Abbe number

Image:Abbe-diagram 2.svg

Image:SpiderGraph Abbe Number-en.svg component additions on the Abbe number of a specific base glass.{{cite web |first = Alexander |last = Fluegel |date = 2007-12-07 |title = Abbe number calculation of glasses |website = Statistical Calculation and Development of Glass Properties (glassproperties.com) |url = http://glassproperties.com/abbe_number/ |access-date = 2022-01-16}}]]

An Abbe diagram, also called 'the glass veil', is produced by plotting the Abbe number $V\_\backslash mathsf\; d$ of a material versus its refractive index $n\_\backslash mathsf\; d\; .$ Glasses can then be categorised and selected according to their positions on the diagram. This can be a letter-number code, as used 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 refractive powers of the elements of achromatic lenses in order to cancel chromatic aberration to first order. These two parameters which enter into the equations for 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).{{cite report |last=Meister |first=Darryl |date=12 April 2010 |title=Understanding reference wavelengths |type=memo |publisher=Carl Zeiss Vision |website=opticampus.opti.vision |url=http://opticampus.opti.vision/files/memo_on_reference_wavelengths.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://opticampus.opti.vision/files/memo_on_reference_wavelengths.pdf |archive-date=2022-10-09 |url-status=live |access-date=2013-03-13}} For example, rather than the standard definition given above, that uses the refractive index variation between the F and C hydrogen lines, one alternative measure using the subscript "e" for mercury's e line compared to cadmium's {{prime|F}} and {{prime|C}} lines is

:$V\_\backslash mathsf\; e\; =\; \backslash frac\{\; n\_\backslash mathsf\; e\; -\; 1\; \}\{\backslash \; n\_\backslash mathsf\{F\text{'}\}\; -\; n\_\backslash mathsf\{C\text{'}\}\backslash \; \}\; ~.$

This alternate takes the difference between cadmium's blue ({{prime|F}}) and red ({{prime|C}}) refractive indices at wavelengths 480.0 nm and 643.8 nm, relative to $\backslash \; n\_\backslash mathsf\; e\backslash$ for mercury's e line at 546.073 nm, all of which are close by, and somewhat easier to produce than the C, F, and e lines. Other definitions can similarly be employed; the following table lists standard wavelengths at which $\backslash \; n\backslash$ is commonly determined, including the standard subscripts used.{{cite book |first1=L.D. |last1=Pye |first2=V.D. |last2=Frechette |first3=N.J. |last3=Kreidl |year=1977 |title=Borate Glasses |publisher=Plenum Press |place=New York, NY}}

Starting from the Lensmaker's equation we obtain the thin lens equation by dropping a small term that accounts for lens thickness, $\backslash \; d\backslash$:{{Cite book |last=Hecht |first=Eugene |title=Optics |date=2017 |publisher=Pearson |isbn=978-1-292-09693-3 |edition=5 ed/fifth edition, global |location=Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich}}

:$P\; =\; \backslash frac\{\; 1\; \}\{\backslash \; f\; ~\}\; =\; (n\; -\; 1)\; \backslash Biggl[\; \backslash frac\{\; 1\; \}\{\backslash \; R\_1\backslash \; \}\; -\; \backslash frac\{\; 1\; \}\{\backslash \; R\_2\backslash \; \}\; +\; \backslash frac\{\backslash \; (n-1)\backslash \; d\; ~\}\{\backslash \; n\backslash \; R\_1\; R\_2\backslash \; \}\; \backslash Biggr]\; \backslash approx\; (n\; -\; 1)\; \backslash left(\; \backslash frac\{\; 1\; \}\{\backslash \; R\_1\backslash \; \}\; -\; \backslash frac\{\; 1\; \}\{\backslash \; R\_2\backslash \; \}\; \backslash right)\backslash \; ,$

when $d\; \backslash ll\; \backslash sqrt\{\backslash \; R\_1\; R\_2\backslash \; \}\; ~.$

