Critical angle

The critical angle is the angle of incidence above which light traveling in a denser medium ($n_1$) can no longer refract into a less dense one ($n_2<n_1$) and is instead totally reflected. It is about 41° for crown glass to air, 48.6° for water to air and 24.4° for diamond to air. Inside a silica fiber whose core and cladding indices differ by less than 1%, it is typically 82° to 85°, so guided rays travel nearly parallel to the axis.

Definition and formula

Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, gives a refraction angle that grows faster than the incidence angle when $n_2<n_1$. The refracted ray grazes the surface (${\theta }_2=90\mathrm{°}$) when

$$\sin {\theta }_c=\frac{n_2}{n_1},{\theta }_c=\arcsin \frac{n_2}{n_1}\mathrm{.}$$

For ${\theta }_1>{\theta }_c$ there is no real refraction angle and the wave undergoes total internal reflection (TIR). No critical angle exists when light goes from low to high index.

Why the reflection is total

The Fresnel equations give a reflection coefficient of unit magnitude for both polarizations above ${\theta }_c$: all incident power returns to the denser medium, and in an ideal lossless interface there is no transmission loss at all. The reflection does impose a polarization-dependent phase shift, which is what Fresnel rhombs use to convert linear to circular polarization, and a small lateral displacement of the reflected beam along the surface (the Goos–Hänchen shift) of order a wavelength.

The evanescent wave. A field still exists in the lower-index medium, decaying exponentially with distance from the surface. Its 1/e field penetration depth is

$$d=\frac{{\lambda }_0}{2\pi \sqrt{n_1^2{\sin }^2{\theta }_1-n_2^2}}\mathrm{.}$$

For BK7 ($n_1=1.5151$ at 633 nm) against air, where ${\theta }_c=41.3\mathrm{°}$, the depth is 604 nm at 42° incidence, 262 nm at 45° and 119 nm at 60°. It diverges as ${\theta }_1$ approaches ${\theta }_c$, so near-critical incidence is where coupling to anything placed near the surface is strongest.

Relation to numerical aperture

In a step-index fiber, a ray is guided if it strikes the core–cladding boundary at more than ${\theta }_c$, that is, if it travels within $90\mathrm{°}-{\theta }_c$ of the axis inside the core. Applying Snell's law at the end face converts this into the numerical aperture:

$$\mathrm{N}\mathrm{A}=n_1\cos {\theta }_c=\sqrt{n_1^2-n_2^2}\mathrm{.}$$

For the silica fiber in the table, $90\mathrm{°}-{\theta }_c=7.2\mathrm{°}$ and NA = 0.184, giving an acceptance angle of 10.6° in air. In a single-mode fiber the ray picture is only qualitative: the V number determines how many modes exist, but the NA it uses is the same quantity.

Measuring it

Abbe and Pulfrich refractometers find the critical angle directly: a liquid or solid sample is placed on a high-index prism, and the boundary between the bright (totally reflected) and dark regions of the field of view sits at ${\theta }_c$, from which $n_2$ follows. Commercial instruments read to about ${10}^{-4}$ in index. In the lab the angle can also be located by rotating a prism on a stage and watching for the transmitted beam to vanish, although the refracted beam becomes grazing and weak as ${\theta }_1$ approaches ${\theta }_c$, and beam divergence blurs the cutoff, which limits the precision of a simple visual reading.

Where it matters

• Optical fiber: guidance by repeated TIR at the core–cladding interface.
• Prism couplers and frustrated TIR: bringing a second high-index body within about one penetration depth lets power tunnel across the gap; this is used for coupling into slab waveguides and in the Otto and Kretschmann configurations for surface plasmon resonance.
• Light pipes and TIR prisms: plastic optical fiber, fiber-optic tapers and retro-reflecting prisms rely on TIR with no reflective coating.
• Photonic crystal and hollow-core fibers: in hollow-core fibers the core has the lower index, no critical angle exists, and confinement comes from band-gap or anti-resonant effects.
• LEDs and photodetectors: light generated inside a high-index semiconductor escapes only within a narrow cone; for $n\approx 3.5$ to air, ${\theta }_c$ is about 16.6°, which limits extraction efficiency from planar chips.

Index contrast. Silicon and silica have an index ratio of 2.41 and a critical angle of 24.5°, whereas the silica fiber above has a fractional index difference of 0.79% and a critical angle of 82.8°. The high contrast in silicon-on-insulator allows sub-micron waveguides and bend radii of a few micrometers; low-contrast fiber needs bend radii of millimeters to centimeters before bend loss becomes significant.

The Snell's law calculator finds the critical angle for any pair of indices and shows the evanescent depth and phase shifts beyond it.

Common questions

Is the critical angle measured from the surface or the normal?

From the normal, like all angles in Snell's law. Some fiber texts quote the complementary angle measured from the axis; for the fiber above that is 7.2°.

Does the critical angle depend on wavelength?

Yes, through the dispersion of both indices. For BK7 to air, ${\theta }_c$ is 0.27° larger at 656 nm than at 486 nm.

What happens exactly at the critical angle?

The transmitted wave runs along the surface and carries no power into the second medium; reflectance is already 100%, and the evanescent penetration depth is formally infinite.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 1 and Ch. 6; Hecht, Optics (5th ed., 2017), Ch. 4; Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 1.