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ATM FOUCAULT TEST - MIRROR SURFACE PROFILER UTILITY v.0.5
Overview
The ATM FOUCAULT TEST - MIRROR SURFACE PROFILER UTILITY v.0.5 is intended for the private use of Amateur Telescope Makers who grind, polish and figure their own precision optics, and who use the Foucault Knife Edge Test in the process, as a way of gauging the surface's conformity to a desired concave shape - usually a paraboloid or a spheroid.
System Requirements
Computer: Developed and tested on Windows XP Pro. Written in (Borland) Delphi 3 which should run on any Windows/Intel - compatible machine which can support a Win32 Operating System: Windows 95 / 98 / ME / 2000 / XP (home/pro) / NT 4.0 or later.
Knife Edge Tester: Moving source (along with knife-edge), or "Slitless" Knife Edge Testers.
(* For Fixed Source / Moving Knife Testers, the Measurement Readings entered may need to be divided by two. Even then, the accuracy may suffer on fast {low f/ ratio} mirrors.)
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Fig. 2 - Zone Scale Indicator placement
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The Zone Indicator Scale is located even with the outer rim of the mirror. Usually this is referred to as an Everest Scale. Though I prefer to use a (laser-) printed scale on a piece of thick paper, laid just below the center of the mirror, and carefully aligned to the edge of the mirror. Accurate placement of the scale (and subsequent readings) are critical to final accuracy when dealing with fast mirrors (f/4.5 and below).
History - Rationale
I built my first telescope at the age of fourteen. Grinding a little 4-1/4" mirror was a daunting task for a teenager with few practices shop skills. The prospect of keeping even that small surface accurate to within a few millionths of an inch seemed impossible. My guidebook was Sam Brown's "All About Telescopes" from then Edmund Scientifics. Most confusing was how the knife edge tester could "amplify the mirror's surface" by a million times or so.
The alarming fact is, that it DOESN'T! Not really. What it DOES do is measure the reflecting angle (or the SLOPE) of the surface, and even then, only indirectly.
A few years ago, I began another mirror project - an 8" f/8, from a kit supplied by Newport Glass Works. This new challenge had four times the surface area of my first 4-1/4", and was much less forgiving in terms of the final curve. In the old days, I had thought that a mirror SURFACE which departed no more than one-quarter-wave from ideal, would yield perfect images.
However, the Rayleigh Criterion specifies that the standard of perfection is the resulting wave front, not the mirror surface. By Rayleigh's Standard, a wave front which deviates by no more than one-quarter wave, from peak-to-valley, will deliver an image as optically perfect as physical optics can deliver.
Certain that the earlier methodology, which I had attempted to learn, and still understood poorly, was flawed, I started searching for a measuring method which was comprehensible, accurate, and within my means of implementing.
History - Numerical Methods
The first improvement came to my attention through the Internet. ATM's had improved upon the genius of the Foucault Test with a version known as the "Slitless Knife-edge Tester". The greatest benefit, to my mind, is that it measures on the optical axis. Each measurement represents the crossing point for a line exactly perpendicular to the slope of the mirror at the spot (zone) being measured. (This is NOT the same as the Center of Curvature, except for the central zone - Zone 0).
Earlier attempts at numeric integration assuming parabolic arcs between successive zone points, proved tantalizingly successful, but only for an already ideal parabola! A similar attempt using circular segments worked somewhat better, but the errors were still not convincingly accurate to small fractions of a wavelength. Hence my mistrust of piecemeal methods.
The parabolic and circular approximations are similar to Simpson's Rule, found in modern Calculus courses. In this case, one is integrating the slope of the curve, at a few discrete points, to ascertain its height at those points. However, with the measurements being taken in thousandths of an inch, (or hundredths of a millimeter) a corresponding error in millionths of an inch is undesirable, if not unavoidable!
