the Lorenz gauge yielding \(\Box A^\nu=0\), of which our vector wave equation is the spatial-component shadow.

Common misconceptions. The wave equation does not replace Maxwell's equations — it is a necessary consequence but not a sufficient one, since it admits longitudinal and divergence-carrying solutions that Maxwell forbids. Likewise, "the speed of light slows in glass" refers to the phase (and group) speed \(v=c/n\) of the macroscopic field; the underlying microscopic fields still propagate at \(c\) between charges, with \(n\) emerging from re-radiation.

Worked examples

Example 1 — Wavenumber and wavelength of green light in crown glass.

Example 2 — Guided mode from the separated Helmholtz equation (X-band waveguide).

Problems

1. Starting from the same source-free Maxwell equations, take the curl of Ampère's law and show that \(\vec{B}\) (equivalently \(\vec{H}\)) obeys \(\nabla^2\vec{B}-\mu\varepsilon\,\partial_t^2\vec{B}=\vec{0}\), the identical wave equation. State where \(\nabla\!\cdot\!\vec{B}=0\) is used.
   Solution

   Take \(\nabla\times\) of \(\nabla\times\vec{H}=\partial_t\vec{D}\): with \(\vec{H}=\vec{B}/\mu\) and \(\vec{D}=\varepsilon\vec{E}\), the left side is \(\frac{1}{\mu}\nabla\times(\nabla\times\vec{B})\) and the right is \(\varepsilon\,\partial_t(\nabla\times\vec{E})=-\varepsilon\,\partial_t^2\vec{B}\) using Faraday. The double-curl identity gives \(\nabla(\nabla\!\cdot\!\vec{B})-\nabla^2\vec{B}\); since \(\nabla\!\cdot\!\vec{B}=0\) always (no magnetic monopoles), the gradient term drops with no extra assumption. Multiplying by \(\mu\): \(\nabla^2\vec{B}-\mu\varepsilon\,\partial_t^2\vec{B}=\vec{0}\).
2. For an inhomogeneous dielectric with \(\varepsilon=\varepsilon(\vec{r})\) (still \(\mu=\mu_0\), source-free), derive the corrected wave equation and identify the coupling term.
   Solution

   Gauss now gives \(\nabla\!\cdot\!(\varepsilon\vec{E})=0\Rightarrow\nabla\!\cdot\!\vec{E}=-\vec{E}\!\cdot\!\nabla\ln\varepsilon\). Repeating steps 3–6, \(\nabla(\nabla\!\cdot\!\vec{E})-\nabla^2\vec{E}=-\mu_0\varepsilon\,\partial_t^2\vec{E}\). Substituting the divergence: \[\nabla^2\vec{E}-\mu_0\varepsilon\,\frac{\partial^2\vec{E}}{\partial t^2}=-\nabla\!\left(\vec{E}\!\cdot\!\nabla\ln\varepsilon\right).\] The right-hand term couples the three components and vanishes only when \(\nabla\varepsilon=0\); it is responsible for reflection, refraction gradients and mode conversion.
3. A HeNe laser (\(\lambda_0=633\ \mathrm{nm}\)) enters water, \(n=1.33\). Find \(v\), \(k\), \(\lambda_{\text{med}}\) and the (unchanged) frequency \(f\).
   Solution

   \(v=c/n=3.00\times10^8/1.33=2.256\times10^8\ \mathrm{m/s}\). \(k=2\pi n/\lambda_0=2\pi(1.33)/633\times10^{-9}=1.320\times10^{7}\ \mathrm{m^{-1}}\). \(\lambda_{\text{med}}=\lambda_0/n=633/1.33=476\ \mathrm{nm}\). Frequency \(f=c/\lambda_0=3.00\times10^8/633\times10^{-9}=4.74\times10^{14}\ \mathrm{Hz}\), identical in vacuum and water.
4. Verify that the plane wave \(\vec{E}=\vec{E}_0\,e^{i(\vec{k}\cdot\vec{r}-\omega t)}\) solves the vector wave equation, and derive the dispersion relation. What extra condition does \(\nabla\!\cdot\!\vec{E}=0\) impose on \(\vec{E}_0\)?
   Solution

   \(\nabla^2\vec{E}=-|\vec{k}|^2\vec{E}\) and \(\partial_t^2\vec{E}=-\omega^2\vec{E}\). Substituting into \(\nabla^2\vec{E}-\mu\varepsilon\,\partial_t^2\vec{E}=0\) gives \((-|\vec{k}|^2+\mu\varepsilon\,\omega^2)\vec{E}=0\), hence \(|\vec{k}|^2=\mu\varepsilon\,\omega^2\), i.e. \(\omega=v|\vec{k}|\) with \(v=1/\sqrt{\mu\varepsilon}\). Then \(\nabla\!\cdot\!\vec{E}=i\,\vec{k}\!\cdot\!\vec{E}=0\Rightarrow\vec{k}\!\cdot\!\vec{E}_0=0\): the wave is transverse, \(\vec{E}_0\perp\vec{k}\).
5. (t3) A medium has real permittivity plus conductivity \(\sigma\), so Ampère reads \(\nabla\times\vec{H}=\sigma\vec{E}+\varepsilon\,\partial_t\vec{E}\). For a monochromatic field, show the field obeys a Helmholtz equation with complex \(k\) and find the attenuation length for a poor conductor \(\sigma\ll\omega\varepsilon\).
   Solution

   With \(\vec{E}\propto e^{-i\omega t}\), \(\sigma\vec{E}+\varepsilon\,\partial_t\vec{E}=(\sigma-i\omega\varepsilon)\vec{E}=-i\omega\big(\varepsilon+i\sigma/\omega\big)\vec{E}\), defining an effective complex permittivity \(\tilde\varepsilon=\varepsilon+i\sigma/\omega\). Repeating the derivation yields \(\nabla^2\vec{E}+\tilde k^2\vec{E}=0\) with \(\tilde k^2=\omega^2\mu\tilde\varepsilon=\omega^2\mu\varepsilon+i\omega\mu\sigma\). Write \(\tilde k=\beta+i\alpha\). For \(\sigma\ll\omega\varepsilon\), \(\tilde k=\omega\sqrt{\mu\varepsilon}\sqrt{1+i\sigma/\omega\varepsilon}\approx\omega\sqrt{\mu\varepsilon}\big(1+i\sigma/2\omega\varepsilon\big)\), so \(\alpha\approx\frac{\sigma}{2}\sqrt{\mu/\varepsilon}\). A field \(e^{i\tilde k z}=e^{i\beta z}e^{-\alpha z}\) decays over the attenuation length \[\delta=\frac{1}{\alpha}=\frac{2}{\sigma}\sqrt{\frac{\varepsilon}{\mu}}.\] The lossless result is recovered as \(\sigma\to0\) (\(\delta\to\infty\)).