Theory of computation

The larger the angle to the normal, the smaller is the fraction of light transmitted rather than reflected, until the angle at which total internal reflection occurs. (The color of the rays is to help distinguish the rays, and is not meant to indicate any color dependence.)

Total internal reflection in a block of acrylic

Total internal reflection is a phenomenon that happens when a propagating wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflectance occurs. This is particularly common as an optical phenomenon, where light waves are involved, but it occurs with many types of waves, such as electromagnetic waves in general or sound waves.

When a wave crosses a boundary between materials with different kinds of refractive indices, the wave will be partially refracted at the boundary surface, and partially reflected. However, if the angle of incidence is greater (i.e. the direction of propagation or ray is closer to being parallel to the boundary) than the critical angle – the angle of incidence at which light is refracted such that it travels along the boundary – then the wave will not cross the boundary and instead be totally reflected back internally. This can only occur where the wave travels from a medium with a higher refractive index (n₁) to one with a lower refractive index (n₂). For example, it will occur with light when passing from glass to air, but not when passing from air to glass.

Total internal reflection in a semi-circular acrylic block

Optical description

Total internal reflection of light can be demonstrated using a semi-circular block of glass or plastic. 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 centre of the flat face will hit the curved surface at a right angle; this will prevent refraction at the air/glass boundary of the curved surface. At the glass/air boundary of the flat surface, what happens will depend on the angle. Where θ_c is the critical angle measurement which is caused by the sun or a light source (measured normal to the surface):

• If θ < θ_c, the ray will split. Some of the ray will reflect off the boundary, and some will refract as it passes through. This is not total internal reflection.
• If θ > θ_c, the entire ray reflects from the boundary. None passes through. This is called total internal reflection.

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

Critical angle

Illustration of Snell's law, $n_{1}\sin \theta _{i}=n_{2}\sin \theta _{t}$.

The critical angle is the angle of incidence above which total internal reflection occurs. The angle of incidence is measured with respect to the normal at the refractive boundary (see diagram illustrating Snell's law). Consider a light ray passing from glass into air. The light emanating from the interface is bent towards the glass. When the incident angle is increased sufficiently, the transmitted angle (in air) reaches 90 degrees. It is at this point no light is transmitted into air. The critical angle $\theta _{c}$ is given by Snell's law,

$n_{1}\sin \theta _{i}=n_{2}\sin \theta _{t}\quad$.

Rearranging Snell's Law, we get incidence

$\sin \theta _{i}={\frac {n_{2}}{n_{1}}}\sin \theta _{t}$.

To find the critical angle, we find the value for $\theta _{i}$ when $\theta _{t}=$90° and thus $\sin \theta _{t}=1$. The resulting value of $\theta _{i}$ is equal to the critical angle $\theta _{c}$.

Now, we can solve for $\theta _{i}$, and we get the equation for the critical angle:

$\theta _{c}=\theta _{i}=\arcsin \left({\frac {n_{2}}{n_{1}}}\right),$

If the incident ray is precisely at the critical angle, the refracted ray is tangent to the boundary at the point of incidence. If for example, visible light were traveling through acrylic glass (with an index of refraction of approximately 1.50) into air (with an index of refraction of 1.00), the calculation would give the critical angle for light from acrylic into air, which is

$\theta _{c}=\arcsin \left({\frac {1.00}{1.50}}\right)=41.8{}^{\circ }$.

Light incident on the border with an angle less than 41.8° would be partially transmitted, while light incident on the border at larger angles with respect to normal would be totally internally reflected.

If the fraction ${n_{2}}/{n_{1}}$ is greater than 1, then arcsine 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 when ${n_{2}}/{n_{1}}$ is less than 1.

Refraction of light at the interface between two media, including total internal reflection.

A special name is given to the angle of incidence that produces an angle of refraction of 90˚. It is called the critical angle.

Derivation of evanescent wave

An important side effect of total internal reflection is the propagation of an evanescent wave across the boundary surface. Essentially, even though the entire incident wave is reflected back into the originating medium, there is some penetration into the second medium at the boundary. The evanescent wave appears to travel along the boundary between the two materials, leading to the Goos-Hänchen shift.

If a plane wave, confined to the xz plane, is incident on a dielectric with an angle $\theta _{I}$ and wavevector $\mathbf {k_{I}}$ then a transmitted ray will be created with a corresponding angle of transmittance as shown in Fig. 1. The transmitted wavevector is given by:

$\mathbf {k_{T}} =k_{T}\sin(\theta _{T}){\hat {x}}+k_{T}\cos(\theta _{T}){\hat {z}}$

If $n_{1}>n_{2}$, then $\sin(\theta _{T})>1$ since in the relation $\sin(\theta _{T})={\frac {n_{1}}{n_{2}}}\sin(\theta _{I})$ obtained from Snell's law, ${\frac {n_{1}}{n_{2}}}\sin(\theta _{I})$ is greater than one.

