Testing Telescope Mirrors
This article covers a range of issues involving the manufacturing and testing of telescope mirrors of medium size aperture (8 to 16 inches) and problems connected with the interpretation of both Ronchigrams and interferograms and the proper understanding of spurious test results that arise from mechanical distortion and atmospheric turbulence in the shop environment. Interpretation of test results from medium size optics can be a subtle and complex exercise that can, if not properly understood, result in misleading analysis. There is an unfortunate notion circulating that each mirror can be supplied with an interferogram that will exactly describe the surface of that optic, a single, unique portrait, so to speak; an interferogram that can be independently analyzed by a third party to verify the analysis supplied with the optic in question. While this may be the case with small optics, this is simply not the case with moderate to large optics and I will explain why as this article progresses.
The Calibrated Ronchi Test
The manufacturing and testing of a mirror is a process distinctly different from the final testing and verification of an optical surface. The manufacturing process must be one that allows the optician to conveniently and without wasting valuable time produce an optical surface of known and dependable quality. While this optic may later be verified by a quantitative test such as interferometry, such tests are often difficult to perform during the course of manufacture. The optician must rely upon a test that is convenient and handy and which will allow him to gradually bring the optic to as near a condition of perfection as he can reasonably attain. Even in these days of easily available interferometry, the practicing optician will most likely employ some kind of knife edge or related test during manufacture. Variants of the knife edge or Foucault test usually include the Ronchi test, wire test, and caustic test. These tests can be further modified for convenient shop use in the testing of aspheric surfaces by the employment of zonal testing masks as well as optical compensators or "nullers" such as the Dall nuller and the Ross nuller. Other tests such as the Hartmann test share with interferometry the difficulty of application that reserves their use for final verification rather than manufacture. My own preference for the production of parabolic mirrors is what is known as the autocollimation test. This test has the advantage that once the optician has the proper setup, the production of parabolic mirrors can be considerably enhanced because the test requires no reduction of data and is a simple null test. The results are obvious and evident at once. Also, the fact that the test is a double pass test makes it a particularly accurate one. My technique is to use it with Ronchi gratings of increasing fineness as the figuring process progresses until the appropriate accuracy is reached. Other workers use the autocollimation test in conjunction with the knife edge test. The problem with all of these shadow or diffraction tests is that they are only a measure of slope error through a single segment of the optic and do not tell the worker about the entire surface and the various other errors that might be present, such as astigmatism, and do not yield a specifically quantified measure of accuracy. Mask-based tests only give an average slope over the area of the mirror exposed in the mask apertures, so narrow peak-to-valley errors are undercounted, sometimes grossly. Peak-to-valley based on knife-edge reduction therefore is a "filtered" value, leading to a false notion of the quality of a mirror, relative to what an interferometer will give for the same mirror. Because interferometers sample at a much higher rate across the surface, as well as seeing the whole surface and not just a cross-section, peak-to-valley derived with an interferometer will usually be much worse than what the knife-edge reduction will give.
It is the utilization of the simple knife edge and Ronchi tests in comparison with known interferometric results and the establishment of boundary norms that can yield repeatable and dependable accuracy. To this end, I have developed a method for testing mirrors that permits the production of high quality objectives in a convenient test environment that insures accurate results on a confidently repeatable basis. The test involves a calibration of the autocollimated Ronchi test with interferometric comparative analysis as well as computer generated Ronchigram boundary images. Discovered in 1923 by the Italian physicist Vasco Ronchi (pronounced Ron-kee), the test, if properly employed, is one of the simplest and most powerful tools available for the evaluation and measurement of an optical wavefront.
The essence of the calibrated autocollimation Ronchi test is the use of an auxiliary master flat to axially reflect parallel rays generated from the point source at the focus of a mirror under test back to the mirror under test once again and finally back to the focal point so as to render a parabolic mirror null when observed through the grating. The optical path of such a test is seen in the figure below. As you can see, the flat gets hit by the light once and the mirror under test twice.
