The observer must practise his judgment in deciding which of the two is darker, as in reading a Bunsen's shadow photometer. If both are equally dark, the knife-edge is at the focus of this zone... (Italics and spelling are Ellison's.)Thus, the article here is an intention to solve the geometrical problem that exists with the regular Foucault testing, namely that of determining the relative dimensions of a particular solid figure, e.g., a portion of a paraboloid of revolution, by estimating relative light intensities (photometry) using the human eye to compare spatially separated illuminated areas (an unfavorable circumstance for precision in visual photometry) and to judge the attainment of that equality of brightness as a function of knife-edge distance along the axis of the mirror. After determining the equality of brightness, and determining the distance from the starting point, relating that radius to an expected (calculated) value for the desired conic.
There have been many attempts to find better ways to survey and control the progress of a mirror as it is being shaped, and to rate the quality of the final result. This test is one simple method of determining the ROC difrerences of the various zones of the mirror under test.
Consider one of the more precise methods, the Gaviola Caustic Test, as first made known to many ATMs in the mid-1950s, (by Irvin Schroeder, in Book Three of the Scientific American ATM triad.) For a beginning ATM, or for those without access to a good workshop, the Caustic Test's need for a rather precise two Cartesian coordinate measuring apparatus may be a bit daunting. What is the worst thing with the Caustic Test's promoters is the advanced workmanship shown with the equipment that the promoters have used. Thus, those paired orthogonal measurements indicating the caustic curve of the actual focus of the zones as used in the caustic test are only one possible path to the ATM's testing goal of finding the shape of the mirror. Since the location of the common crossover point on the mirror's axis for the two cones of light from horizontally paired zonal mask apertures, shown in the image to the right and if known to sufficient precision, provides enough information to determine the slope of the mirror at the zone being investigated. This point is the point at which the Foucault test is supposedly measuring the shape of the mirror. The question then arises, how might we conveniently locate the crossover point with sufficient precision without a lot of delicate measuring of the caustic curve? The knife-edge of the Foucault test is insufficiently accurate for that purpose as the darkness of the mask aperture is a very subjective thing. The traditional method in the Caustic Test is to take a wire and use it to obstruct the image of the pinhole being returned. This again is a bit subjective as, while the minimum can sort of be found, that minimum is often somewhat broad in its postiion.
If you think of each Couder Mask aperture as delimiting a small reflecting telescope's primary mirror, which images the pinhole near (remember that the pinhole is actually below the center line of the mirror's axis) the C-of-C as its object, which the telescope brings to a primary image focus near that pinhole. If the object/source is displaced a small amount vertically, the pin hole's image can be accessed for examination with an ocular magnifier near the lighted pinhole's location. The setup will still provide a projected normal to the mirror's center, as its vertical plane of reflection intersects the horizontal plane of the test along a line corresponding to the mirror's axis of symmetry. This is the normal setup with the slitless version of the Foucault Test as the Light Source is directly below the returned image, with the slight astigmatism produced by the method showing only in the vertical direction and none in the horizontal direction where the testing is done.
For each zonal aperture, we will effectively have a small aperture, very large length-to-diameter-ratio (100:1 or greater) reflecting telescope, focused on a small-diameter bright object. Although there is no need for a Newtonian secondary mirror and its spider support to enable use of an microscope, suppose that we nevertheless install "wire support spiders" at each mask aperture. The small bright source's near-focus images will then show something resembling a Newtonian reflector's familiar diffraction spikes. Furthermore, if the "wiresz" are in a bilaterally symmetric arrangement crossing on the center of the aperture as the spider of the Newtonian secondary's support, these spikes can be used to visually determine the centerline of each light cone more precisely than if we try to estimate the centers of the somewhat defocused small diameter light-source's images, absent such spikes. (Human astrographic-plate measurers have traditionally used diffraction spikes to get better fixes on the bloated star images of the photographs.)
More importantly, as we shift the focal plane of the microscope longitudinally near the crossing point of the cones of light from the zonal apertures, with suitably distinctive patterns of wires and hence distinguishable diffraction-spike patterns, we will be able to easily recognize the direction of change required for feedback to find the crossing point of the two beams, and thus to find the longitudinal location of their transverse plane of intersection (by their symmetric superposition) more easily and more exactly.
(The diffraction rings from a small, circular source-aperture may sometimes be used to see superposition, but lack of any feedback-direction information makes it more difficult to set precisely. However, when observed, symmetrical ring-alignment may lend confirmation to our spike-mediated null determinations.)
In Figure 1, is a sketch of the two mask-apertures equipped with cross wires. Note that one is vertical/horizontal and the other is at 45 degrees left and right and that the apertures are round rather than the regular arc of the Couder Mask used with the Foucault Test. Below that are stylized representations of the images produced by the left aperture (A)and by the right aperture (B). In (C), the appearance of the images in a plane transverse to the mirror axis somewhat inside crossover is shown. (Outside crossover, the [B]-image would be to the left of the [A]-image.) Their symmetrical superposition at crossover is represented in (D) which occurs when the two images overlay each other at the focal point of the curvature of the mirror at the locations where the apertures are..
