# Refraction

In Physical of the Wave S — in particular in Optical, Acoustic and Seismology — the phenomenon of refraction is the deviation of a Onde when the speed of this one changes between two mediums. Typically, that occurs with the interface between two mediums, or during a change of Impédance of the medium.

One can represent such a wave by two approaches:

• by its Wave front: it is the line which a wave in water describes (undulatory Optique and Sismologie);
• by a ray: it is the direction of propagation wave, perpendicular to the wave front (geometrical Optique).

The two models are equivalent in the case of the refraction, however one will prefer the first for to explain the phenomenon, and the second for the to quantify .

## Description

The Lumière is deviated when it passes from a transparent medium to another (for example: Air with the Water, or the opposite…). It is this phenomenon which one observes when one looks at a straw in glass: this one appears broken. This apparent fracture is at the origin of the word refraction .

The light known as “is refracted” and the property which characterizes the various transparent mediums is the “refringency”, which results in a numerical value: the “Index of refraction”.

## Geometrical approach: Law of Snell-Descartes

Only the second Loi of Snell-Descartes relates to the refraction. Each transparent medium is characterized by its Index of refraction noted ni . One calls Dioptre surface separating the two mediums.

The laws of the refraction, stated by Snell and Descartes, make it possible to give an account of the phenomenon quantitatively. For the refraction, the Lois of Snell-Descartes specify that:

• the refracted ray is located in the plan of incidence (defined by the incidental ray and the normal in the diopter in the movement of incidence), incidental ray and refracted ray being on both sides normal;
• the angles of incidence and refraction (θ1) and (θ2), measured compared to the normal are such as:

One can then notice that:

• Plus the index of refraction n2 is large, plus the refracted ray approaches the normal, and vice versa ;
• When the index of refraction n2 is smaller than n1 (for example: glass with the air), one can exceed an incidence known as “critical angle” beyond which there is total Réflexion.

### Interpretation of the experiment of the “broken pencil”

The explanation of the experiment of the broken pencil rests on two important points: the Laws of Snell-Descartes, and the property of approximate Stigmatisme of the plane Diopter permitted by the eye, which intercepts only one fine brush of refracted light.

It is noted whereas, the rays being refracted while deviating from the normal, since the index of the air is lower than the index of the water, the light which arrives in the eye of the observer seems to come from a higher point.

The diagram opposite illustrates for a point of the end of the pencil. It would be necessary to make thus for each point to have the image (within the meaning of the approximate stigmatism) of the pencil. (The effect of a diopter is also to give a deformed image).

One can state certain note:

• the image of the end of the pencil is not with the vertical of that of the pencil itself (contrary to certain fast schematizations);
• the image of the end of the pencil depends on the position of the observer (immediate consequence of nonthe stigmatism of the Dioptre).

### Angle limits refraction

If N 1 > N 2 (for example drainage duct towards the air), then according to the law of Snell-Descartes:
$n_1 \ sin \left\{\ theta_1\right\} = n_2 \ sin \left\{\ theta_2\right\} \,$
thus:
$\ sin \left\{\ theta_1\right\} = \ frac \left\{n_2\right\} \left\{n_1\right\} \ sin \left\{\ theta_2\right\}$

For values of sin (θ2) close to 1, i.e. for shaving incidences (incidental ray near to surface), the law of Snell-Descartes gives a value of sin (θ1) higher than 1. One leaves indeed his field of validity: that corresponds to situations where there is no refraction but only of the reflection, one speaks about “total Réflexion”.

The limiting angle of refraction is thus:

$\ theta_ \left\{lim\right\} = \ mathrm \left\{Arcsin\right\} \ left \left(\ frac \left\{n_2\right\} \left\{n_1\right\} \ right\right) \,$

This property is made profitable in certain reflecting systems (prism with total reflection) and the fiberoptics.

### Construction of Descartes

The relation of Snell-Descartes can be translated geometrically. This allows a simple geometrical construction (known as of Descartes) of the refracted ray.

This construction rests on the layout of the “circles of the indices”. One traces the two circles of radius $\ rho_1 = n_1$ and $\ rho_2 = n_2$ centered on the point of incidence ( I ). The incidental ray (coming from medium 1) is prolonged in medium 2 and cuts circle 1 in a point has whose projection H is such as, by construction, IH = $n_1 \ sin i$.

To satisfy the relation of Snell-Descartes, the refracted ray must cut circle 2 in a point B having even projection. It is thus enough to prolong the line (AH) until its intersection with circle 2.

## Undulatory approach: Principle of Huygens-Fresnel

The celerity of the light is not the same one in the two mediums. This change of value is enough to interpret the change of management of the wave. It is Christiaan Huygens which, the first, gave a model, by associating the light propagation with the propagation of a wave front.

