Abbe number

 ( Dimensionless Numbers Of Physics ) 

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 $V\_\backslash text\{d\}$ of a material is defined as:

$$V\_\backslash text\{d\}\; \backslash equiv\; \backslash frac\{\; n\_\backslash text\{d\}\; -\; 1\; \}\{\; n\_\backslash text\{F\}\; -\; n\_\backslash text\{C\}\; \},$$ where $n\_\backslash text\{C\}$, $n\_\backslash text\{d\}$, and $n\_\backslash text\{F\}$ are the refractive indices of the material at the wavelengths of the Fraunhofer lines C, d, and F (656.3 nanometre, 587.56 nm, and 486.1 nm, respectively). This formulation only applies to visible spectrum; 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 $$V\; \backslash equiv\; \backslash frac\{\; n\_\backslash text\{center\}\; -\; 1\; \}\{\; n\_\backslash text\{short\}\; -\; n\_\backslash text\{long\}\; \},$$ where $n\_\backslash text\{short\}$, $n\_\backslash text\{center\}$, and $n\_\backslash text\{long\}$ 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 have relatively small Abbe numbers $V$ less than 55, whereas the lower dispersion crown glasses have larger Abbe numbers. Values of $V\_\backslash text\{d\}$ 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 , 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 ), the full dispersion relation is used (i.e., refractive index as a function of wavelength).

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Abbe diagram An Abbe diagram (sometimes referred to as "the glass veil") is produced by plotting the refractive index of a material $n\_\backslash text\{d\}$ as a function of Abbe number $V$. Glasses can then be categorized and selected according to their positions on the diagram. This categorization could be in the form of a letter-number code, as used for example 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 of the elements of in order to cancel chromatic aberration to first order. These two parameters, which enter into the equations for the 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). For example, rather than the standard definition given above, which uses the refractive index variation between the F and C hydrogen lines, one alternative measure is to use mercury's e-line compared to cadmium's - and -lines: $$V\_\backslash text\{e\}\; =\; \backslash frac\{\; n\_\backslash text\{e\}\; -\; 1\; \}\{\; n\_\{\backslash text\{F\}\text{'}\}\; -\; n\_\{\backslash text\{C\}\text{'}\}\; \}\; .$$ This formulation takes the difference between cadmium's blue () and red () refractive indices at wavelengths 480.0 nm and 643.8 nm, respectively, relative to $n\_\backslash text\{e\}$ for mercury's e-line at 546.073 nm, all of which are in close proximity to—and somewhat easier to produce—than the C, F, and d-lines. Other definitions can be similarly employed; the following table lists standard wavelengths at which $n$ is commonly determined, including the Fraunhofer lines used.

UV-A

violet

blue

blue

blue

green

yellow

yellow

red

red

red

Infrared light

Infrared light

Infrared light

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Derivation of relative change Starting with the Lensmaker's equation, we obtain the thin lens equation by neglecting the small term that accounts for lens thickness $d$:

Eugene Hecht (2025). 9781292096933, Pearson. ISBN 9781292096933

$$P\_0\; =\; \backslash frac\{1\}\{f\}\; =\; (n\; -\; 1)\; \backslash Biggl\; \backslash approx\; (n\; -\; 1)\; \backslash left(\; \backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\; \backslash right)\; ,$$ when $d\; \backslash ll\; \backslash sqrt\{\; R\_1\; R\_2\; \}$.

The change in refractive power $P\_0$ between two wavelengths $\backslash lambda\_\backslash text\{short\}$ and $\backslash lambda\_\backslash text\{long\}$ is given by $$\backslash Delta\; P\_0\; =\; P\_\backslash text\{short\}\; -\; P\_\backslash text\{long\}\; =\; (\; n\_\backslash text\{s\}\; -\; n\_\backslash ell\; )\; \backslash left(\; \backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\; \backslash right)\; ,$$ where $n\_\backslash text\{s\}$ and $n\_\backslash ell$ are the short and long wavelengths' refractive indexes, respectively.

The difference in power can be expressed relative to the power at a center wavelength $\backslash lambda\_\backslash text\{c\}$: $$P\_\backslash text\{c\}\; =\; (n\_\backslash text\{c\}\; -\; 1)\; \backslash left(\; \backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\; \backslash right)\; ,$$ with $n\_\backslash text\{c\}$ having an analogous meaning as above. Now rewrite $\backslash Delta\; P\_0$ to make $P\_\backslash text\{c\}$ and the Abbe number at the center wavelength $V\_\backslash text\{c\}$ accessible: $$\backslash Delta\; P\_0\; =\; \backslash left(\; n\_\backslash text\{s\}\; -\; n\_\backslash ell\; \backslash right)\; \backslash left(\; \backslash frac\{\; n\_\backslash text\{c\}\; -\; 1\; \}\{\; n\_\backslash text\{c\}\; -\; 1\; \}\; \backslash right)\; \backslash left(\; \backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\; \backslash right)\; =\; \backslash left(\; \backslash frac\{\; n\_\backslash text\{s\}\; -\; n\_\backslash ell\; \}\{\; n\_\backslash text\{c\}\; -\; 1\; \}\; \backslash right)\; P\_\backslash text\{c\}\; =\; \backslash frac\{\; P\_\backslash text\{c\}\; \}\{\; V\_\backslash text\{c\}\; \}\; .$$ The relative change is therefore inversely proportional to $V\_\backslash text\{c\}$: $$\backslash frac\{\backslash Delta\; P\_0\}\{\; P\_\backslash text\{c\}\; \}\; =\; \backslash frac\{1\}\{V\_\backslash text\{c\}\; \}\; .$$

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See also

• Abbe prism
• Abbe refractometer
• Calculation of glass properties, including Abbe number
• Glass code
• Sellmeier equation, a more comprehensive and physical model of dispersion

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External links

• Abbe graph and data for 356 glasses from Ohara, Hoya, and Schott

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