The Eikonal Equation as the Short-Wavelength Limit

Derivation

The Eikonal Equation as the Short-Wavelength Limit

D-196 Home PU-206 Threads light · waves Depends on Optical Wave and Helmholtz Equations from Maxwell

Statement

Starting from the scalar Helmholtz equation \(\nabla^2 U + k_0^2 n^2(\mathbf{r})\,U = 0\) for a monochromatic field in a slowly varying medium, we insert the amplitude–phase ansatz \(U = A(\mathbf{r})\,e^{i k_0 S(\mathbf{r})}\) and take the short-wavelength limit \(\lambda \to 0\) (\(k_0 \to \infty\)). At leading order the phase function \(S\) — the eikonal, an optical path length — obeys the eikonal equation \(\lvert\nabla S\rvert^2 = n^2\), and rays are defined as the curves everywhere orthogonal to the wavefronts \(S=\text{const}\), i.e. tangent to \(\nabla S\).

Why it matters

The eikonal equation is the bridge between wave optics and geometrical optics. It shows that ray tracing is not a separate postulated theory but the leading term of a systematic asymptotic expansion of Maxwell's equations in powers of the wavelength. Every statement in ray optics — Fermat's principle, Snell's law, the lens-maker's equation, the ray equation \(\frac{d}{ds}(n\,\frac{d\mathbf{r}}{ds})=\nabla n\) — descends from this single first-order partial differential equation.

The same mathematics recurs across physics: the eikonal is the optical analogue of the classical action, and the passage \(\lambda\to 0\) mirrors the semiclassical limit \(\hbar\to 0\) that turns the Schrödinger equation into the Hamilton–Jacobi equation. Understanding it once illuminates WKB theory, high-frequency acoustics, seismic ray tracing, and gravitational lensing.

Assumptions

Monochromatic, time-harmonic field.If the field is polychromatic, each frequency carries its own \(k_0\) and its own eikonal; dispersion couples them and no single ray family describes the pulse — one must track group velocity and the frequency-dependent index separately. Isotropic, source-free, non-magnetic scalar medium.Drop scalarity and the two polarization components mix through \(\nabla(\ln\varepsilon)\) terms; the clean scalar Helmholtz starting point is lost and one needs the vector eikonal with polarization transport. The index \(n(\mathbf{r})\) and amplitude \(A(\mathbf{r})\) vary slowly on the scale of a wavelength.This is the crux: if \(n\) or \(A\) changes appreciably over one wavelength, the neglected term \(\nabla^2 A/(k_0^2 A)\) is no longer small, the asymptotic ordering collapses, and diffraction, reflection, and tunnelling — none of which the eikonal captures — dominate. The expansion is asymptotic, not convergent.Treating the \(1/k_0\) series as a convergent power series and pushing to high order gives divergent nonsense; the eikonal is only the leading term and is meaningful strictly in the \(k_0\to\infty\) sense.

Derivation

\[ \nabla^2 U + k_0^2\, n^2(\mathbf{r})\, U = 0, \qquad k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda_0} \]

Scalar Helmholtz equation, obtained from Maxwell's equations for a time-harmonic field \(\mathbf{E}\propto e^{-i\omega t}\) in a source-free medium; \(k_0\) is the vacuum wavenumber and \(n\) the local refractive index. A

\[ U(\mathbf{r}) = A(\mathbf{r})\, e^{i k_0 S(\mathbf{r})}, \qquad A,\,S \ \text{real} \]

Amplitude–eikonal ansatz. Any complex field can be written this way with real amplitude \(A\ge 0\) and real phase; separating a large factor \(k_0\) out of the phase is the choice that exposes the short-wavelength ordering. A

\[ \nabla U = \left(\nabla A + i k_0 A\, \nabla S\right) e^{i k_0 S} \]

Product and chain rule on the gradient. The phase factor differentiates to \(i k_0 \nabla S\), the amplitude to \(\nabla A\). A

\[ \nabla^2 U = \left[\nabla^2 A + 2 i k_0\, \nabla A\!\cdot\!\nabla S + i k_0 A\, \nabla^2 S - k_0^2 A\, \lvert\nabla S\rvert^2\right] e^{i k_0 S} \]

