4.5.2. Foucault test

Invented by the French scientist Leon Foucault in 1858, this ingenious test uses point source of light placed at the center of curvature of a concave mirror (in practice, slightly to the side, so that the image can be observed), as illustrated on FIG. 34. The combination of simplicity and accuracy has made it the single most used test in the amateur telescope makers' circles.

FIGURE 34: The principle of the Foucault test. Light reflected from mirror surface carries the information on geometric properties of the reflecting surface: if it is perfect spherical, the light from the entire surface will converge to a single aberration-free focus. If surface deviates from spherical, focus location will vary with the zonal height. An opaque thin plate with straight, sharp edge (usually some type of metal blade, called knife edge, or KE for short) moving perpendicularly across the focusing light in the proximity of focus location produces a shadow moving across the surface; shape of the shadow tells instantly whether a surface is spherical, with one unique focus for the entire surface, or not.

If it is a sphere, a straight-edge shadow moves over the surface as the KE cuts through the light converging to or diverging from the focus (A'); or the uniform, light-grayish shadow spreads over the entire surface when the KE intercepts converging cone at the focus, producing so called null (A). This makes the test particularly well suited for quick and reliable tests of spherical reflecting surfaces. Non-spherical surfaces produce defocused, commensurate to their conic, creating various shadow forms as the KE moves through the converging light near aberrated focus (B), with only a single zone nulled, and the rest of surface area split between darker and brighter areas (B', edge zone nulled, patterns generated by Mike Lindner's Foucault Simulator).

While long ago replaced with newer testing technologies in the professional circles, Foucault test is still widely used by the amateurs. The standard reference is How to make a telescope by Texereau; more recent, A manual for amateur telescope makers, Leclaire. Online, David Harbour's description is among the most detailed of the Foucault test. Programs for analyzing Foucault test data, such as SIXTEST by Jim Burrows, are a computer-era enhancement to the test's proven value. More recent variants of the test include replacing KE with a wire (wire test), the Foucault zonal mask by (Everest) pin-stick, or the eye by a camera, taking shots of the entire mirror surface for selected blade locations, which then can be analyzed with computer software (automated, or Robo-Foucault, such as the "semi-automated" version detailed by Mike Peck).

Due to inherent aberrations of the spherical mirror surface in imaging distant objects, Foucault test is most often used in making paraboloidal mirrors. Ideally, paraboloid should be tested with collimated beam (i.e. for object at infinity), which is a null test for this conic. However, due to practical difficulty of producing collimated beam, it is usually tested with light source at the center of curvature (vertex radius). Since for non-spherical surface in such setup every zone has somewhat different focus location, the test consist in extracting the degree of deviation of the actual measured zonal foci from those corresponding to a perfect paraboloid.

Longitudinal aberration for object at the center of curvature is given by LA=KDρ2/8F, where K is the conic, D mirror diameter, ρ the zonal (radius) height normalized to 1 for D/2, and F the mirror focal ratio ƒ/D, ƒ being the focal length. It is measured with respect to the paraxial focus, with the minus sign for K<0 indicating that outer zones focus farther away than the central zone.

Specific manner in which the data obtained by measuring deviations from perfect zonal foci is used varies somewhat from one source to another. In general, the row data - the actual measured deviations from the zonal foci of a corresponding paraboloid - are first offset by the longitudinal aberration of a perfect paraboloid. The result is called residuals. Specifically, if the longitudinal aberration of a paraboloid for four zonal heights are L1P, L2P, L3P and L4P, and actual measured values L1A, L2A, L3A and L4A, the residuals are (L1A-L1P), (L2A-L2P), (L3A-L3P) and (L4A-L4P).

The purpose of deriving residuals is to filter out the error in placing the reference point for the measurements vs. actual mirror center of curvature, which is hard to pinpoint. The next step is so called reduction, which is deducting residual of a selected zone from the rest of residual values. Its purpose is to reduce the value of residuals to an actual longitudinal differential between the measurements. Selection of the zonal value to be deducted is arbitrary; it is usually either the inner zone, the ~70% (ρ~0.7) or the outer zone.

Obviously, the reduced LA value for selected zone is zero, which means that it is assumed to be a part of the perfect reference surface. If the actual measurements are identical to those for the perfect paraboloid, except for the reference point differential, the residuals for all zones would equal the reference point differential, and the reduced LA value would be zero for all.

