Total Internal Reflection

Total internal reflection (TIR) is a phenomenon in optics in which light traveling through a transparent medium is completely reflected at its boundary with a medium of lower refractive index, provided that the angle of incidence exceeds a critical angle. Unlike ordinary partial reflection, ideal TIR transmits no optical power away from the interface into the second medium. Nevertheless, an electromagnetic field extends a short distance beyond the boundary. (openstax.org)

Conditions and critical angle

For two ordinary transparent, isotropic media with refractive indices $n_1$ and $n_2$ , total internal reflection requires:

Light incident from the higher-index medium: $n_1>n_2$ .
An incidence angle $\theta_i$ greater than the critical angle $\theta_c$ .

Angles are measured from the normal, the line perpendicular to the interface, rather than from the surface. “Higher optical density” in this context means higher refractive index, not necessarily greater mass density. (openstax.org)

The critical angle follows from Snell’s law, which describes refraction:

n_1\sin\theta_i=n_2\sin\theta_t,

where $\theta_t$ is the transmitted angle. At the critical angle, the limiting transmitted direction is parallel to the interface, so $\theta_t=90^\circ$ . Consequently,

\boxed{\theta_c=\arcsin\!\left(\frac{n_2}{n_1}\right)},\qquad n_1>n_2.

Above this angle, Snell’s law would require $\sin\theta_t>1$ , so no propagating transmitted ray exists. The reflected ray remains in the first medium and obeys the usual law of reflection: its angle equals the incidence angle. (openstax.org)

For a water–air boundary, the critical angle is approximately $48.6^\circ$ ; for diamond–air, it is approximately $24.4^\circ$ . These values depend on the refractive indices used. Below the critical angle, reflection and transmission generally coexist; the critical angle itself is the limiting case between propagating transmission and an exponentially decaying transmitted field. (openstax.org)

Electromagnetic explanation

A ray diagram describes the direction of light but not the entire field at the boundary. Applying Maxwell’s equations and the electromagnetic boundary conditions shows that the field in the lower-index medium does not vanish during TIR. Instead, it becomes an evanescent wave: its amplitude decreases exponentially with distance perpendicular to the interface. (ocw.mit.edu)

For a planar boundary, let $z$ measure distance into the second medium and let $\lambda_0$ be the vacuum wavelength. The field amplitude has the spatial dependence

E(z)\propto e^{-\kappa z}, \qquad \kappa=\frac{2\pi}{\lambda_0} \sqrt{n_1^2\sin^2\theta_i-n_2^2}.

The amplitude decay length is $1/\kappa$ , while the squared field magnitude falls by a factor $e$ over $1/(2\kappa)$ . Penetration increases as the angle approaches the critical angle from above. In the ideal lossless, single-interface problem, this field carries no time-averaged energy flux away from the boundary in the normal direction. Thus, “total” describes reflected power, not the absence of a field outside the first medium. (ocw.mit.edu)

Phase and polarization

The Fresnel equations give the reflected field amplitudes for the two independent polarizations: s, with the electric field perpendicular to the plane of incidence, and p, with it parallel to that plane. Under ideal TIR, both reflection coefficients have unit magnitude, but their phases generally differ. Complete power reflection therefore need not preserve the incident polarization state. (ocw.mit.edu)

A Fresnel rhomb exploits this phase difference. Two suitably chosen internal reflections produce a quarter-wave relative phase shift between the polarization components. With an appropriate input orientation, the device converts linearly polarized light into circularly polarized light without relying on a birefringent wave plate. (farside.ph.utexas.edu)

Frustrated total internal reflection

If another higher-index medium is brought sufficiently close to the reflecting boundary, the evanescent field can couple across the intervening lower-index gap. Some light then propagates in the third medium, reducing the reflected power. This is called frustrated total internal reflection (FTIR). Its strength depends strongly on the gap width and the optical geometry. (ocw.mit.edu)

FTIR is mathematically analogous to quantum tunneling through a barrier: an exponentially decaying wave connects two regions that permit propagation. The optical effect, however, can be explained entirely by classical electromagnetism. Contact-induced disruption of TIR has also been used in optical touch-screen systems. (ocw.mit.edu)

Applications

Optical guidance. In an optical fiber, a higher-index core surrounded by lower-index cladding confines light through internal reflection. Light pipes and other optical waveguides use the same principle to transport light with low loss. (openstax.org)

Prisms and retroreflectors. Uncoated prism faces can redirect light by TIR rather than by a metallic mirror coating. Corner-cube prisms use successive reflections to return an incident beam toward its source. Although reflection can be highly efficient, the associated phase shifts can alter polarization. (thorlabs.us)

Surface-selective microscopy. Total internal reflection fluorescence microscopy uses the evanescent field to excite fluorescent molecules close to a glass–sample interface. Excitation is concentrated in a shallow region, often roughly 100 nanometres deep, suppressing fluorescence from more distant material and enabling observations near cell surfaces. (onlinelibrary.wiley.com)

Infrared analysis. In attenuated total reflectance (ATR), a sample contacts a high-index optical element. Absorption from the evanescent field reduces the internally reflected infrared beam at characteristic wavelengths. The resulting spectrum is used in infrared spectroscopy to investigate chemical composition. In this application, reflection is deliberately attenuated rather than perfectly total. (thermofisher.com)

Idealization and practical limits

Perfect TIR assumes transparent, lossless media and a boundary arrangement that does not allow the evanescent field to transfer power elsewhere. An absorbing sample can remove energy from that field, while a nearby transmitting medium can enable FTIR. Consequently, satisfying the critical-angle condition alone does not guarantee that a real optical device has zero loss. (thermofisher.com)

The critical-angle formula also assumes ordinary isotropic media described by scalar refractive indices. More general electromagnetic interfaces require analysis of their permitted wave modes and boundary conditions. Even in the elementary case, complete reflected power should not be confused with unchanged phase, unchanged polarization, or a vanishing external field. (ocw.mit.edu)

References

4 Total Internal Reflection — University Physics Volume 3openstax.org
4 Total Internal Reflection — College Physicsopenstax.org
Electromagnetic Waves and Interfaces — MIT OpenCourseWareocw.mit.edu
Total Internal Reflection — University of Texas at Austinfarside.ph.utexas.edu
Phase Shifts in Total Internal Reflection — MIT OpenCourseWareocw.mit.edu
Lecture 2, MIT 2.71 Optics, Spring 2009ocw.mit.edu
Lecture 6: Lightfields Part 2 — MIT OpenCourseWareocw.mit.edu
2-D Input Device Based on Frustrated Total Internal Reflection — MIT Media Labmedia.mit.edu
Mounted TIR Retroreflector Prisms — Thorlabsthorlabs.us
The Attenuated Total Reflection (ATR) Technique for Analyzing Plastics — Thermo Fisher Scientificthermofisher.com