4.8.1. Tests for optical surface accuracy: double pass test, interferometry

Among many different types of tests for optical surface quality, a few that gained popularity among amateurs are the Foucault test, Ronchi test, Waineo null test, Hindle sphere test, Dall and Ross null test, Offner null test, double pass test and, more recently, interferometric tests.

Addressing briefly the last two, the double pass test uses large flat mirror to reflect back to a surface, or to a system under test, a collimated beam produced by placing point-source of light at the surface/system infinity focus. This results in the light passing twice through a system, which doubles the effect of surface errors and, by that, test sensitivity.

Contrary to the common belief, the testing flat doesn't have to be exceptionally flat. A smooth, rotationally symmetric curve over this reflecting surface will mainly cause slight change of the final focus location, without inducing appreciable effect on wavefront quality. The P-V wavefront error of spherical aberration (best focus) induced by such testing flat deformation can be expressed with the general relation for a spherical concave mirror, W=D/2048F3, with D being the flat diameter and F its focal ratio. In a more specific form, after substituting f/D for F, R/2 for f, and R=D2/8z, W=2z3/D2, z being the center depth (sagitta) of the test surface. Even as much as 1mm - or 1818 waves of 550nm wavelength - deep curve on a 250mm flat will produce only 1/17 wave P-V (550nm wavelength) of spherical aberration at the best focus (with F=D/16z, this "flat" is a 250mm f/15.6 sphere).

But this is only the error produced in the first reflection, that of the collimated beam reflecting off flat's surface. This beam is reflected back to the system tested, which will process non-collimated beam. This much of wavefront bending would produce much more significant error in most any system optimized for object at infinity, by effectively placing the point-object at the distance equal to the focal length of the "flat" (in other words, by making the light reflected toward system under test slightly divergent, or convergent). In this case, the beam reflected back to the system under test is as if coming from a point at as little as 3.9m. It would induce significant error (spherical aberration) to most optical surfaces, or systems.

A hundred times longer focal length - i.e. an 18 waves deep flat - would induce negligible error to most systems. For a paraboloid, the object distance induced spherical aberration, as the P-V wavefront error at the best focus, is W=(1-ψ)ψD/512F3~ψD/512F3, with ψ being the inverse of object distance in units of mirror focal length. It would take D=500mm f/4 paraboloidal mirror to produce 1/7 wave P-V of spherical aberration for that object distance, which is already unacceptably larger error (which means that faster and larger paraboloids generally require higher quality testing flat).

In terms of "flat's" sagitta z, the P-V wavefront error induced to a setup due to the effective point-object distance is given by W={[K+(1-32zF/D)2]D}/2048F3, where K is the mirror conic. For a paraboloid and K=-1, W~z/32F2.

Rotationally asymmetrical surface deviations of the testing flat - specifically the astigmatic type - are much less forgiving with the double pass test. Any surface error here will double in the wavefront, and a good flat shouldn't have significantly more than ~1/20 wave P-V surface error of this type of deformation. Similar applies to random surface deviations and roughness, which would prevent the tester to judge smoothness of the test surface(s).

Interferometric testing of optical surface quality makes clever use of the interference of light which, under specific conditions, can produce visual patterns disclosing surface "topography" down to a fraction of a wavelength. In general, the interferometer is an optical device combining two wavefronts - one reference, perfect, and the other produced by the test surface - in order to produce the interference pattern making test surface literally visible at well below the sub-wavelength size level. The simplest, and probably the oldest interferometer consists of two flat surfaces positioned at a slight angle one to another (FIG. 50 A below). As light passes through the two pieces of glass (refraction is negligible at the actual tilt angles for principal rays/wavefront), at every section where the gap increases by about 1/2 wave, waves tend to interfere destructively, forming dark lines, so called interference fringes. The shape of lines depends directly on the surface shape.

FIGURE 50: If both surfaces are flat, the interference lines are straight (A). If one or both surfaces are curved, the dark lines of destructive light interference will form circular, with rotationally symmetrical surface (B, C). Surface irregularities will show as deviation of the interference lines from either straight, curved or circular line form, and can be measured to a small fraction of wavelength.

Since conic aberrations cause different form of wavefront deformations, they also show distinctly different interferometric patterns (and since wavefront deformation relative to a reference sphere varies with focus point within the aberrated focus zone, interferogram patterns will also be different for best focus vs. other focus points for each particular aberration). The fringe spacing in a single-pass interferometer corresponds to λ/2 differential on the surface, or λ on the wavefront; in a double-pass interferometer fringe spacing corresponds to half as large surface/wavefront differential.

