The end of polishing on each surface should be gentle and prolonged. You're trying to get the best, most spherical surfaces that you can. Once you finish, then it's time to test. The best way to test an achromat is in autcollimation against a good flat. The flat only needs to be slightly larger than the clear aperture of the lens, and it doesn't even need to be very flat (several waves of curvature are fine), but it needs to be smooth and even without conic deformation.
It lies beyond the scope of this webpage to detail how to make a useful flat. Two sources where you can begin to learn that are: Texerau, chapter 3 ("The Plane Diagonal Mirror"); and T. Waineo, "Making and Testing Optical Flats," in "Amateur Telescope Making Journal," vol. 8, pp. 4-8. Additionally, the original ATM volumes contain several useful articles on flat production, especially volume III. See R.E. English, "Optical Flats," pp. 156ff.
If you cannot obtain a useful flat, then you can still test by means of the star test or by means of testplates as is discussed in Appendix 2. For star testing the best sources are H.D. Taylor "The Adjustment and Testing of Telescope Objectives," (Newcastle upon Tyne, 1946), chapters 1-8, and Richard Suiter's "Star Testing Astronomical Telescopes," (Willmann-Bell, 1994). It will not be easy to perform the test, but if you persevere you will learn an enormous amount and become both a better ATM and a better astronomer. You will want to examine the state of the spherical aberration, the color correction, lateral color, and astigmatism. Taylor is extremely useful for this, though Suiter has more complete illustrations. It is also possible to do a knife-edge test or Ronchi test at the eyepiece, though it may prove difficult and inconclusive.
If you do have a flat, then once you are polished out, you will want to assemble your lens in a cell or temporary holder as detailed in the next section. You should put your knife-edge tester or Ronchi tester at the focal position for the objective and put your objective in front of the mirror with R1 facing the flat. Remember that this test works in "double-pass," meaning that the light goes through the lens twice: once on its way to the flat, where it doubles back through the lens to focus. This double traverse doubles the apparent errors in the lens. And since the lens is now working as if it were pointed at a star, the correct figure should be like that for a spherical mirror. Therefore the test is EXTREMELY sensitive, much more so than the usual center of curvature test on a mirror. Slight figure errors will appear glaringly, as if they were much worse than they in fact are.
It is not necessary to make a zonal test mask and take measurements, though you are free to do so. Remember that you are testing against a flat so you will be looking for a spherical surface indication and any other figure is going to be not correct for this version of the Foucault or Ronchi test. It's optically just like testing a paraboloid mirror with an autcollimation flat. The TEX measurement software available for free over the internet has the capacity to analyze spherical surfaces. You can use that if you like. Nor is it necessary to fuss over slight zones or a slight departure from sphericity on the wavefront. These errors will only be barely visible in a star test, if at all. The autcollimation test is so sensitive that it can reveal errors which remain invisible or just barely visible in the field. As long as your wavefront looks decently smooth and nearly spherical, your lens will perform flawlessly. Of course, a perfectionist might strive for more. By all means try! Oftentimes the error may be in the glass itself and the old masters used to correct for errors of the poor glasses that they had at the time by correcting the shape of the lenses to remove most of the error.
It is usual to view the knife-edge cutoff or the Ronchi fringes through a narrow passband filter centered on the d- or e-line of the spectrum. Suitable filters can be purchased from Edmund Industrial Optics. If you're really serious about achromats and have some money to throw away, you might also get a filter for the C-line and the F-line. These are handy for assessing the state of the color correction. Even using the deep colored Kodak filter series will help in the inspection of the lenses. The sets intended for EPs are also a fairly cheap source although they don't filter as well as the good narrowband filters. It might also be noted here that you can't use a LED for the illumination light for these tests. You need to use an incadesent lamp otherwise you won't have the necessary colors being generated in the light. White LEDs really aren't a broadband white light source and so, can't be used. Go down to the hobby shop or electronics shop and buy a small low voltage lamp and use it.
But even if you don't have these filters you can figure your mirror well enough. Set your tester up near the presumed focus, find the return and try to cut it off or look closely at the Ronchi fringes (a 100 line grating should suffice) with only 5 or 6 visible. You will probably recoil the first time you do this, because you'll see vivid colors on the lens like you never saw on a mirror before! At best focus, the right half of the lens will appear sort of putrid green, and the left half reddish purple in the knife edge test. With the Ronchi test, the bands will fade away and become indistinct close to focus showing multiple colors as you move the grating.
