# Report on finding best focus of slow systems

### Experimental setup:

A Point Source Microscope (PSM) was mounted on a motorized vertical stage above a 25 mm diameter, 200 mm efl lens sitting above a plane mirror. A square black paper mask with an 8 mm diameter hole was placed over the lens as shown in the figure below to give an f/25 aperture.

### Background:

I was asked, “How well can the PSM find best focus of slow optical system?” In particular, could best focus be determined to ± 10 μm on an f/25 system? Conventional wisdom would say this is not possible as the depth of focus is on the order of λ(f/#)^2, something like 400 μm in this case. On the other hand, conventional theory claimed sub-micron microscopy in the visible was not possible until the invention of confocal microscopy. The answer for me was to try an experiment.

(The black ring mirror mount surrounding the lens and mask where used in another experiment.) The PSM had a 4x microscope objective attached.

After centering the lens to view the back focus in the center of the PSM video monitor, the PSM was scanned vertically over a range of about 1 mm while collecting and storing data about the reflected image every 4 μm for a total of about 264 points per scan.

The centroiding algorithm in the PSM looks at pixels more intense than a user set threshold. The threshold was set to about 125 out of an 8 bit range of 256. When the PSM is used for alignment, the centroiding algorithm calculates the center of gravity of those pixels above the threshold. The software also records the number of pixels above the threshold at each point in the scan. The shutter speed or exposure was user set so there were about 35 pixels above the threshold near best focus.

The limits of the 1 mm scan were then set to give a roughly symmetrical distribution of pixels around best focus so that a typical scan gave a curve in Excel like that below.

Using the curve fitting options in Excel, a quadratic curve fits well as expected. If the criterion for best focus is taken as the distance where there are the greatest number of pixels above the threshold, then taking the derivative of the equation and setting it to zero gives x = 0.2842 mm. The data are a bit noisy but the results look promising.

Much to my pleasant surprise, the other 9 repeated scans looked very much the same. The results are summarized in Table 1.

 3 y = -84.213x2 + 48.431x + 29.412 0.2876 4 y = -82.682x2 + 46.997x + 29.391 0.2842 5 y = -82.427x2 + 46.602x + 29.297 0.2827 6 y = -84.332x2 + 47.958x + 29.28 0.2843 7 y = -81.882x2 + 46.543x + 29.005 0.2842 8 y = -82.67x2 + 46.447x + 29.301 0.2809 9 y = -82.649x2 + 46.619x + 29.445 0.282 10 y = -84.167x2 + 47.557x + 29.56 0.2825 11 y = -81.648x2 + 46.481x + 29.519 0.2846 12 y = -81.92x2 + 46.969x + 29.705 0.2867

Table 1 The scan number, equation of fit to the number of pixels above threshold vs focus position and the position of best focus in mm

Ten scans through focus were taken under identical conditions, that is, over the same distance with the same threshold and exposure. The data from the scans were fit to second order polynomials as in the Scan 7 picture, and the coefficients of the fit are displayed in Table 1. Solving for the best focus position, defined as the position where there are the most pixels above the threshold gave an average best focus position of 0.2840 ±0.0020 mm.

These encouraging results must be viewed with two caveats; the test for finding best focus was a double pass test off a plane return mirror so single pass results will be less precise by a factor of two. The other caveat is the definition of what constitutes best focus, this may not be best focus in terms of best resolution for a particular spatial wavelength. However, it is a useful definition if it is desired to see if multiple examples of the same optical system behave similarly to a measurement of best focus.

The conclusion of this simple experiment is that the PSM can find best focus for an f/25 lens in double pass transmission with a standard deviation of 2.0 μm when axially stepped multiple times through the region of focus. Each scan took about 5 seconds, and the fitting and analysis of the data from multiple scans are easily automated.

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#### Robert Parks

Robert Parks received a BA and MA in physics from Ohio Wesleyan University and Williams College, respectively. His career started at Eastman Kodak Company as an optical engineer and then went on to Itek Corp. as an optical test engineer.

He learned about optical fabrication during a 4 year stay at Frank Cooke, Inc. This experience led to a position as manager of the optics shop at the College of Optical Sciences at the Univ. of Arizona and where he worked for 12 years and had a title of Assistant Research Professor. During that time he had the opportunity to write about the projects in the shop and the optical fabrication and testing techniques used there including papers about absolute testing and the installation and used of a 5 m swing precision optical generator.

Mr. Parks left the University in 1989 to start a consulting business specializing in optical fabrication and testing. Among the consulting projects was one working for the Allen Board of Investigation for the Hubble Telescope where he stayed in residence at HDOS for the duration of the investigation. In 1992 he formed Optical Perspectives Group, LLC as a partnership with Bill Kuhn, then a PhD student at Optical Sciences.

The consulting and experience with Optical Perspectives provided many more opportunities to publish work on optical test methods and applications. While still at Optical Sciences, Mr. Parks became involved in standards work and for twenty years was one of the US representatives to the ISO Technical Committee 172 on Optics and Optical Instruments. For two years he was the Chairman of the ISO Subcommittee 1 for Fundamental Optical standards. Recently Mr. Parks temporarily rejoined Optical Sciences part time helping support optical fabrication projects and teaching as part of the Opto-Mechanics program.

Bob is a member of the Optical Society of America, a Fellow and past Board member of SPIE and a member and past President of the American Society for Precision Engineering. He is author or co-author of well over 100 papers and articles about optical fabrication and testing, and co-inventor on 6 US patents. He remains active in development of new methods of optical testing and alignment.

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University of Toronto

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