Take the divergence of step 3 and apply the product rule again. The four terms carry powers \(k_0^0, k_0^1, k_0^1, k_0^2\) respectively — the seed of the ordering. B

\[ \nabla^2 A + i k_0\!\left(2\,\nabla A\!\cdot\!\nabla S + A\,\nabla^2 S\right) - k_0^2 A\,\lvert\nabla S\rvert^2 + k_0^2 n^2 A = 0 \]

Substitute step 4 into step 1 and cancel the common nonzero factor \(e^{i k_0 S}\). A

\[ \underbrace{k_0^2 A\!\left(n^2 - \lvert\nabla S\rvert^2\right)}_{\mathcal{O}(k_0^2),\ \text{real}} + \underbrace{i k_0\!\left(2\,\nabla A\!\cdot\!\nabla S + A\,\nabla^2 S\right)}_{\mathcal{O}(k_0),\ \text{imag}} + \underbrace{\nabla^2 A}_{\mathcal{O}(1),\ \text{real}} = 0 \]

Group by power of \(k_0\). Because \(A,S,n\) are real, the real and imaginary parts must vanish separately; the powers of \(k_0\) sort the equation into a hierarchy. B

\[ n^2 - \lvert\nabla S\rvert^2 + \frac{1}{k_0^2}\,\frac{\nabla^2 A}{A} = 0 \]

Divide the real part of step 6 by \(k_0^2 A\). This is exact — no approximation yet — and isolates the small parameter \(1/k_0^2 = (\lambda_0/2\pi)^2\) as the coefficient of the diffraction term \(\nabla^2 A/A\). B

\[ \left|\frac{1}{k_0^2}\frac{\nabla^2 A}{A}\right| \sim \left(\frac{\lambda_0}{2\pi\,\ell_A}\right)^2 \ll n^2 \quad\Longrightarrow\quad \lim_{k_0\to\infty}\frac{1}{k_0^2}\frac{\nabla^2 A}{A}=0 \]

Estimate the diffraction term by the amplitude scale length \(\ell_A\) (where \(\nabla^2 A/A \sim 1/\ell_A^2\)). When the wavelength is tiny compared with \(\ell_A\), this term is negligible; the short-wavelength limit \(\lambda\to 0\) sends it to zero. This is the physical content of "geometrical optics." C

\[ \boxed{\ \lvert\nabla S\rvert^2 = n^2(\mathbf{r})\ } \]

The leading-order (real) balance with the diffraction term dropped. This is the eikonal equation: a first-order nonlinear PDE for the phase surfaces \(S=\text{const}\). A

\[ 2\,\nabla A\!\cdot\!\nabla S + A\,\nabla^2 S = 0 \quad\Longleftrightarrow\quad \nabla\!\cdot\!\left(A^2\,\nabla S\right)=0 \]

The imaginary part of step 6 (the next order). Multiplying by \(A\) and recognizing \(2A\nabla A\cdot\nabla S + A^2\nabla^2 S = \nabla\cdot(A^2\nabla S)\) gives the transport equation: the energy flux \(A^2\nabla S\) is divergence-free, i.e. conserved in ray tubes. B

\[ \frac{d\mathbf{r}}{ds} = \frac{\nabla S}{\lvert\nabla S\rvert} = \frac{\nabla S}{n} \qquad\Longrightarrow\qquad \frac{d}{ds}\!\left(n\,\frac{d\mathbf{r}}{ds}\right) = \nabla n \]

Define a ray as the integral curve of \(\nabla S\), parametrized by arc length \(s\). Rays are orthogonal to wavefronts because \(\nabla S \perp \{S=\text{const}\}\). Differentiating \(n\,d\mathbf{r}/ds = \nabla S\) along the ray and using the eikonal equation yields the ray equation — Newton's second law for light. C

Result

\[ \lvert\nabla S(\mathbf{r})\rvert^2 = n^2(\mathbf{r}) \qquad\text{with rays}\ \ \frac{d}{ds}\!\left(n\frac{d\mathbf{r}}{ds}\right)=\nabla n \]

Reading. The magnitude of the phase gradient equals the local refractive index everywhere. Geometrically, the wavefronts \(S=\text{const}\) are surfaces of constant optical path from the source, and light energy streams along the rays — the orthogonal trajectories that thread these surfaces. Where \(n\) is uniform the rays are straight; where \(n\) varies they bend toward higher index. The accompanying transport equation \(\nabla\cdot(A^2\nabla S)=0\) conserves power in each ray tube, fixing how brightness changes as rays converge or diverge.