### The principle of Huygens-Fresnel

The Principe of Huygens-Fresnel stipulates that with an interface, all the points reached by a wave coming from a first medium re-emit a wave in the second medium. One can then interpret the refraction like the deviation of the wave front related to the lower speed (or more rapid) of these re-emitted waves.

Huygens — being thus opposed to Newton — considered that the light was a Onde, being propagated gradually in the transparent mediums. He imagined the wave front like the superposition of ondelettes, so that in the passing of a diopter, celerity being different on both sides, the size of the ondelettes was changed as much and the consequently deviated face. The report/ratio of the indices of the mediums seems then simply the report/ratio of celerities:

$\ frac \left\{n_1\right\} \left\{n_2\right\} = \ frac \left\{v_2\right\} \left\{v_1\right\}$

One can also use this same principle to give an account of the reflection (it is indeed enough to consider the part of the ondelettes spreading itself in the first medium) and of the Diffraction.

### Construction of Huygens of the refracted ray

This interpretation also allows a geometrical construction. This one is similar to that of Descartes, but it is based on the comparison of celerities.

The rays to be traced are then in 1/n1 and 1/n2 and the geometrical reasoning rests on the common intersection of the plans of wave (not B ), which, by nature must be tangent with the ondelettes.

The largest ondelette corresponds on the figure to the position of the wave front if there were no diopter (here n2 > n1), while the smallest circle thus corresponds to the face of the diffracted wave.

The refracted ray is thus well according to (IC) ( I being the point of incidence).

## Approaches the “less course”: principle of Fermat

A particularly astonishing aspect is the possibility of also interpreting these laws of Snell-Descartes in term of less course, and more precisely in term of less time.

It is Fermat which introduced this interpretation, source all at the same time for him of fundamental questionings on the “reason” of this least course, and a very powerful theoretical approach known as of less action.

Statement of the Principle of Fermat:

the light is propagated of a point to another on trajectories such as the duration of the course is stationary . ”

There still, a “mechanical” analogy can help to include/understand why lasted of course and broken trajectory are closely dependant.

Let us consider an athlete now having to start from a point of the beach and as quickly as possible to join a buoy located in water. There still, the athlete runs more quickly on the beach than it does not progress in water. Consequently, it is thus advisable not to go in straight line towards the buoy, but to lengthen the distance covered on sand (and to decrease that to traverse in water). But obviously, one should not either too much lengthen on sand…

One can then seek which is the way which corresponds at least of time. It is a way such as the point of arrival at the edge of water is neither the intersection with the straight line, nor the case where the distance in water is weakest (while swimming perpendicular to bank) but a point between the two, and which is such as:

$\ frac \left\{1\right\} \left\{v_1\right\} \ sin \left\{\ theta_1\right\} = \ frac \left\{1\right\} \left\{v_2\right\} \ sin \left\{\ theta_2\right\}$

One finds the expression of the law of the refraction.

### Chemin optics

When a ray traverses a distance D in a medium of index N , one calls optical Chemin, and one notes L , the product of the distance and the index:

$\ mathcal \left\{L\right\} = n.d = D. \ frac \left\{v\right\} \left\{C\right\}$

If a ray changes medium and traverses a distance D 1 in a medium of index N 1 and a distance D 2 in a medium of index N 2, then the traversed optical way is:

$\ mathcal \left\{L\right\} = n_1.d_1 + n_2.d_2 \,$

It is noticed whereas the way that traverses a ray to go from a point to another always corresponds to a extrémum of L (minimum or sometimes maximum): straight line in a medium given, and refraction according to the law of Snell-Descartes when it changes medium. It is what one calls a Principe of less action.

Note that it is of a observation, a consequence, and not a cause. The luminous ray does not have strategy, it does not decide to borrow such or such way, and the point of arrival is not given in advance! But this principle is very powerful and can be generalized with many approaches of physics. In optics, it makes it possible to calculate the way in a variable medium of index.

#### Mediums of variable index

One up to now considered homogeneous and isotropic mediums, in which speed of light was the same one everywhere and in all the directions. But there exist mediums in which speed of light, therefore the index of refraction, varies in a continuous way, for example air.

If the ground is hot, then the temperature of the air decreases when one rises in altitude. The density of the air varies, and speed of light, therefore the index, also (Gradient of index); thus the bitumen in very hot weather deforms the images or reveals imaginary water puddle pools reflecting the sky (concavity to the top of the luminous way) and that one can see a Oasis in the desert although it is behind a dune (concavity downwards in this case), but the term of “Mirage” also applies to the effect of the sun on the imagination of the traveller.

