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An Abbe diagram plots refractive index $n_d$ against Abbe number $V_d$ to show available optical glass families. Many diagrams put increasing $V_d$ to the left. Ordinary crown glasses have larger $V_d$ than ordinary flint glasses, while modern compositions occupy a broader region.

For two thin lenses in contact, achromatism requires ${\Phi }_1/V_1+{\Phi }_2/V_2=0$, where $V_i$ are their Abbe numbers. With total power $\Phi =1/f$, the component powers are ${\Phi }_1=\Phi V_1/(V_1-V_2)$ and ${\Phi }_2=-\Phi V_2/(V_1-V_2)$. A positive doublet using a larger-$V$ crown glass element and smaller-$V$ flint glass element therefore uses a converging crown and a diverging flint.

An achromatic lens combines elements with different Abbe numbers to make two specified wavelengths share a focal length. Other wavelengths can retain secondary chromatic error.

Write ${\Phi }_i=1/f_{id}$ and $\Phi =1/f_d$. The achromatic doublet power balance isSolving these two linear equations gives ${\Phi }_1=\Phi V_{1d}/(V_{1d}-V_{2d})$ and ${\Phi }_2=-\Phi V_{2d}/(V_{1d}-V_{2d})$. Taking reciprocals gives the required focal lengths:The design requires distinct Abbe numbers; identical relative dispersion cannot yield a nonzero total achromatic power from this two-element thin model.

At the intermediate $d$ reference wavelength, the lensmaker's equation gives $1/f_{id}={\rho }_i(n_{id}-1)$. Therefore the curvature-factor ratio from the previous part impliesand henceThe last equality uses the Abbe number definition and expresses the achromatic doublet power balance in terms of component focal lengths.

$V_d$ is the Abbe number, a measure of inverse relative optical dispersion. Ordinary crown glasses have comparatively large $V_d$ and lower refractive index, while ordinary flint glasses have smaller $V_d$ and often higher index. The Abbe diagram below spans typical optical-glass values; the shaded family ranges are schematic, and the marked examples are catalogue data. Modern glass compositions broaden and overlap the simple crown/flint picture.

For a positive doublet with $V_{1d}>V_{2d}$, the preceding formulas require $f_{1d}>0$ and $f_{2d}<0$: a converging crown glass element and a diverging flint glass element. Their dispersion corrections cancel, while their net optical power is positive. A small difference in Abbe numbers requires large opposing component powers, which makes the design harder to correct for other optical aberrations.