You need to keep the grating further back so that you can see the bands clearly. An e- or d-line filter clears all the extraneous colors away, leaving a bright green or yellow colored optic, and a gray cut-off in the knife-edge test and clear dark Ronchi bands.
The garish light show you've just enjoyed is the "secondary spectrum." Blue and red focus further away than yellow and green. This makes testing a lens a bit more challenging at first, but definitely more colorful than those drab gray Newtonian mirrors!
Still another point to remember is that very probably you don't have your lens perfectly collimated with the flat. Chances are very good that it is slightly tilted. So long as the tilt isn't extreme (you can put a ruler up to the lens's cell in several places around its periphery and make sure that it is positioned reasonably square to the flat--get it as good as you can), then you'll have no difficulty figuring an f/15 lens. But in the knife-edge and Ronchi tests you may see what looks like astigmatic bendings. If this were a small Pyrex mirror of 1:6 aspect ratio, then you'd have reason for concern: the error would almost certainly be in the surface. But with your achromat, it is extremely easy to see slight astigmatic bendings which are simply the result of how the lens is tilted with respect to the flat. Remember that an achromat like a Newtonian mirror is not corrected for astigmatism off-axis. And if the lens is tilted to the flat, then you're viewing it off-axis.
So as long as the bendings seem small, don't worry. If you've ground and polished carefully, then chances are excellent that your lens will be fine. The star test, once the lens is installed in its tube, will give a decisive answer. In figuring, just pay attention to the "equatorial" region of the lens as you're viewing it. The "polar" regions (top and bottom as you look at it) are where the bendings or stray light are likely to occur. I've seen these bendings often and yet none of my lenses has had intrinsic astigmatism. If you've built a non-aplanatic objective, then by the Ronchi test you may also see image coma. This manifests itself as asymmetrically spread out Ronchi bands. The bands may be oddly stretched in a vertical or horizontal direction, almost as if the plane of the bands has been tilted toward or away from you in various orientations. If you've ever autocollimated a paraboloidal mirror then chances are that you've seen this effect. It can complicate somewhat the figuring, which is another point in favor of a Fraunhofer coma-corrected objective. But with practice you can overcome the problem, either by mentally compensating, or by collimating the lens better. You could even go whole-hog and build a lens holder with tip-tilt adjustments. I haven't found this necessary, since I generally don't try to quantify the wavefront of my amateur lenses.
And now the real minutiae of lens testing (take a deep breath!): the first and most important thing to remember about lens testing is that your interpretation of "high" and "low" zones on the lens must be exactly REVERSED from how you did it on a mirror. In other words, if you saw a shadow pattern on your lens that looked like a hole-well, it is a hole in the wavefront (i.e. an area that focuses light too close to the objective). BUT YOU MUST TREAT IT AS A HILL WHEN YOU FIGURE! It is absolutely essential that you stop and think about how the refractive surfaces in your lens are acting, before you accidentally to make them worse by practicing your well-worn mirror figuring techniques.
So consider: in a concave REFLECTIVE optic (mirror), a central hole occurs when a saucer-like depression forms in the glass. That seems to make common sense: the wavefront looks like the optical surface. But in a concave REFRACTIVE optic, light is DIVERGED, not CONVERGED. So if you were going to figure the wavefront hole in your lens by retouching the concave of the flint, you should realize that a wavefront hole could not be caused by a surface hole in the glass. A surface hole would produce stronger negative refraction and so diverge the light more and send it off to a more distant focus: a surface hole in the flint would cause a HILL in the wavefront.
Equally, in the convex refractive surfaces of the crown, a surface hole would produce weaker positive refraction, again sending the light to a more distant focus, not a closer one. So a hole causes a hill! Yikes!
You may reasonably feel befuddled. There are two ways to deal with the problem. One is to simply imagine that the lighting in your tester or that your Ronchi band patterns are reversed from what you grew accustomed to in mirror making: what used to look like a hole, now interpret as a hill. What you trained yourself to see as undercorrection is now overcorrection. This is what Texerau advises, and it will work so long as you rigidly adhere to the convention. If the light source on your tester lies on the right side of the knife edge, imagine that it is illuminating the lens from the left side (or vice versa). Then you can make your corrections with confidence. Just be sure that you forget about surface holes focusing closer and surface hills focusing further away: in a lens, surface holes focus further away, and surface hills focus closer to the optic.
