The Abbe number describes how colors of light travel through something clear at different speeds. For a lower Abbe number, the colors travel over a wider range of speeds. The speed difference means that the colors will not arrive at the same point or at the same time, which is usually bad.

The color of light is measured by its wavelength. Blue has a wavelength of ~0.48 μm, green has a wavelength of ~0.58 μm, and red has a wavelength of ~0.65 μm. While the eye can see colors from purple with a wavelength of ~0.40μm to deep red with a wavelength of ~0.70μm, there are many more colors of light that eyes can't see. These colors may be invisible, but they are used every day in phones, factories, cars, and more. There are clear materials for wavelengths shorter than 0.2μm and longer than 10μm, but this site focuses mostly on glass for wavelengths from 0.264μm to 2.3254μm. Also, while the site supports many different wavelengths of light, some wavelengths are easier to use than others: See 'Recommended Test Wavelengths'.

The speed of a color in something clear is described by comparing it to the speed of light in nothingness (a perfect vacuum). The ratio between these different speeds is called the index of refraction. Air has an index of refraction of ~1.001. Water has an index of refraction of ~1.33. Different types of glass have refractive indices ranging from ~1.45 to ~2.1. With that said, the index of refraction of a material can change for many reasons: different colors of light, temperatures, pressures, and more.

Finding the Abbe number for a material starts by selecting three colors of light to measure the index of refraction at. The three different indices of refraction should then be sorted by the color's wavelength: shortest wavelength $\lambda_{short}$, center wavelength $\lambda_{center}$, longest wavelength $\lambda_{long}$. Then Abbe number, $V$, can be found with this equation, where $n_\lambda$ is used to represent the index of refraction of the material:

$V$ = Abbe number
$n_{short}$ = Index of refraction at $\lambda_{short}$
$n_{center}$ = Index of refraction at $\lambda_{center}$
$n_{long}$ = Index of refraction at $\lambda_{long}$
$$V \equiv \frac{n_{center}-1}{n_{short} - n_{long}}$$

German physicist Ernst Abbe defined the Abbe number in his research. He studied ways to make lenses that fixed the problems caused by the speed difference of colors in glass: lateral and chromatic focal shift. Lateral focal shift is when multiple colors from a single point focus to different places after going through a lens. Chromatic focal shift is when only a handful of colors are in focus at once. Solving these issues requires addressing the speed differences of clear materials, generally by using multiple types of glass in a lens system. The Abbe number and Abbe diagram were a direct result of Ernst Abbe trying to solve chromatic issues. He also developed the first apochromatic lens (a lens with multiple transparent materials that corrects chromatic problems) and the first refractometer (a device that measures the index of refraction of a material).

Where $\lambda_{short} < \lambda_{center} < \lambda_{long}$
$V$ = Abbe number
$n_{short}$ = Index of refraction at $\lambda_{short}$
$n_{center}$ = Index of refraction at $\lambda_{center}$
$n_{long}$ = Index of refraction at $\lambda_{long}$
$$V \equiv \frac{n_{center}-1}{n_{short} - n_{long}}$$

The Abbe number can be used for calculating chromatic focal shift of a thin lens; the TLDR? use this equation:
$f_c$ = focal length for central wavelength
$V$ = Abbe number for range of wavelengths and lens material
$\Delta f$ = The chromatic focal shift $$\frac{f_c}{V} \approx \Delta f$$

Chromatic Focal Shift Derivation:

The first half of the derivation comes from this great Wikipedia article:

We start with the lensmaker's equation
$P$ = refractive power
$f$ = focal length
$n$ = index of refraction
$R$ = radius of curvature
$d$ = center thickness
$$P = \frac{1}{f} = (n - 1)[\frac{1}{R_1}-\frac{1}{R_2}+\frac{(n-1)d}{n R_1 R_2}]$$
and then apply the thin lens assumption where $d \ll \sqrt{R_1 R_2}$ $$P \approx (n-1)(\frac{1}{R_1}-\frac{1}{R_2})$$
if $\Delta P_0$ is defined as the change in refractive power between a short and long wavelength $\lambda_{short} $ and $ \lambda_{long}$ where $n_s$ and $n_l$ are their respective refractive index's. $$\Delta P = P_{short} - P_{long} = (n_s-n_l)(\frac{1}{R_1}-\frac{1}{R_2})$$
Additionally, the difference in power can be expressed relative to the power at a center wavelength $\lambda_c$ $$P_c = (n_c-1)(\frac{1}{R_1}-\frac{1}{R_2})$$
Now we can rewrite the above equations to bring in the Abbe number equation $$\Delta P = (n_s-n_l)(\frac{n_c-1}{n_c-1})(\frac{1}{R_1}-\frac{1}{R_2}) = (\frac{n_s-n_l}{n_c-1})P_c = \frac{P_c}{V}$$
And then we can find that the change in refractive power is inversely proportional to the Abbe number, $V$ $$\frac{\Delta P}{P_c} = \frac{1}{V}$$
This relationship is useful on its own, but we can also create a version for focal length by expanding $\Delta P_0$ and using the refractive power to focal length relation. $$\frac{\Delta P}{P_c} = \frac{P_{short} - P_{long}}{P_c} = \frac{\frac{f_l-f_s}{f_s f_l}}{\frac{1}{f_c}} = \frac{1}{V}$$
If we then assume that $f_s f_l \approx f_c^2$ $$\frac{1}{V} \approx \frac{f_l-f_s}{f_c}$$ $$\frac{f_c}{V} \approx \Delta f$$
This is a very useful napkin method for estimating the chromatic focal shift of a thin lens. Consider a 200mm focal length, thin, plano convex, N-BK7 ($V$ = 64) lens; the chromatic focal shift from blue to red can be found to be around 3.1mm. $$\Delta f = \frac{f_c}{V} = \frac{200mm}{64} = 3.1mm$$

Yep! The abbe number can be used to create achromats, apochromats, and more. See this Wikipedia article.

We collect data from manufacturer catalogs, so our accuracy depends on their accuracy. Additionally, we calculate Abbe numbers and indices of refraction using continuous dispersion formulas, such as the Sellmeier equation. Because the curve-fitting of these equations is imperfect, you may see minor inconsistencies. We strongly encourage you to confirm your results with the manufacturer before proceeding with critical projects.

While you can use any wavelength, suppliers generally test their glass at only a select number of wavelengths. When specifying Abbe numbers or index of refraction values, the wavelengths in this list are strongly recommended.

Preferred Test Wavelengths (μm):
0.2483 - UV-C/MUV
0.2804 - UV-B/MUV
0.2967 - UV-B/MUV
0.3126 - UV-B/NUV
0.3341 - UV-A/NUV
0.3650 - UV-A/NUV
0.4046 - Purple
0.4358 - Blue
0.442   - Blue
0.4799 - Blue
0.4861 - Blue
0.5460 - Green
0.5875 - Yellow
0.5893 - LPS/Yellow
0.6328 - HeNe/Red
0.6438 - Red
0.6562 - Red
0.7065 - Dark Red
0.7682 - IR-A/NIR
0.8521 - IR-A/NIR
1.0139 - IR-A/NIR
1.060   - Nd:YAG/IR-A/NIR
1.129   - IR-A/NIR
1.530   - Telecom/IR-B/SWIR
1.970   - IR-B/SWIR
2.325   - IR-B/SWIR