Null tests are usually considered the best and most accurate of tests inasmuch as they require no further complex analysis or reduction. The results are immediately self evident. In the present case, the accuracy of the mirrors and the flats used as autocollimators was tested interferometrically by making several parabolic mirrors in autocollimation using the flats and then examining these parabolic mirrors at the radius of curvature with an interferometer completely independent of the auxiliary test flats and reducing the data by means of mathematical algorithms. The mirrors were fabricated using grating of 133, 200 and 500 lines per inch in double pass. These mirrors were then tested on an interferometer with no intervening nulling optics and the measured aberrations, such as spherical aberration, plotted against predicted amounts. Mirrors down to f/5 were tested in this manner and found to be well within the desired wavefront accuracy. Once the resolution limit of accuracy of the autocollimation test was established computer-generated images of predicted Ronchi test results were developed to form boundaries from which to work.
The following are sample boundary analysis synthetic Ronchigrams computed from Fourier analysis. They are monochromatic and coherent, so they look just like what you would see if you used a laser for the light source and a grating of about 133 lines per inch in double pass or simply viewed through a telescope pointed at a star with a 266 line grating at the focus.
The following Ronchigrams are the same as above except that they are modeled assuming a slit of the same width as the grating, thus simulating the actual appearance of the shop test, which uses the same grating a optician looks through to cover the light source.
The following are actual Ronchigrams taken of a 10", f/6 mirror in autocollimation using a 133 line per inch grating. These are uncropped CCD images exactly as I see them under test. The only enhancement used in this presentation was an increase in contrast.
Much of the mottling is due to atmospheric turbulence. One can see how even with a double pass null test the difference between a 10th and an 8th wave is not great, but well within the ability of a trained worker.
An even greater level of error can be seen by using gratings of higher frequency. Although the results are not aesthetically pleasing, I routinely use finer gratings for special levels of examination. Finer gratings are needed when testing optics of smaller focal ratio. A 133 line grating is appropriate for an f/6 or f/8 mirror but 200 lines would more appropriate for an f/4 mirror, while 85 lines would be appropriate for an f/15 mirror or refractor objective.
The following are from a 12.5", f/6 mirror subjected to a 200 line per inch grating. It must be emphasized that this mirror is better than .03 RMS or .965 Strehl. The bending of the bands at the top of the inside focus Ronchigram and the bottom of the outside focus Ronchigram are due to atmospheric effects within the shop area.
It is generally not realized that the Ronchi test can become enormously powerful when used in a null configuration and employing gratings of very high frequency. Much of the criticism directed against the test is due to its frequent misapplication as a center of curvature test and an attempt to render the test quantitative in that configuration. While the Foucault test in a similar embodiment (zonal testing at center of curvature) can yield highly accurate results in the hands of a skilled worker, the Ronchi test does not yield the same degree of accuracy, but when the Ronchi grating is used in a null configuration, amazingly high accuracy can be attained.
Unlike the knife edge test and the geometric Ronchi test, which only look at a single vertical slice of the optic under test, the interferometer is able to look at the entire optical surface at a single glance. In this way, a wide variety of aberrations unnoticed in the former tests become immediately visible and precisely measurable. It is the ability to quantitatively measure these wide variety of aberrations that make interferometry such a powerful and valuable tool to the practicing optician. While not as handy and quick in use as a knife edge or Ronchi test, the interferometer is an indispensable tool for the final verification of an optical surface or system. The interferometer used in my tests is a Shack type interferometer of simple design. In essence, it consists of a beam splitter cube and a reference element of verified high accuracy cemented to the front side of the cube. It is this reference surface which forms a comparison or reference wavefront by which all the optical surfaces under test are compared. The method I utilize is to take interferograms at the radius of curvature without intervening compensating optics. These yield interferograms having curved fringes that describe the degree of asphericity present in a parabolic mirror or a surface of any asphericity. The interferograms are then processed through a appropriate software that compares the actual wavefront as described by the fringes against desired norms for the given optic under test. This type of analysis is direct and relies entirely upon mathematical reduction so that there is little chance of error creeping into the analysis as a result of the use of compensating or nulling optics.
