Abbe number

The Abbe number, or V-number of a glass (not to be confused with the fiber V-number), condenses how strongly the refractive index changes across the visible into a single figure:

where $n_d$, $n_F$ and $n_C$ are the indices at the helium d line (587.6 nm), the hydrogen F line (486.1 nm) and the hydrogen C line (656.3 nm). The numerator measures how strongly the glass bends light and the denominator how much that bending varies from blue to red. A high Abbe number means low dispersion.

Typical values

N-BK7 has $n_d$ = 1.51680, $n_F$ = 1.52238 and $n_C$ = 1.51432, giving $V_d$ = 64.17 in the Schott catalogue. Optical glasses are divided at about $V_d$ = 50 (55 for glasses with $n_d$ below 1.60) into crowns, above, and flints, below. Fused silica has an Abbe number near 68, calcium fluoride about 95, dense flints go into the low 20s, and many optical polymers sit between 30 and 58. A glass is usually identified by a six-digit code combining the two numbers, 517642 for N-BK7: $n_d$ = 1.517 and $V_d$ = 64.2. Glass manufacturers plot their catalogue as a glass map of $n_d$ against $V_d$, and designers choose pairs from it.

A variant, $V_e$, uses the mercury e line at 546.1 nm and the cadmium F′ and C′ lines at 480.0 and 643.8 nm, and is common in European specifications; the two differ slightly for the same glass.

Why it matters

The axial chromatic aberration of a thin singlet between F and C is about $f/V$, 0.78 mm for a 50 mm N-BK7 lens. An achromatic doublet pairs a crown and a flint with widely different Abbe numbers, so that a strong positive crown and a weaker negative flint cancel their colour errors while keeping net power; the greater the difference in $V$, the weaker the individual elements need to be. Correcting the residual secondary spectrum needs glasses with anomalous partial dispersion, which the Abbe number alone does not describe.

The Abbe number describes the visible only. For infrared optics the same ratio is formed from indices at infrared wavelengths, and for fiber and waveguide work the relevant quantity is the chromatic dispersion or the group index.

Measurement

The Abbe number is computed from refractive indices measured at the three lines, usually with a refractometer or a minimum-deviation prism goniometer; catalogue values come from melt data, and each glass melt ships with its own measured $n_d$ and $V_d$.

References: SCHOTT Optical Glass data sheets and TIE-29, Refractive index and dispersion; ISO 7944:1998, Optics and optical instruments: Reference wavelengths; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).