Alternatively, instead of thinking about hills and holes, you can take a more analytical approach and examine the longitudinal distances at which various zones are focusing. Then consider how you must reshape the REFRACTIVE surfaces to remove the zones and cause the light to focus at one point. This is what I do to avoid confusion.
Generally, it's a good idea to concentrate the figuring action on just one surface. R1 is convenient and that's where I do my figuring. If, however, you know positively that another surface contains a correctable error, then by all means correct it. You can directly examine R3 of the flint at its radius of curvature by reflection. Here you're on familiar ground: since this examination works by reflection, the shadows are uncolored and analysis and correction work exactly as in mirror making. You should certainly examine this surface by reflection before figuring the system as a whole. But don't imagine once that's done, that you can continue treating R3 like a mirror surface once you begin to figure the lens in transmission. Comforting as that concave surface may look, it has become its optical inverse: a convex refractive optic, instead of a concave reflective one!
So if you have a turned up outer zone in the wavefront, let us say, or rather in our new language of testing: "a wavefront zone which focuses too close to the lens," then you can no more plane down the outer surface zone of R3 to correct that defect than you can do likewise to R1. That zone is caused either by too much positive refraction in the crown (i.e. the outer zones on the glass are turned down), or too little negative refraction in the flint (i.e. R3 or R4 is turned down). This is why it may be more useful (if you can keep the conventions straight) to imagine the mirror maker's hills to be in fact holes, and vice versa, as was discussed above.
Once you get over that difference, the actual figuring action is much the same as in mirror making. Another reason why it's preferable to figure R1 instead of R3 is that the crown element is stronger and stiffer than the flint. It can take more force before it bends an unacceptable amount under the polishing pressures. Indeed, because during actual use a lens is much less likely to have its images damaged by flexure than a mirror is, it is attractive and possible to make high performance lenses from thinner substrates than has traditionally been done with mirrors. Thus the Clarks along with their equally skilled successors the Lundins typically used blanks with aspect ratios as low as 1:20. The Yerkes' 40" lenses, for example, are only 2" thick. This was bold work for the 1890s, but C.A.R. Lundin and his son who actually made the objective were successful.
Therefore, a skilled lens maker could fashion your lens from blanks only 7.5mm thick, instead of 15mm, thus halving the objective's weight. But the extra thickness provides you with 8X as much stiffness and will make your job much easier. Later on, if the refractor bug really bites, you could try the much thinner rubbery blanks. Good luck!
Armed with your trusty d- or e-line filter you can now proceed to figure your lens. Once again I suggest that you not to worry about turned edges, unless they extend into the clear aperture. Fixing them is pointless and may cause other problems. It may also be noted that, unlike a mirror, the refractor lens does have it's edges obstructed by the mounting system which will hide most, if not all, of the turned edge.
As to general figure errors, in the case of my own lenses, I have several times recently ended up with a general undercorrection of the figure at the end of polishing: the edge zone focuses closer than the center zone. Since you cannot stroke the center of R1 over the edge of the lap, as you might in a mirror, to correct this or it will get worse, I have successfully fixed the figure by using a combination of an inverse-star lap and a full-sized smoothing lap. First, I cut a star pattern out of paper with the solid center area having the same dimension as the stroke length I will use and the arms extending almost to the edge of the lap. Then I warmed the pitch lap under water or on the hot plate, and pressed in the paper star by laying the star on the center of the pitch, covering the star and the whole lap with a piece of foil or paper, laying the lens on top of the assembled body and putting 5 lbs (2 kg) or so of weights on that. Of course, exercise caution if you do this or you could damage your lens with the weights.
After 20 min. I remove the weights, lens, etc., and then the paper star. Voila: an inverse star lap. Next, wash the lap in cool water, apply some rouge and lay the clean lens on top of it. It's best to mount the lens in its appropriate support base with shelf-liner and tape, and then to grasp the holder rather than the lens itself during figuring, so that the warmth of your hands doesn't heat lens, cause it to expand, and damage the figure. But if it's not possible then don't worry too much. Just keep your rouge stained hands from rubbing the back surface of the glass and accidentally figuring that side!