The primary aberrations taken into consideration are spherical aberration and astigmatism. Other aberrations such as defocus, coma, and tilt are usually cycled out of the analysis routine. During manufacture spherical aberration or correction is the principal aberration considered inasmuch as it is the setting of the proper correction through parabolization that is the primary task of the optician. Astigmatism must also be examined because it is a defect which can creep into an optical surface if careful production techniques are not adhered to or if the optics under production is inherently subject to astigmatic defects. Such an astigmatism prone optic might be, for example, a very thin mirror, or even worse, a thin large mirror. In any case, astigmatism must be accounted for during the course of optical production, though it is typically not visible in the knife edge type tests except when quite large. Such defects as zones (broad rolled edges or internal high or low rings) and turned edges, will also manifest themselves in interferometric analysis and have an impact on the final RMS value and Strehl ratio.
Defining Reality - Artificial Distortions
In the manufacture of medium to large size mirrors, particularly those of less than generally accepted "full thickness" proportions, false astigmatism generated by test supporting devices can be a problem that can result in an optic being unfairly measured. This problem does not usually manifest itself in optics below 8 inches in size and of standard thickness ratio, but in optics of ten inches and over this can be a very serious problem. In normal use mirrors are not usually held on their edges but flat on their backs. Tests performed in my shop indicate that mirrors ten inches and over, having a standard diameter thickness ratio of 6 to 1, and supported vertically, can produce false astigmatism to the extent of 1/4 wave and greater in the wavefront reading. In larger mirrors this problem will manifest itself in exponential proportions. In order to fully understand and intelligently compensate for this problem, a variety of tests were run to determine the exact extent of this false astigmatism. Two different supporting mechanisms were utilized in these tests, each generating a their own unique astigmatic patterns. The two pictures seen below are of a 12.5" f/6 mirror, 2" thick. The pictures at the left show the mirror held with two pads at the bottom sides spaced approximately 30 degrees apart. In in the two pictures at the right, the mirror was held with a strap. The differences in the wavefronts as the result of the two supporting mechanisms are easily seen. In both cases the interferometer was defocused outside of focus so as to produce concentric rings. These should have been perfectly circular.
As can be seen, the mirror is compressed or squashed so as to generate egg-shaped (with the pads) and oval (with the chain sling) patterns in the rings. Because the interferometer is defocused outside of focus, the rings show a reverse pattern with the major axis oriented horizontally. The aberrations generated appear to be about 1/3 wave when visually examined. In order to confirm that the astigmatism seen is related to the manner in which it is supported and not a part of the optic itself, the mirror was rotated to four positions 90 degrees apart and viewed interferometrically. In all cases the astigmatism remained essentially stationary. There is no question that the mirror is being artificially distorted.
Various other things can occur which can produce anomalous false astigmatism patterns in an interferogram. Other than distortion due to gravitational forces, as described above, these principally include spurious refractions due to air currents. These can be particularly destructive and produce the effect of false astigmatism to the extent of 1/4 wave or better in extreme cases. The test must be carried out in the temperature controlled environment not only to ensure that the optic under test is not in the process of expansion or contraction but that formation of air currents is inhibited.
What should begin to become evident in all of this is that taking a single interferogram of any optic, and in particular a large optic, can lead to misleading characterization of the actual optical surface and its resulting wavefront. Verification of larger optics to high accuracy is a difficult and tricky proposition and must be handled with extreme care. Practitioners of interferometry understand that the certification of an optic requires the taking of many interferograms in succession, understanding and omitting aberrant information, and reducing the analysis to an accurate statistical mean. It's not just snapping a picture and measuring the fringes.
