Notes
Building a tolerance budget for a small imaging lens
October 14, 2026
A tolerance budget turns shop limits into one number you can set against an image-quality allowance. Each limit has a sensitivity from the design model. The contribution is that sensitivity times the tolerance, divided by the sensitivity step. Those contributions then combine. This page is an example for a 3-element lens. Every input and every result is illustrative. It is not a measured lens and not a client prescription.
Root sum square
The unit is waves RMS. For element 2 decenter, 0.025 waves RMS per 0.01 mm at a tolerance of 0.02 mm contributes 0.05. The budget figure is the root sum square, the square root of the sum of squares. A worst-case sum adds the absolute contributions and assumes every error hits its limit together. The root sum square treats the errors as independent, which is the usual small-lens budget unless one error is known to stack. The example allowance is 0.1 waves RMS.
Focus is the compensator. Element 1 power, carried as defocus, is 0.04 waves RMS per 0.01 mm at 0.02 mm, a contribution of 0.08. The row stays in the table and stays out of the reported root sum square, because a focus adjustment removes that defocus at assembly.
Example rows
| Item | Change | Sensitivity | Tolerance | Contribution | Compensator |
|---|---|---|---|---|---|
| Element 1 | power, as defocus | 0.04 per 0.01 mm | 0.02 mm | 0.08 | focus |
| Element 1 | center thickness | 0.012 per 0.01 mm | 0.03 mm | 0.036 | none |
| Element 2 | center thickness | 0.009 per 0.01 mm | 0.03 mm | 0.027 | none |
| Air gap after element 1 | spacing | 0.014 per 0.01 mm | 0.02 mm | 0.028 | none |
| Air gap after element 2 | spacing | 0.01 per 0.01 mm | 0.02 mm | 0.02 | none |
| Element 2 | decenter | 0.025 per 0.01 mm | 0.02 mm | 0.05 | none |
| Element 2 | tilt | 0.015 per 0.05 deg | 0.1 deg | 0.03 | none |
| Element 3 | refractive index | 0.006 per 0.001 index | 0.001 index | 0.006 | none |
| Element 3 | wedge | 0.018 per 0.01 deg | 0.015 deg | 0.027 | none |
What the rollup says
Including focus, the root sum square is 0.117 waves RMS, over the allowance. After focus is removed, it is 0.086, and the margin is 0.014. The worst-case sum of the compensated rows is 0.224, and with focus included it is 0.304. Both worst-case sums miss the allowance. The combination rule is part of the budget, not a detail to add later.
Element 2 decenter is the largest compensated contributor, at 0.05, and it is 33.9 percent of the compensated sum of squares. Cutting that tolerance from 0.02 mm to 0.01 mm drops the compensated root sum square from 0.086 to 0.074. The index row, at 0.006, barely moves the same total. Tighten the driver.
Where the sensitivities come from
On a real lens the sensitivities are perturbations of the named model. Zemax OpticStudio, Quadoa, LensHH, or OpTaLiX is where those derivatives are usually taken, and the scoping note names the file. This page only rolls the example numbers up in a custom script. The allowance, the compensators, and which errors may stack should be written down before the root sum square is treated as a pass or a fail.