Units check. \(U=A e^{i k_0 S}\) requires \(k_0 S\) dimensionless; with \([k_0]=\mathrm{m^{-1}}\) this forces \([S]=\mathrm{m}\), so \(S\) is an optical path length. Then \([\nabla S]=\mathrm{m/m}\) is dimensionless, and \(\lvert\nabla S\rvert^2\) is dimensionless — matching \(n^2\), which is dimensionless. Both sides balance.

Limiting cases

Uniform medium \(n=\text{const}\): \(\lvert\nabla S\rvert=n\) with \(\nabla n=0\) gives \(\nabla S=n\hat{\mathbf{k}}\), \(S=n\,\hat{\mathbf{k}}\!\cdot\!\mathbf{r}\) — plane wavefronts and straight rays (\(d^2\mathbf{r}/ds^2=0\)).
Point source in vacuum \(n=1\): spherically symmetric \(S=r\), so \(\lvert\nabla S\rvert=1\); wavefronts are spheres, rays are radii, and the transport equation gives \(A\propto 1/r\) (inverse-square intensity).
Planar interface \(n_1\to n_2\): matching the tangential component of \(\nabla S\) (continuity of the wavefront's trace) across the boundary reproduces Snell's law \(n_1\sin\theta_1=n_2\sin\theta_2\).
Slow transverse gradient: for a ray nearly along \(\hat{\mathbf{z}}\) with \(n=n(x)\), the invariant \(n\sin\theta=\text{const}\) makes rays curve toward increasing \(n\) — the origin of mirages and GRIN focusing.
Formal \(\lambda_0\to 0\): the diffraction term \(\nabla^2A/(k_0^2A)\to 0\) exactly, so ray optics is the rigorous zero-wavelength limit; finite \(\lambda\) reintroduces diffraction as the first correction.

Breaks when

Caustics and foci. Where neighbouring rays cross, the ray-tube cross-section shrinks to zero, so the transport equation forces \(A\to\infty\). The amplitude scale length \(\ell_A\to 0\) violates the slow-variation assumption; the true field is finite (Airy-function structure) and only a diffraction integral captures it.
Sharp index features. Near an interface, edge, or any structure varying over a distance \(\lesssim\lambda\), the neglected \(\nabla^2 A/(k_0^2A)\) term is order unity. Reflection, diffraction, evanescent tunnelling, and thin-film interference — all invisible to the eikonal — take over.
Turning points / total internal reflection. Where \(\lvert\nabla S\rvert\to 0\) or a ray grazes tangentially, \(S\) can turn complex; the real eikonal fails and one needs the WKB connection formulae (evanescent, exponentially decaying fields).
Coherent wave phenomena. Interference between two overlapping ray families, speckle, and any effect depending on the relative phase of distinct paths lie beyond a single-valued \(S\); a sum of eikonal branches (or full wave optics) is required.

Failure modes

Confusing \(S\) with the total phase. \(S\) is the eikonal (optical path, units of metres); the physical phase is \(k_0 S\). Forgetting the \(k_0\) makes the dimensional check and the ordering-by-\(k_0\) argument incoherent.
Setting the diffraction term to zero "because it is small" rather than because \(\lambda\to0\). The term is small only when \(\lambda\ll\ell_A\); at a caustic or sharp feature it is not, and dropping it silently is exactly where the physics is lost.
Writing \(\lvert\nabla S\rvert = n\) with the wrong sign or as \(\nabla S=n\). \(\nabla S\) is a vector of magnitude \(n\); it equals \(n\hat{\mathbf{t}}\) with \(\hat{\mathbf{t}}\) the ray direction, not the scalar \(n\).
Believing rays carry energy independently of the transport equation. The eikonal fixes ray paths; the amplitude along them is set separately by \(\nabla\cdot(A^2\nabla S)=0\). Students often quote ray optics but never conserve flux, then mispredict intensities.
Assuming a single-valued \(S\) everywhere. After a focus the eikonal becomes multivalued (multiple rays reach a point); insisting on one branch produces spurious infinities and misses the Gouy/Maslov phase jumps.

