Total internal reflection (TIR)

Total internal reflection occurs when light traveling in a medium of refractive index $n_1$ strikes the interface with a medium of lower refractive index $n_2 < n_1$ at an angle of incidence greater than the critical angle. Under these conditions, no light is transmitted into the second medium — 100% of the incident power is reflected back into the first medium.

Critical angle. Snell's law states $n_1 \sin\theta_1 = n_2 \sin\theta_2$. For refraction into a lower-index medium, $\sin\theta_2 > \sin\theta_1$, so $\theta_2 > \theta_1$. As $\theta_1$ increases, $\theta_2$ approaches 90°. The critical angle $\theta_c$ is the angle of incidence at which $\theta_2 = 90°$:

$$\sin\theta_c \;=\; \frac{n_2}{n_1}.$$

For incidence angles $\theta_1 > \theta_c$, no real solution for $\theta_2$ exists — the wave cannot propagate into the second medium and is fully reflected.

Typical critical angles.

Interface	$n_1 / n_2$	$\theta_c$
Crown glass to air	1.52 / 1.00	41.1°
Fused silica to air	1.45 / 1.00	43.6°
Water to air	1.33 / 1.00	48.6°
Diamond to air	2.42 / 1.00	24.4°
SMF-28 core to cladding	1.467 / 1.463	85.7°
Silicon to SiO₂ (1550 nm)	3.48 / 1.45	24.6°
GaAs to AlGaAs	3.55 / 3.32	69.2°

Fresnel coefficients above critical angle. Above $\theta_c$, the Fresnel reflection coefficients have complex values with unit magnitude:

$$|r_s|^2 \;=\; |r_p|^2 \;=\; 1,$$

confirming 100% reflectance. However, the s- and p-polarizations experience different phase shifts upon reflection. The phase difference $\delta = \phi_p - \phi_s$ varies with $\theta_1$ and is the basis of the Fresnel rhomb — an achromatic quarter-wave retarder built entirely from TIR.

Evanescent wave. Although no real propagating wave exists in medium 2 above the critical angle, an evanescent wave penetrates a short distance into medium 2 with exponentially decaying amplitude. The penetration depth is:

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

typically $\lambda_0 / 4$ to $\lambda_0$ at moderately supercritical angles. The evanescent wave carries no time-averaged power across the boundary (in the steady state), but it is the basis of frustrated total internal reflection — where bringing a second medium within the penetration depth allows light to "tunnel" through the gap.

TIR-based devices.

• Optical fibers rely entirely on TIR at the core-cladding interface to guide light
• Slab waveguides and ridge waveguides in photonic ICs use TIR at the core-cladding (or core-air) boundaries
• TIR prisms (right-angle, Porro, roof) redirect light without metallic coatings
• Retroreflectors (corner cubes) use three orthogonal TIR surfaces to return light parallel to incidence
• TIR microscopy (TIRF) uses the evanescent field above the critical angle to excite fluorescence only within $\sim 100$ nm of the substrate
• Frustrated TIR beam splitters vary the gap between two glass surfaces to control reflection-vs-transmission

Loss mechanisms. TIR is theoretically 100% but real surfaces add small losses:

• Surface roughness scattering: $\sim 10^{-5}$ – $10^{-3}$ loss per bounce in polished optical fiber
• Material absorption: from impurity ions or color centers in the bulk material near the interface
• Cladding contamination: dust, oils, or moisture at the surface couples evanescent light out

For polished optical fiber, residual TIR loss is so small that bulk Rayleigh scattering (0.2 dB/km at 1550 nm in SMF-28) and OH absorption dominate the total transmission loss.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 2 (electromagnetic optics, reflection at interfaces); Hecht, Optics (5th ed., 2017), Ch. 4 for the geometric treatment.