One ATM math web page hinted at doing a fourth order Runge-Kutta approach, found in Differential Equations, in order to obtain the necessary accuracy. But the necessity to make several measurements covering small intervals (between test zones), seemed burdensome. It would be more work than I or many ATM's are willing to undertake, especially if we didn't understand the process in the first place.
All was not lost. Soon the Ghost of Calculus Passed visited me with yet another gift - the series expansion of functions! Such methods provide a very good way to achieve accurate approximating curves for the mirror's surface profile, and their use benefit greatly from the computing power of modern PC's.
Yet, since we can only determine the slope of the mirror (the first derivative) at a few test zones, a way had to be found to accurately determine the higher order derivatives required by Taylor and MacLaurin Series. Using averages among adjacent zones would have worked only if the curve to be determined was already a parabola. Otherwise errors would creep into the process, and no improvement in accuracy or certainty would result.
Splines and Approximating Polynomials
In the end, higher order derivatives are not required. It turns out that knowing both the (approximate) X and Y' values will allow the creation of an approximating polynomial of the slope ( dy / dx) curve y' = f'(x). The process is similar to that used to calculate cubic splines which allow fitting smooth curves to relatively few data points.
As it turns out, the slope curve can itself be integrated to produce another approximating polynomial - that of the Mirror Surface Profile itself, y = f(x). At last, the actual surface heights can be known, and direct assessment of surface accuracy is assured. *
(* Accurate determination of surface height rests on the assumption that the curve of the mirror surface is relatively smooth between measurement points. Surface roughness, irregularities or sudden jumps can invalidate the results, which is true of most other Foucault analysis methods as well.)
The Process
In short, here's the algorithm:
1. Enter system parameters and measurement data
Accuracy is improved if data for multiple test sets are entered (for the same test zones, of course!) The data are averaged for each zone and the offset of the central zone is subtracted, to produce the final Measurements.
2. X-Adjustment (Parallax)
Calculate the Parallax between the Zone Scale and the actual mirror surface where the readings apply. (A circle with a radius of twice the focal length is used. Later, an iterative approach refines these X-adjustments to even greater precision.)
3. Calculate Slopes
Calculate the slopes of the measured Zone points.
4. Calculate Coefficients
Calculate the coefficients of the Slope Approximating Polynomial. This is a Gauss-Jordan matrix manipulation algorithm for solving systems of linear equations. The terms for the matrix are a power series for the values of X, approximated earlier, equating to the slopes for the given zone.
5. Refine Parallax X-Adjustment
Using the newly obtained coefficients for the slope polynomial, calculate the approximated measurement point for each zone. Subtract the original measurement point, and divide the difference by the slope of the Measurement Line to get the adjustment to X, which will be used in the next iteration.
6. Iteration Loop
Repeat steps 4 and 5 until the desired number of iterations is complete. Fairly accurate results are obtained in fewer than five such loops.
7. Surface Height Profile
Integrate the slope polynomial to yield the Mirror Surface Profile. (This always assumes that the height of mirror at zone 0 is 0.000...).
Instructions - How To Use The Profiler
1. Enter the Mirror Parameters
Diameter and Focal Length.
2. Enter Measurement Readings
The Data Entry mode must be selected. Enter Zone Radii for each zone tested.
Leave untested zones (and readings) blank, on right.
Enter readings for each test set in the corresponding row. The software will automatically average the readings, and subtract the Zone 0 offset. Do not enter partial test sets. They will not be used.
3. Click on the "Calculate" button
The display will turn to the Parabola comparison mode. A graph at the bottom of the form, will display the difference between the Mirror's Surface Height and the Surface Height for the Ideal Parabola having the same focal length.
4. Save Data to File
If you desire to save the calculation results to a text file, select [File -> Save] from the menu. Choose a folder and filename, then click on [OK]. The output file is in standard ASCII text format (no graph, sorry!) But it can be imported into any spreadsheet using a fixed width format filter, for whatever calculations and graphing isn't provided in this release.
Other Display Modes
Show Parabola Data - Compares mirror surface height to Ideal Parabola of same focal length.
