Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Abbe number
open-in-new
<h2 id="abbe-diagram">Abbe diagram</h2> <p>An Abbe diagram, also called 'the glass veil', is produced by plotting the Abbe number V d {\displaystyle V_{\mathsf {d}}} of a material versus its refractive index n d . {\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 <a href="/facts/Schott_Glass/YI29pUgU">Schott Glass</a> catalogue, or a 6 digit <a href="/facts/Glass_code/BRrmLkFY">glass code</a>. </p><p>Glasses' Abbe numbers, along with their mean refractive indices, are used in the calculation of the required <a href="/facts/Refractive_power/jmIWyX25">refractive powers</a> of the elements of <a href="/facts/Achromatic_lens/6R4A63JY">achromatic lenses</a> in order to cancel <a href="/facts/Chromatic_aberration/Fy1hIg1Y">chromatic aberration</a> 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. </p><p>Due to the difficulty and inconvenience in producing sodium and hydrogen lines, alternate definitions of the Abbe number are often substituted (<a href="/facts/ISO/IacYR7Yg">ISO</a> 7944).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> For example, rather than the standard definition given above, that uses the refractive index variation between the F and C <a href="/facts/Fraunhofer_lines/rVSJ6nlH">hydrogen lines</a>, one alternative measure using the subscript "e" for <a href="/facts/Mercury_(element)/WlxfoHva">mercury</a>'s e line compared to <a href="/facts/Cadmium/wXkV2eIX">cadmium</a>'s F′ and C′ lines is </p> V e = n e − 1 n F ′ − n C ′ . {\displaystyle V_{\mathsf {e}}={\frac {n_{\mathsf {e}}-1}{\ n_{\mathsf {F'}}-n_{\mathsf {C'}}\ }}~.} <p>This alternate takes the difference between cadmium's blue (F′) and red (C′) refractive indices at wavelengths 480.0 nm and 643.8 nm, relative to n e {\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 n {\displaystyle \ n\ } is commonly determined, including the <a href="/facts/Fraunhofer_lines/rVSJ6nlH">standard subscripts</a> used.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <table><tbody><tr><th>λ(nm)</th><th><a href="/facts/Fraunhofer_lines/rVSJ6nlH">Fraunhofer'ssymbol</a></th><th>Lightsource</th><th>Color</th></tr><tr><td>365.01</td><td>i</td><td><a href="/facts/Mercury_(element)/WlxfoHva">Hg</a></td><td><a href="/facts/Ultraviolet_light/kncBUqn5">UV-A</a></td></tr><tr><td>404.66</td><td>h</td><td><a href="/facts/Mercury_(element)/WlxfoHva">Hg</a></td><td>violet</td></tr><tr><td>435.84</td><td>g</td><td><a href="/facts/Mercury_(element)/WlxfoHva">Hg</a></td><td>blue</td></tr><tr><td>479.99</td><td>F′</td><td><a href="/facts/Cadmium/wXkV2eIX">Cd</a></td><td>blue</td></tr><tr><td>486.13</td><td>F</td><td><a href="/facts/Hydrogen/BoYHjl0J">H</a></td><td>blue</td></tr><tr><td>546.07</td><td>e</td><td><a href="/facts/Mercury_(element)/WlxfoHva">Hg</a></td><td>green</td></tr><tr><td>587.56</td><td>d</td><td><a href="/facts/Helium/yRul93TG">He</a></td><td>yellow</td></tr><tr><td>589.3</td><td>D</td><td><a href="/facts/Sodium/eYuSPu2V">Na</a></td><td>yellow</td></tr><tr><td>643.85</td><td>C′</td><td><a href="/facts/Cadmium/wXkV2eIX">Cd</a></td><td>red</td></tr><tr><td>656.27</td><td>C</td><td><a href="/facts/Hydrogen/BoYHjl0J">H</a></td><td>red</td></tr><tr><td>706.52</td><td>r</td><td><a href="/facts/Helium/yRul93TG">He</a></td><td>red</td></tr><tr><td>768.2</td><td>A′</td><td><a href="/facts/Potassium/FWPwS7Rj">K</a></td><td><a href="/facts/Infrared_light/44XdJNcc">IR-A</a></td></tr><tr><td>852.11</td><td>s</td><td><a