Instructions

Introduction
Test Setup
Taking Images
Using the Program
Interpreting Results

Introduction

These instructions assume that you understand the basics of knife-edge testing. If not, there is extensive literature, both on-line and off, that explains the theory and the procedure of the test in great detail. Just about every book related to amateur telescope making, such as Texereau, Berry or Willman-Bell's collection, has a chapter dedicated to the test.

Online, try a Google search or one of these excellent resources:

http://bobmay.astronomy.net/harbour1.htm

http://www.stellafane.com/atm/atm_foucault_tester/atm_tester_main.htm

If you'd like to learn more about the mathematics used to interpret the results of a Foucault test, Jim Burrows has an excellent overview available at

http://www.halcyon.com/burrjaw/atm/atm_math.lwp/atm_math.htm

Test Setup

This program analyzes images from a moving-source Foucault test. This means that the light source (pin-hole, slit or another section of the same knife edge) moves together with the knife. The most popular type of moving-source tester nowadays seems to be the slitless tester. The simplest way to make one is to attach a large and bright LED to one side of a utility-knife blade, so that only one half of the LED illuminates the mirror. The camera is then placed near another section of the blade on the same side as the LED, so the reflection of the LED can be "cut off" by tilting the blade. It is essential that the two sections of the blade be parallel to each other, since they need to effectively form a slit as far as photons are concerned; in other words, the knife edge must be straight. The widely available and inexpensive multi-segmented utility knives tend to be quite satisfactory in this regard.

The medhod employed to analyze test images is a slight variation of the common Couder mask method. When the knife is inside the mirror's radius of curvature and touching the axis of the mirror, the image of the knife will appear to shade one half of the mirror's surface. On the outside of the radius of curvature the image will flip upside-down, so the knife will shade the other half. The distance where this flip happens tells us the radius of curvature of the mirror. When using a Couder mask, one divides the mirror radially into zones, and then finds the average radius of curvature of each zone by finding the exact knife position where the rate of darkening of the zone on the opposide sides of the mirror is equal. In contrast, this program uses "zones" one pixel wide, quantifies the illumination of the two mirror halves at many different knife positions, and then extrapolates the knife position where the illuminations get equal.

Because the method analyzes data from many images at once, accurate measurement of knife position for any one image is not critical. In fact, I was able to achieve good enough accuracy by sticking the LED/knife assembly into a block of wood, with the knife oriented horizontally, and moving the assembly by nudging it along a ruler attached to a flat surface. Eyeballing the relative position of the knife on the ruler is quite sufficient--no micrometers required! If you use this method, you'll need a way to elevate and tilt the flat surface so that the knife edge slides along the mirror's axis. The simplest way of achieving this is to run three screws through a piece of smooth plywood or plastic kitchen cutting board. Of course, this crude method is unsuitable for very large or very thin mirrors, since the mirror will sag under its own weight enough to significantly skew test results.

The number of knife offsets and images required to make a good measurement will vary with your mirror's focal ratio. A good starting point is to begin about a centimeter inside the average radius of curvature and take twenty or so images one millimeter apart. If you are using a tester with a slit or a pin-hole, the shadow flip will occur over a smaller distance so you can reduce knife travel accordingly.

Taking Images

To get meaningful results from image analysis, you need to take some precautions to ensure a good quality of images without large systematic effects. Here are some tips to ensure an accurate measurement:

Place your camera as close to the knife as possible. You will be taking a sequence of measurements with the knife at different offsets, so for initial setup place the knife at the rearmost location and adjust the camera lens so it almost touches the knife.
Fix the aperture, duration of exposure, focus and white balance of your camera throughout an entire measurement. You might be able to get meaningful results with a fully automatic ("idiot") camera, but it is certainly not worth the doubt if you can buy or borrow a camera with a "full manual" mode.
Try to keep the camera steady throughout a measurement. If you nudge it so the image of the mirror shifts a few pixels in the images that's OK, it can be taken care of in the program. If, on the other hand, you inadvertently change the zoom or the distance of the objective from the mirror, discard the measurement and start again.
Set a long enough exposure to get good contrast and to escape low-level random noise inherent in CCD devices, but not so long that you saturate pixels. To give you an idea, a half-second exposure seems to work well with a typical consumer camera and a three-candle green LED from Radio Shack.
Minimize background light, especially reflected sunlight if working during the day. On a partly cloudy day variations in sunlight intensity can render a measurement useless. Of course, the ideal setting is working in complete darkness with a black screen behind the mirror, but you don't have to go overboard to get accurate readings.
Use a large LED or a long slit. When the knife is far from the radius of curvature you'll be taking images not of a single point but rather of a relatively large section of the light source, as different points get reflected by different areas on the mirror.
Use "optical" and "digital" zoom and smaller image sizes so you can fit more and finer-grained measurements on your hard disk. If your camera won't zoom that close, you can also crop useless regions of images in an image editing program to save space. These image collections tend to get quite large over time...

