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 a material's dispersion (change in refractive index as a function of wavelength), with high Vd values 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.

The Abbe number ${\displaystyle V_{\text{d}}}$ of a material is defined as: $${\displaystyle V_{\text{d}}\equiv {\frac {n_{\text{d}}-1}{n_{\text{F}}-n_{\text{C}}}},}$$ where ${\displaystyle n_{\text{C}}}$, ${\displaystyle n_{\text{d}}}$, and ${\displaystyle n_{\text{F}}}$ 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 human vision; outside this range, alternative spectral lines are required. For non-visible spectral lines, the term "V-number" is more commonly used. The more general formulation is $${\displaystyle V\equiv {\frac {n_{\text{center}}-1}{n_{\text{short}}-n_{\text{long}}}},}$$ where ${\displaystyle n_{\text{short}}}$, ${\displaystyle n_{\text{center}}}$, and ${\displaystyle n_{\text{long}}}$ 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 ${\displaystyle V}$ less than 55, whereas the lower dispersion crown glasses have larger Abbe numbers. Values of ${\displaystyle V_{\text{d}}}$ 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 useful in the design of achromatic lenses, as their reciprocal is proportional to dispersion (slope of refractive index versus wavelength) in the domain where the human eye is most sensitive (see above figure). For other wavelength regions, or for higher precision in characterizing a system's chromaticity (such as in the design of apochromats), the full dispersion relation is used (i.e., refractive index as a function of wavelength).