Abbe number

In optics and lens design, the Abbe number, also known as the Vd-number or constringence of a transparent material, is an approximate measure of the material's dispersion (change of refractive index versus wavelength), with high values of Vd indicating low dispersion.

It is named after Ernst Abbe (1840–1905), the German physicist who defined it.

The term Vd-number should not be confused with the normalized frequency in fibers.

This formulation only applies to the human vision.

Outside this range requires the use of different spectral lines.

For non-visible spectral lines the term "V-number" is more commonly used.

are the refractive indices of the material at three different wavelengths.

Abbe numbers are used to classify glass and other optical materials in terms of their chromaticity.

For example, the higher dispersion flint glasses have relatively small Abbe numbers

whereas the lower dispersion crown glasses have larger Abbe numbers.

Abbe numbers are used in the design of achromatic lenses, as their reciprocal is proportional to dispersion (slope of refractive index versus wavelength) in the wavelength region where the human eye is most sensitive (see graph).

For different wavelength regions, or for higher precision in characterizing a system's chromaticity (such as in the design of apochromats), the full dispersion relation (refractive index as a function of wavelength) is used.

An Abbe diagram, also called 'the glass veil', is produced by plotting the Abbe number

of a material versus its refractive index

Glasses can then be categorised and selected according to their positions on the diagram.

This can be a letter-number code, as used in the Schott Glass catalogue, or a 6 digit glass code.

Glasses' Abbe numbers, along with their mean refractive indices, are used in the calculation of the required refractive powers of the elements of achromatic lenses in order to cancel chromatic aberration to first order.

These two parameters which enter into the equations for design of achromatic doublets are exactly what is plotted on an Abbe diagram.

Due to the difficulty and inconvenience in producing sodium and hydrogen lines, alternate definitions of the Abbe number are often substituted (ISO 7944).

[3] For example, rather than the standard definition given above, that uses the refractive index variation between the F and C hydrogen lines, one alternative measure using the subscript "e" for mercury's e line compared to cadmium's F′ and C′ lines is This alternate takes the difference between cadmium's blue (C′) and red (F′) refractive indices at wavelengths 480.0 nm and 643.8 nm, relative to

Other definitions can similarly be employed; the following table lists standard wavelengths at which

is commonly determined, including the standard subscripts used.

[4] Starting from the Lensmaker's equation we obtain the thin lens equation by dropping a small term that accounts for lens thickness,

are the short and long wavelengths' refractive indexes, respectively, and

and regrouping, get The relative change is inversely proportional to

Refractive index variation for SF11 flint glass, BK7 borosilicate crown glass, and fused quartz, and calculation for two Abbe numbers for SF11.
Most of the human eye's wavelength sensitivity curve, shown here, is bracketed by the Abbe number reference wavelengths of 486.1 nm (blue) and 656.3 nm (red)
An Abbe diagram, also known as 'the glass veil', plots the Abbe number against refractive index for a range of different glasses (red dots). Glasses are classified using the Schott Glass letter-number code to reflect their composition and position on the diagram.
Influences of selected glass component additions on the Abbe number of a specific base glass. [ 2 ]