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 ${\theta }_2=90\mathrm{°}$:

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

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.

Fresnel coefficients above critical angle. Above ${\theta }_c$, the Fresnel reflection coefficients have complex values with unit magnitude:

$$\mathrm{\mid }{r}_s{\mathrm{\mid }}^2=\mathrm{\mid }{r}_p{\mathrm{\mid }}^2=1,$$

confirming 100% reflectance. However, the s- and p-polarizations experience different phase shifts upon reflection. The phase difference $\delta ={\varphi }_p-{\varphi }_s$ varies with ${\theta }_1$ and is the basis of the Fresnel rhomb, an achromatic quarter-wave retarder built entirely from TIR. The Fresnel equations calculator computes both phases at any angle.

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 ${\lambda }_0\mathrm{/}4$ 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: 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 (most of the roughly 0.2 dB/km at 1550 nm in SMF-28) and material absorption dominate the total transmission loss.

The Snell's law calculator shows the onset of total internal reflection and the depth of the evanescent field for chosen indices and wavelength.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 6 (polarization optics, reflection and refraction); Hecht, Optics (5th ed., 2017), Ch. 4 for the geometric treatment.