href="/facts/Cesium/dXKN1S5k">Cs</a></td><td><a href="/facts/Infrared_light/44XdJNcc">IR-A</a></td></tr><tr><td>1013.98</td><td>t</td><td><a href="/facts/Mercury_(element)/WlxfoHva">Hg</a></td><td><a href="/facts/Infrared_light/44XdJNcc">IR-A</a></td></tr></tbody></table> <h2 id="derivation">Derivation</h2> <p>Starting from the <a href="/facts/Lens/ejlH6n5c">Lensmaker's equation</a> we obtain the <a href="/facts/Lens/ejlH6n5c">thin lens equation</a> by dropping a small term that accounts for lens thickness, d {\displaystyle \ d\ } :<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> P = 1 f = ( n − 1 ) [ 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ] ≈ ( n − 1 ) ( 1 R 1 − 1 R 2 ) , {\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)\ ,} <p>when d ≪ R 1 R 2 . {\displaystyle d\ll {\sqrt {\ R_{1}R_{2}\ }}~.} </p><p>The change of <a href="/facts/Refractive_power/jmIWyX25">refractive power</a> P {\displaystyle \ P\ } between the two wavelengths λ s h o r t {\displaystyle \ \lambda _{\mathsf {short}}\ } and λ l o n g {\displaystyle \ \lambda _{\mathsf {long}}\ } is given by </p> Δ P = P s h o r t − P l o n g = ( n s − n ℓ ) ( 1 R 1 − 1 R 2 ) , {\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)\ ,} <p>where n s {\displaystyle \ n_{\mathsf {s}}\ } and n ℓ {\displaystyle \ n_{\mathsf {\ell }}\ } are the short and long wavelengths' refractive indexes, respectively, and n c , {\displaystyle \ n_{\mathsf {c}}\ ,} below, is for the center. </p><p>The power difference can be expressed relative to the power at the center wavelength ( λ c e n t e r {\displaystyle \ \lambda _{\mathsf {center}}\ } ) </p> P c = ( n c − 1 ) ( 1 R 1 − 1 R 2 ) ; {\displaystyle \ P_{\mathsf {c}}\ =(n_{\mathsf {c}}-1)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\,;} <p>by multiplying and dividing by n c − 1 {\displaystyle \ n_{\mathsf {c}}-1\ } and regrouping, get </p> Δ P = ( n s − n ℓ ) ( n c − 1 n c − 1 ) ( 1 R 1 − 1 R 2 ) = ( n s − n ℓ n c − 1 ) P c = P c V c . {\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}}}}~.} <p>The relative change is <a href="/facts/Inversely_proportional/xDQmfh6t">inversely proportional</a> to V c : {\displaystyle \ V_{\mathsf {c}}\ :} </p> Δ P P c = 1 V c . {\displaystyle {\frac {\ \Delta P\ }{P_{\mathsf {c}}}}={\frac {1}{\ V_{\mathsf {c}}\ }}~.} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Abbe_prism/FgzzIIPX">Abbe prism</a></li> <li><a href="/facts/Abbe_refractometer/wccEcLJy">Abbe refractometer</a></li> <li><a href="/facts/Calculation_of_glass_properties/S8YzGwtw">Calculation of glass properties</a>, including Abbe number</li> <li><a href="/facts/Glass_code/BRrmLkFY">Glass code</a></li> <li><a href="/facts/Sellmeier_equation/LJMlvtlA">Sellmeier equation</a>, more comprehensive and physically based modeling of dispersion</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="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</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="978-3-642-63349-2" target="_blank">978-3-642-63349-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="http://opticampus.opti.vision/files/memo_on_reference_wavelengths.pdf" target="_blank">http://opticampus.opti.vision/files/memo_on_reference_wavelengths.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Pye, L.D.; Frechette, V.D.; Kreidl, N.J. (1977). Borate Glasses. New York, NY: Plenum Press. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="978-1-292-09693-3" target="_blank">978-1-292-09693-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>