Image format doesn't seem to make any practical difference. Lossy formats such as JPEG tend to lose information in an unbiased manner, so given a choice by your camera use any format that lets you store enough images to complete an entire sequence without having to download and erase images in the middle of a measurement. The program currently supports JPEG, PNG and GIF formats.

As a guideline, this is what a good sequence of images looks like:

If your camera can achieve this kind of contrast, uniformity and resolution, you're all set.

Using the Program

Now that you're armed with a good set of images of your mirror, it's time to fire up the mighty analysis engine. By this point you should have successfully downloaded and installed the software. Before proceeding any further, you should select a place on a hard drive with lots of free space where you'll keep all the images and the project file. For the sake of this discussion, let's say this location is c:\mymirror.

You will probably be making quite a few measurements as the figuring progresses. To avoid confusion, it's a good idea to keep the images of each measurement in a separate directory. Therefore, create a directory c:\mymirror\measurement-1 and download into it the images of the first measurement from your camera.

When you first start the software, you will be greeted by the project pane. Before doing anything else, fill in the title, description and parameters of your mirror, then select Save As from the File menu and save your project to a file such as c:\mymirror\mirror.xml. It's a good idea to keep the project file one directory level above the image files; that will save you some time when entering new measurements in the future, and makes it easy to back up the entire project. Once you save your project, the program will remember its location and open it automatically the next time you invoke it.

All the functionality of the program is accessed through four panes.

The project pane is displayed whenever the program is started. Here you can specify mirror parameters and describe your project in more detail. This pane also contains a list of all measurements. Note that measurements are listed most recent first.
The measurement pane shows details of a single measurement. The name of the measurement defaults to the date and time it is entered, although you can change it to any string. In the description field it's a good idea to write a few sentences describing the actions that led to this stage, for instance "two hundred parabolizing strokes" and so on. This information will be indispensable later on when you need to fine-tune the shape of the mirror or correct a specific defect.
The image pane allows you to select the region of an image to use for analysis, as well as specify the relative position of the knife edge at which the image was taken.
Finally, the analysis pane contains several tabs that display the results of analysis in different formats: as numbers, surface deviation from parabola, radius of curvature, data for a single zone, apparatus alignment and image contrast.

Images are on the disk, project file is saved, let's analyze.

While on the project pane click the New button to create a new measurement. The measurement pane should come up with a default measurement name and an empty description and list of images.
Name and describe the new measurement as appropriate, then click the Add button to add images to it. A file selection dialog should open in the c:\mymirror directory. Navigate to c:\mymirror\measurement-1, select all the images and click Open. The list of images will fill up with files sorted alphabetically and each image will be assigned a default offset starting from zero and incrementing by one millimeter.
Double-click on the first image (or click Edit while it is selected) to bring up the image pane. Use the buttons along the top to align the mirror outline with the image of the mirror in the photo. The straight line should be oriented like the knife edge, so if your knife is horizontal rotate the line by ninety degrees. Once you're satisfied with the results click Set as Default to copy these parameters to all the other images in this measurement. You can also exclude individual images from the measurement, for example if you want to keep Ronchi patterns with the rest of the measurement for comparison.
Check the rest of the images by clicking Next to ensure that the outline still covers the mirror, i.e. the camera didn't shift during the measurement. If you notice a small shift on certain images you can adjust the outline location only (not the radius or the knife orientation) for each image separately. These individual settings override measurement defaults and will show up in the list of images. To delete such an override edit the image and click Default. Also make sure that the knife offset, shown in the box below the buttons, is correct for each image (remember: millimeters).
Once you're satisfied with the settings for all images, return to the measurement pane and click Reduce All to begin data reduction. On a reasonably recent computer this process shouldn't take more than a few tens of seconds.
Click Analyze and wait for the numbers to show up. Read the next secion for aid in interpreting results.