An Actual Interferometric Analysis
As an example of the kind of analysis necessary I offer the following 10" f/6 mirror as a case study. This mirror may be taken as a representative example and its accuracy as typical of what one gets from me when purchasing of one of my standard mirrors. It is not a perfect mirror, but one that one would reasonably expect to receive. The mirror is actually that which is represented by the Ronchigrams shown above for the 10" f/6 mirror. This same mirror was supported with a chain sling and interferograms taken with the mirror rotated clockwise to four positions: 0 degrees, 90 degrees, 180 degrees and 270 degrees. A set of ten interferograms were taken at each of the four positions for a total of 40 interferograms. These interferograms were then separately analyzed and the individual analyses averaged together. In conducting this analysis each group of ten, with the exception of the 0 degree position, were rotated in reverse of their actual orientation so that they assume an effective 0 degree position. This is a commonly applied technique that has the fact of zeroing out any false astigmatism due to mounting the optic. For example, the 90 degree position was digitally rotated 90 degrees counterclockwise, the 180 degree position digitally rotated 180 degrees counterclockwise, etc. False wavefront errors generated by atmospheric disturbance, including false astigmatism are zeroed out in the normal averaging process irrespective of rotation.
A typical interferogram at each position is seen below along with the average of the readings at that position. The small, black rectangle seen at the edge of the mirror is a vertical reference mark always at the 0 degree position at the time the interferogram is taken. All interferograms were taken in He Ne light at 632.8 nm. The aberrations listed are those normally considered important to the analysis of an optical surface. They are Spherical Aberration Focus Balanced (FB), RMS wavefront, and Astigmatism. Spherical aberration is presented as focused balanced rather than peak to valley because that is how the wavefront is observed by either the eye or camera; images are observed at their best focus position. Each interferogram pictured below is a single representative image.
One can see the obvious effects of averaging inasmuch as the values for all aberrations are moderately to significantly reduced. This is the result of a great many data points acting upon each other and cycling out those false aberrations which are not consistent within the optic itself but an artifact of mounting stress or atmospheric turbulence. The RMS numbers for the 4 angle sets are higher than the final because the mount astigmatism is still contained and expressed in the data for each set of averages, but has been averaged out in the final number. The astigmatism in the 0 and 180 degree position interferograms have the mount distortion vertical, whereas in the 90 and 270 degree positions it is horizontal. The wavefront maps from which these numbers are derived, containing the mount astigmatism, as they do, when added together cause the mount astigmatism to cancel out, leaving only the mirror astigmatism.
After arriving at the final values for the principal aberrations, the only remaining calculation that needs to be made is that for the Strehl ratio. This is based entirely upon the RMS value and is found to be .976. After adjusting for the difference between He Ne light at 632.8 nm (red) and helium d line at 587 nm (mid visual range), a factor of 1.078, we arrive at the following values for the mirror:
It is interesting to note that the raw peak to valley aberration value is just under 1/5 of a wave. Yet, the optic itself is functioning at a Strehl ratio of .972, a relatively high value. The overall spherical correction is 1/31.3 wave. Astigmatism is better than 1/10 wave. The raw peak to valley aberration value, though it would appear to suggest by common standards only a mediocre mirror, has relatively little impact on the final performance of the optic. This is do largely to the peaks and valleys representing relatively tiny proportions of the entire optical surface. The picture below is an actual topographic representation of this optical surface.
Casual visual examination shows that most of the problem with extreme peaks occurs at the very edge of the optic. The vast majority of the surface appears to be within 1/10 wave, or so. Other forms of visual or heuristic analysis are also useful for viewing optical surfaces. The following is a standard contour plot diagram of the same surface.
One can easily see that there is present some astigmatism in the order of perhaps 1/10 wave in the optic itself. Defects? Yes, and visible, but well within tolerance. No optic is perfect, but just how imperfect are they? That's the question.
