# Abbe number

In the Abbe diagram, the Abbe number is plotted against the refractive index.

The Abbe number , also Abbe's number , is a dimensionless quantity to characterize the optical dispersive properties of optical glasses , i.e. how much their refractive index changes with the wavelength of light. The greater the relative dispersion of the glass, the smaller its reciprocal, the Abbe number.

The Abbe number is named after the German physicist Ernst Abbe . As a formula symbol is common. ${\ displaystyle \ nu}$

## Basics

The strength of the refraction of light depends on its wavelength (one can also say: on the light color). This is known as dispersion and is, for example, the reason for the splitting of a white light beam at a prism into a colored spectrum . All materials used for optical components (e.g. lenses ) exhibit a more or less strong dispersion. Optical devices such as camera lenses therefore generally display the various colors of light differently, which can be shown by colored edges on the images of object edges or colored halos around bright light sources. This aberration is called chromatic aberration .

For a complete description of the dispersion of a material (e.g. a type of glass ), it is indicated how the refractive index of the material changes when the frequency or wavelength of the light varies . The function or must therefore be specified. To do this, you can either specify the refractive index at different specified wavelengths or a set of coefficients of a dispersion formula such as the Cauchy or Sellmeier equation . For simple calculations, however, it is often sufficient to describe the dispersion in the range of visible light by a single parameter, namely the Abbe number. ${\ displaystyle n}$${\ displaystyle f}$${\ displaystyle \ lambda}$${\ displaystyle n (f)}$${\ displaystyle n (\ lambda)}$

## definition

Wavelength in nm Fraunhofer
line
Light source colour
365.0146 i Ed UV
404.6561 H Ed violet
435.8343 G Ed blue
479.9914 F ' CD blue
486.1327 F. H blue
546.0740 e Ed green
587.5618 d Hey yellow
589.2938 D. N / A yellow
643.8469 C ' CD red
656.2725 C. H red
706.5188 r Hey red
768.2 A ' K red
852.11 s Cs NIR
1013.98 t Ed NIR

The dimensionless Abbe number is defined as ${\ displaystyle \ nu}$

${\ displaystyle \ nu = \ nu _ {d} = {\ frac {n _ {\ rm {d}} - 1} {n _ {\ rm {F}} - n _ {\ rm {C}}}}}$ (old definition)

or

${\ displaystyle \ nu _ {e} = {\ frac {n _ {\ rm {e}} - 1} {n _ {\ rm {F '}} - n _ {\ rm {C'}}}}}$ (new definition) ,

where n d , n F etc. are the refractive indices of the material at the wavelengths of the corresponding Fraunhofer lines . The table opposite lists the wavelengths of some of these Fraunhofer lines for which the refractive index is usually determined. For example, n d is the index of refraction at a wavelength of 587.6 nm.

A material with low dispersion has a high Abbe number. The reciprocal value of the Abbe number is also known as the relative dispersion .

The typical values ​​of the Abbe numbers for the most frequently used types of glass range from approx. 20 ( flint glass ) to 60 ( crown glass ). The limit for the designation of glasses as flint glass or crown glass is an Abbe number of 50.Special types of glass (fluorite crown glass) have indicators around 85. Magnesium fluoride even reaches an Abbe number of 95, so its dispersion is particularly low.

## application

Influences of the addition of selected glass components on the Abbe number of a special base glass.

The Abbe number is important when designing achromatic lenses . These are lens systems in which the chromatic aberration is particularly small. For example, an achromatic lens consisting of two thin lenses that are a short distance apart has the same focal length for Fraunhofer lines F and C, if

${\ displaystyle \ nu _ {1} \, f_ {1} + \ nu _ {2} \, f_ {2} = 0}$

is, where are the Abbe numbers and the focal lengths of the two lenses. Such a lens system images blue and red light of the wavelengths 486 nm (F) and 656 nm (C) onto the same point. The remaining color error is significantly less than if only one type of glass had been used. Achromatic lenses were the basis for the construction of large lens telescopes in the 19th century. ${\ displaystyle \ nu _ {i}}$${\ displaystyle f_ {i}}$

The color error still remaining with the achromatic lens (the so-called secondary spectrum ) is often expressed in a purple border around light objects. In order to reduce the chromatic aberration even further, the refractive index must be taken into account for more than two wavelengths (if the light is mapped onto a point at three wavelengths, you have an apochromat ). Nevertheless, the Abbe number helps to roughly classify types of glass.

In the infrared and ultraviolet range , the Abbe number, which is defined for wavelengths in the range of visible light, is unsuitable.

## literature

• Gottfried Schröder: Technical optics , Chapter 2.1.1: Optical glasses , page 23, 5th edition, Vogel-Buchverlag, Würzburg, 1986, ISBN 3-8023-0067-X