Abbe number

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.

Influences of selected glass component additions on the Abbe number of a specific base glass.[2]

An Abbe diagram, also called 'the glass veil', is produced by plotting the Abbe number ${\displaystyle \ V_{\mathsf {d}}\ }$ of a material versus its refractive index ${\displaystyle \ n_{\mathsf {d}}~.}$ 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).[3] 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 F′ and C′ lines is

${\displaystyle V_{\mathsf {e}}={\frac {n_{\mathsf {e}}-1}{\ n_{\mathsf {F'}}-n_{\mathsf {C'}}\ }}~.}$

This alternate takes the difference between cadmium's blue (C′) and red (F′) refractive indices at wavelengths 480.0 nm and 643.8 nm, relative to ${\displaystyle \ n_{\mathsf {e}}\ }$ 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 ${\displaystyle \ n\ }$ is commonly determined, including the standard subscripts used.[4]

Starting from the Lensmaker's equation we obtain the thin lens equation by dropping a small term that accounts for lens thickness, ${\displaystyle \ d\ }$:

${\displaystyle P={\frac {1}{\ f~}}=(n-1){\Biggl [}{\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ (n-1)\ d~}{\ n\ R_{1}R_{2}\ }}{\Biggr ]}\approx (n-1)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ ,}$

when ${\displaystyle d\ll {\sqrt {\ R_{1}R_{2}\ }}~.}$

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

${\displaystyle \Delta P=P_{\mathsf {short}}-P_{\mathsf {\ \!long}}=(n_{\mathsf {s}}-n_{\mathsf {\ell }})\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ ,}$

where ${\displaystyle \ n_{\mathsf {s}}\ }$ and ${\displaystyle \ n_{\mathsf {\ell }}\ }$ are the short and long wavelengths' refractive indexes, respectively, and ${\displaystyle \ n_{\mathsf {c}}\ ,}$ below, is for the center.

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

${\displaystyle \ P_{\mathsf {c}}\ =(n_{\mathsf {c}}-1)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\,;}$

by multiplying and dividing by ${\displaystyle \ n_{\mathsf {c}}-1\ }$ and regrouping, get

${\displaystyle \Delta P=\left(n_{\mathsf {s}}-n_{\mathsf {\ell }}\right)\left({\frac {\ n_{\mathsf {c}}-1\ }{n_{\mathsf {c}}-1}}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)=\left({\frac {\ \ n_{\mathsf {s}}-n_{\mathsf {\ell }}\ }{n_{\mathsf {c}}-1}}\right)P_{\mathsf {c}}={\frac {\ P_{\mathsf {c}}\ }{V_{\mathsf {c}}}}~.}$

The relative change is inversely proportional to ${\displaystyle \ V_{\mathsf {c}}\$ :}

${\displaystyle {\frac {\ \Delta P\ }{P_{\mathsf {c}}}}={\frac {1}{\ V_{\mathsf {c}}\ }}~.}$

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

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