False Astigmatism due to Mounting of Optic
But how much of that astigmatism was due to distortion in the mount? Even using a flat chain sling to hold the mirror, recognized as one of the best methods available, some false astigmatism is injected into the interferogram and the exact amount extracted. A simple analysis consisting of the following will show this amount. First, note that the astigmatism measured with the mirror at 0 degrees contains both the mirror and the mount astigmatism. By rotating the mirror about its optical axis by 90 degrees, then reorienting the interferogram digitally so that the mirror top is again at the top of the image, the mount astigmatism will be seen to have rotated by 90 degrees relative to the mirror. The average of these two results therefore will eliminate the mount astigmatism from the result by cancellation. All reading are at 587 nm.
Average of 0-180 Degree Astigmatism: .041
Astigmatism due to
As you can now see, reading the interferogram directly and without complex analysis would suggest astigmatism in the order of 1/5 wave! And the rest of the analysis would be concomitantly degraded.
Final Conclusions on Interferometery
I hope I have demonstrated that one cannot simply hand out a single interferogram and an analysis of that interferogram and call it a mirror. This should also help to make it clear as to why the precise certification of optics for amateur telescopes would place the cost of such optics, were the certification done correctly, much higher than they currently are. The present certification required the production of 40 interferograms, all of which had to be individually analyzed. Even though I possess modern, interactive software that speeds up this process considerably, it still requires considerable time and effort. Therefore, I've chosen to produce my mirrors by method which ensures that they will meet my advertised criteria without having to engage in extremely time-consuming and expensive interferometric analysis for each optic. The calibrated autocollimation Ronchi test has proven time and time again to produce mirrors of consistently high-quality.
Comparison of RMS and Strehl with Knife Edge Peak to Valley Wavefront Standards
Just how do the above RMS and Strehl measurements compare with commonly accepted peak to valley measurements and measurement standards? While there is no direct relationship between the two there is a way to rationalize the relationship when one understands that the commonly accepted wavefront standards are really related to knife edge readings as applied from the Foucault test and not from interferometery. As stated at the outset of this article, the problem with all shadow or diffraction tests is that they are only a measure of slope error through a single segment of the optic and do not tell the worker about the entire surface and the various other errors that might be present, such as astigmatism. Mask-based tests only give an average slope over the area of the mirror exposed in the mask apertures, so narrow peak-to-valley errors are undercounted. Peak-to-valley based on knife-edge reduction therefore is a "filtered" value omitting many small peaks and valleys. Because interferometers sample at a much higher rate across the surface, as well as seeing the whole surface and not just a cross-section, peak-to-valley derived with an interferometer will usually be much worse than what the knife-edge reduction will give.
The following analysis, on the same 10" mirror tested above, takes the data set map (wavefront error as expressed in all the individual data points) of an interferogram and performs an analysis in imitation of a person using a knife edge test. It measures a group of data points within hypothetical mask areas similar to the standard Couder mask. The data within that mask is averaged (as knife edge user would) and the slope error for that mask area measured. Moreover, the program balances the data between each opposite side opening just as a worker would do, to "balance the shadow and find the intersect point of the opposing cones of light". Thus, asymmetric errors that might exist (they always do; it's a question of whether their magnitude is significant) are "balanced out" by the very process of taking longitudinal aberration readings. Asymmetric errors are thereby rendered null. In effect, they are completely ignored by the knife-edge test. (This is of course not true of interferometry.) The present analysis used five one inch mask windows set at radius points of .5" (central aperture), 1.5", 2.5", 3.5" and 4.5". The results of this analysis indicated a P-V wavefront error of 1/47.4 at 587 nm. A fiftieth wave mirror! But after much rejection of data, rejection that the knife-edge test by its very design always imposes. It's really that .972 Strehl that's important.
All of my standard mirrors are guaranteed to have the following optical characteristics as a minimum: