Energy carried by electromagnetic wave

Part 2 built the machinery: Maxwell's four equations, the plane-wave solution, and the realisation that light is a coherent pattern of electric and magnetic field oscillations propagating at \(c\). The question it deferred: those oscillating fields must carry energy — but how much, and where does it flow?

The energy: \(E = hf\) and the Poynting vector

Oscillating fields carry energy. There are two complementary statements to make about that energy, one quantum-mechanical and one purely classical. They have to be consistent, and the consistency turns out to be exact.

\(E = hf\) — the quantum input

Each photon of frequency \(f\) carries energy

\[ E_{\text{photon}} = h f, \]

where \(h \approx 6.626 \times 10^{-34}\) J·s is Planck's constant. This relation does not come from Maxwell's equations. It emerged from Planck's 1900 fit to the spectrum of blackbody radiation and Einstein's 1905 explanation of the photoelectric effect — two experimental facts that classical electromagnetism categorically cannot account for. \(E = hf\) is the bridge by which quantum mechanics enters: it discretises the energy of an electromagnetic wave into integer multiples of \(hf\).

The Poynting vector — the classical output

On the classical side, Maxwell's equations completely determine both how much energy is stored in the field and where it flows. The energy per unit volume stored in the field is

\[ u = \frac{\varepsilon_0}{2}\, |\mathbf{E}|^2 + \frac{1}{2\mu_0}\, |\mathbf{B}|^2, \]

and the rate per unit area at which that energy flows is the Poynting vector,

\[ \mathbf{S} = \frac{1}{\mu_0}\, \mathbf{E} \times \mathbf{B} \quad [\text{W} / \text{m}^2]. \]

Both definitions are not postulates. If you dot Faraday's law with \(\mathbf{B}\), dot Ampère–Maxwell with \(\mathbf{E}\), and subtract, the algebra collapses into a continuity equation:

\[ \frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}. \]

This is Poynting's theorem. Read in one sentence: the energy stored in a region can change in only two ways — energy flows in or out across its boundary (the \(\nabla \cdot \mathbf{S}\) term), or the field does work on the charges inside (the \(\mathbf{J} \cdot \mathbf{E}\) term). In vacuum, \(\mathbf{J} = \mathbf{0}\) and the equation is a clean statement of energy conservation: whatever leaves one region arrives somewhere else.

The bridge

Take the plane wave from Part 2 and time-average the Poynting magnitude over one oscillation period. Working in the complex form makes this a one-liner:

\[ \langle | \mathbf{S} | \rangle = \frac{\varepsilon_0 c}{2}\, |\mathbf{E}_0|^2, \qquad |\mathbf{E}_0|^2 = a_x^2 + a_y^2. \]

That is the time-averaged energy flux of the wave: joules per square metre per second. Now invoke \(E = hf\). If the wave is built out of photons of frequency \(f\), and each carries energy \(hf\), then the rate at which photons cross a unit area must be

\[ N = \frac{\langle | \mathbf{S} | \rangle}{h f} \quad [\text{photons} / \text{m}^2 / \text{s}]. \]

That is the precise relationship: the classical energy flux (which Maxwell gives us in closed form) divided by the per-photon energy quantum (which quantum mechanics gives us) equals the photon current density. The classical wave and the quantum particle picture stop being separate stories and become the same story expressed in different units.

One more sanity check: for the plane wave from Part 2, \(\mathbf{S}\) and \(\hat{\mathbf{z}}\) point the same way. Plugging \(\mathbf{B} = (\hat{\mathbf{z}} \times \mathbf{E})/c\) into \(\mathbf{S} = \mathbf{E} \times \mathbf{B} / \mu_0\) gives \(\mathbf{S} = (|\mathbf{E}|^2 / \mu_0 c)\, \hat{\mathbf{z}}\) — the photon's energy flows in the direction of propagation, as expected. What was previously a piece of intuition is now a derived consequence.