Another common experiment consists to take an aquarium filled with water and to put salt at the bottom: the salt concentration is more important at the bottom than on the surface, and the index of refraction varies according to this concentration; can a laser beam passing in an aquarium containing a little fluorescein then give (??) curve (and either a straight line).

Lastly, in the fiberoptics, one voluntarily makes vary the index of refraction according to the distance compared to the center of fiber; in this case the variation of index is used “to trap” the luminous ray which undulates and follows fiber rather than to reflect itself on the edges.

In these mediums, the index N thus depends on the point considered, N is a function of the position ( X , there , Z ) (See the function Gradient).

#### Chemin optics total

The way of the luminous ray is a curve C in the medium. Let us consider a small way of the point S at the point s+ds on which the index can be regarded as constant ( S is the curvilinear X-coordinate on C , i.e. the distance covered while following the curve since the starting point). The optical way is locally:

$d \ mathcal \left\{L\right\} = N \ left \left(S \ right\right) ds$

the total optical way is thus:

$\ mathcal \left\{L\right\} = \ int_ \left\{C\right\} D \ mathcal \left\{L\right\} = \ int_ \left\{C\right\} N \ left \left(S \ right\right) ds$

According to the principle of less action, the way followed by the luminous ray corresponds to that which with the minimal value L . This makes it possible to calculate the trajectory of the ray.

## Refraction of the mechanical waves

In a general way, and thus in mechanics, the propagation follows the same fundamental laws, in particular the fact that celerity depends only on the medium: its elasticity and its inertia. The phenomena of reflection, refraction, diffraction and interferences thus exist also for these waves. According to the number of space size offered by the medium to the propagation, whole or part of these phenomena are visible.

Thus for a unidimensional propagation (a wave along a cord for example) it is easy to observe the reflection, and possible to try out the transmission (with partial reflection) between two cords of different linear densities. For the waves at water surface, the phenomena of reflection, refraction and diffraction are easy to observe. As for the acoustic waves which surround us, their propagation in three dimensions to our ear is very often the fruit of all these phenomena at the same time.

### Waves at water surface

The equations governing the waves in water are not linear. The celerity of the waves of surface depends on the depth of the liquid, the current velocity and of their amplitude. In particular celerity is lower if the depth is lower. This makes it possible to have the first interpretation owing to the fact that the peaks of the waves become almost parallel to the beach when they approach the shore: the part of the deeper water peak is propagated more quickly than the part out of not very deep water and the peak turns towards the beach.

This change of celerity is thus precisely the cause of the change of management of propagation of a plane wave as the principle of Huygens evoked explains it above.

Observation of this phenomenon perhaps realized on a “tank with waves”: a flat vat contains a low height of water (about the centimetre). On part of the bottom of the vat, one places a plate which thus causes an abrupt variation depth. A plane wave (caused by the vibration of a bar which levels water) is then deviated in the passing of this diopter.

### Refraction of the sound waves

The sound waves undergo also such a deviation. In the atmosphere the speed of sound varies with the pressure and the temperature (thus altitude), the moisture and the speed of the wind. The phenomenon of variation in the temperature with altitude, called Gradient of temperature, causes to curve the sound rays upwards in normal weather, i.e. when the temperature decreases with altitude, and downwards at the time of a inversion of temperature. For this reason the sound goes up the slopes, phenomenon often audible in mountain.

In the same way when the speed of the wind increases with altitude the sound rays are refracted downwards in the direction of the wind, and upwards in the direction opposed to the wind. Therefore the wind “carries the sound”. It is the variation of the wind with the altitude (the gradient of wind) which is important and not the speed of the wind it even (much weaker than the speed of sound).

### Refraction of the seismic waves

The propagation velocity of the seismic waves depends on the density, therefore the depth and of its composition. It thus occurs:

• a refraction with the transition between two geological layers, in particular between the coat and the core;
• in the coat, a deviation: it is a medium with variable indices. bojour

## Refraction of the radio waves

As a luminous ray is deviated when it passes from a medium of index of refraction N 1 to another of index N 2, a radio wave can undergo a change of management depending at the same time on its frequency and the variation of the index of refraction. This phenomenon is particularly important in the case of the ionospheric propagation, the reflection which a decametric wave in the ionosphere undergoes is in fact a continuous succession of refractions. It is possible to reproduce with a radio wave of which the wavelength is of a few centimetres to a few decimetres the phenomenon observed with a lens or a prism in traditional optics.

## Optical observations

• Corsica seen of Nice by optical phenomenon of refraction (scientific explanations)

## See too

### External bonds

• Simulation of the ondelettes of Huygens;
• Demonstration in Java

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