Tolerancing is about accounting for manufacturing error, so that the 'as built' design meets its intended specification. In order to tolerance successfully, you should know what manufacturing and test methods will be used in the production of the lens, so that your model of the production process matches reality.

There are many diferent methods used to manufacture and test lenses, and ZEMAX gives you ways to model all of them. In this case, we will assume that the lenses are made through traditional hand-polishing methods, and make further assumptions as stated below.

Manufacturing tolerances are entered through Editors...Tolerance Data and then clicking on Tools...Default Tolerances in that editor. The Default Tolerances dialog allows you to enter system-wide tolerances, which you can then edit by hand if necessary:



'Surface' tolerances are those tolerances placed on an optical surface itself, e.g radius, irregularity etc. 'Element' tolerances are positional tolerances on the element as it is placed in the optical system.

The default tolerances chosen to start this tolerancing run with are as follows:

Radii of curvature tolerances will be measured by using a test plate and looking at interference fringes in double-pass, using a He-Ne light source. This is a standard manufacturing test method. Let's assume we choose to measure radius tolerances in fringes, and to allow a maximum of 1 fringe difference between the surface under test and the test plate.

As this measurement is performed at He-Ne wavelengths, set the test wavelength to 0.6328. Note that this is not the wavelength the system itself operates at (which is the Ar⁺ line at 0.5145 microns): this is the wavelength at which the surface is tested. 

The default center thickness tolerance os 0.2 mm: we also need to account for how the lens is mounted, and this will be discussed further later in this article.

ZEMAX then allows you to enter the surface decentration and tilt. Care is needed here. An aspheric surface may have separate decentration and tilt. For example, a parabolic mirror may be seperably decentered and tilted, as it has only a single axis of rotational symmetry. A sphere has an infinite number of axes of symmetry, and tilts and decenters are therefore degenerate. 

It is usual in lens manufacture to measure the wedge of a lens holding the lens in a vacuum chuck and either (a) projecting a laser beam through the center, and observing the precession of the beam on a distant wall and the lens is rotated, or (b) using a dial gauge to measure the variation in lens edge thickness as the lens is rotated. The lens is then centered and edged to minimize either variation.

In this case, I assume that the manufacturer will use edge-thickness variation as the test method. I have entered a maximum tilt in both x and y of 0.2mm, so that the maximum radial tilt is SQRT(0.2² + 0.2²) = 0.28 mm.

Surface irregularity can be defined as a sum of spherical and astigmatism (which is usually what is done when test-plate interferograms are 'eyeballed') or using Zernike coefficients (which is usual when measurements are made with an interferometer). In this case I assume the 'eyeball' method, and set a surface irregularity of l/5 initially.

Element tolerances measure how misaligned the elements are with respect to the 'mechanical axis' of the system.

Index tolerances describe how different the actual glass used is, compared to the catalog values. As this design is used at only one wavelength, we don't need the Abbe tolerance and simply tolerance the refractive index value itself.
 
ZEMAX also offers to build a back-focus compensator. As this lens is afocal this is not needed, and so it is unchecked. When the OK button is pressed, a set of default tolerances is produced.

In the next section we will discuss mechanical mounting and compensation.