Total internal reflection
In physics, total internal reflection is the phenomenon in which waves arriving at the interface from one medium to another are not refracted into the second medium, but completely reflected back into the first medium. It occurs when the second medium has a higher wave speed 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.
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, 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.
File:Total internal reflection by fluorescence.jpg|thumb|Fig.2:Repeated total internal reflection of a 405nm laser beam between the front and back surfaces of a glass pane. The color of the laser light itself is deep violet; but its wavelength is short enough to cause fluorescence in the glass, which re-radiates greenish light in all directions, rendering the zigzag beam visible.
Refraction is generally accompanied by partial reflection. When waves are refracted from a medium of lower propagation speed to a medium of higher propagation speed —e.g., from water to air—the angle of refraction is greater than the angle of incidence. 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, continuous, and of infinite extent, but can be conspicuously less than total if the evanescent wave is absorbed by a lossy external medium, or diverted by the outer boundary of the external medium or by objects embedded in that medium. Unlike partial reflection between transparent media, total internal reflection is accompanied by a non-trivial phase shift for each component of polarization, 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, 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 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 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 to the interface is called the angle of incidence. 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 is greater than the angle of incidence. For the moment, let us call the angle of incidence θ and the angle of refraction θt. As θ increases and approaches a certain "critical angle", denoted by θc, the angle of refraction approaches 90°, and the refracted ray becomes fainter while the reflected ray becomes brighter. 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. 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 ; none of it is transmitted.
Critical angle
When a wavefront is refracted from one medium to another, the incident and refracted portions of the wavefront meet at a common line on the refracting surface. Let this line, denoted by L, move at velocity across the surface, where is measured normal to L. 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.
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. However, if we now suppose that the properties of the media are isotropic, 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.
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 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. 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 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.
The medium with the higher refractive index is commonly described as optically denser, and the one with the lower refractive index as optically rarer. 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. The brightness of the reflected image – just as bright as the "direct" view – can be startling.A similar effect can be observed by opening one's eyes while swimming just below the water's surface. If the water is calm, the surface outside the critical angle appears mirror-like, reflecting objects below. The region above the water cannot be seen except overhead, where the hemispherical field of view is compressed into a conical field known as Snell's window, whose angular diameter is twice the critical angle. The field of view above the water is theoretically 180° across, but seems less because as we look closer to the horizon, the vertical dimension is more strongly compressed by the refraction; e.g., by Eq., for air-to-water incident angles of 90°, 80°, and 70°, the corresponding angles of refraction are 48.6°, 47.6°, and 44.8°, indicating that the image of a point 20° above the horizon is 3.8° from the edge of Snell's window while the image of a point 10° above the horizon is only 1° from the edge.
Fig.7, for example, is a photograph taken near the bottom of the shallow end of a swimming pool. What looks like a broad horizontal stripe on the right-hand wall consists of the lower edges of a row of orange tiles, and their reflections; this marks the water level, which can then be traced across the other wall. The swimmer has disturbed the surface above her, scrambling the lower half of her reflection, and distorting the reflection of the ladder. But most of the surface is still calm, giving a clear reflection of the tiled bottom of the pool. The space above the water is not visible except at the top of the frame, where the handles of the ladder are just discernible above the edge of Snell's window – within which the reflection of the bottom of the pool is only partial, but still noticeable in the photograph. One can even discern the color-fringing of the edge of Snell's window, due to variation of the refractive index, hence of the critical angle, with wavelength.
The critical angle influences the angles at which gemstones are cut. The round "brilliant" cut, for example, is designed to refract light incident on the front facets, reflect it twice by TIR off the back facets, and transmit it out again through the front facets, so that the stone looks bright. Diamond is especially suitable for this treatment, because its high refractive index and consequently small critical angle yield the desired behavior over a wide range of viewing angles. Cheaper materials that are similarly amenable to this treatment include cubic zirconia and moissanite ; both of these are therefore popular as diamond simulants.