The reduced value LR are used to obtain relative transverse aberration, from RTA=-ρLR/2F. It is termed relative because it is based on longitudinal aberration relative to a chosen zonal focus. It is usually measured against, or expressed in units of the Airy disc diameter. The minus sign determines RTA value as positive for the longitudinal aberration LR relative to the reference focus extending away from the mirror, and vice versa.

RTA is not to be confused with the transverse spherical aberration, because the two are generally different, i.e. relative transverse aberration in Airy disc diameters is generally smaller, possibly significantly, than the transverse spherical aberration for given value of conic deviation.

Consider 300mm f/5 prolate ellipsoid with K=-0.5 tested at the center of curvature. Taking, for simplicity, median zonal heights (ρ) as 0.9, 0.7, 0.5 and 0.3, the measured zonal foci will deviate from those of the perfect paraboloid by half its longitudinal aberration, which from LA=KDρ2/8F, comes to 3.04mm, 1.84mm, 0.89mm and 0.34mm, respectively.

Choosing the inner zone for the reduction gives the respective reduced values 2.7mm, 1.5mm, 0.54mm and zero. They give the respective RTA as -0.243mm, -0.105mm, -0.045mm and zero. In units of the Airy disc diameter (2.44λF, or 0.0134mm for 550nm wavelength and the effective F=10), it is 18.1, 10, 3.6 and zero, in the same order. For comparison, full longitudinal spherical aberration for this mirror (for infinity focus) is -3.75mm, and its paraxial blur diameter is 0.375mm, or 27.9 Airy disc diameters.

If the 70% zone is chosen for reduction, the relative longitudinal aberration values are 1.2mm, zero, -0.95mm and -1.5mm, and RTA is 0.108mm, zero, 0.048mm and 0.045mm (8, zero, 3.5, and 3.4 in Airy disc diameters), respectively. Nominally much better, despite being for the same surface. Obviously, the choice of reduction zone significantly influences nominal indicators of surface quality with the Foucault test, and the two test results are comparable only if using identical reduction zone. Choice of a different reduction zone will also give a different indication of which portion of mirror surface needs to be addressed. Choosing central zone indicates that all (other) zones focus short of the reference paraboloid, the higher zone the more so, hence that the surface needs to be corrected by polishing off nearly the entire surface, increasing toward the outer area. On the other hand, with reduction relative to the 70% zone, relative longitudinal aberration indicates the outer zone focusing shorter, and the two inner zones focusing longer. That indicates that the outer and inner zones need to be worked on, leaving 70% zone area out.

With the outer zone being chosen for reduction, the indication would be that surface deviation increases toward mirror center.

Since RTA can be used to construct the wavefront profile in zonal segments, with the wavefront zonal angular deviation from the reference wavefront's slope closely approximated by β=RTA/2R, R being the mirror radius of curvature, and the corresponding linear deviation of the aberrated wavefront Zβ, Z being the zonal width, the choice of reduction zone will also determine the resulting wavefront profile. Consequently, the best fit parabola for the outer, 70% and inner zone, will be one centered at the marginal focus (approximately), best focus and paraxial focus (approximately).

While commonly used in figuring paraboloidal mirrors, Foucault test can be used for any other conic shape. Longitudinal zonal defocus Δ with respect to paraxial focus relates to the conic K and the vertex radius of curvature R as

Δ = -K(ρd)2/R,

with ρ being the zonal height normalized to 1 for the pupil radius d, for fixed light source, and half as much when both, source and edge, are moving together (the minus sign of defocus for K=-1 indicates that the outer zones focus beyond paraxial focus, the result of over-correction). It is arrived at by setting 2ψ=1 in Eq. 9.1, and using the resulting peak aberration coefficient, Eq. 9.3, in the relation for longitudinal aberration in Eq. 10 (note that when substituting for F in the latter, F=-R/D due to the doubled mirror-to-image separation). It is particularly convenient when testing prolate ellipsoids with reasonably close far focus, when the light source is placed at the near focus, and null is observed at the far focus. Obviously, this defocus figure is based on the lower-order surface approximation.

The higher-order longitudinal aberration term, Δ'~-(2+K)K2(ρd)4/R3 (for fixed source, half as much for moving source), is negligible in nearly all amateur mirrors. Note that its sign is opposite to that for the lower-order aberration, for K>-2, which indicates smaller combined longitudinal aberration (for K=-2, the higher-order term does not affect the lower-order longitudinal aberration, and begins to add to it for K<-2).