Why the straight line wins

Scope note

For the rest of this section we work strictly inside Maxwell's world of classical electromagnetism: light is a spatial and temporal variation of the electric and magnetic field vectors governed by Maxwell's equations, transporting energy at a rate set pointwise by the Poynting vector \(\mathbf{S} = \mathbf{E} \times \mathbf{B} / \mu_0\) derived above. The Born rule, the photon, and any "sum over photon histories" that you may have heard from science documentaries and pop culture sit outside this section by construction.

What do we actually mean

The question is purely classical: For light originating at point \(A\), why does its time-averaged energy flux at a distant point \(B\) concentrate sharply along the straight segment \(\overline{AB}\)? But first let's be precise with exactly what we mean: the energy flux at point \(B\) is given by the Poynting vector which is a scaled cross product of electric and magnetic field vectors at point \(B\); since the magnetic field vector is completely determined by the electric field vector (via Faraday's law), we can focus our attention on the electric field vector at \(B\): it's a quantity that can also be expressed as an integral sum of contributions from all smooth paths connecting \(A\) to \(B\), each path's local contribution can be either reinforced or cancelled out by a nearby path due to their relative phase alignment.

Let's also be precise about what we mean by "light originating at point \(A\)". In classical EM light is all but spatial and temporal variations of electromagnetic field vectors; to initiate such a disturbance, we require the acceleration of charged particles in a small but non-zero volume of space. Think of this as stirring a lake with a stick, the stick doesn't need to move much in order to initiate a wave that propagates across the entire lake. Therefore when we say "light originates at a point" what we really mean is there's a disturbance in electromagnetic field caused by the acceleration of charged particles in a neighborhood around point \(A\) that is small compared to its distance to the point of observation. That smallness is the far-field condition: from \(B\)'s perspective, the source is effectively a point.

The TLDR explanation

The electric field at \(B\) can be computed as a sum of contributions from all smooth paths connecting \(A\) to \(B\). Each path contributes the same magnitude but a different phase. Paths near the straight line have nearly equal lengths — and therefore nearly equal phases — so their contributions add constructively. Paths far from the straight line have neighbors with wildly different lengths: for any such path, a slight adjustment produces a neighbor whose contribution points in nearly the opposite direction, and the two cancel in the sum. The farther a path deviates from the straight line, the smaller the adjustment needed to flip the phase, until essentially every contribution is paired with a nearly opposite one nearby. The time-averaged energy flux at \(B\) is proportional to the squared magnitude of this total sum, so it is dominated by the straight-line neighborhood — that is essentially why from point \(B\)'s perspective the energy flux is concentrated along the straight line from \(A\) to \(B\).

The field at \(B\) and the Helmholtz equation

Take a compactly supported, time-harmonic charge–current distribution \((\rho, \mathbf{J})\) localised near \(\mathbf{r}_A\), oscillating at angular frequency \(\omega\) with wavenumber \(k = \omega/c\). "An electromagnetic wave originates at \(A\)" means exactly this: a localised source term in the inhomogeneous Maxwell equations. The unique retarded solution fixes the fields everywhere via the retarded potentials

\[ \mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}', t - |\mathbf{r} - \mathbf{r}'|/c)} {|\mathbf{r} - \mathbf{r}'|}\, d^3 r', \qquad \mathbf{E} = -\nabla\phi - \partial_t \mathbf{A}, \quad \mathbf{B} = \nabla \times \mathbf{A}, \]

with the scalar potential \(\phi\) defined analogously. Away from the source, every scalar component \(u\) of \(\mathbf{E}\) or \(\mathbf{B}\) satisfies the wave equation \(\nabla^2 u - c^{-2}\partial_t^2 u = 0\). For a time-harmonic field \(u(\mathbf{r}, t) = \mathrm{Re}[\tilde{u}(\mathbf{r})\, e^{-i\omega t}]\), this reduces to the Helmholtz equation

\[ (\nabla^2 + k^2)\, \tilde u(\mathbf{r}) = 0 \quad \text{in the source-free region.} \]

The complex spatial amplitude \(\tilde u(\mathbf{r}_B)\) is the entire object of computation. In the locally plane-wave far field of the source, the magnetic field reduces algebraically to \(\mathbf{B} = (1/c)\hat{\mathbf{k}} \times \mathbf{E}\), so the time-averaged Poynting magnitude — the physical observable encoding "how much energy reaches \(B\)" — is

\[ \langle|\mathbf{S}(\mathbf{r}_B)|\rangle = \frac{\varepsilon_0 c}{2}\,|\tilde u(\mathbf{r}_B)|^2, \]

derived earlier in this article. The straight-line question is therefore a question about the spatial structure of \(\tilde u(\mathbf{r}_B)\).

