Editing Abbe number

{{Short description|Material dispersion property}}
In [[optics]] and [[lens design]], the '''Abbe number''', also known as the '''Vd-number''' or '''constringence''' of a [[Transparency (optics)|transparent]] material, is an approximate measure of the material's [[dispersion (optics)|dispersion]] (change of [[refractive index]] versus wavelength), with high values of ''Vd'' 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 (fiber optics)|normalized frequency in fibers]].

[[File:Abbe_number_calculation.svg|right|thumb|300px|Refractive index variation for SF11 flint glass, BK7 borosilicate crown glass, and fused quartz, and calculation for two Abbe numbers for SF11.]]
The Abbe number,<ref>{{cite book |title=The Properties of Optical Glass |year=1998 |series=Schott Series on Glass and Glass Ceramics |language=en |publisher=[[Schott Glass]] |doi=10.1007/978-3-642-57769-7 |isbn=978-3-642-63349-2 |url=https://link.springer.com/book/10.1007/978-3-642-57769-7 |editor1=Bach, Hans |editor2=Neuroth, Norbert }}</ref> <math> V_\mathsf d\ ,</math> of a material is defined as

:<math> V_\mathsf d \equiv \frac{ n_\mathsf d - 1 }{\ n_\mathsf F - n_\mathsf C\ }</math>,

where <math>n_\mathsf C,</math> <math>n_\mathsf d,</math> and <math>n_\mathsf F</math> are the [[refractive indices]] of the material at the wavelengths of the [[Fraunhofer lines|Fraunhofer's]] C, d, and F [[spectral line]]s (656.3&nbsp;[[nanometre|nm]], 587.56&nbsp;nm, and 486.1&nbsp;nm respectively). This formulation only applies to the [[visible spectrum|human vision]]. Outside this range requires the use of different spectral lines. For non-visible spectral lines the term "V-number" is more commonly used. The more general formulation defined as,

:<math> V \equiv \frac{ n_\mathsf{center} - 1 }{ n_\mathsf{short} - n_\mathsf{long} }</math>,

where <math>n_\mathsf{short},</math> <math>n_\mathsf{center},</math> and <math>n_\mathsf{long},</math> are the refractive indices of the material at three different wavelengths. The shortest wavelength's index is <math>n_\mathsf{short}</math>, and the longest's is <math>n_\mathsf{long}</math>.

Abbe numbers are used to classify [[glass]] and other optical materials in terms of their [[chromaticity]]. For example, the higher dispersion [[flint glass]]es have relatively small Abbe numbers <math>V < 55</math> whereas the lower dispersion [[crown glass (optics)|crown glass]]es have larger Abbe numbers. Values of <math>V_\mathsf d</math> 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.

[[File:Eyesensitivity.svg|thumb|Most of the human eye's wavelength sensitivity curve, shown here, is bracketed by the Abbe number reference wavelengths of 486.1&nbsp;nm (blue) and 656.3&nbsp;nm (red)]]
Abbe numbers are used in the design of [[achromatic lens]]es, as their [[reciprocal (mathematics)|''reciprocal'']] is proportional to dispersion (slope of refractive index versus wavelength) in the wavelength region where the human eye is most sensitive (see graph). For different wavelength regions, or for higher precision in characterizing a system's chromaticity (such as in the design of [[apochromat]]s), the full dispersion relation (refractive index as a function of wavelength) is used.

== Abbe diagram ==
[[Image:Abbe-diagram 2.svg|right|thumb|380px|An Abbe diagram, also known as 'the glass veil',  plots the Abbe number against refractive index for a range of different glasses (red dots). Glasses are classified using the Schott Glass letter-number code to reflect their composition and position on the diagram.]]
[[Image:SpiderGraph Abbe Number-en.svg|250px|thumb|Influences of selected [[glass]] component additions on the Abbe number of a specific base glass.<ref>{{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}}</ref>]]
An '''Abbe diagram''', also called 'the glass veil', is produced by plotting the Abbe number <math>V_\mathsf d</math> of a material versus its refractive index <math>n_\mathsf d .</math> 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&nbsp;digit [[glass code]].

Glasses' Abbe numbers, along with their mean refractive indices, are used in the calculation of the required [[refractive power]]s of the elements of [[achromatic lens]]es 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).<ref>{{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}}</ref> For example, rather than the standard definition given above, that uses the refractive index variation between the F and C [[Fraunhofer lines#Naming|hydrogen lines]], one alternative measure using the subscript "e" for [[mercury (element)|mercury]]'s e&nbsp;line compared to [[cadmium]]'s {{prime|F}} and {{prime|C}}&nbsp;lines is
:<math> V_\mathsf e = \frac{ n_\mathsf e - 1 }{\ n_\mathsf{F'} - n_\mathsf{C'}\ } ~.</math>
This alternate takes the difference between cadmium's blue ({{prime|F}}) and red ({{prime|C}}) refractive indices at wavelengths 480.0&nbsp;nm and 643.8&nbsp;nm, relative to <math>\ n_\mathsf e\ </math> for mercury's e&nbsp;line at 546.073&nbsp;nm, all of which are close by, and somewhat easier to produce than the C, F, and e&nbsp;lines. Other definitions can similarly be employed; the following table lists standard wavelengths at which <math>\ n\ </math> is commonly determined, including the [[Fraunhofer lines|standard subscripts]] used.<ref>{{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}}</ref>