You can apply a color "filter" to each measurement to increase image contrast--see the radiobuttons at bottom right of the measurement pane. What this actually means is that during data reduction only a single color of each pixel will be taken into account. How pixel colors map to the physical wavelength(s) of light used depends on the camera and the light source. For example, a green LED that I use excites green and red pixels in Canon G1 about equally, blue pixels hardly at all.

You have a choice between saving a little bit of space and a little bit of time. Any reduced data present when you save a project gets saved in the project file. For large images and a large number of measurements this can result in a very large project file. If you have a fast computer so you'd rather re-run the reduction every time you wish to repeat the analysis, click Compact on the measurement pane to discard any reduced data for that measurement before saving.

Interpreting Results

As you begin taking and analyzing measurements, spend some time adjusting your test apparatus so the knife travels along the mirror's axis. In other words, look at the Axis Alignment graph and get the data line as close to zero as possible by finely adjusting the orientation of you tester. Misalignment will produce inaccurate readings that get worse toward the mirror's center. [Note: there is probably a way to compensate for this from the available data on a purely geometric basis. If you have some spare time and a little programming skill, this improvement would make a nice high-school-level project.]

Another piece of information that you want to watch early on is the effective image contrast. Adjust the settings of your camera to maximize it, but do not saturate too many pixels since that will reduce the dynamic range at the other end.

As you get proficient in taking photos of the test, you will likely need to watch only the numeric results and the surface deviation plot; all the other graphs are shown only to let you fine-tune your test setup and as a sanity check.

The surface deviation measurement tells you what to do with the mirror to bring it closer to ideal. Any regions above the zero line need to have material taken off by polishing; the ones below zero should be polished less. The exact actions required to correct specific defects depend on too many factors to even start explaining here. The books mentioned near the top of this page will give you many ideas, but keep in mind that there is no single correct answer: any method that takes glass off in the right places will ultimately work. Also, your particular polishing tool will have its own "character." It is a very good idea to describe well every measurement, so that later on you will be able to find and reapply a method that in the past produced a certain desirable result with the same tool.

Finally, the numeric results of an analysis tell you when to stop working on the mirror. There are three quantitative measures of mirror quality:

Strehl ratio > 0.8
Strehl ratio is the fraction of starlight that ends up in the central peak of the Airy disk. A perfect mirror will reflect about 85% of the light from a point source into the central diffraction peak. Any deviations from parabola will smear some part of this light throughout the surrounding rings, and therefore reduce the resolution of your mirror. Strehl ratio better than 0.8 is the most commonly used threshold for a satisfactory mirror.
RMS < 1/14
The "root of the mean of squares" is a measure of how close a parabola can be fit to the measured shape of the mirror's surface. This is the quantity that is minimized when finding the closest parabola. This measure is an improvement over the Raleigh criterion (which states that the deviations should not exceed a quarter of the wavelength of light used) in that it weighs the relative contribution of deviations from parabola by the area of the surface segment. This criterion, unlike Strehl ratio, assumes a Gaussian likelihood function for the best fit, which doesn't take into account the wave nature of light. Use Strehl ratio instead.
Peak-Valley < 1/4 wavelength
This is the original Raleigh criterion for an effectively perfect mirror. The problem with this measure is that it doesn't take into account exactly how much of a mirror's surface deviates by a quarter wave, so different mirrors that satisfy this criterion can produce images of a dramatically different quality. This number is given only as an additional sanity check. Use Strehl ratio instead.

And when you reach that perfect mirror, please do send me an email describing your impressions about the program and suggestions for improvement. Remember, this code is free software, meaning not only free as in money, but also free as in you're free to study it and improve it yourself!

Maintained by Dejan Vucinic <dvucinic (at) users.sourceforge.net>
Last updated on May 28th, 2002.

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