The field at \(B\) as a sum over paths

We now derive the TLDR's central claim: the field amplitude at \(B\) equals a sum of contributions from every smooth path connecting \(A\) to \(B\), each contributing the same magnitude but a different phase.

We want to compute \(\tilde u(\mathbf{r}_B)\) without knowing the field everywhere in space — just on a surrounding surface. Green's second identity gives exactly this. Let \(S\) be any closed, piecewise-smooth surface enclosing \(\mathbf{r}_B\) but not the source. The outgoing free-space Green's function of the Helmholtz equation is

\[ G(\mathbf{r}', \mathbf{r}_B) = \frac{e^{i k |\mathbf{r}' - \mathbf{r}_B|}} {|\mathbf{r}' - \mathbf{r}_B|}, \qquad (\nabla'^2 + k^2)\, G = -4\pi\, \delta^{(3)}(\mathbf{r}' - \mathbf{r}_B). \]

Applying Green's second identity to the pair \((\tilde u, G)\) on the volume enclosed by \(S\) and using the sifting property of \(\delta^{(3)}\) yields the Helmholtz–Kirchhoff representation:

\[ \tilde u(\mathbf{r}_B) = \frac{1}{4\pi} \oint_S \left[ \tilde u(\mathbf{r}')\, \partial_{n'} G(\mathbf{r}', \mathbf{r}_B) - G(\mathbf{r}', \mathbf{r}_B)\, \partial_{n'} \tilde u(\mathbf{r}') \right] dS'. \]

Read in one sentence: the value of a Helmholtz field at any interior point is determined entirely by the field and its normal derivative on a surrounding surface. Each surface element at \(\mathbf{r}'\) contributes a spherical wavelet \(e^{ikR}/R\) with \(R = |\mathbf{r}' - \mathbf{r}_B|\), weighted by the local field — the rigorous classical statement of Huygens' principle, derived from Maxwell's equations alone.

Now iterate. Insert \(N\) evenly-spaced flat surfaces between \(A\) and \(B\) and apply the theorem at each stage: the field on the \(j\)-th surface is itself a boundary integral of the field on the \((j{-}1)\)-th. The chain of \(N + 1\) boundary integrals collapses into a single integral over the \(N\)-tuple of intermediate points \(\mathbf{x} = (x_1, \ldots, x_N) \in (\mathbb{R}^3)^N\), with each tuple defining a polyline \(p_{\mathbf{x}} = A \to x_1 \to \cdots \to x_N \to B\) and contributing a phasor \(e^{ikL[p_{\mathbf{x}}]}\) where \(L[p_{\mathbf{x}}]\) is the total length of that polyline. This construction — inserting \(N\) flat surfaces, integrating the phasors over all polylines whose interior points lie on those surfaces, and taking the limit \(N \to \infty\) — is precisely the definition of the path integral:

\[ \tilde u(\mathbf{r}_B) = \int_{\Gamma_{AB}} \mathcal{D}\gamma \; e^{i k L[\gamma]}, \qquad L[\gamma] = \int_0^1 |\dot\gamma(t)|\, dt, \qquad k = \frac{2\pi}{\lambda}. \]

This is the mathematical form of the TLDR's first two claims. Each path \(\gamma\) from \(A\) to \(B\) contributes a unit-modulus phasor \(e^{ikL[\gamma]}\) — same magnitude, different phase \(kL[\gamma]\). Within classical EM this is not a Feynman sum over photon histories; no photon has been introduced. It is the continuum limit of the iterated Helmholtz–Kirchhoff identity — bookkeeping for a high-dimensional ordinary integral whose integrand factors through the single scalar \(L[p_{\mathbf{x}}]\).

