﻿ Focal length

# Focal length

The focal length is the distance between an optical lens or a curved mirror and the focus ( focal point). More specifically, it is the distance between the principal plane of the lens or of the mirror and the focal point. A converging lens focuses a parallel beam incident in the past for her focal point ( figure on the right, first image). For diverging lenses, the focus is off the lens, and a parallel incident beam is scattered, as if the individual beams all of this focus came (second picture ).

In a concave mirror, a parallel beam incident in the past before the mirror focal point converges (third picture). In a convex mirror, a parallel incident beam is scattered, as if the individual beams all of the behind the mirror focus ancestral ( fourth image ).

For a number of lenses and / or mirrors existing systems - such as lenses of cameras or microscopes - have defined analogously focal lengths. Here, let the positions of the principal planes (two per system ) specify not as easy as with a single lens ( in it) or a single mirror (on top of his head ).

The focal length is a concept of paraxial optics. They therefore only refers to radiation having a small angle and a small distance from the optical axis of the imaging system.

Large focal lengths caused by flat, slightly curved surfaces. Small focal lengths caused by sharp bends. Especially for the individual lenses of the reciprocal of the focal length or refracting power value is called. In converging lenses and concave mirrors, the focal length is defined as a positive value at diverging lenses and convex mirrors as a negative value.

The focal length is used in the application of the lens equation. In photography, the focal length of the lens determines, together with the recording format of the image angle ( see also form factor ). This also applies to the intermediate image in the microscope. With telescopes and binoculars, the focal lengths of the objective and eyepiece together determine the magnification.

## Power

The reciprocal of the focal length is called power. It is given for spectacle lenses in the derived SI unit dioptre.

## Measurement of the focal length

According to the mapping equation is in a sharp optical image by a thin lens of the reciprocal of the focal length is equal to the sum of the reciprocal values ​​of the object distance and the image distance:

This can be used to determine the focal length of the lens. If the subject of the picture is very far away, the relationship is particularly simple. The focal length is approximately equal to the image distance and can be read directly from the distance of the image from the lens.

A method that does not require a far object to is the auto-collimation. Here, the distant object is replaced by a plane mirror. The Bessel method for determining the focal length of thin lenses exploits the fact that two positions of the lens produce a sharp image at a fixed distance between object and image. From the distance between these two positions and the distance between object and image, the focal length of the lens can then be calculated.

For thick lenses and imaging systems with multiple optical components, the distance between the principal planes may not usually be neglected. Then estimating the magnification ratio can provide more accurate results. With the Abbe method, a sentence is taken from positions in which images the imaging system objects sharp. These points satisfy a linear equation. From the parameters of the straight line is the focal length and the position of the principal planes can be determined.

Optician determine the focal length aspheric lenses and over the surface varying refractive power of progressive lenses by wavefront analysis. In this case, usually a Hartmann- Shack sensor is used. The automated devices are called for historical reasons Lensmeter.

## Calculation of the focal length

### Breaking surface

A refractive surface is defined as the interface between two optical media having different refractive indices. If the light beam from the left, n is the refractive index on the left side, and n ' is the refractive index on the right side of the interface. The curvature of the interface is described by the curvature radius r. If the center of the circle describing the interface, on the side remote from the incident light side, r is positive, negative otherwise. A non- curved boundary surface has the radius of curvature.

The focal length of the other side is obtained by interchanging the indices of refraction as the light now coming from the right of n ' is excreted by n:

### Lens

The refraction of a lens of thickness d can be calculated from the openings of its two spherical interfaces. With the focal lengths and the two areas and their distance does

The image-side focal length of the lens. With the above equations, the focal surface is obtained with

The image-side lens focal length as a function of the radii of curvature, and the refractive indices n and n '. As in the illustration is the focal length f measured ' from the principal plane H'. Object -side and image-side focal lengths are the same size when the lens is bordered on both sides by media with equal refractive index n. The latter equation is also known as lens grinder formula.

### Thin lens

The approximation is satisfied for. This approach is referred to as a thin lens, and the main planes of the two boundary surfaces coincide (namely, to the median plane ). The equation of the focal length simplifies to

Being measured again away from the center plane.

In geometrical optics, ie front surface power and back surface power. The above equation can thus be in the form

. Write The optical effect of spectacle lenses is expressed by the lensmeter.

### System of two thin lenses

The system of two thin lenses is the system " lens of two refracting surfaces ," similar in principle ( see figure to the above it ). As to the object of and the image side focal length of each thin lens are the same size, applies

It is against Board and image-side focal length of the lens system also equal to:

Dependence of the focal lengths of the lens system of two thin lenses on the refractive indices and radii of curvature can be reached when applying for and the above lens grinder formulas for thin lenses.

When together of thin lenses. The spacing can be neglected. The focal length of such a system is approximately equal to

This equation is used, for example for two thin lenses cemented together. Such a double lens is usually made from two different types of glass, thus less aberration than in the case of consisting of only one type of glass can be achieved with the same lens focal length, such as the achromatic lens.

## Aberrations with direct connection to the focal length

The focal length is strictly defined only in the paraxial optics. However arise under certain conditions and especially for real non-parabolic lenses various so-called aberrations, resulting in an ( apparently partly ) changing focal length.

In the paraxial optics, it is always possible to approximate a spherical surface as a paraboloid. Real lenses are often designed as spherical surfaces, since they are easier to produce as aspherical surfaces. You will still be associated with a focal length that is actually valid only for rays close to the optical axis. For achsfernere rays shifted foci arise. This is a lens error is called spherical aberration.

Further, the focal distance depends on, inter alia, on the refractive index of the lens material, which in turn depends on the wavelength of light. When light falls of different wavelengths ( for example, white light), so this is dependent on the wavelength focused to a lens on various points. This is called chromatic aberration.

If the shape of a lens is not rotationally symmetrical with respect to the optical axis, but ellipsoidal, then fan-shaped focused light beam, depending on their orientation in different image sizes. Full light beams are not focused to a point but in two consecutive internal lines in the directions of two principal axes of the ellipsoid. This aberration is called axial astigmatism.

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