Total internal reflection

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$The larger the angle to the normal, the smaller is the fraction of light transmitted, until the angle when total internal reflection occurs. (The colour of the rays is to help distinguish the rays, and is not meant to indicate any colour dependence.)$

Total internal reflection is an optical phenomenon that occurs when light strikes a medium boundary at a steep angle. If the refractive index is lower on the other side of the boundary no light can pass through, so effectively all of the light is reflected. The critical angle is the angle of incidence above which the total internal reflection occurs.

When light crosses a boundary between materials with different refractive indices, the light beam will be partially refracted at the boundary surface, and partially reflected. However, if the angle of incidence is shallower (closer to the boundary) than the critical angle, the angle of incidence where light is refracted so that it travels along the boundary, then the light will stop crossing the boundary altogether and instead totally reflect back internally. This can only occur where light travels from a medium with a higher refractive index to one with a lower refractive index. For example, it will occur when passing from glass to air, but not when passing from air to glass.

1 Optical description
2 Critical angle
3 Frustrated total internal reflection
4 Applications
5 See also
6 References
7 External links

[edit] Optical description

Total internal reflection can be demonstrated using a semi-circular glass block. A "ray box" shines a narrow beam of light (a "ray") onto the glass. The semi-circular shape ensures that a ray pointing towards the center of the flat face will hit the surface at right angles. This prevents refraction at the air/glass boundary.

At the glass/air boundary what happens will depend on the angle. Where θ_c is the critical angle:

If θ < θ_c, as with the red ray in the above figure, the ray will split. Some of the ray will reflect off the boundary, and some will refract as it passes through.
If θ > θ_c, as with the blue ray, all of the ray reflects from the boundary. None passes through.

The second situation is total internal reflection.

This physical property makes optical fibres useful, and rainbows and prismatic binoculars possible. It is also what gives diamonds their distinctive sparkle, as diamond has an extremely high refractive index.

An important side effect of total internal reflection is the propagation of an evanescent wave across the boundary surface. This wave may lead to a phenomenon known as frustrated total internal reflection.

[edit] Critical angle

The critical angle is the minimum angle of incidence at which total internal reflection occurs. The angle of incidence is measured with respect to the normal at the refractive boundary. The critical angle <math>\theta_c</math> is given by:

<math>\theta_c = \arcsin \left( \frac{n_2}{n_1} \right), </math>

where <math>n_2</math> is the refractive index of the less dense medium, and <math>n_1</math> is the refractive index of the denser medium. This equation is a simple application of Snell's law where the angle of refraction is 90°.

If the incident ray is precisely at the critical angle, the refracted ray is tangent to the boundary at the point of incidence. For visible light travelling from glass into air (or vacuum), the critical angle is approximately 41.5°. The critical angle for diamond is about 24.4°, which means that light is much more likely to be internally reflected within a diamond. Diamonds for jewelry are cut to take advantage of this; in particular the brilliant cut is designed to achieve high total reflection of light entering the diamond, and high dispersion of the reflected light (known to jewelers as fire).

If the fraction: <math>\frac{n_2}{n_1}</math> is greater than 1, then arcsin is not defined--meaning that total internal reflection does not occur even at very shallow or grazing incident angles. So the critical angle is only defined for <math>\frac{n_2}{n_1}\leq1</math>.

[edit] Frustrated total internal reflection

Under "ordinary conditions" it is true that the creation of an evanescent wave does not affect the conservation of energy, i.e. the evanescent wave transmits zero net energy. However, if a third medium with a higher refractive index than the second medium is placed within less than several wavelengths distance from the interface between the first medium and the second medium, the evanescent wave will be different from the one under "ordinary conditions" and it will pass energy across the second into the third medium. (See evanescent wave coupling.)

A common example in everyday use is a beam splitter. A transparent, low refractive index material is sandwiched between two prisms of another material. This allows the beam to "tunnel" through from one prism to the next in a process very similar to quantum tunneling while at the same time altering the direction of the incoming ray.

[edit] Applications

Optical fibres, which are used in endoscopes and telecommunications, operate based on total internal reflection of light. Mirages are also formed by total reflection of light traveling from a denser medium (cool air) to a less dense medium (warm air near the ground).

New York University Courant Institute of Mathematical Sciences researcher Jefferson Y. Han is currently doing research in the field of FTIR touch sensing for the purpose of Human-Computer Interaction.<ref>FTIR Touch Sensing. Jefferson Y. Han (February 15, 2006).</ref>