For f/6 optics with a 100 line/inch grating, taking as the wavelength the yellow-green color that the human eye likes best, the maximum tolerable distortion with 2.5 lines showing is about 0.04.

Anyone who has ever used one of these gratings knows that the sharpness of Fig. A-6 present an unrealistic view of what is happening. The actual shadows are fuzzy and indistinct. A 4% bowing of the lines in the presence of this fuzziness is nearly impossible to see. Figure A-8 depicts the actual pattern that results from a 1/4 wavelength of undercorrection, still without the blurring at the edges of the lines.3

2 The Airy disk in this diagram is greatly exaggerated in size. Hence, the crossing point of the dotted line appears nowhere near 2x from focus. However, in real situations with very small Airy disks, an inner crossing distance twice that of the edge is an excellent approximation.

3 This pattern does not use the approximation in Eq. A.3 but is more accurately calculated using an adaptation of the method of Prugna (1991) to include correction error.

very slight bowing very slight bowing

Fig. A-8. The Ronchi distortion expected of a A-wavelength undercorrected f/6 telescope when a 100 line/inch grating is placed near focus and the telescope is directed toward a star. The thin exterior lines are the borders of the first- and second-order interference.

An arbitrary cutoff can be imposed. This limit is somewhat artificial, but let's set it conservatively at 8%. If we can't set up a test with at least an 8% maximum bowing of the lines, we are not testing the optics with anything close to the kind of precision necessary. One suspects the actual cutoff should be higher, but for the purpose of argument, let's give the geometric Ronchi test the benefit of doubt.

Examination of Eq. A.3 reveals how the Ronchi test could recover enough sensitivity to barely achieve the limit. If the grating were moved closer to focus, reducing the number of lines seen on the aperture to 1.25, this mirror will have lines that bow 8%. If the number of lines is increased to 200/inch, the pattern will also distort 8%.

These solutions contain inherent problems. Although the geometric Ronchi test is derived with a ray approximation in mind, the real world doesn't care about the designer's thinking. It follows wave physics. Interference between the gaps on the grating result in superimposed offset artifacts called diffraction-order images. These side images make the edges of the bars less certain, obscure behavior at the true edge, and generally muddle the view. They grow less important as fewer and fewer lines are strongly illuminated, but they can't be eliminated entirely because the outer portions of the image remain dimly lit.

White-light diffraction expresses itself as a blur at the edge of the lines, so if a line is expanded by moving the grating closer (the first solution), the blur is increased as well. Increasing the frequency of the lines (the second solution) doubles the angles at which the higher-order images appear, making the edges even more difficult to see. The brute force methods of either moving the grating to show fewer lines or using a finer grating are limited in their ability to improve sensitivity.

Again consulting Eq. A.3, distortion increases when testing optics of high focal ratio. At f/12, we have reached the 8% arbitrary limit with no other changes. The test has not suddenly become more sensitive, but the tolerances are wider with slow optics. For high focal ratios, the Ronchi test can indeed discriminate the difference between bad and good systems.

An additional method doubles sensitivity. A Ronchi null test is conducted not on a distant star but a point source at the focus. Light exits the instrument in the reverse direction, bounces against a full-aperture optical flat, and travels back through the instrument in the reverse direction. It is intercepted near focus by a Ronchi grating. This autocollimation mode doubles the aberration because the optics have been traversed twice. A well-known Schmidt-Cassegrain manufacturer tests telescopes in this fashion. Because the aberrations are doubled, and the f/10 focal ratio is high to begin with, we can see that this manufacturer's test is an adequately sensitive one. The problem for ordinary individuals is the same as it is for all autocollimation tests—huge optical flats are expensive.

What about a Ronchi test at the center of curvature? Several amateur authors have suggested that the geometric Ronchi test can be successful in a bench test of paraboloids at the center of curvature (Mobsby 1974; Terebizh 1990; Prugna 1991 and Schultz 1980). The problem with most of these tests is that they never calculate the sensitivity of their methods. The way of properly showing sensitivity is to compute the shape of a pattern for both perfect optics and optics which display a % wavelength of correction error. One must demonstrate that the two patterns are sufficiently different that a distinction can be made.

In Fig. A-9, an example pattern of a perfect 16-inch f/4.5 mirror is calculated together with the pattern of the same mirror if it were a full '/2 wavelength undercorrected. Both are calculated as viewed in a 150 line/inch grating. The Ronchi screen is set at slightly different locations that present similar appearances near the mirror's center. One pattern is for a mirror better than any optical surface has ever been made. The other pattern is for a mirror of little or marginal usefulness in an astronomical instrument. Figure A-10 is a photograph of precisely this situation. The photograph shows, as no theoretical argument ever could, the difficulties of the geometric Ronchi test at the center of curvature.