After 10 or 15 minutes of cold pressing, stroke the lens over the lap in the usual "W" pattern. Try this for 10 minutes and then test the lens again and see what happened. The stroke and lap are designed to flatten out the outer zones of the lens without affecting the center much. This should reduce the undercorrection. You will probably also need a full sized lap to smooth out the results --it's a good idea to prepare two full-sized polishing laps for R1 during the polishing-out period and break them both in with rouge: then during figuring you can keep one intact as a smoother and make a star lap or whatever you need out of the other. After a couple of such figuring sessions you'll need to smooth out R1 with the unaltered lap for a few minutes. Very likely a bump (a real bump!) will form in the center of R1, or some zoniness in the outer area. Both these problems can be addressed with the full-sized polisher. Just think about what you're trying to achieve, and figure accordingly. Finally, you may never fully correct the undercorrection all the way out to the edge. But remember that your test is working in double-pass at f/15. Very likely the residual error will be quite small. An actual star test will decide the matter. You will certainly want to do a full-scale star test several times at least before declaring the lens finished. And remember that not even Alvan Clark's lenses were always perfect. Get a decent figure and your achromat should perform amazingly well.
In general I find the Ronchi test more convenient than the knife-edge for this work. Since I'm shooting for straight fringes, it's pretty easy to tell when the figure looks good. You don't even need a filter for this job, since working with 7 or so dark Ronchi bands and a 100 line grating will keep the color error from confusing your interpretation. A knife-edge and filter, however, does make for more precise interpretations.
But even without a filter it is also possible to figure using a knife-edge tester. The main difficulty is that the left and right halves of the lens will show vivid and contrasting colors: near best focus the right half (if the light source is on the left) will look yellowish green when you cutoff the return beam, and the left half will look reddish purple. Both side will appear of roughly equal intensity. When you move away from best focus the coloration changes and far from focus returns to a basically gray tone. If zonal errors are present, when you insert the knife-edge near best focus, you can still see grayish arcs, just as in mirror testing, but the arcs are now immersed in the brightly colored extraneous light. You could never take zonal measurements this way, but you don't need to. All you are after is a spherical wavefront. So with perseverance, it should be possible (I've never tried this) to figure the gray zones away just as you would if you wanted a spherical mirror. An excellent cutoff would show the right side of the lens go uniform dark, and no dark zones on the left side but only the bright reddish purple of the unfocused secondary spectrum (you'd see this color all over the mirror, but the knife-edge has cut it off on the right side). Of course, if your light source is on the right and knife-edge on the left then you have to reverse this colorful scheme! I might also note that this is a place where LEDs for light sources works well as you don't have to worry about the color problem!
One last word about figuring: in less than first class pieces of optical glass, there can exist areas of slight inhomogeneity which will not have appeared clearly during the preliminary transmission test (if you used that) which is specified in Appendix 3. For a 6" objective made with current glass this should not be a problem. But on the other hand, very often when you're testing a lens either in transmission or by looking at its one concave surface by reflection, you don't see the velvety smooth texture to the wavefront which is the pride and joy of a mirror maker. The lens may look somewhat blotchy as though you have ripple.
Don't worry about this. It's difficult to get crown and flint as smooth-looking as Pyrex, in the first place, and secondly by reflection their images are more plagued with thermal effects, since they are not low expansion glasses. Moreover, in the autcollimation test you're looking in double-pass at an f/15 wavefront. It's never going to look as smooth as a single-pass center of curvature test on an f/6 or shorter mirror. It's much too sensitive. Your mirror wasn't really so smooth anyway: the knife-edge test just wasn't powerful enough to reveal the residual roughness. But real roughness in a refractor would look horrible by the autcollimation test. Try a star test, or look at the Moon, when you finish figuring. Do these objects seem less sharp than in an equal-sized Newtonian? Chances are that if the figure is decent, you'll be amazed at how razor sharp the images looks.
As for the inhomogeneities, if you find any they should be small. You can try to do what Alvan Clark did: put some rouge on your fingertips and figure them gently away. It lies beyond my scope to discuss finger polishing, but you may get some useful hints from Texerau, chapter 10-7 ("Polishing and Retouching"). Personally, I have never so-far seen a disk that I could successfully figure by finger polishing. Good luck!
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