Discussion

The derivation is a template for every "classical limit" in physics. The ansatz \(U=Ae^{ik_0S}\) plays the role of \(\psi=Ae^{iW/\hbar}\) in quantum mechanics; the eikonal equation \(\lvert\nabla S\rvert^2=n^2\) is the exact analogue of the time-independent Hamilton–Jacobi equation \(\lvert\nabla W\rvert^2=2m(E-V)\), with the refractive index playing the part of \(\sqrt{2m(E-V)}\). Rays are the optical particles, wavefronts are surfaces of constant action, and the passage \(\lambda\to0\) is the passage \(\hbar\to0\). This is why the same WKB machinery, the same caustics, and the same Maslov-index phase corrections appear in both subjects.

Physically, the two equations that fall out of the expansion partition the information cleanly. The eikonal equation is geometry: it says nothing about how bright the light is, only where it goes. The transport equation is energetics: it propagates the intensity \(A^2\) along the rays fixed by the geometry, conserving power in each ray tube. Together they reconstruct the full leading-order field. The elegance is that a second-order wave equation splits, at leading order, into a first-order equation for phase and a first-order equation for amplitude — a decoupling that only survives while diffraction is negligible.

The eikonal equation is also the microscopic justification of Fermat's principle. Because \(\nabla S\) is orthogonal to wavefronts and has magnitude \(n\), the optical path \(\int n\,ds\) between two points is stationary along a ray — rays are the geodesics of the optical metric \(ds_{\text{opt}}^2 = n^2\,d\mathbf{r}^2\). Curved rays in a graded medium are then literally straight lines in a curved optical geometry, a viewpoint that generalizes directly to light bending in a gravitational field, where the metric is set by spacetime curvature rather than by \(n\).

A subtle point of rigour: the \(1/k_0\) expansion is asymptotic, not convergent. Adding more terms improves accuracy only up to an optimal order beyond which the series diverges; the eikonal is exact only in the strict \(k_0\to\infty\) limit. Moreover the naive expansion breaks at caustics, where a uniform asymptotic treatment (Airy functions, and the theory of the Maslov index counting caustic crossings) is required to keep the amplitude finite and to supply the \(\pi/2\) phase jumps that a raw ray sum misses. These corrections are what reconcile geometrical optics with the diffraction pattern actually observed near a focus.

Common misconceptions. (i) "Geometrical optics ignores the wave nature of light" — no; it is derived from the wave equation and is its rigorous short-wavelength limit. (ii) "Rays are physical objects" — they are the characteristics of a PDE, mathematical trajectories orthogonal to the real wavefronts. (iii) "The eikonal equation predicts intensity" — it does not; intensity comes from the transport equation. (iv) "Small \(\lambda\) always means the eikonal is valid" — validity depends on \(\lambda/\ell\) where \(\ell\) is the medium/amplitude scale, and near caustics \(\ell\to0\) no matter how small \(\lambda\) is.

Worked examples

Example 1 — Spherical wave from a point source in glass.

\[ n=1.50\ \text{(uniform)},\qquad S(\mathbf{r})=n\,r \]

Symmetry: a point source in a uniform medium radiates spherical wavefronts, so \(S\) depends only on the radial distance \(r\). Try \(S=nr\) and verify. A

\[ \nabla S = n\,\hat{\mathbf{r}}, \qquad \lvert\nabla S\rvert^2 = n^2 \]

\(\nabla r = \hat{\mathbf{r}}\), so \(\nabla S = n\hat{\mathbf{r}}\); its squared magnitude is \(n^2\), satisfying the eikonal equation identically. Rays are the radii \(\hat{\mathbf{r}}\). A

\[ \nabla\!\cdot\!(A^2\nabla S)=0 \ \Rightarrow\ \frac{1}{r^2}\frac{d}{dr}\!\left(r^2 A^2 n\right)=0 \ \Rightarrow\ A=\frac{A_0}{r} \]

Transport equation in spherical symmetry: \(r^2A^2n=\) const, so \(A\propto1/r\) and intensity \(I\propto A^2\propto 1/r^2\) — the inverse-square law. B

\[ \Delta S = n\,\Delta r = 1.50\times(0.0200-0.0050)\,\text{m} = 0.0225\ \text{m} \]

Numbers: optical path accumulated between \(r_1=5.0\,\text{mm}\) and \(r_2=20.0\,\text{mm}\). The physical phase advance is \(k_0\Delta S\). A

\[ \Delta S = 22.5\ \text{mm},\qquad \frac{I(r_1)}{I(r_2)}=\left(\frac{r_2}{r_1}\right)^2 = 16 \]

Reading. The wavefront at \(20\,\text{mm}\) lags the one at \(5\,\text{mm}\) by \(22.5\,\text{mm}\) of optical path (a physical phase \(k_0\Delta S\)); the light is \(16\times\) fainter, exactly as the transport equation dictates.