As a result of this $\cos(\theta _{T})$ becomes complex:

$\cos(\theta _{T})={\sqrt {1-\sin ^{2}(\theta _{T})}}=i{\sqrt {\sin ^{2}(\theta _{T})-1}}$

The electric field of the transmitted plane wave is given by $\mathbf {E_{T}} =\mathbf {E_{0}} e^{i(\mathbf {k_{T}} \cdot \mathbf {r} -\omega t)}$ and so evaluating this further one obtains:

$\mathbf {E_{T}} =\mathbf {E_{0}} e^{i(\mathbf {k_{T}} \cdot \mathbf {r} -\omega t)}=\mathbf {E_{0}} e^{i(xk_{T}\sin(\theta _{T})+zk_{T}\cos(\theta _{T})-\omega t)}$

and

$\mathbf {E_{T}} =\mathbf {E_{0}} e^{i(xk_{T}\sin(\theta _{T})+zk_{T}i{\sqrt {\sin ^{2}(\theta _{T})-1}}-\omega t)}$.

Using the fact that $k_{T}={\frac {\omega n_{2}}{c}}$ and Snell's law, one finally obtains

$\mathbf {E_{T}} =\mathbf {E_{0}} e^{-\kappa z}e^{i(kx-\omega t)}$

where $\kappa ={\frac {\omega }{c}}{\sqrt {(n_{1}\sin(\theta _{I}))^{2}-n_{2}^{2}}}$ and $k={\frac {\omega n_{1}}{c}}\sin(\theta _{I})$.

This wave in the optically less dense medium is known as the evanescent wave. Its characterized by its propagation in the x direction and its exponential attenuation in the z direction. Although there is a field in the second medium, it can be shown that no energy flows across the boundary. The component of Poynting vector in the direction normal to the boundary is finite, but its time average vanishes. Whereas the other two components of Poynting vector (here x-component only), and their time averaged values are in general found to be finite.

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Frustrated total internal reflection

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Phase shift upon total internal reflection

A lesser-known aspect of total internal reflection is that the reflected light has an angle dependent phase shift between the reflected and incident light. Mathematically this means that the Fresnel reflection coefficient becomes a complex rather than a real number. This phase shift is polarization dependent and grows as the incidence angle deviates further from the critical angle toward grazing incidence.

The polarization dependent phase shift is long known and was used by Fresnel to design the Fresnel rhomb which allows to transform circular polarization to linear polarization and vice versa for a wide range of wavelengths (colors), in contrast to the quarter wave plate. The polarization dependent phase shift is also the reason why TE and TM guided modes have different dispersion relations.

Applications

Mirror like effect

• Total internal reflection is the operating principle of optical fibers, which are used in endoscopes and telecommunications.
• Total internal reflection is the operating principle of automotive rain sensors, which control automatic windscreen/windshield wipers.
• Another application of total internal reflection is the spatial filtering of light.
• Prismatic binoculars use the principle of total internal reflections to get a very clear image.
• Some multi-touch screens use frustrated total internal reflection in combination with a camera and appropriate software to pick up multiple targets.
• Gonioscopy employs total internal reflection to view the anatomical angle formed between the eye's cornea and iris.
• A gait analysis instrument, CatWalk XT, uses frustrated total internal reflection in combination with a high speed camera to capture and analyze footprints of laboratory rodents.
• Optical fingerprinting devices use frustrated total internal reflection in order to record an image of a person's fingerprint without the use of ink.
• A Total internal reflection fluorescence microscope uses the evanescent wave produced by TIR to excite fluorophores close to a surface. This is useful for the study of surface properties of biological samples.

Examples in everyday life

Total internal reflection can be seen at the air-water boundary.

Total internal reflection can be observed while swimming, when one opens one's eyes just under the water's surface. If the water is calm, its surface appears mirror-like.

One can demonstrate total internal reflection by filling a sink or bath with water, taking a glass tumbler, and placing it upside-down over the plug hole (with the tumbler completely filled with water). While water remains both in the upturned tumbler and in the sink surrounding it, the plug hole and plug are visible since the angle of refraction between glass and water is not greater than the critical angle. If the drain is opened and the tumbler is kept in position over the hole, the water in the tumbler drains out leaving the glass filled with air, and this then acts as the plug vanished. Viewing this from above, the tumbler now appears mirrored because light reflects off the air/glass interface.

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See also

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• Goos-Hänchen effect
• Perfect mirror
• Snell's window

References

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• FTIR Touch Sensing
• Multi-Touch Interaction Research
• Georgia State University
• Total Internal Reflection by Michael Schreiber, Wolfram Demonstrations Project
• Total Internal Reflection – St. Mary's Physics Online Notes
• Template:Cite web

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