Abbe number
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{d}} of a material is defined as:[1] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{d} \equiv \frac{ n_\text{d} - 1 }{ n_\text{F} - n_\text{C} },} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{C}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{d}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{F}} are the refractive indices of the material at the wavelengths of the Fraunhofer's C, d, and F spectral lines (656.3 nm, 587.56 nm, and 486.1 nm, respectively). This formulation only applies to human vision; 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \equiv \frac{ n_\text{center} - 1 }{ n_\text{short} - n_\text{long} },} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{short}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{center}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\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 flint glasses have relatively small Abbe numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} less than 55, whereas the lower dispersion crown glasses have larger Abbe numbers. Values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\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 achromatic lenses, 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 apochromats), the full dispersion relation is used (i.e., refractive index as a function of wavelength).
Abbe diagram
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An Abbe diagram (sometimes referred to as "the glass veil") is produced by plotting the refractive index of a material Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{d}} as a function of Abbe number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 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 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).[3] 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 Template:Prime- and Template:Prime-lines: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{e} = \frac{ n_\text{e} - 1 }{ n_{\text{F}'} - n_{\text{C}'} } .} This formulation takes the difference between cadmium's blue (Template:Prime) and red (Template:Prime) refractive indices at wavelengths 480.0 nm and 643.8 nm, respectively, relative to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is commonly determined, including the standard subscripts used.[4]
| λ (nm) | Fraunhofer's symbol | Light source | Color |
|---|---|---|---|
| 365.01 | i | Hg | UV-A |
| 404.66 | h | Hg | violet |
| 435.84 | g | Hg | blue |
| 479.99 | Template:Prime | Cd | blue |
| 486.13 | F | H | blue |
| 546.07 | e | Hg | green |
| 587.56 | d | He | yellow |
| 589.30 | D | Na | yellow |
| 643.85 | Template:Prime | Cd | red |
| 656.27 | C | H | red |
| 706.52 | r | He | red |
| 768.20 | Template:Prime | K | IR-A |
| 852.11 | s | Cs | IR-A |
| 1013.98 | t | Hg | IR-A |
Derivation of relative change
[edit | edit source]Starting with the Lensmaker's equation, we obtain the thin lens equation by neglecting the small term that accounts for lens thickness Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} :[5] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0 = \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \ll \sqrt{ R_1 R_2 }} .
The change in refractive power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0} between two wavelengths Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_\text{short}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_\text{long}} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta P_0 = P_\text{short} - P_\text{long} = ( n_\text{s} - n_\ell ) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) ,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{s}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_\text{c}} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\text{c} = (n_\text{c} - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) ,} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_\text{c}} having an analogous meaning as above. Now rewrite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta P_0} to make Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\text{c}} and the Abbe number at the center wavelength Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{c}} accessible: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta P_0 = \left( n_\text{s} - n_\ell \right) \left( \frac{ n_\text{c} - 1 }{ n_\text{c} - 1 } \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) = \left( \frac{ n_\text{s} - n_\ell }{ n_\text{c} - 1 } \right) P_\text{c} = \frac{ P_\text{c} }{ V_\text{c} } .} The relative change is therefore inversely proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{c}} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\Delta P_0}{ P_\text{c} } = \frac{1}{V_\text{c} } .}
See also
[edit | edit source]- Abbe prism
- Abbe refractometer
- Calculation of glass properties, including Abbe number
- Glass code
- Sellmeier equation, a more comprehensive and physical model of dispersion
References
[edit | edit source]- ↑ Bach, Hans; Neuroth, Norbert, eds. (1998). The Properties of Optical Glass. Schott Series on Glass and Glass Ceramics. Schott Glass. doi:10.1007/978-3-642-57769-7. ISBN 978-3-642-63349-2.
- ↑ Fluegel, Alexander (2007-12-07). "Abbe number calculation of glasses". Statistical Calculation and Development of Glass Properties (glassproperties.com). Retrieved 2022-01-16.
- ↑ Meister, Darryl (12 April 2010). Understanding reference wavelengths (PDF). opticampus.opti.vision (memo). Carl Zeiss Vision. Archived (PDF) from the original on 2022-10-09. Retrieved 2013-03-13.
- ↑ Pye, L.D.; Frechette, V.D.; Kreidl, N.J. (1977). Borate Glasses. New York, NY: Plenum Press.
- ↑ Hecht, Eugene (2017). Optics (5 ed/fifth edition, global ed.). Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich: Pearson. ISBN 978-1-292-09693-3.