Units check. \(n\) dimensionless \(\times\) length \(=\) length for \(\Delta S\); intensity ratio dimensionless. Consistent.

Example 2 — Bending of a horizontal ray in a hot-road mirage.

\[ \frac{d}{ds}\!\left(n\frac{d\mathbf{r}}{ds}\right)=\nabla n,\qquad n\,\frac{d\hat{\mathbf{t}}}{ds}+\frac{dn}{ds}\hat{\mathbf{t}}=\nabla n \]

Ray equation from step 11, expanded with \(\hat{\mathbf{t}}=d\mathbf{r}/ds\) the unit tangent. We want the curvature of an initially horizontal ray. B

\[ n\,\kappa = \lvert\nabla_{\!\perp} n\rvert \quad\Longrightarrow\quad R=\frac{1}{\kappa}=\frac{n}{\lvert\nabla_{\!\perp} n\rvert} \]

Take the component perpendicular to the ray. \(d\hat{\mathbf{t}}/ds=\kappa\hat{\mathbf{n}}\) (curvature \(\kappa\), radius \(R\)); only the transverse gradient \(\nabla_{\!\perp}n\) curves the ray. C

\[ n\approx 1.000,\qquad \left|\frac{dn}{dz}\right| = 4.0\times10^{-7}\ \text{m}^{-1}\ \ (\text{vertical, hot ground}) \]

Numbers: near a sun-heated road the air is hottest (least dense, lowest \(n\)) at the ground, so \(n\) increases upward with this typical gradient. For a horizontal ray the gradient is fully transverse. A

\[ R=\frac{1.000}{4.0\times10^{-7}\ \text{m}^{-1}} = 2.5\times10^{6}\ \text{m} \]

Substitute into \(R=n/\lvert\nabla_\perp n\rvert\). The ray bends concave-upward (toward higher \(n\)), so a downward-curving line of sight appears to come from below the road — the shimmering "water" of a mirage. A

\[ R \approx 2.5\times10^{3}\ \text{km} \]

Reading. The radius of curvature is enormous compared with the eye's few-hundred-metre sightline, so the bend is tiny — a sag of order \(L^2/2R\approx(200\,\text{m})^2/(5\times10^6\,\text{m})\approx 8\,\text{mm}\) over \(200\,\text{m}\) — yet enough to lift a patch of sky into the roadway. Rays curve toward higher index, exactly as the eikonal-derived ray equation predicts.

Units check. \(n\) dimensionless \(/\ \text{m}^{-1} = \text{m}\) for \(R\). Consistent.