A word of caution: the integral sign \(\int \mathcal{D}\gamma\) does not denote integration against a measure on \(\Gamma_{AB}\) in the Lebesgue sense — no such measure exists. The path integral is defined purely as a limit of finite-dimensional integrals, and that limit exists only as an asymptotic series in \(1/k\), not as a convergent number. Readers interested in why the traditional \(\varepsilon\)-\(\delta\) definition of an integral fails here, what replaces it, and why the proof that follows remains rigorous should expand the technical note below.

Why \(\int\mathcal{D}\gamma\) has no \(\varepsilon\)-\(\delta\) definition, what replaces it, and why the proof still works

Measurable sets exist — the measure doesn't. The space \(\Gamma_{AB}\) has a perfectly good \(\sigma\)-algebra: the one generated by cylinder sets

\[ \bigl\{\gamma \in \Gamma_{AB} : \bigl(\gamma(t_1), \ldots, \gamma(t_n)\bigr) \in B\bigr\}, \]

where \(t_1, \ldots, t_n \in [0, 1]\) are fixed sample times and \(B \subset (\mathbb{R}^3)^n\) is a Borel set. A cylinder set pins the path at finitely many times and leaves it free elsewhere — the same construction that underpins probability on stochastic processes. So "measurable subset of path space" is well-defined. The obstacle is the measure itself: no \(\sigma\)-additive measure on this \(\sigma\)-algebra is both translation-invariant and assigns finite nonzero values to open sets.

The reason is dimensional. In \(\mathbb{R}^n\) Lebesgue measure works because the unit cube \([0, 1]^n\) has finite volume \(1\). In the infinite-dimensional space \(\Gamma_{AB}\), the analogous "unit cube" — paths whose coordinates at every \(t\) lie in \([0, 1]\) — would need "volume" \(1^\infty\), which has no meaningful value. The Lebesgue construction simply has no infinite-dimensional analogue. (Wiener measure, used in stochastic calculus, sidesteps this by giving up translation invariance: it is a Gaussian measure concentrated on continuous paths, weighting them by \(e^{-S}\) with a real, positive-definite action. But our integrand \(e^{+ikL}\) is oscillatory, not damped — Wiener measure does not help here.)

The time-slicing workaround. Since no measure on \(\Gamma_{AB}\) exists, the path integral is defined as a limit of finite-dimensional integrals where honest Lebesgue measure is available. Fix a partition \(0 = t_0 < t_1 < \cdots < t_N < t_{N+1} = 1\); for each tuple \(\mathbf{x} = (x_1, \ldots, x_N) \in (\mathbb{R}^3)^N\), let \(p_{\mathbf{x}}\) be the piecewise-linear path joining \(A, x_1, \ldots, x_N, B\). Then

\[ \int_{\Gamma_{AB}} \mathcal{D}\gamma\; F[\gamma] \;\stackrel{\text{def}}{=}\; \lim_{N \to \infty} C_N \int_{(\mathbb{R}^3)^N} F[\,p_{\mathbf{x}}\,]\, d^3 x_1 \cdots d^3 x_N. \]

Each finite-\(N\) expression on the right is a genuine Lebesgue integral over \((\mathbb{R}^3)^N\) — no foundational issues there. Each tuple \(\mathbf{x}\) determines a polyline \(p_{\mathbf{x}}\), attaches the phasor \(F[p_{\mathbf{x}}] = e^{ikL[p_{\mathbf{x}}]}\), and we sum those phasors across every placement of the interior points.

The normalisation \(C_N\). Each interior slot adds three dimensions, so the bare phasor sum diverges as \(N\) grows; \(C_N = (c_N)^{N+1}\) rescales it, with one factor \(c_N\) per polyline segment. Each \(c_N\) is fixed by demanding the limit reproduce the spherical wavelet \(e^{ikr}/r\) at \(r = |B - A|\) — the unique outgoing Green's function the source must produce. The renormalised limit is well-defined as an asymptotic series in \(1/k\) — meaning the first few terms of the \(1/k\) expansion stabilise as \(N \to \infty\), even though the full series typically diverges.

