Abbe number

In optics and lens design, the Abbe number, also known as the V-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 V indicating low dispersion. It is named after Ernst Abbe (1840–1905), the German physicist who defined it. The term V-number should not be confused with the normalized frequency in fibers.

The Abbe number,[1] ${\displaystyle V_{\mathsf {d}}\ ,}$ of a material is defined as

${\displaystyle V_{\mathsf {d}}\equiv {\frac {n_{\mathsf {d}}-1}{\ n_{\mathsf {F}}-n_{\mathsf {C}}\ }}\ ,}$

where ${\displaystyle \ n_{\mathsf {C}}\ ,}$ ${\displaystyle \ n_{\mathsf {d}}\ ,}$ and ${\displaystyle \ n_{\mathsf {F}}\ }$ are the refractive indices of the material at the wavelengths of the Fraunhofer's C, d, and F spectral lines (656.3 nm, 587.56 nm, and 486.1 nm respectively). 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. The more general formulation defined as,

${\displaystyle V\equiv {\frac {n_{\mathsf {center}}-1}{n_{\mathsf {short}}-n_{\mathsf {long}}}},}$

where ${\displaystyle \ n_{\mathsf {short}}\ ,}$ ${\displaystyle \ n_{\mathsf {center}}\ ,}$ and ${\displaystyle \ n_{\mathsf {long}}\ ,}$ are the refractive indices of the material at three different wavelengths. The shortest wavelength's index is ${\displaystyle \ n_{\mathsf {short}}\ ,}$ and the longest's is ${\displaystyle \ n_{\mathsf {long}}~.}$

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 ${\displaystyle \ V<55\ }$ whereas the lower dispersion crown glasses have larger Abbe numbers. Values of ${\displaystyle \ V_{\mathsf {d}}\ }$ range from below 25 for very dense flint glasses, around 34 for polycarbonate plastics, up to 65 for common crown glasses, and 75 to 85 for some fluorite and phosphate crown glasses.

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.