# Main level (optics)

Figure 1: Main planes and main points in a relatively thick lens : In the working model paraxial optics , the rays at the interfaces are continued unbroken (dashed) up to the respective main plane and only shown there broken (center ray 3 only offset parallel).

Main planes are two planes defined in an imaging system in the working model paraxial optics , in which the refractions of the light rays are assumed in a simplified manner. In the space between the main levels, the light rays are thought to run parallel to the optical axis . The advantage of the main levels is that a multi-lens system, e.g. B. a lens , can be expressed by just a thin lens with only one equivalent focal length and two main planes. The main planes can generally be calculated using the matrix optics.

The main planes are the reference planes for distance measurements , one in object space , the other in image space . Corresponding distances are the focal length and the object and image distance, which are linked to one another in a simple manner (linear equation) in the lens equation .

A thin lens and a slightly curved spherical mirror largely meet the requirements for paraxial optics and the application of the lens equation. The tangential planes at the vertices of the surfaces can already be assumed to be the reference planes necessary for their imaging properties. Since the two surfaces almost coincide with a thin lens, one speaks here of only one main plane assumed in the middle.

When working with the lens equation, with the exception of chromatic aberration, all other aberrations are not taken into account. The greater the angle between the rays and the optical axis, the more pronounced these are. Therefore, the concept of paraxial optics, in which the main levels belong, can only be extended to compact optical systems if the lens errors are corrected with the help of more complex working models. This interpretation of paraxial optics is called Gaussian optics .

The points of intersection of the main planes with the optical axis are the main points , which are designated with H (object-side) and H '(image-side main point) (in the adjacent figure H 1 and H 2 ).

In the case of complex imaging systems, the main plane on the image side can also be "in front of" the object-side. An afocal lens system has no main planes, or these are at infinity .

## Main plane construction for a lens

Figure 2: Construction of the image-side focal point F 'and the image-side main plane H'
Lens consists of the refractive surfaces H 1 and H 2
1. The focal point F 'is the mapping of the focal point F' 1 of the area H 1 through the area H 2 . An axially parallel ray (red in the adjacent figure) entering the lens passes the focal point F 'on the image side.
2. The concept of the main levels means that the image scale between them is 1. A point in H 1 has the same distance from the optical axis in H ': h' = h.

The main plane H 'contains the intersection of the axially parallel incident beam (red) deflected by the lens into the focal point F' with the undeflected elongated axially parallel incident beam (broken line in the adjacent figure).

${\ displaystyle a '= {\ overline {H_ {2} H'}} = {\ frac {f '_ {2} d} {d-f' _ {1} -f_ {2}}}}$ .

The same applies to the property side:

${\ displaystyle {\ overline {H_ {1} H}} = {\ frac {f_ {1} d} {d-f '_ {1} -f_ {2}}}}$ .

If the lens is relatively thin (  when the thin lens is by definition ), these distances are zero. The main planes remain on the vertices of the surfaces that coincide in one plane . ${\ displaystyle d << f_ {1}, f_ {2}}$${\ displaystyle d = 0}$${\ displaystyle H}$

## Main plane construction for a system of two thin lenses

Figure 3: System of two thin lenses (H 1 and H 2 )
Construction of the image-side focal point F 'and the image-side main plane H'

The construction is analogous to that for a lens (see above). It is only to be noted that the main planes H 1 and H 2 each represent two refractive surfaces and that each has two equally large focal lengths (object and image side). It is also possible for the two thin lenses to consist of different materials (n ' 1 ≠ n' 2 ).

The two equations above change slightly (but with fundamentally different expressions for the focal lengths ) to: ${\ displaystyle f}$

${\ displaystyle a '= {\ overline {H_ {2} H'}} = {\ frac {f_ {2} d} {d-f_ {1} -f_ {2}}}}$ ,
${\ displaystyle {\ overline {H_ {1} H}} = {\ frac {f_ {1} d} {d-f_ {1} -f_ {2}}}}$ .

When the thin lenses ( ) move closer together, the distances also approach zero. The main plane H 1 or H 2 or a plane H assumed instead of both represents the system of, for example, two thin lenses cemented together . ${\ displaystyle d << f_ {1}, f_ {2}}$

By combining the individual constructions described with the help of Figures 2 and 3, one finds the main plane construction for a system of two lenses with a distinct thickness (“ thick lens ”).

## Notes and individual references

1. The next more precise and complex working model is geometric optics
2. Heinz Haferkorn: Optics: Physikalisch-Technischen Basics and Applications , Barth, 1994, ISBN 3-335-00363-2 , pp. 198 and 207.
3. Heinz Haferkorn: Optics: Physikalisch-Technischen Basics and Applications , Barth, 1994, ISBN 3-335-00363-2 , p. 200. In contrast to oat grain, the focal lengths are written here without a sign.