Why the proof still works. The stationary-phase estimates that follow — the integration-by-parts bound on \(U_\varepsilon\) and the Gaussian saddle-point evaluation of the Fresnel zone — are statements about the finite-dimensional integral at each \(N\), where everything is a rigorous Lebesgue integral. These estimates hold uniformly in \(N\) and survive the \(N \to \infty\) limit. The proof's validity does not depend on the path integral being an integral against a measure on path space; it depends on the finite-dimensional stationary-phase theorem, applied at each \(N\).

The path-space topology. We endow \(\Gamma_{AB}\) with the sup-norm topology: the distance between two paths is

\[ \|\gamma_1 - \gamma_2\|_\infty = \sup_{t \in [0, 1]} \bigl|\gamma_1(t) - \gamma_2(t)\bigr|, \]

the largest Euclidean separation between corresponding points. The \(\varepsilon\)-tube around a path \(\gamma_0\) is the open ball \(\{\gamma \in \Gamma_{AB} : \|\gamma - \gamma_0\|_\infty < \varepsilon\}\). A set \(U \subset \Gamma_{AB}\) is bounded away from \(\gamma_0\) if there exists \(\varepsilon > 0\) such that \(\|\gamma - \gamma_0\|_\infty \ge \varepsilon\) for every \(\gamma \in U\).

Sup-norm closeness is not arc-length closeness: a low-amplitude, high-frequency wiggle around a straight line lies inside any \(\varepsilon\)-tube yet can exceed the straight line's arc length by an arbitrary amount. The metric that controls the iterated Kirchhoff integral is closeness in length to within a wavelength, \(k\,|L[\gamma_1] - L[\gamma_2]| \lesssim 1\) — the regime in which the phasors \(e^{ikL[\gamma_1]}\) and \(e^{ikL[\gamma_2]}\) are nearly equal. The negation, \(k\,|L[\gamma_1] - L[\gamma_2]| \gg 1\), is the precise meaning of "two paths differ wildly in length": their phase factors point in uncorrelated directions in the complex plane.

What controls accumulation versus cancellation is how fast \(L\) changes locally in path space. Where small adjustments to a path barely change its length, neighbouring phasors point in similar directions and add. Where a small adjustment shifts \(L\) by a wavelength or more, the phasors spin rapidly and cancel pairwise within each small neighbourhood. The next three sections make this precise.

The straight line is the unique stationary path

Among all paths from \(A\) to \(B\), which ones produce phasors that don't spin under small perturbations? Those are the stationary paths of the arc-length functional \(L\) — and they turn out to be exactly the paths whose nearby neighbours all have nearly equal phase. A path \(\gamma_0\) is stationary if its first variation vanishes:

\[ \delta L[\gamma_0]\,\eta \equiv \left. \frac{d}{d\varepsilon}\, L[\gamma_0 + \varepsilon\eta] \right|_{\varepsilon = 0} = 0 \quad \text{for every variation } \eta \text{ with } \eta(0) = \eta(1) = 0. \]

Computing the variation and integrating by parts gives

\[ \delta L[\gamma_0]\,\eta = -\int_0^1 \frac{d}{dt}\!\left( \frac{\dot\gamma_0(t)}{|\dot\gamma_0(t)|} \right) \cdot \eta(t)\, dt. \]

By the fundamental lemma of the calculus of variations, the right side vanishes for every admissible \(\eta\) if and only if the unit tangent \(\dot\gamma_0/|\dot\gamma_0|\) is constant in \(t\). In free Euclidean space this identifies a unique \(\gamma_0\): the straight line from \(A\) to \(B\), with \(L[\gamma_0] = |B - A|\). The straight line is not merely stationary — it is the unique global minimum of \(L\) over \(\Gamma_{AB}\). This is Fermat's principle in vacuum, derived here as a consequence of the variational structure, not posited.

