Evanescent wave

An evanescent wave is an electromagnetic field that exists adjacent to an interface or guiding structure but does not propagate as a wave — instead, its amplitude decays exponentially with distance from the boundary. Evanescent fields appear in two principal situations:

1. In the lower-index medium during total internal reflection when light exceeds the critical angle at an interface
2. Outside the guiding region of an optical waveguide as the spatial extension of the bound mode field

Mathematical form. Consider light at incidence angle $\theta_1 > \theta_c$ at the boundary between media of indices $n_1$ and $n_2 < n_1$. The field in medium 2 has the form:

$$E_2(x, z, t) \;=\; E_0 \, e^{-\kappa z} \, e^{i(k_x x - \omega t)},$$

where the surface is at $z = 0$, the wave propagates parallel to the surface in the $x$-direction with wavenumber $k_x = (n_1 \omega / c) \sin\theta_1$, and decays into medium 2 with decay constant:

$$\kappa \;=\; \frac{\omega}{c} \sqrt{n_1^2 \sin^2\theta_1 - n_2^2}.$$

The penetration depth $d_\text{pen} = 1/\kappa$ at which the field falls to $1/e$:

$$d_\text{pen} \;=\; \frac{\lambda_0}{2\pi \sqrt{n_1^2 \sin^2\theta_1 - n_2^2}}.$$

For typical conditions (TIR at glass-air interface, $\lambda_0 = 633$ nm, $\theta_1 = 60°$):

$n_1$	$n_2$	$\theta_1$	$d_\text{pen}$
1.52 (BK7)	1.00 (air)	60°	96 nm
1.52	1.00	50°	269 nm
1.52	1.00	45°	850 nm
1.45 (silica)	1.00	60°	105 nm
1.467 (SMF core)	1.463 (SMF clad)	LP01 mode propagation	7 μm
3.48 (Si)	1.45 (SiO₂)	TIR at Si/SiO₂	50 – 100 nm

The evanescent field penetration is generally a sub-wavelength to wavelength-scale distance — much smaller than the geometric optics scale.

No average power flow. The Poynting vector of the evanescent wave has a time-averaged component parallel to the surface (carrying the wave forward) but zero time-averaged component perpendicular to the surface. No net power flows across the boundary in steady state. The energy that enters during a portion of the optical cycle is returned to the high-index medium during the next portion — exactly balancing.

Frustrated total internal reflection. If a second high-index medium is brought within $\sim d_\text{pen}$ of the TIR surface, the evanescent wave can couple to a propagating wave in the second medium — essentially "tunneling" through the gap. This is the operating principle of:

• Evanescent-coupled prisms: variable-ratio beam splitters
• Integrated photonic directional couplers: parallel waveguides exchange power via overlapping evanescent fields
• Quantum tunneling analogy: the optical version of quantum-mechanical barrier tunneling

Applications.

• TIR fluorescence microscopy (TIRF): excites only fluorophores within 100 – 200 nm of a glass slide; standard tool for studying cell-membrane processes
• Near-field scanning optical microscopy (NSOM): a sub-wavelength tip scans a sample's surface, using evanescent coupling to achieve sub-100-nm resolution
• Fiber-optic evanescent-field sensors: removing fiber cladding over a short section exposes the evanescent field to the environment; refractive index changes from chemicals or biomolecules in the surrounding medium modulate the transmitted signal
• Surface plasmon resonance: evanescent wave couples to plasmons in a thin metal film; resonance condition is exquisitely sensitive to refractive index near the surface
• Whispering-gallery modes: light propagates around the equator of a microsphere or microdisk via repeated TIR; evanescent field couples in/out via a nearby waveguide
• Photonic-crystal slab modes: out-of-slab fields are evanescent; coupling to/from these requires evanescent matching

Evanescent coupling to waveguides. The fundamental method for moving light between waveguides on a photonic IC. Two parallel waveguides with sufficiently small separation have overlapping evanescent fields, allowing power to transfer between them at a rate set by the field overlap. This is the basis of:

• Directional couplers: 2-port to 2-port power splitters
• Multimode interference couplers: extended to multi-port via self-imaging
• Ring resonator coupling: light evanescent-couples between bus waveguide and ring
• Grating coupling: surface gratings convert between in-plane evanescent fields and out-of-plane propagating beams
• Adiabatic coupling: gradually tapering waveguide widths transfers power between mode shapes

Why evanescent fields matter for photonic-IC design. The evanescent extension of a guided mode determines the minimum separation between adjacent waveguides without crosstalk. For silicon photonic strip waveguides at 1550 nm, a $\sim$ 100 nm evanescent decay constant means waveguides $> 1$ μm apart have negligible crosstalk. Tighter packing requires careful design with deliberate coupling control.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 2 (TIR analysis) and Ch. 8 (waveguide coupling); Born & Wolf, Principles of Optics (7th ed.), Ch. 13 (surface waves); Novotny & Hecht, Principles of Nano-Optics (2nd ed., 2012) for near-field treatment.