The change of refractive power $\backslash \; P\backslash$ between the two wavelengths $\backslash \; \backslash lambda\_\backslash mathsf\{short\}\backslash$ and $\backslash \; \backslash lambda\_\backslash mathsf\{long\}\backslash$ is given by

:$\backslash Delta\; P\; =\; P\_\backslash mathsf\{short\}\; -\; P\_\backslash mathsf\{\backslash \; \backslash !long\}\; =\; (n\_\backslash mathsf\; s\; -\; n\_\backslash mathsf\; \backslash ell)\; \backslash left(\; \backslash frac\{\; 1\; \}\{\backslash \; R\_1\backslash \; \}\; -\; \backslash frac\{\; 1\; \}\{\backslash \; R\_2\backslash \; \}\; \backslash right)\backslash \; ,$

where $\backslash \; n\_\backslash mathsf\; s\backslash$ and $\backslash \; n\_\backslash mathsf\; \backslash ell\backslash$ are the short and long wavelengths' refractive indexes, respectively, and $\backslash \; n\_\backslash mathsf\; c\backslash \; ,$ below, is for the center.

The power difference can be expressed relative to the power at the center wavelength ($\backslash \; \backslash lambda\_\backslash mathsf\{center\}\backslash$)

: $\backslash \; P\_\backslash mathsf\; c\backslash \; =\; (n\_\backslash mathsf\; c\; -\; 1)\; \backslash left(\; \backslash frac\{\; 1\; \}\{\backslash \; R\_1\backslash \; \}\; -\; \backslash frac\{\; 1\; \}\{\backslash \; R\_2\backslash \; \}\; \backslash right)\backslash ,\; ;$

by multiplying and dividing by $\backslash \; n\_\backslash mathsf\; c\; -\; 1\backslash$ and regrouping, get

:$\backslash Delta\; P\; =\; \backslash left(\; n\_\backslash mathsf\; s\; -\; n\_\backslash mathsf\backslash ell\; \backslash right)\; \backslash left(\; \backslash frac\{\backslash \; n\_\backslash mathsf\; c\; -\; 1\backslash \; \}\{\; n\_\backslash mathsf\; c\; -\; 1\; \}\; \backslash right)\; \backslash left(\; \backslash frac\{\; 1\; \}\{\backslash \; R\_1\backslash \; \}\; -\; \backslash frac\{\; 1\; \}\{\backslash \; R\_2\backslash \; \}\; \backslash right)=\; \backslash left(\; \backslash frac\{\backslash \; \backslash \; n\_\backslash mathsf\; s\; -\; n\_\backslash mathsf\backslash ell\backslash \; \}\{\; n\_\backslash mathsf\; c\; -\; 1\; \}\; \backslash right)\; P\_\backslash mathsf\; c\; =\; \backslash frac\{\backslash \; P\_\backslash mathsf\; c\backslash \; \}\{\; V\_\backslash mathsf\; c\; \}\; ~.$

The relative change is inversely proportional to $\backslash \; V\_\backslash mathsf\; c\backslash \; :$

:$\backslash frac\{\backslash \; \backslash Delta\; P\backslash \; \}\{\; P\_\backslash mathsf\; c\; \}\; =\; \backslash frac\{\; 1\; \}\{\backslash \; V\_\backslash mathsf\; c\backslash \; \}\; ~.$

• Abbe prism
• Abbe refractometer
• Calculation of glass properties, including Abbe number
• Glass code
• Sellmeier equation, more comprehensive and physically based modeling of dispersion

{{reflist}}

• [https://web.archive.org/web/20151011033820/http://www.lacroixoptical.com/sites/default/files/content/LaCroix%20Dynamic%20Material%20Selection%20Data%20Tool%20vJanuary%202015.xlsm Abbe graph and data for 356 glasses from Ohara, Hoya, and Schott]

{{DEFAULTSORT:Abbe Number}}

Category:Dimensionless numbers of physics

Category:Optical quantities

Category:Glass physics