The problem with the Ronchi test conducted at the center of curvature is that it is swamped by the overcorrection of aspherical mirrors operated far from their natural focus at infinity. The Ronchi screen is so obviously responding to something that one forgets that the difference between the response to a bad mirror and a good mirror may be slight. The under-

a) 1/2 wavelength undercorrected b) perfect

Fig. A-9. Theoretical patterns are generated for a Ronchi ruling of 150 lines/inch placed slightly inside the center focus of a 16-inch f74.5 mirror: a) if the mirror is V wavelength undercorrected; b) if the mirror is perfect. Distances: a) —0.06 inches and b) —0.05 inches.

Fig. A-10. A photograph of a Ronchi test on a real mirror as calculated in Fig. A-9. Is the mirror perfect or terrible? The mirror was Foucault-tested and found to be l/8 wavelength undercorrected. The Foucault patterns appear in Fig. A-3. (Photograph by William Herbert.)

corrected prolate spheroid, the perfect paraboloid, and the overcorrected hyperboloid all look badly overcorrected at the center of curvature. All but the highest focal ratios show this condition.

For the 16-inch f/4.5 mirror at the center of curvature, the length of the blurry focus region is about 0.444 inches (11.3 mm). That's what the

Ronchi screen is acting on. However, the length of that region is 0.428 inches if the mirror is undercorrected right at the 74-wavelength Rayleigh limit or 0.460 inches if it is overcorrected. This difference of 0.016 inches is slight, only a tiny adjustment of those grossly distorted Ronchi patterns.

Sensitivity could be recovered by using a kinematic measurement platform and matching a number of theoretical patterns, all the time being careful to record the longitudinal motion of the Ronchi screen (as in Prugna 1991). Unfortunately, the test is almost never undertaken in this manner, probably because it so closely resembles the Foucault test that the user was trying to avoid in the first place.

Professional optical workers have dealt with the sensitivity of the Ronchi test. Cornejo and Malacara (1970) write:

The Ronchi test is a very powerful test for spherical as well as aspherical mirrors. However, this test is of an accuracy limited by diffraction to a value such that the resulting surface can be used to form images, but not for wavefronts to be analyzed interferometrically. [Italics added.]

In other words, the test can generate surfaces accurate enough for use in cameras or other coarse imaging devices but not surfaces that are so precise that they can be tested with an interferometer. Such accuracy is demanded in astronomical telescopes.

Ronchi himself commented on the accuracy of the geometric test that carries his name (Ronchi 1964). In an excellent review article he says,

As long as the grating employed had a very low frequency, like the ones that we had used at first and that had also been used by other authors treating the same argument, the geometric reasoning corresponded quite well with the results of the experiments and measurements; but at the same time the method did not lead to results as fine as desired. It was evident that in order to increase this sensitivity it would be necessary to use gratings of the highest frequency possible, but then the results decidedly deviated from those predicted from geometrical reasoning. [Italics added.]

Here Ronchi describes his justification for abandoning the geometric test in the 1920s. He goes on to describe the techniques to use the overlap between diffraction orders as an interferometric test. The evaluation is made by the complicated interpretation of two equally-aberrated wavefronts somewhat displaced from one another. In this true wave-optics Ronchi test, the separation of the shadows is not determined by geometry but by the interference of light.

The geometrical Ronchi test has some uses, however. It is an excellent way of seeing sharp zones. It can detect seriously defective optics— curvature at stellar focus is often enough to reject an instrument out of hand. It is a good way of testing camera optics or any optics used far from the diffraction limit. But this test has a variable sensitivity that has been unappreciated or ignored by many advocates.

Readers may notice that if a checkerboard grid of squares replaced the Ronchi ruling, the Ronchi test would have a superficial resemblance to the Hartmann test. The corners of the squares would be equivalent to the hole positions. Yet no complaint was made about the Hartmann test's sensitivity. The critical difference is that the Hartmann screen is rigidly fixed on the aperture instead of floating somewhere near the focus. Plates are exposed and data are taken with high-precision measuring devices. Systematic errors are reduced by making plate measurements of dot positions from multiple directions. These raw measurements are followed by a sophisticated mathematical reduction procedure.

Many amateurs, particularly those who promote the use of the geometric Ronchi test at the center of curvature for aspherical optics, present the test as the simple comparison of patterns. They have stripped the mathematics away, and with it goes the error-detection ability of the test. The siren song of the geometric Ronchi test is that people can just look at a pattern and avoid the difficulty of measurement. Unfortunately, the measurements contain the sensitivity.

In summation, the geometric Ronchi test is not recommended for telescope evaluators for the following reasons:

1. The sensitivity of the test is variable and depends on the focal ratio of the tested optical system and the frequency of the grating used. Also, results vary depending on whether the test is done at the focus or the center of curvature. Certain combinations are very sensitive; other combinations are fatally insensitive.

2. When conducted at the center of curvature, the geometric Ronchi test is commonly used as a simple comparison of patterns, but the bending of such patterns can differ little from the patterns with unacceptable correction errors.

3. It requires a sufficient acquaintance with the theory of the test so that a "worst case" unacceptable pattern can be calculated. Before declaring tested optics to have passed, one must have an appreciation of what failure looks like. A good place to start is to calculate the anticipated test pattern with and without :/4 wavelength of low-order spherical aberration on the wavefront.

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