Problems

(A) Plane wave. A collimated beam in a uniform medium of index \(n\) travels along \(\hat{\mathbf{k}}\). Propose \(S=n\,\hat{\mathbf{k}}\cdot\mathbf{r}\) and verify it satisfies the eikonal equation; identify the wavefronts and rays.
Solution
\(\nabla S = n\hat{\mathbf{k}}\) (since \(\nabla(\hat{\mathbf{k}}\cdot\mathbf{r})=\hat{\mathbf{k}}\)), so \(\lvert\nabla S\rvert^2=n^2\hat{\mathbf{k}}\cdot\hat{\mathbf{k}}=n^2\). Eikonal satisfied. Wavefronts \(S=\text{const}\) are the planes \(\hat{\mathbf{k}}\cdot\mathbf{r}=\text{const}\), perpendicular to \(\hat{\mathbf{k}}\); rays run along \(\nabla S/n=\hat{\mathbf{k}}\), i.e. straight parallel lines. With \(\nabla n=0\) the ray equation gives \(d^2\mathbf{r}/ds^2=0\), confirming straight rays.
(A) Straight rays in a uniform medium. Using the ray equation \(\frac{d}{ds}(n\,d\mathbf{r}/ds)=\nabla n\), prove that rays are straight lines wherever \(n\) is constant, and state what determines the ray's speed of parametrization.
Solution
If \(n=\text{const}\) then \(\nabla n=0\) and \(n\) pulls out of the derivative: \(n\,d^2\mathbf{r}/ds^2=0\Rightarrow d^2\mathbf{r}/ds^2=0\). Integrating, \(d\mathbf{r}/ds=\hat{\mathbf{t}}_0\) (constant unit vector) and \(\mathbf{r}(s)=\mathbf{r}_0+s\,\hat{\mathbf{t}}_0\) — a straight line. The parameter \(s\) is geometric arc length (\(\lvert d\mathbf{r}/ds\rvert=1\)); the optical path is \(n s\), and the physical phase advances at \(k_0 n\) per unit length.
(B) Transport / intensity. Show \(2\nabla A\cdot\nabla S + A\nabla^2 S = \frac{1}{A}\nabla\cdot(A^2\nabla S)\), and use it to find how intensity varies along a ray tube of cross-sectional area \(\sigma(s)\).
Solution
Expand \(\nabla\cdot(A^2\nabla S)=\nabla(A^2)\cdot\nabla S + A^2\nabla^2 S = 2A\nabla A\cdot\nabla S + A^2\nabla^2 S = A(2\nabla A\cdot\nabla S + A\nabla^2 S)\). Dividing by \(A\) gives the transport equation, which states \(\nabla\cdot(A^2\nabla S)=0\), i.e. the flux vector \(A^2\nabla S = nA^2\hat{\mathbf{t}}\) is solenoidal. Integrating over a thin ray tube (Gauss's theorem, no flux through the sides), \(n A^2\sigma = \text{const}\) along the tube. Hence intensity \(I\propto A^2\propto 1/(n\sigma)\): where rays converge (\(\sigma\downarrow\)) the beam brightens, and it diverges to infinity at a caustic where \(\sigma\to0\).
(B) GRIN slab. In a medium \(n^2(x)=n_0^2-\beta^2 x^2\) with a ray in the \(x\)–\(z\) plane, use the eikonal in the paraxial form to show the ray oscillates sinusoidally about the axis, and find the spatial period.
Solution
Seek \(S=\alpha z + f(x)\) with \(\alpha\) constant (translational invariance in \(z\)). Eikonal: \((\partial_x S)^2+(\partial_z S)^2=n^2\Rightarrow (f')^2=n_0^2-\beta^2x^2-\alpha^2\). The ray direction obeys \(dx/dz=\partial_x S/\partial_z S=f'/\alpha\). For paraxial rays \(\alpha\approx n_0\); differentiating and using the ray equation \(\frac{d}{ds}(n\,dx/ds)=\partial_x n\) with \(n\,\partial_x n=-\beta^2 x\) and \(s\approx z\) gives \(n_0\,d^2x/dz^2 = -\beta^2 x/n_0\), i.e. \(d^2x/dz^2 = -(\beta/n_0)^2 x\). Solution \(x(z)=x_0\cos(\beta z/n_0)+ (x_0'n_0/\beta)\sin(\beta z/n_0)\): sinusoidal. Spatial period \(\Lambda = 2\pi n_0/\beta\). This is the self-focusing pitch of a GRIN lens/fibre.
(C) Validity estimate. Green light \(\lambda_0=500\,\text{nm}\) enters a biological tissue whose index varies over a scale \(\ell=1.0\,\mu\text{m}\) (cell-membrane structure) with \(n\approx1.4\). Estimate the fractional size of the neglected term in the eikonal expansion and judge whether ray optics is trustworthy.
Solution
The neglected term relative to \(n^2\) is \(\dfrac{\lvert\nabla^2A/(k_0^2A)\rvert}{n^2}\sim\dfrac{1}{n^2}\left(\dfrac{\lambda_0}{2\pi\ell}\right)^2\) with \(\ell\) the amplitude/index scale. Numerically \(\lambda_0/(2\pi\ell)=500\times10^{-9}/(2\pi\times1.0\times10^{-6})=0.0796\), squared \(=6.3\times10^{-3}\), divided by \(n^2=1.96\) gives \(\approx3.2\times10^{-3}\). This is a few tenths of a percent — small, so the eikonal is marginally valid for smooth features at this scale, but structure varying on \(\ell\lesssim\lambda_0\) (or sharp membrane boundaries) pushes the ratio toward unity, where diffraction and scattering dominate and ray tracing fails. Conclusion: usable as a leading estimate, unreliable for sub-micron detail.