Tolerancing surface irregularity is difficult because surface irregularity is not deterministic. Often times, the RMS surface error introduced on an optic during manufacturing is specified by the lens supplier by taking the average of RMS surface error for a single sample or a statistical batch of samples. It is common to hear that a surface is "flat to l/10 or l/20" As we shall see, the spatial frequency of the irregularity is also important.

Tolerancing for RMS surface error is straightforward using the TEZI tolerance operand, assuming :

- The original surface type is Standard or Even Aspheric
- Zernike polynomials can reasonably represent the physical surface error. This is usually a  good assumption if an interferometer is used for surface testing, as the interferometer software can normally list the error between the real surface sag and that produced by a specified number of Zernike terms.

The RMS amplitude of the surface irregularity does not define the shape of the irregularity. ZEMAX replaces the Standard or Even Aspheric surface used to describe the nominal system with a  Zernike Standard Sag surface for tolerancing. This has the same basis surface shape as the Standard or Even Asphere surface, and Zernike coefficients represent the deviation from nominal surface shape. For more information about the Zernike Standard Sag surface, please refer to the user manual.

We will illustrate the use of the TEZI operand through a simple example. First, open the included lens file TEZI.ZMX. This file uses the afocal image space mode.





and it represents looking at a flat window. Of course, the surface can be any shape supported by the Even Asphere surface, but we will use a flat surafce for simplicity. Set the tolerance data editor as following.



The tolerance is set to 1mm RMS surface error on surface 2. The min tolerance value is automatically set to the negative of the max value; this is done to yield both positive and negative coefficients on the Zernike Standard Sag surface. The resulting RMS is of course always a positive number whose magnitude is equal to the max tolerance value.

The number of Zernike terms is given by MAX# and MIN# parameters. Generally speaking, if lower order terms are used, the irregularity will be of low frequency, with fewer "bumps" across the surface. If higher order terms are used, there will be higher frequency irregularity, with more "bumps" across the surface. The maximum and minimum number to be used can be chosen by looking at pieces produced by the manufacturer with an interferometer, and setting the minimum and maximum terms required to give good fit to the surfaces produced by the manufacturing process. This will be discussed in more detail later. For this example, we will use between term 2 and term 9

The 1st term of the Zernike standard polynomial corresponds to piston which ZEMAX always ignores; therefore, the smallest possible Min# parameter value is 2. The maximum possible Max# parameter value is 231, although such high terms are almost never required: terms up to 28 or so are usually the highest required.

Open the tolerancing window under Tools > Tolerancing > Tolerancing and set set the following values.


Run the tolerancing by clicking OK.

The tolerance reports shows the statistical results for the criteria; RMS wavefront error.



To see the saved monte carlo file, (we only saved one, but we could have saved all of them), open the file named MC_T0001.ZMX located in the same directory as the TEZI.ZMX lens file.

Notice how in the Lens Data Editor the surface #2 type is set to Zernike Standard Sag surface.

Open the Surface Sag analysis under Analysis > Surface > Surface Sag and select surface #2.



If we re-run the tolerance with the Max# parameter set to 27 you would get a sag similar to shown below. Notice how there are more bumps across the surface. The number of Zernike terms can control the frequency of the peaks and valleys (bumps).



Now this is an important point. As we polish a surface from l/5 to l/10 to l/20 to l/50, the RMS surface deviation decreases, but usually the spatial frequency of the irregularity increases. Surfaces polished to say l/5 are often quite "slow" in terms of the spatial frequency of the irregularity, whereas super-polished surfaces often have a very high spatial freqeuncy of irregularity.  The optical perfromance of a surface depends not only on the RMS amplitude of the irregularity but also on the frequency of those peaks and valleys, because it is the slope of the surface that bends rays. To illustrate this, open the attached "periodic surface.ZMX" file. The surface #2 type is Periodic with a periodic structure in Y direction only.  The 3D layout shows the difference in the ray trace results when the frequency of the periodic structure is increased while keeping the amplitude constant.



The Universal plot shows how the RMS spot size in Y direction changes as a function of the structure frequency.




This is why it is important to model the spatial frequency of the irregularity, as well as its RMS amplitude. See Reference 1 for more details.

This article has demonstrated the use of the TEZI operand for surface irregularity tolerance. In summary:

• Monte Carlo tolerancing should be used  for manufacturing-related surface irregularity tolerance
• The spatial frequency of the irregularity as well as its RMS amplitude must be modelled

References

1. Optical System Design, R. E. Fischer and B. Tadic-Galeb, McGraw-Hill, ISBN0-07-134916-2, Chapter 16