Achromat Designer
Design multi-element achromatic lens systems using the thin-lens approximation and achromatic condition. Select glass materials and focal lengths for up to 4 lens elements, then evaluate chromatic correction over your chosen wavelength range. This is a simplified, first-order model and should only be used as a starting point for your design.
FAQ
An achromat, apochromat, or super achromat are all different types of lenses that correct for differences in how colors of light focus differently.
The difference in focus is called chromatic aberration. with achromats, apochromats, and super achromats typically providing increasing levels of chromatic correction.
An achromat is a lens that corrects for chromatic aberration for two wavelengths, an apochromat is a lens that corrects for chromatic aberration for
three wavelengths, and a super achromat is a lens that corrects for chromatic aberration for four wavelengths.
Typically, this improved performance is achieved by using multiple types of glass in the lens system and this tool can help you test your glass choices.
You can read more about achromats here: Wikipedia article.
You can read more about apochromats here: Wikipedia article.
The recommended focal lengths are calculated using the thin-lens approximation and achromatic condition.
For the total power of the system, the equation is:
$$P = \sum {P_i}$$
where $P$ is the total power of the system, $P_i$ is the power of the $i$th lens, and $P = \frac{1}{f}$ calculates the focal length of the lens.
For the achromatic condition, the equation is:
$$C = \sum \frac{P_i}{V_{i, \lambda}}$$
where $C$ is the achromatic condition, $V$ is the Abbe number, and $f$ is the focal length.
Then the power of each lens is calculated by the solving a series of linear equations. Below is the system of equations for 4 lenses.
$$
\left\{
\begin{aligned}
P &= P_1 + P_2 + P_3 + P_4, \\
0 &= \frac{P_1}{V_{{1, \lambda_1}}} + \frac{P_2}{V_{{2, \lambda_1}}} + \frac{P_3}{V_{{3, \lambda_1}}} + \frac{P_4}{V_{{4, \lambda_1}}}, \\
0 &= \frac{P_1}{V_{{1, \lambda_2}}} + \frac{P_2}{V_{{2, \lambda_2}}} + \frac{P_3}{V_{{3, \lambda_2}}} + \frac{P_4}{V_{{4, \lambda_2}}}, \\
0 &= \frac{P_1}{V_{{1, \lambda_3}}} + \frac{P_2}{V_{{2, \lambda_3}}} + \frac{P_3}{V_{{3, \lambda_3}}} + \frac{P_4}{V_{{4, \lambda_3}}}.
\end{aligned}
\right.
$$
The recommended focal lengths are then calculated by solving the equation for $P$ and calculating $f = \frac{1}{P}$.
Notably abbe numbers must be chosen to cover different sections of the design range.
For example, if the design range is from 400 nm to 700 nm, the abbe number must be chosen to cover the section from 400 nm to
587.56 nm alongside the section from 400 nm to 700 nm in the case of a triplet lens system.
This is required because the abbe number is a measure of the dispersion of the glass, and the dispersion is different for different sections of the design range.
More information about the achromatic condition can be found in this Wikipedia article.
For 2 lenses, pick two glasses with Abbe numbers that differ by a substantial amount and meets other design constraints. N-BK7 and N-SF11 are popular Schott glass choices in the visual range. For other design ranges, we recommend consulting our Abbe diagram to find glasses that have suitable Abbe numbers.
For 3 lenses, use our partial dispersion diagram to help choose glasses: in the 3D plot, the three glasses form a triangle, and you want that triangle to have the largest possible area (points spread out). The Deviation of Partial Dispersion tab shows how far each glass is from the best-fit line; glasses far from the line help form a larger triangle and better correction at three wavelengths. As a rule of thumb, include one glass in the fluorite/extra-low-dispersion family (e.g. N-PK51 or N-PK52A by Schott, or calcium fluoride).
For 4 lenses, use our partial dispersion diagram: the four glasses form a tetrahedron in 3D partial-dispersion space, and you want the largest possible volume (four points spread out, not all on one plane). The Plane Deviation tab shows which glasses lie off the best-fit plane; you need at least one such glass for the fourth vertex of a strong super achromat. Try different combinations and compare focal-length curves; the diagram helps narrow down promising candidates.
Spherochromatism is a combination of spherical aberration and chromatic aberration.
For advanced, fast lens designs spherochromatism is typically the limiting factor on the color performance of the lens.
One of the best ways to mitigate spherochromatism is slow bending of light with gentle radii.
To that end, we provide a very rough gauge of the spherochromatism of a lens by the thin-lens approximation.
If the radius required for a given focal length will be substantially small for a plano lens, then we display a warning.
If you are struggling with small radii, but not tiny radii, consider lens splitting in your design software.
If you have suggestions for improving spherochromatism in this tool, please contact us.
If you see a '-' there was a mathematical error in the calculation. This can be for a variety of reasons, such as being outside of the wavelength range of the glass or a division by zero error. Try adjusting your choice of glass or design range.
Design Analysis
Primary chromatic aberration metric: $C = \Sigma{\frac{1}{Vf_i}} = 0.00061269 mm^{-1}$ Not well-corrected (|C| = 0.000613)
Over design range [0.350–1.000 μm]: max $\left|\frac{f(\lambda)}{f_{target}} - 1\right| = 0.041483$ Poorly corrected (max = 0.041483)
Thermal expansion mismatch (−30–70 °C): |ΔTCE| = 0.0 ×10−6 K−1 Good thermal match
Spherochromatism (plano-convex approx.): worst |R| = 51.7 mm (N-BK7) Limited (|R| ≥ ftarget/4 = 25.0 mm)
| Glass | Manufacturer | f (mm) | $\frac{1}{Vf}$ | Rel. Cost | TCE −30–70 °C (×10−6/K) | |R| (mm) |
|---|---|---|---|---|---|---|
| N-BK7 | Schott | 100.00 | 0.00061269 | 1.0 | 7.1 | 51.7 OK |