Figure 2 shows one possible modification of the "wire" arrangements, which can enhance the setting precision, as illustrated with the 4 images, [A] with the left aperture, [B] the right aperture, [C] with the two away from the focus position in the scope and [D] with the two apertures at the focus position.
An enclosed High Intensity LED behind a pinhole aperture (a lens may be used to focus the light on the pinhole from the LED if so desired in the same fashion that a bright light source may be done for the Foucault test) may be placed above or below an microscope magnifier mounted on a longitudinal measuring-slide similar to those used for other C-of-C mirror tests like the Foucault tester. The pinhole and microscope may move together, or separately as desired as with the various designs of the Foucault tester with a moving or stationary source. If separate, the mathematical analysis must, of course, be adjusted appropriately as the motion of the viewing stage is going to be twice that of the pinhole and microscope together. The measuring slide's mounting should also be capable of adjustments, which will permit bringing the pinhole, and the microscope's axis into alignment with the mirror's axis of symmetry. (This will also have the effect of setting the microscope magnifier's field perpendicular to the test axis in the horizontal aspect.)
The unfortunate thing about this test is that a pinhole must be used as we are going to be viewing the returned image of that pinhole with the microscope. Thus the slit or half slit that would be fine with the Foucault test won't work right.
For example: My current testing microscope's eyepiece was scavenged long ago from a broken pre-WW2-vintage binocular. The objective is a 3/8 inch diameter, ~ 0.75 inch focal length, plano-convex simple lens, that I made (out of plate glass) nearly 20 years ago, as practice, while trying out a new small-lens-spindle setup. For the monochromatic light from LEDs, achromatism of the lens is not an issue so a single piece of glass for the lens is quite fine. The microscope's aluminum tube was cut from material salvaged from a discarded lawn chair or you can go all kinds of fancy with a lathe and make a nice tube for a small microscope as desired. Those who work in wood can always make one from that.
Besides offering magnification of the pinhole images and their diffraction spike labels, the compound microscope can reduce uncertainty in the axial location of the crossover. This is a consequence of the optical property that longitudinal magnification varies as the square of transverse magnification. In Figure 3, assume focal length of the objective lens is f' = 0.7500 in. The tube length, L', is 8 in.; so, by the thin lens formula:
Since optical systems are reversible, we can find that the diameter of the visible Test Focal Field is equal to the diameter of the Eyepiece Focal Field times L/L', or that the visible Test Field is 1/9.666667 of the Eyepiece Field's diameter. However, the depth of field at the test plane will be equal to 1/(9.666667 ^2 ) times the depth of field of the eyepiece. So if the estimated depth of the eyepiece field by the above "stacked strips" procedure is 0.003 in., then the estimated depth of field at the test plane is 0.003 in. x 1/93.444451, or only 0.0000321 in.!
Now assume that in testing, the viewer accommodates 0.003 in. closer to his or her eye, making L' equal to 8.003 in. instead of 8 in., we can see how much effect this has on the measurement (and also check up on the square law of longitudinal magnification used previously,) by substituting 8.003 as L' in the thin lens optical law, so that
Materials for masks might include: light-weight wall paneling, aluminum flashing, good quality cardboard, or one of my favorites --- pieces of the thin plywood used to make crates to ship the citrus fruit known as "Clementines." The important thing is to make the apertures round and centered on the midpoint of the zone being tested. The diffraction string needs to be crossing at the middle of each zone and is the marker in the test of the actual center of the zone. A rectangular zone mask will, of course, make another set of diffraction spikes which may confuse your eyes in the testing.
Circular mask apertures can be located by conventional layout techniques to within a few thousandths of an inch, by using a machinists scale graduated to hundredths (mentally subdivided to thousandths,) an eye loupe or simple lens magnifier, and a sharp scriber, followed by careful application of a good center punch. Use of center and pilot drills will help maintain accuracy in drilling the actual apertures.
In DAFT Figure 3, the wires (actually elastic thread) are wrapped around short pegs press-fitted and glued into holes located in the above fashion, and the thread is then also glued where it is clove-hitched around the pegs. (I use Elmers Glue-All for this, as it can be softened and removed with warm soapy water, for repairs or adjustment.)
Wm. S. Maddux
*We oldsters do sometimes have an advantage over those of fewer years. In this particular case, not owing to our more mature wisdom, merely to our more mature presbyopia.
**Of possible interest to ATMs, the behavior of longitudinal magnification offers an interesting way to make precise measurements of differential longitudinal position, by employing a sliding eyepiece moving along a relatively crudely graduated longitudinal scale. (A simple card-and-fabric bellows can join the eyepiece holder to the objective mount as an "extendible light-shielding tube.") In the above example, the longitudinal magnification starts out as 0.003 in. resultant enlarged scale interval, for an object shift of about 0.0000321 in., and of course magnification increases with further movement in the same direction. Aside from figure testing, this principle might be applicable to spherometers, wedge measurers, etc., where high precision results, without the need for highly precise instrument construction, can be a boon to the amateur, (or to the professional.)