Glossary entry

Critical angle

The minimum incidence angle for total internal reflection at the interface between two media of different refractive indices. The geometric basis of all dielectric optical waveguiding.

When light travels from a higher-index medium ( $n_1$ ) into a lower-index medium ( $n_2$ ), Snell's law gives the refraction angle:

n_1 \sin\theta_1 \;=\; n_2 \sin\theta_2.

When $\theta_1$ increases, $\theta_2$ also increases. At the critical angle $\theta_c$ , the refraction angle reaches 90° (grazing along the interface):

\sin\theta_c \;=\; \frac{n_2}{n_1}, \qquad \theta_c \;=\; \arcsin(n_2 / n_1).

For $\theta_1 > \theta_c$ , no refracted wave exists — the incident wave is fully reflected back into the higher-index medium. This is total internal reflection (TIR), and it is the geometric basis for all dielectric optical waveguiding.

Interface	$\theta_c$
Water (1.33) to air	48.6°
Crown glass (1.52) to air	41.1°
Diamond (2.42) to air	24.4°
Silica fiber core (1.467) to fiber cladding (1.4554)	81.7°
Silicon (3.48) to silica cladding (1.444)	24.5°

Why TIR is total. The Fresnel equations predict that $|R| = 1$ for $\theta_1 > \theta_c$ , meaning all incident power returns to the high-index medium. There is no absorption or transmission loss in the ideal case.

Evanescent wave. Although no power is transmitted across the interface, a non-zero field penetrates the lower-index medium. This evanescent wave decays exponentially with distance from the interface:

\text{decay length} \;\sim\; \frac{\lambda_0}{2\pi \sqrt{n_1^2 \sin^2\theta_1 - n_2^2}}.

For incidence near critical angle, the evanescent wave extends $\sim$ wavelengths into the low-index medium; for incidence far from critical, decay length is sub-wavelength.

Engineering implications:

Optical fiber confinement — light propagates by repeated TIR at the core-cladding interface; the V number determines how many modes can simultaneously satisfy TIR
Prism couplers — frustrate TIR by bringing two high-index media to within wavelengths of each other, coupling power across the gap (basis of in-coupling to slab waveguides for SPR sensors)
Light pipes / total internal reflection illumination — TIR is used to confine light in plastic optical fiber, fiber-optic taper plates, and direct-write waveguides
Photonic crystal cladding — band-gap confinement supplements TIR in PCF and provides confinement where simple TIR would fail (hollow-core fibers)

The critical angle is dictated by the index contrast between core and cladding; higher index contrast produces a smaller critical angle and stronger confinement. This is why silicon-on-insulator waveguides confine light so tightly compared to silica fiber — the silicon-to-silica index ratio of 2.4× produces a 24° critical angle, whereas silica fiber's 0.8% index contrast yields an 82° critical angle requiring much shallower bends to maintain confinement.