Tool · Optical materials

Abbe Number Calculator

How strongly does a glass disperse light, and what does that do to a lens? From three refractive indices, or from the Sellmeier coefficients of N-BK7, N-SF11, fused silica, calcium fluoride or sapphire, the calculator returns the Abbe number, the partial dispersion, the glass family, the chromatic focal shift of a singlet and the element focal lengths of a thin achromatic doublet. Background: Abbe number, chromatic aberration and achromatic doublet.

Glass

Indices from

n_d at 587.56 nm

n_F at 486.13 nm

n_C at 656.27 nm

Choose a material to compute its indices from published Sellmeier coefficients, or type the three indices from a glass catalog.

Lens

Focal length (mm)

Second glass of a doublet

Its Abbe number V₂

A thin cemented achromat brings the F and C lines to one focus when the two elements have powers in the ratio −V₁/V₂.

Presets

Readouts

Glass map: n_d against V_d

Abbe number runs from right to left, as on glass catalogs. The dashed line is the crown and flint boundary. Gray dots are the other Sellmeier materials; the orange dot is the current glass.

Index across the visible

The vertical lines are the F, d and C lines. V_d is n_d − 1 divided by the index drop between the F and C lines.

Learn with it

Two short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The Abbe numbers computed from the Sellmeier coefficients are compared with the values the coefficient sources give, and the lens formulas with values worked out by hand.

Check	Expected	Computed	Tolerance

The expected Abbe numbers are those listed by SCHOTT for N-BK7 and N-SF11 and the values that follow from Malitson’s silica and CaF₂ fits; the lens results follow the thin-lens achromat condition φ₁/V₁ + φ₂/V₂ = 0. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

The Abbe number compares a glass’s refractivity with its dispersion across the visible,

V_{d} = \frac{n_{d} - 1}{n_{F} - n_{C}}

with $n_{d}$ at the helium d line, 587.56 nm, and $n_{F}$ , $n_{C}$ at the hydrogen F and C lines, 486.13 and 656.27 nm. $V_{e}$ uses the e, F′ and C′ lines instead (546.07, 479.99 and 643.85 nm), and the partial dispersion $P_{g, F} = (n_{g} - n_{F}) / (n_{F} - n_{C})$ describes how the dispersion is shared within the band. A thin singlet of focal length $f$ has power $ϕ \propto n - 1$ , so its focus moves between the F and C lines by

f_{C} - f_{F} \approx \frac{f}{V_{d}}

Two thin lenses in contact with powers $ϕ_{1}$ and $ϕ_{2}$ have the same focus at F and C when $ϕ_{1} / V_{1} + ϕ_{2} / V_{2} = 0$ , which with $ϕ_{1} + ϕ_{2} = 1 / f$ gives

f_{1} = f \frac{V_{1} - V_{2}}{V_{1}}, f_{2} = - f \frac{V_{1} - V_{2}}{V_{2}}

The Sellmeier indices come from the coefficients listed with each material in the refractive index calculator. The doublet result corrects only the primary color; the residual secondary spectrum depends on the difference of the partial dispersions, and the element shapes that also correct spherical aberration and coma are a further design step.

Worked example

N-BK7 has $n_{d} = 1.51680, n_{F} = 1.52238, n_{C} = 1.51432$ , so $V_{d} = 0.51680 / 0.008054 = 64.17$ , a crown. A 100 mm singlet in it focuses blue F light 1.558 mm closer than red C light. Paired with N-SF11 ( $V_{d} = 25.68$ ), a 100 mm achromat needs a crown element of $f_{1} = 100 (64.17 - 25.68) / 64.17 = 59.98$ mm and a flint element of $f_{2} = - 149.9$ mm.

Related: the refractive index calculator gives the index and dispersion at any wavelength, the Sellmeier equation entry explains the fits, and the thin lens calculator handles the imaging.

References: SCHOTT AG, Optical glass data sheets (N-BK7, N-SF11). I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965). W. J. Smith, Modern Optical Engineering, 4th ed., McGraw-Hill (2008).