Optics, electromagnetic theory Definition

Evanescent field

Definition

Let a homogeneous lossless medium fill the half-space $z > 0$ , and let $k > 0$ be the wavenumber of free waves of angular frequency $\omega$ in it. Let $A \neq 0$ be a constant, $\beta > k$ real, and $\kappa = \sqrt{\beta^2 - k^2} > 0$ . Consider the function $u(x, y, z, t) = A \, e^{i(\beta x - \omega t)} \, e^{-\kappa z}.$ It solves the Helmholtz equation $\nabla^2 u + k^2 u = 0$ , and it is called an evanescent wave in this region. An evanescent field is a superposition of such waves. More generally, it is a field near a surface that oscillates along the surface and decays exponentially away from it.

Remarks

Check: substituting gives $-\beta^2 + \kappa^2 + k^2 = 0$ , which holds by the choice of $\kappa$ . The wave vector is formally the complex vector $(\beta, 0, i\kappa)$ , whose "length squared" $\beta^2 - \kappa^2$ equals $k^2$ . Example: consider total internal reflection from index $n_1$ to $n_2$ at angle $\theta_i$ above the critical angle, with vacuum wavenumber $k_0$ . The field in medium $2$ has $\beta = n_1 k_0 \sin\theta_i > n_2 k_0 = k$ and decay rate $\kappa = k_0\sqrt{n_1^2 \sin^2\theta_i - n_2^2}$ . Non-example: if $0 \le \beta < k$ , the same ansatz needs $\kappa$ purely imaginary, and $u$ is a propagating plane wave that carries power away from the surface.

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Plane wave concept
Evanescent field definition

Tag	`807P2AS5WD`
URI	`m2://807P2AS5WD`
Kind	definition
Writing	W0 stub
MSC	78A40, 78A25
Defines	Evanescent field
Uses	Plane wave
Changed	3 November 2026

Evanescent field

Definition

Remarks

Reading path

Further reading

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