Thin lens

A thin lens is an optical lens is less thick. The curvature of its marginal area is small, so that the two surfaces are in close proximity.

In the paraxial optics -thin lens is a concept that after the last real thick lens is replaced by a plane. The two breaking operations of a light beam at the interfaces are combined into a crushing process in this plane. If the actual lens has a symmetrical cross-section ( for example, bi-convex or bi-concave ), its central plane is used for breaking plane. A light beam which is incident on the center of the plane, is at an unchanged direction and parallel misalignment without passing through the lens.

The concept of the thin lens is an idealization of the last real thick lens and a good approximation for large focal lengths. The formal reduction to a layer does not mean that the lens has not refractive index no or curved boundaries, because of both depends on their focal length.

Optical imaging by means of a thin lens

The two main levels and a general optical system coincide in their spare plane ( midplane ) in the thin lens. The thin lens has only one main level. Both focal lengths and are measured from the main plane of. The ray optical design of the optical imaging is somewhat simpler and is as follows.

The image point B ' with the help of two of the three major rays parallel beam ( 1), center beam ( 3) or focal beam ( 2) ( Figure 1 from top to bottom, numbering refers to Figure 2), run out from the object point B, found. The beams are only once - namely on the center / main level - broken ( jet ( 3) remains unbroken ). The parallel beam is refracted such that it passes through the image-side focal point F '. The focal point of the beam passes through the object-side focal point F and is broken at the mid-plane such that it becomes a parallel beam on the image side.

Theory for thick lenses

In the following, the theory of the thick lens is represented and derived by a limiting transition to a thin lens, the lens equation.

A (thick ) lens is made of a transparent material having refractive index. The lens forms with its surroundings two interfaces. Normally, the ambient air and thus the refractive index. A beam of light coming from the left ( see Figure 2) is broken at each of the two interfaces according to Snell's law of refraction. The two openings of the light beam are to compute the focal length of a lens viewed successively.

Refraction at a curved surface

First, the refraction of the light beam is on the left interface seen ( Figure 3).

Pure geometric applies

For paraxial rays, the angles are small and it is approximated well. Moreover applies. This results in

By Snell's law, as well as results

The image-side focal length of a curved interface for paraxial rays thus depends only on the refractive indices of the two materials, and the curvature of the surface. If we suppose now that the light is coming from the right, runs to the left and then broken, so one can apply the above formula also. The refractive indices in the formula must now be exchanged, since the light beam from the medium runs after coming with.

Is called the object side focal length.

It can also now be a relationship between the image distance and the object distance derived.

There are the following angular relationships: as well. Therefore, in the small angle approximation for the Snelliusche refraction law applies:

Also, applies

Insertion into the small angle approximation of Snell's law, all approximations are treated as equations yields,

The right hand side can now be expressed in terms of the above- derived Focal length:

Refraction at a thick lens

The refraction of paraxial rays in a lens is equivalent to two successive refractions at curved interfaces. It is believed that the light is incident from the left. If it is a convex lens (as shown in the upper diagram), the radius of curvature of the first interface of the radius of curvature of the second interface, however, is positive, negative. For a diverging lens ( concave lens ) the situation is reversed.

For the second refractive taking the image of the first refraction as the subject. It is the object distance to the first refraction and the object distance of the second opening. In the same agreement applies to the image distances and the radii of curvature.

For the image width according to the first refractive applies:

Now it is the image distance as the object distance in the formula for the second refraction one: with the thickness of the lens ( as shown in the first figure given). The shift must be performed, since the derivation is assumed above assumes that the curved surface passing through the origin. Of course, the refractive indices must be reversed. For the second refraction applies:

Inserting results

Addition of the two refractions results:

This will be counted from the left side of the lens, and from the right side of the lens.

Approximation for thin lens

One can now introduce the object distance and the image distance for the entire lens: . It refers to a lens as thin if and valid. Thus, the above formula simplifies to

This is the well-known lens equation and describes the imaging paraxial rays with a thin lens. For paraxial rays ( the object is at infinity ). Now On the image side, the rays must pass through the focus. Therefore, the focal distance of a thin lens:

For a bi-convex lens with

Insertion into the imaging equation yields the Abbbildungsgleichung thin lenses

Now In the end, the lens-grinder formula was derived:

• Paraxial optics