In physics, total internal reflection (TIR) is the phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected back into the first ("internal") medium. It occurs when the second medium has a higher wave speed (i.e., lower refractive index) than the first, and the waves are incident at a sufficiently oblique angle on the interface. For example, the water-to-air surface in a typical fish tank, when viewed obliquely from below, reflects the underwater scene like a mirror with no loss of brightness (Fig.1). A scenario opposite to TIR, referred to as total external reflection, occurs in the extreme ultraviolet and X-ray regimes.
TIR occurs not only with electromagnetic waves such as light and microwaves, but also with other types of waves, including sound and water waves. If the waves are capable of forming a narrow 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 associated wavefronts. The total internal reflection occurs when critical angle is exceeded.
Refraction is generally accompanied by partial reflection. When waves are refracted from a medium of lower propagation speed (higher refractive index) to a medium of higher propagation speed (lower refractive index)—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 a certain threshold, called 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 from water to air, and about 42° for incidence from common glass to air.
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 used 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 image-erecting Porro/roof prisms for monoculars and binoculars.
Optical description
Although total internal reflection can occur with any kind of wave that can be said to have oblique incidence, including (e.g.) microwaves and sound waves, it is most familiar in the case of light waves.
Total internal reflection of light can be demonstrated using a semicircular-cylindrical block of common glass or acrylic glass. In Fig.3, a "ray box" projects a narrow beam of light (a "ray") radially inward. The semicircular cross-section of the glass allows the incoming ray to remain perpendicular to the curved portion of the air/glass surface, and then hence to continue in a straight line towards the flat part of the surface, although its angle with the flat part varies.
Where the ray meets the flat glass-to-air interface, the angle between the ray and the normal (perpendicular) to the interface is called the angle of incidence.Jenkins & White, 1976, p.11. If this angle is sufficiently small, the ray is partly reflected but mostly transmitted, and the transmitted portion is refracted away from the normal, so that the angle of refraction (between the refracted ray and the normal to the interface) is greater than the angle of incidence. For the moment, let us call the angle of incidence θ and the angle of refraction θt (where t is for transmitted, reserving r for reflected). As θ increases and approaches a certain "critical angle", denoted by θc (or sometimes θcr), the angle of refraction approaches 90° (that is, the refracted ray approaches a tangent to the interface), and the refracted ray becomes fainter while the reflected ray becomes brighter.Jenkins & White, 1976, p.527. (The refracted beam becomes fainter in terms of total power, but not necessarily in terms of visibility, because the beam also becomes narrower as it becomes more nearly tangential.) As θ increases beyond θc, the refracted ray disappears and only the reflected ray remains, so that all of the energy of the incident ray is reflected; this is total internal reflection (TIR). In brief: If θ < θc, the incident ray is split, being partly reflected and partly refracted; If θ > θc, the incident ray suffers total internal reflection (TIR); none of it is transmitted.
Critical angle
The critical angle is the smallest angle of incidence that yields total reflection, or equivalently the largest angle for which a refracted ray exists.Jenkins & White, 1976, p.26. For light waves incident from an "internal" medium with a single refractive index , to an "external" medium with a single refractive index , the critical angle is given by and is defined if . For some other types of waves, it is more convenient to think in terms of propagation velocities rather than refractive indices. The explanation of the critical angle in terms of velocities is more general and will therefore be discussed first.
When a wavefront is refracted from one medium to another, the incident (incoming) and refracted (outgoing) portions of the wavefront meet at a common line on the refracting surface (interface). Let this line, denoted by L, move at velocity across the surface,Born & Wolf, 1970, pp.12–13. where is measured normal to L (Fig.4). Let the incident and refracted wavefronts propagate with normal velocities and respectively, and let them make the dihedral angles θ1 and θ2 respectively with the interface. From the geometry, is the component of in the direction normal to the incident wave, so that Similarly, Solving each equation for and equating the results, we obtain the general law of refraction for waves:
But the dihedral angle between two planes is also the angle between their normals. So θ1 is the angle between the normal to the incident wavefront and the normal to the interface, while θ2 is the angle between the normal to the refracted wavefront and the normal to the interface; and Eq.() tells us that the sines of these angles are in the same ratio as the respective velocities.Huygens, 1690, tr. Thompson, p.38.
This result has the form of "Snell's law", except that we have not yet said that the ratio of velocities is constant, nor identified θ1 and θ2 with the angles of incidence and refraction (called θi and θt above). However, if we now suppose that the properties of the media are isotropic (independent of direction), two further conclusions follow: first, the two velocities, and hence their ratio, are independent of their directions; and second, the wave-normal directions coincide with the ray directions, so that θ1 and θ2 coincide with the angles of incidence and refraction as defined above.Birefringent media, such as calcite, are non-isotropic (anisotropic). When we say that the extraordinary refraction of a calcite crystal "violates Snell's law", we mean that Snell's law does not apply to the extraordinary ray, because the direction of this ray inside the crystal generally differs from that of the associated wave-normal (Huygens, 1690, tr. Thompson, p.65, Art.24), and because the wave-normal speed is itself dependent on direction. (Note that the cited passage contains a translation error: in the phrase "conjugate with respect to diameters which are not in the straight line AB", the word "not" is unsupported by Huygens' original French and is geometrically incorrect.)
Obviously the angle of refraction cannot exceed 90°. In the limiting case, we put and in Eq.(), and solve for the critical angle:
In deriving this result, we retain the assumption of isotropic media in order to identify θ1 and θ2 with the angles of incidence and refraction.For non-isotropic media, Eq.() still describes the law of refraction in terms of wave-normal directions and speeds, but the range of applicability of that law is determined by the constraints on the ray directions (Buchwald, 1989, p.29).
For electromagnetic waves, and especially for light, it is customary to express the above results in terms of refractive indices. The refractive index of a medium with normal velocity is defined as where c is the speed of light in vacuum.Born & Wolf, 1970, p.13; Jenkins & White, 1976, pp.9–10. This definition uses vacuum as the "reference medium". In principle, any isotropic medium can be chosen as the reference. For some purposes it is convenient to choose air, in which the speed of light is about 0.03% lower than in vacuum (Rutten and van Venrooij, 2002, pp.10, 352). The present article, however, chooses vacuum. Hence Similarly, Making these substitutions in Eqs.() and (), we obtain
and
Eq.() is the law of refraction for general media, in terms of refractive indices, provided that θ1 and θ2 are taken as the dihedral angles; but if the media are isotropic, then and become independent of direction, while θ1 and θ2 may be taken as the angles of incidence and refraction for the rays, and Eq.() follows. So, for isotropic media, Eqs.() and () together describe the behavior in Fig.5.
According to Eq.(), for incidence from water () to air (), we have , whereas for incidence from common or acrylic glass () to air (), we have .
The arcsin function yielding θc is defined only if Hence, for isotropic media, total internal reflection cannot occur if the second medium has a higher refractive index (lower normal velocity) than the first. For example, there cannot be TIR for incidence from air to water; rather, the critical angle for incidence from water to air is the angle of refraction at grazing incidence from air to water (Fig.6).Jenkins & White, 1976, p.25.
The medium with the higher refractive index is commonly described as optically denser, and the one with the lower refractive index as optically rarer.Jenkins & White, 1976, pp.10, 25. Hence it is said that total internal reflection is possible for "dense-to-rare" incidence, but not for "rare-to-dense" incidence.
Everyday examples
When standing beside an aquarium with one's eyes below the water level, one is likely to see fish or submerged objects reflected in the water-air surface (Fig.1). The brightness of the reflected image – just as bright as the "direct" view – can be startling.