FIGURE 51: Interferograms to the left (generated by OSLO) show deviations of the actual wavefront away from perfect reference wavefront for three Seidel point-image quality aberrations, spherical, coma and astigmatism, each at ~1/2 wave P-V. Since the interferogram is set to show fringe generated for every 1/2 wave OPD on the wavefront, the separation between either two maximas (light) or minimas (dark) indicates 1/2 wave wavefront deformation generated in between. Two common forms of interferogram presentation are: (1) with the planes of test plate (i.e. reference surface) and wavefront under test parallel or coinciding, and (2) with the test plate tilted with respect to the wavefront plane. The top row shows interferograms orthogonal to the chief ray of the wavefront (i.e. with test plate plane parallel or coinciding with the surface/wavefront plane) and the bottom row interferograms with test plate tilted in the plane of wavefront's axis of aberration. Tilting an interferometric plate (bottom) can result in better defined wavefront form from the standpoint of its three-dimensional characteristics; the fringes become more numerous as a result of tilt, and they also unfold as a result of the decreased wavefront deviation relative to its distance from the measuring plate (number of lines indicates the amount of tilt in waves; note that the angle of tilt, as well as the wavefront error, is grossly exaggerated). This makes the direction of deviation directly visible, unlike the pattern with zero tilt, where the direction of deviation is undetermined (for comparison, 1 wave P-V of defocus and λ/2 wave P-V of spherical aberration have very similar interferogram, while the forms of their wavefront deviation is quite different); on the other hand, the latter shows spatial extent of the deviation more clearly. Test plate can be tilted in various planes, optimizing the form of interferometric output to a specific need. With radially symmetrical aberrations it does not change the pattern itself, only the orientation, while with asymmetric aberrations change in the plane of tilt affects the form of interferometric lines (for instance, if the plate is tilted around the axis perpendicular to the shown tilt, it will produce S-like lines with coma, the larger error, the more so; with astigmatism, it would produce identical pattern to th eone shown above, only of different orientation, while tilting equally in both planes would produce oblique straight lines).

INTERFEROGRAMS AND ZERNIKES
Most programs for analysing interferograms of optical surfaces and systems use Zernike terms as wavefront descriptors. Since less than perfectly aligned interferometer, optical system, or both, induce aberrations not inherent to the subject of testing - primarily defocus, coma and astigmatism - Zernike terms for these three aberrations are routinelly excluded. However, there is no way to know whether those terms are caused by misalignment or they originate in the actual surface/system. Also, longitudinal misalignment can induce spherical aberration, yet it is never excluded. As an example, here's interferometric analysis of a 300mm f/9.8 Schupmann telescope. Due to its tilted surfaces, residual aberrations form asymmetrical wavefront, more or less like those of actaual, imperfect optical surfaces.

Top left shows system's longitudinal aberrations and ray spot plots for blue (F, 486nm), green (e, 546nm) and helium-neon line (He-Ne, 633nm), the last being the line interferometers commonly work with. Next to it is the center point wavefront in OSLO. To the right are synthetic i-gram of the wavefront in e-line (center) and helium-neon line with e-line best focus (top right). Evidently, if a system with axial chromatic error is tested in the helium-neon line, it will not show the error at the e-line focus. Instead, it will show the helium-neon line error, either somewhat defocused (with best e-line focus), or at the best helium-neon line focus, generally different from the error at the e-line focus. Hence, such systems, or elements, need to be tested in their optimized spectral line.
But this is only a small digression; on the bottom bottom is shown how exluding defocus (insignificant in this case), coma, and astigmatism, alters the actual wavefront, resulting in inaccurate RMS wavefront error and Strehl. For deducting Zernike terms is used Atmos Fringe, which is in good agreement with OSLO as to what the RMS error and wavefront shape correspond to the OSLO synthetic interferogram. The reconstructed wavefront map is quite similar to the original; presence of astigmatism is obvious, less so for coma, even if it is according to the terms about as much significant.
Due to the low system aberration level, the Strehl error due to exclusion of coma and astigmatism (0.99 vs. 0.97) is smaller than what it would be with largere errors. For instance, if the starting error is doubled to 0.056 wave RMS - which is still generally acceptable - the corresponding Strehl would be 0.88 before, and 0.96 after coma/astigmtism terms exclusion.
It is interesting that OSLO and ATMOS give different values for specific Zernike terms, but the final result agrees. The RMS wavefront error is obtained as a sum of all Zernike coefficients - equaling the RMS error in their absolute value - given as the term divided with its corresponding normalization factor (sq.rt[5] for primary spherical, sq.rt.[6] for primary astigmatism and sq.rt.[8] for primary coma). The final RMS error is given as a square root of the sum of the individual coefficients squared. When done for the first eight terms (erroneously named "coefficients" in ATMOS), it agrees to within +/-0.01 wave RMS with the sum for all terms, despite different term values. It implies that there are different Zernike algorithms; the one used by ATMOS appears to be using the higher order terms (beyond #8) more than the one used by OSLO.
The error due to removal of these Zernike terms can can be either positive, or negative (i.e. can result in a better, or worse than the actual Strehl), depending on their sign.

Interferometer can be fairly easily built by an amateur. While high optical quality can be achieved, or confirmed, with more traditional tests, like Foucault, interferometric test has the advantage of being able to test and quantify the entire surface. Also, coupled with a computer software program, it allows for efficient, flexible fringe analysis. Various forms of interferometer adopted to amateur's means and needs have been created by amateurs' ingenuity. Some of the examples include Bath inteferometer, Shearing interferometer, and Fizeau interferometer.

Follows more detailed description of other tests mentioned above: Foucault's, Ronchi, Waineo, Hindle, Dall, Ross and Offner null tests.

◄ 4.8. Fabrication errors ▐ 4.8.2. Foucault test ►

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