By establishing focus locations of annular zonal openings for non-spherical surfaces, the surface shape can be approximated with high level of accuracy. Common consensus seems to be that the general limit to a repeatable accuracy with the Foucault test is ~1/10 wave P-V on the wavefront, assuming needed testing skills. In practice, the accuracy limit vary with the type of deformation: it is to expect that it is higher for the rotationally symmetrical overall figure, where needed surface accuracy is only half that showing in the wavefront.

On the other hand, Foucault's accuracy is generally lower for zonal, local and rotationally asymmetrical figure errors, particularly astigmatism. It also becomes less reliable for relative apertures significantly larger than ~ƒ/4.

Significant changes in the wavefront and ray geometry for the object at the center of curvature versus infinity, brings on quite a bit of change in surface aberrations. The P-V wavefront error of lower-order spherical aberration at best focus, from Eq. 7 and 9, is given by

Ws = -KD4/256R3 = KD/2048F3,

with K being the mirror conic, and F being the mirror focal number for object at infinity. For prolate ellipsoids, paraboloid and hyperboloids, the negative sign determined by the conic indicates over-correction. Comparing it with Eq. 66 shows that paraboloid with object at the center of curvature exerts the same amount of spherical aberration as a comparable sphere for object at infinity, only of opposite sign.

However, since the effective focal ratio F has doubled, the geometric aberration has changed: as specified earlier, longitudinal spherical is larger by a factor of four vs. that for the corresponding sphere and object at infinity (L=D/32F for the diameter D in mm), while the transverse spherical is now doubled, keeping the same proportion to the Airy disc. The former changes in proportion to the square, and the latter in proportion to the cube of the zonal height.

The wavefront aberration, transverse and longitudinal aberrations change in proportion to the fourth, third and second power of the zonal height, respectively.

For the stop at the surface, coma with object at the center of curvature is cancelled, regardless of the conic. The wavefront error of astigmatism, on the other hand, doesn't change with object distance, remaining as given by Eq. 71.1. It quickly increases with angular separation between the source and mirror focus, which needs to be kept at a minimum in order to preserve best possible focus quality. Since this separation is the incidence angle (with respect to the axis passing through mirror center) doubled, the value corresponding to off-axis height h in the equation is s/2, s being the . source-to-focus separation. Due to doubled mirror-to-image distance, the effective angle is yet another two times smaller, corresponding to s/4 off axis height at the infinity focus. Thus, replacing h with s/4 gives the RMS wavefront error of astigmatism induced as ω=s2/627DF3, for the aperture diameter D and infinity focal ratio F (in units of 550nm wavelength, ω=2.9s2/DF3; as little as 20mm source-to-focus separation with 200mm ƒ/4 mirror induces 0.09 wave RMS error of lower-order astigmatism).

Unlike the wavefront aberration, the geometric aberrations of astigmatism does change due to the doubled effective focal ratio F for mirror with object at its center of curvature, with the longitudinal aberrations larger by a factor of four, and the transverse aberration nominally doubled. The wavefront error changes in proportion to the square of the zonal height, transverse aberration with the zonal height, and the longitudinal is, as expected, constant.

Contrary to the common belief, mirror astigmatism can be detected with the Foucault setup (as well as Ronchi), even when quite low. Both, geometric and diffraction analysis (Astigmatism under the Foucault test, Linfoot, click on "print this article" for PDF) predict that astigmatism produces uneven intensity distribution along one of the two perpendicular (or nearly so, for less symmetrical forms) astigmatic axes, which has one side brighter, and the other darker than the rest of illuminated surface. Depending on its orientation in the setup, and the point of interception, this illumination asymmetry results on more or less obvious apparent rotation of the shadow (in the Ronchi, also depending on the orientation, it will cause either gradual change in line width, similar to the effect of spherical aberration - when the grating orientation coincides with that of astigmatic axes - or S-like line deformation when astigmatic axes are at 45 angle vs. grating). John Sherman's spot test web pages describe some practical approaches for detecting astigmatism in the Foucault or Ronchi setup, along with graphic illustrations. Quantifying astigmatism with either of the two tests is, however, more difficult.

◄ 4.5.1. Testing optical quality ▐ 4.5.3. Ronchi test ►

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