Near the straight line: the Fresnel zone

To quantify the TLDR's claim that nearby paths "add constructively", we expand the arc length \(L\) to second order around the stationary path. For \(\gamma = \gamma_0 + \eta\) with an admissible variation \(\eta\),

\[ L[\gamma] = L[\gamma_0] + \tfrac{1}{2}\, \delta^2 L[\gamma_0](\eta, \eta) + O(\|\eta\|^3). \]

The linear term vanishes because \(\gamma_0\) is stationary. The second variation \(\delta^2 L[\gamma_0]\) is a positive-definite quadratic form on transverse variations (the straight line is a minimum, not merely a saddle), so paths near \(\gamma_0\) are all longer than the straight line, and the phase excess \(k(L[\gamma] - L[\gamma_0])\) grows quadratically in the deviation.

The coherent zone is the set of paths for which this phase excess stays below about one radian:

\[ k\, \delta^2 L[\gamma_0](\eta, \eta) \lesssim 1. \]

Within this zone neighbouring phasors are nearly aligned and their contributions add constructively. To get a concrete radius, consider a single transverse displacement of amplitude \(\xi\) at a point whose distances from \(A\) and \(B\) are \(r_A\) and \(r_B\). Geometry gives \(\delta^2 L \approx \xi^2(1/r_A + 1/r_B)\), so the coherent condition becomes \(k\xi^2(1/r_A + 1/r_B) \lesssim 1\), yielding the first Fresnel zone radius

\[ \xi_F \;\sim\; \sqrt{\frac{\lambda\, r_A\, r_B}{r_A + r_B}}. \]

This is the rigorous answer to "how straight is straight": the tube of paths that contribute coherently to \(\tilde u(\mathbf{r}_B)\) has transverse width \(\sim \sqrt{\lambda \cdot \text{geometric length}}\), which shrinks to zero as \(\lambda \to 0\).

Away from the straight line: pairwise cancellation

The TLDR's claim was: "for any path far from the straight line, a slight adjustment produces a neighbour whose contribution points in nearly the opposite direction." We now make that precise.

For paths outside the Fresnel zone, \(k(L[\gamma] - L[\gamma_0]) \gg 1\): the phase \(kL[\gamma]\) changes by many radians over any small neighbourhood in path space, so the phasors \(e^{ikL[\gamma]}\) spin rapidly and cancel pairwise. To turn this into a quantitative bound we use a systematic tool for estimating oscillatory integrals — the stationary-phase integration-by-parts argument.

Consider first the finite-dimensional model \(\int f(x)\, e^{i k \phi(x)}\, dx\) with smooth phase \(\phi\). Wherever \(\nabla\phi \ne 0\), the first-order differential operator

\[ \mathcal{L} = \frac{\nabla\phi}{i k\,|\nabla\phi|^2} \cdot \nabla \]

satisfies \(\mathcal{L}\,e^{i k \phi} = e^{i k \phi}\) (verify by expanding: \(\mathcal{L}\) feeds \(ik\nabla\phi\) into its own \(\nabla\phi / (ik|\nabla\phi|^2)\), which yields \(|\nabla\phi|^2/|\nabla\phi|^2 = 1\)). Substituting \(e^{ik\phi} = \mathcal{L}\,e^{ik\phi}\) and integrating by parts shifts \(\mathcal{L}\) onto the smooth factor \(f\), producing one factor of \(1/k\) from the denominator of \(\mathcal{L}\). Iterating \(N\) times gives

\[ \int f\,e^{i k \phi}\, dx = O(k^{-N}) \]

on any compact set where \(|\nabla\phi|\) is bounded away from zero. The argument breaks down at the stationary set \(\{\nabla\phi = 0\}\), where \(\mathcal{L}\)'s denominator blows up — which is exactly why the stationary neighbourhood dominates (the next section).

Now apply this to the path integral with \(\phi = L\) and \(\nabla \to \delta/\delta\gamma\). Since the straight line is the unique stationary path (previous section), the functional gradient \(\delta L / \delta\gamma\) is bounded below in magnitude on any set bounded away from \(\gamma_0\). Concretely, for every \(\varepsilon > 0\) define

\[ U_\varepsilon = \{\gamma \in \Gamma_{AB} : \|\gamma - \gamma_0\|_\infty \ge \varepsilon\} \]

— the set of paths at sup-norm distance at least \(\varepsilon\) from the straight line. On \(U_\varepsilon\) the integration-by-parts argument applies at every order, giving

\[ \left| \int_{U_\varepsilon} \mathcal{D}\gamma\; e^{i k L[\gamma]} \right| = O(k^{-N}) \quad \text{for every } N \ge 1. \]

In words: the collective contribution from all paths that deviate from the straight line by at least \(\varepsilon\) is smaller than any power of \(1/k\). It is not that individual paths "don't matter" — each still carries a unit-magnitude phasor — but within every small neighbourhood of such a path, phasors span all directions and their vector sum collapses.