Show Sphere Data - Compares mirror surface height to Ideal Sphere of same radius.
Show Check Data - Indicates validity, accuracy and error range of latest calculations. Note that the row labeled "Ideal Measure" indicates zone readings for Mirror having profile of Ideal Parabola. (Readings for Ideal Sphere are always 0.0 for all zones.)
(* Accurate determination of surface height rests on the assumption that the curve of the mirror surface is relatively smooth between measurement points. Surface roughness, irregularities or sudden jumps can invalidate the results, which is true of most other Foucault analysis methods as well.)
Notes on Unit Conversions and Decimal Separator
Consistency - as long as the same units of measure are used for all numbers entered, the results (other than wave values) will be in those same units.
Conversions
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Entries
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Non-Wave Results
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Wave Result Correction
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INCHE S
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Inches
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Inches
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read results directly
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Millime ters
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Millime ters
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Millimeters
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divide by 25.4 to get accurate wave results
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Centim eters
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Centim eters
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Centimeters
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divide by 2.54 to get accurate wave results
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Decimal Separator - The standard decimal separator for the U. S. A. is the period. This utility recognizes only the period at the moment. The comma (virgule / virgola) is standard decimal separator for most other industrialized nations, Yet, its use here is not supported at this time. Please use the period until revised software is released.
Notes, Licenses, Caveats, Warnings, Disclaimers, etc.
This version (0.5) is FREEWARE!
It is made available FREE of charge, "As Is", without warranty or guarantee of suitability for any particular use.
The author maintains ownership and copyright of the intellectual property involved, but grants a General Public License to anyone who may wish to download, use, copy, share or distribute for noncommercial purposes without charge or expectation of a profit for doing so.
Commercial interests (optical shops, commercial telescope makers, etc. ) are invited to obtain and use the software in order to determine its suitability for use in their routine business operations.
Feedback - This is intended to be the first release of the Surface Profiler, not the last. If you download and use the utility, and have some comment about how to improve its functionality or usefulness, or additional questions about its use, accuracy or future releases, please contact the author at the link provided below.
Determining Accuracy - This utility was tested on an Idealized 10-inch, f/3 mirror curve, for both sphere and paraboloid figures. Fast (short-focus) mirrors usually show the greatest potential for mistakes. The X-Zone (Parallax) Adjustment compensates for some of the errors introduced by most zone-masks. Given the idealized numbers for such a fast mirror, the resulting surface errors were still below 1 / 1,000th of a wave. Of course, in the real world, measurements are seldom more accurate than 0.0005 inches.
The user may wish to play around with the numbers to see how drastically truncation errors can affect certainty of the final figure. This is especially true of the inner zones. Outer zone measurements are much more forgiving. And do not forget to vary the mirror's focal length up to an inch or so in either direction. For fast mirrors, a mirror-to-knife-edge distance (Zone 0 radius) should be measured to within 1/8 inch or better.
Bibliography and Resources
Books
- All About Telescopes - Sam Brown (Edmund Scientific)
- Amateur Telescope Making - Vol. 1 (of 3) - Ingalls ? (Willman-Bell)
- How to Make a Telescope - 2nd Ed. - Jean Texareau (Willman-Bell)
- Numerical Methods for Engineers - 3rd Ed. - Chapra and Canale - WBC / McGraw-Hill
Download & Setup
- Download - Foucault.exe Click Here!
- Save the downloaded program to the desired destination folder for execution.
- No Installer is required. The program is stand-alone, and does not need VBRUNxxx.dll or VCL*.* support files.
- If the program is helpful, you may wish to create shortcuts to in for your desktop or Start Program menu.
- Filesize = 471,552 bytes.
- File Date: Wednesday, January 26, 2005, 12:51:56 AM
- Requirements: Windows 95 or better, 16 MB RAM ?, VGA Monitor
- Instructions / Help File: This webpage! (Save/print a copy!)
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