Diopter

In Optical, a diopter is a surface separating two transparent mediums from different indexes of refraction. If the light is propagated in straight line in a homogeneous and isotropic medium, it is deviated at the time of the passage of a diopter: there is Réfraction.

In a general way, there are at the same time refraction and reflection: part of the light is thought of surface of the diopter and the other part is refracted at the time of its passage in the other medium.

The change of management on the level of the diopter is described by the Lois of Snell-Descartes which found the geometrical Optique. These laws can be represented graphically by applying them to a single ray - known as incident - intercepting the diopter in a point does not say an incidence. To include/understand the effect of a diopter on the light, it is necessary to consider a minimal number of rays in order to represent the Faisceau light.

Plane diopter

Question of stigmatism

One of the consequences of the laws of Snell-Descartes is that the plane diopter is a not-stigmatic system. The illustration above shows that the light resulting from a point placed in an aquarium, for example, gives rays refracted in the air which have directions without common point.

However, when a fish is looked at, it well is seen! It is thus that the eye of fish, for example, constitutes a luminous object which forms an image on the retina of the eye of the observer. This is not possible that because the beam of light is sufficiently narrow so that the spot on the retina seems a point. One is well then in a case of approximate stigmatism.

It is this phenomenon which makes it possible to explain the experiment of the “broken stick” that one shows in general to illustrate the Réfraction.

Limiting refraction and total reflection

It is seen that if N 1 > N 2 (for example drainage duct towards the air), then for values of sin (θ1) close to 1, i.e. for shaving incidences (incidental ray near to surface), one obtains by this Formula One a value of sin (θ2) higher than 1! This is obviously impossible, that corresponds to situations or there is no refraction but only of the reflection; one speaks about total reflection

The limiting angle of refraction is thus such as:

sin (θlim) = n2/n1

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

Spherical diopter

General case

The application of the laws of Snell-Descartes also makes it possible to treat the nonplane case of the diopters. It is enough to consider the normal with the diopter locally not incidence of each ray contributing to the beam.

Again, by geometrical construction, one notes that the spherical diopter is not stigmatic, except obviously for his center, since each ray arriving perpendicular to the diopter is not deviated. The image of the center is then the center itself. (In fact, it is also stigmatic for two other particular points of the optical axis, called points of Weierstrass).

Spherical diopter under the conditions of Gauss

When a weak part of the diopter is used or, another way of saying, when the radius of curvature is very large in front of dimensions related to the object (size, distance), one can be placed under the conditions known as of Gauss: it is not considered whereas the rays which pass close to the axis and which are not very tilted. The mathematical consequence is the possibility of comparing the sines to the value of the angles (in radian) and the physical consequence is that one is then under the conditions of an approximate stigmatism: consequently, with a point object, one can associate a point image.
This is particulièremet important for the manufacture of the lenses (see hereafter).

In particular, one can define a hearth, image of one object ad infinitum, i.e. otherwise, convergence point (or of divergence) of an incidental beam parallel with itself and parallel with the axis. And more generally, one can write a Relation of conjugation between a point has axis and his A' image given by the diopter.

The spherical diopters are then represented in a conventional way:

The relation of conjugation below makes it possible to define the positions of the hearths. According to the curve (concave/convex) and according to the report/ratio of the indices (n'> N or < N) the hearths are real or virtual.

For a point has on the axis (directed), the point A' image is such as: \ frac {} {\ bar {SA'}} - \ frac {N} {\ bar {SA}} = \ frac {- N} {\ bar {SC}} and gamma= \ frac {A' B} {AB}

Applications

The applications are, in a general way, the optical instruments. Those consist of objects refracting which have necessarily at least 2 faces. Schematically, one can consider the association of two nonparallel plane diopters: there is then a prism, whose properties of réfexion total or dispersion make of it an object largely used. And to in addition consider the association of 2 diopters of which one (at least) is not plane: the lenses, for which one naturally finds the conditions of Gauss to be able to associate an image with an object.

See too

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