{| class="wikitable"
|-
! {{mvar|λ}}<br/>(nm) !! [[Fraunhofer lines|Fraunhofer's<br/>symbol]] !! Light<br/>source !! Color
|-
| 365.01 ||align=center| i  ||  [[Mercury (element)|Hg]]  || [[Ultraviolet light|UV-A]]
|-
| 404.66 ||align=center| h  ||  [[Mercury (element)|Hg]] || violet
|-
| 435.84 ||align=center| g  ||  [[Mercury (element)|Hg]] || blue
|-
| 479.99 ||align=center| {{prime|F}} ||  [[Cadmium|Cd]] || blue
|-
| 486.13 ||align=center| F  ||  [[Hydrogen|H]] || blue
|-
| 546.07 ||align=center| e  ||  [[Mercury (element)|Hg]] || green
|-
| 587.56 ||align=center| d  ||  [[Helium|He]]  || yellow
|-
| 589.3  ||align=center| D  ||  [[Sodium|Na]]  || yellow
|-
| 643.85 ||align=center| {{prime|C}} ||  [[Cadmium|Cd]]  || red
|-
| 656.27 ||align=center| C  ||  [[Hydrogen|H]] || red
|-
| 706.52 ||align=center| r  ||  [[Helium|He]]  || red
|-
| 768.2  ||align=center| {{prime|A}} ||  [[Potassium|K]]  || [[Infrared light|IR-A]]
|-
| 852.11 ||align=center| s  ||  [[Cesium|Cs]]  || [[Infrared light|IR-A]]
|-
| 1013.98 ||align=center| t  ||  [[Mercury (element)|Hg]]  || [[Infrared light|IR-A]]
|}

==Derivation==
Starting from the [[Lens#Lensmaker's equation|'''Lensmaker's equation''']] we obtain the [[Lens#Thin lens approximation|'''thin lens''' equation]] by dropping a small term that accounts for lens thickness, <math>\ d\ </math>:<ref>{{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}}</ref>
:<math> P = \frac{ 1 }{\ f ~} = (n - 1) \Biggl[ \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } + \frac{\ (n-1)\ d ~}{\ n\ R_1 R_2\ } \Biggr] \approx (n - 1) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ ,</math>
when <math> d \ll \sqrt{\ R_1 R_2\ } ~.</math>

The change of [[refractive power]] <math>\ P\ </math> between the two wavelengths <math>\ \lambda_\mathsf{short}\ </math> and <math>\ \lambda_\mathsf{long}\ </math> is given by
:<math> \Delta P = P_\mathsf{short} - P_\mathsf{\ \!long} = (n_\mathsf s - n_\mathsf \ell) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\ ,</math>
where <math>\ n_\mathsf s\ </math> and <math>\ n_\mathsf \ell\ </math> are the short and long wavelengths' refractive indexes, respectively, and <math>\ n_\mathsf c\ ,</math> below, is for the center.

The power difference can be expressed relative to the power at the center wavelength (<math>\ \lambda_\mathsf{center}\ </math>)
: <math>\ P_\mathsf c\ =  (n_\mathsf c - 1) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)\, ;</math>
by multiplying and dividing by <math>\ n_\mathsf c - 1\ </math> and regrouping, get
:<math> \Delta P = \left( n_\mathsf s - n_\mathsf\ell \right) \left( \frac{\ n_\mathsf c - 1\ }{ n_\mathsf c - 1 } \right) \left( \frac{ 1 }{\ R_1\ } - \frac{ 1 }{\ R_2\ } \right)= \left( \frac{\ \ n_\mathsf s - n_\mathsf\ell\ }{ n_\mathsf c - 1 } \right) P_\mathsf c = \frac{\ P_\mathsf c\ }{ V_\mathsf c } ~.</math>
The relative change is [[inversely proportional]] to <math>\ V_\mathsf c\ :</math>
:<math> \frac{\ \Delta P\ }{ P_\mathsf c } = \frac{ 1 }{\ V_\mathsf c\ } ~.</math>

==See also==
*[[Abbe prism]]
*[[Abbe refractometer]]
*[[Calculation of glass properties]], including Abbe number
*[[Glass code]]
*[[Sellmeier equation]], more comprehensive and physically based modeling of dispersion

==References==
{{reflist}}

==External links==
*[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]]