Tolerancing is the process by which the effects of manufacturing defects and alignment errors are considered. This article is an overview of the entire process, and is intended to be the first article a new user would read to start to understand the process. Other Knowledge Base articles give more specific advice.

Let us assume that our task is to design a laser beam expander for volume production. An Argon-ion laser beam of 2.5 mm 1/e² full width is to be expanded x3, and the resulting wavefront error must not exceed l/20 RMS over two full beam widths (i.e. measured over a total beam diameter of 5 mm).

This file (which can be downloaded from the zip archive on the last page of this article) uses two plano-vex lenses:



The nominal system performance greatly exceeds that required in production:



Note that the maximum scale is 0.01 waves, so the peak-to-valley  OPD error is less than l/200. The RMS wavefront error (found by building a default wavefront merit function) is 2.10^-3 waves, which may be lower than you might expect by visual inspection of the OPD plot. Note however that the aperture definition of this system uses Gaussian apodization. The entrance pupil diameter is 5 mm, and an apodization factor of 2 is used, so at the edge of the pupil the beam intensity has fallen to 1/e⁴ = 1.8% of peak intensity.

That is why the specifications of the manufactured system matters so much. The RMS OPD of the system when illuminated by a laser beam of 5 mm 1/e² diameter (2.10^-2 waves) is different to that of the same system uniformly illuminated (3.10^-2 waves) because of the fast-varying spherical at the edge of the pupil, which is less important with Gaussian illumination than with uniform illumination.

We will now go on to consider the effects that manufacturing tolerances have on this system's performance.