Total internal reflection

Total internal reflection (TIR) is the optical phenomenon when waves travelling in one medium strike at sufficiently oblique incident angle (called the critical angle) against the boundary with another medium of lower refractive index, instead of transmitting into the second ("external") medium at a refracted angle, the waves all get reflected back into the first ("internal") medium. An example would be the water surface (the boundary between water and air) in a fish tank, which when viewed obliquely from below often reflects the underwater scenery like a mirror with no loss of brightness (Fig.  1).
TIR not only occurs with electromagnetic waves such as light and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam (Fig. 2), the reflection tends to be described in terms of "rays" rather than waves; in a medium whose properties are independent of direction, such as air, water or glass, the "rays" are perpendicular to the associated wavefronts.

Refraction is generally accompanied by small amount of partial reflection. When waves are refracted from a medium of lower propagation speed (higher refractive index) to one of higher speed (e.g., from water to air), the angle of refraction (between the outgoing ray and the surface normal) is greater than the angle of incidence (between the incoming ray and the normal). As the angle of incidence approaches the critical angle, the angle of refraction approaches 90°, at which the refracted ray becomes parallel to the boundary surface. As the angle of incidence increases beyond the critical angle, the conditions of refraction can no longer be satisfied, so there is no refracted ray, and the partial reflection becomes total. For visible light, the critical angle is about 49° for incidence at the water-to-air boundary, and about 42° for incidence at the common glass-to-air boundary.
Details of the mechanism of TIR give rise to more subtle phenomena. While total reflection, by definition, involves no continuing flow of power across the interface between the two media, the external medium carries a so-called evanescent wave, which travels along the interface with an amplitude that falls off exponentially with distance from the interface. The "total" reflection is indeed total if the external medium is lossless (perfectly transparent), continuous, and of infinite extent, but can be conspicuously less than total if the evanescent wave is absorbed by a lossy external medium ("attenuated total reflectance"), or diverted by the outer boundary of the external medium or by objects embedded in that medium ("frustrated" TIR). Unlike partial reflection between transparent media, total internal reflection is accompanied by a non-trivial phase shift (not just zero or 180°) for each component of polarization (perpendicular or parallel to the plane of incidence), and the shifts vary with the angle of incidence. The explanation of this effect by Augustin-Jean Fresnel, in 1823, added to the evidence in favor of the wave theory of light.
The phase shifts are utilized by Fresnel's invention, the Fresnel rhomb, to modify polarization. The efficiency of the total internal reflection is exploited by optical fibers (used in telecommunications cables and in image-forming fiberscopes), and by reflective prisms, such as the image-erecting Porro/roof prisms for monoculars and binoculars.

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