Assembly: light travels in a straight line

We can now combine the two preceding results. Split the full path integral into the Fresnel zone (paths within sup-norm \(\varepsilon\) of \(\gamma_0\)) and its complement \(U_\varepsilon\):

\[ \tilde u(\mathbf{r}_B) = \underbrace{\int_{\Gamma_{AB} \setminus U_\varepsilon} \mathcal{D}\gamma\; e^{ikL[\gamma]}}_{\text{Fresnel zone}} \;+\; \underbrace{\int_{U_\varepsilon} \mathcal{D}\gamma\; e^{ikL[\gamma]}}_{O(k^{-N})}. \]

The second integral vanishes faster than any power of \(1/k\) (pairwise cancellation, previous section). The first integral is a Gaussian saddle-point integral: inside the Fresnel zone the phase is well-approximated by its quadratic expansion around \(\gamma_0\), and the Gaussian integral evaluates to a prefactor \(\mathcal{N}(k)\) times the stationary-point phasor. The result is the stationary-phase theorem:

\[ \tilde u(\mathbf{r}_B) = \mathcal{N}(k)\, e^{i k L[\gamma_0]}\,\bigl(1 + O(1/k)\bigr). \]

As \(k \to \infty\) the field amplitude at \(B\) is dominated entirely by contributions from the straight line \(\overline{AB}\) — and therefore, via \(\langle|\mathbf{S}|\rangle \propto |\tilde u(\mathbf{r}_B)|^2\), so is the time-averaged energy flux. That is "light travels in a straight line" stated and proved as a theorem about Maxwell's equations in the short-wavelength limit, with no quantum input.

Fermat's principle in matter ("light follows the path of stationary optical length") is the same theorem with \(L\) replaced by the optical path length \(\int n(\mathbf{r})\, ds\) — obtained by repeating the derivation with the Helmholtz equation \((\nabla^2 + n^2 k^2)\tilde u = 0\) for slowly varying refractive index \(n(\mathbf{r})\) — and \(\gamma_0\) the corresponding extremal. Snell's law of refraction and the geometric optics of high-school physics are the \(k \to \infty\) (eikonal) limit of the path-integral representation of Maxwell's wave equation. Rectilinear propagation is not a starting assumption; it is the asymptotic shadow that wave optics casts when the wavelength is small compared to every other length scale in the problem.

Teaser for next article

Part 2 introduced polarisation as a classical concept — the trajectory the tip of \(\mathbf{E}\) traces in the transverse plane, parameterised by the four real numbers \(a_x, a_y, \alpha_x, \alpha_y\) of the plane-wave ansatz. Maxwell's equations admit every such trajectory and prefer none; classically, polarisation is simply a free choice. The quantum theory of light gives that choice a deeper origin. Just as the energy carried in a mode of frequency \(f\) is partitioned into discrete quanta of size \(hf\), the polarisation of each photon is the classical shadow of two intrinsic angular-momentum properties: spin angular momentum (eigenvalues \(\pm\hbar\) along \(\hat{\mathbf{k}}\), corresponding to the two circular polarisations) and orbital angular momentum (carried by helical phase fronts in the mode profile). Poynting vector and polarisation are both continuous classical attributes that reveal, under closer inspection, a discrete quantum scaffolding underneath.

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References

1. Wikipedia contributors. Poynting vector. Wikipedia, The Free Encyclopedia. en.wikipedia.org/wiki/Poynting_vector
2. Wikipedia contributors. Huygens–Fresnel principle. Wikipedia, The Free Encyclopedia. en.wikipedia.org/wiki/Huygens–Fresnel_principle