◄ 4.8.1. Testing optical quality   ▐    4.8.3. Ronchi test ►
 

4.8.2. 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 mirror focus is separated from the source, and focusing light can be intercepted without cutting off source of illumination), as illustrated on FIG. 52. The combination of simplicity and accuracy has made it the single most used test in the amateur telescope makers' circles.

Due to inherent aberrations of the spherical mirror surface in imaging distant objects, paraboloid is preferred surface for telescope mirrors. Consequently, 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.

These expressions are valid for most amateur mirrors. For very fast mirrors, inclusion of correction terms may be needed to maintain test accuracy.

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).

The following is a graphic example of the main Foucault test parameters and relationships, illustrated with the actual surface being a sphere. If the sphere is, for instance, 150mm f/8 mirror, thus with radius of curvature R=2400mm, its longitudinal aberration at infinity focus is D/32F=0.586mm. With the object at the center of curvature, the wavefront incident on mirror surface coincides with it, and it is reflected straight back to the center of curvature. Thus, the longitudinal aberration is zero. However, if paraboloidal surface is desired, then the reference is a paraboloid of identical vertex radius, which would produce longitudinal aberration four times greater than that of the sphere at infinity focus, or 2.34mm. It varies with the square of zonal height, so taking for simplicity three off-center zones, 30-50, 50-65 and 65-75, hence of the respective zonal widths z1=20mm, z2=15mm and z3=10mm, with median heights (h) of 40, 57.5 and 70mm, their longitudinal aberrations (LA) are 0.67, 1.38 and 2.04mm, respectively.

Since the measured zonal sections of the sphere are perpendicular to the mid-zone radius, and all the radii project into the center of curvature, the shape of reconstructed wavefront approximates mirror surface. However, if the actual surface is desired paraboloid, projected zonal segments are perpendicular to the mean zonal radius (i.e. local radius of curvature of the paraboloid) do not converge to a single point (with moving source, Foucault test directly measures foci locations of these local, i.e. mean zonal radii, which in turn determine averaged zonal slope at the surface). As a result, sections of wavefront from any single point do not reflect off simultaneously. Since paraboloid's surface spreads out toward the edge, the incident wavefront originating, for instance, at the center of curvature, reaches outer zones latter, and those sections lag behind the inner wavefront portion. In effect, the wavefront reflected off the paraboloid flattens out relative to paraboloid's surface, and the differential between it and the wavefront produced by a sphere is double their zonal sagitta differential.

There is a direct relationship between the degree of deviation of these two wavefronts (the actual one, and that of a perfect reference surface) and measured zonal deviations.

From TA=(h/R)LA, the corresponding transverse aberrations for the three zonal LA readings with the above sphere, in the same order, are 0.011, 0.033 and 0.0595mm, and their zonal slope deviations, s=(TA/R)z, are 0.0000917, 0.000206 and 0.00025mm. In units of λ=0.00055mm wavelengths, it is 0.17, 0.037 and 0.45λ. The total deviation is a simple sum of the three, or 0.99λ.

This is nearly double the maximum surface differential between this sphere and its reference parabola of 0.52 wave (the figure is generally somewhat lower, due to the outer zone being averaged out). Assuming null at the zone 1, the corresponding relative transverse aberration is greatest at the zone 3, as RTA3=(TA3-TA1)/2.44λF (for the doubled Airy disc with object at the center of curvature, with F being the infinity focal ratio) - a disastrous 4.5 Airy disc radii. But this mirror is, in fact, near diffraction-limited. Taking mid-zone for the null nearly reduces the RTA in half, as does nulling at the outer zone.

The 0.99λ wavefront error at the wavefront constructed from the measurements is not the actual error. In this case - and nearly any other - there is a parabola of slightly different radius, whose wavefront deviates less - often significantly so - from the constructed wavefront. It is called "best fit parabola". In this case, best fit paraboloid has slightly stronger curvature than one with vertex radius equal to that of the actual surface. It has identical sagitta to that of the sphere, touching the latter at the vertex and at the edge. The error vs. this parabola is reduced by a factor of 0.25, from 0.99λ down to 0.25λ. This is the final test result.

The only difference between sphere and a conic closer to a paraboloid is that the latter does not have zonal foci coinciding at the center of curvature. These foci are separated as for paraboloid, either in somewhat smaller (prolate ellipsoid) or greater degree (hyperboloid with conic less than -2), thus with smaller longitudinal aberration at each zone.

Correction terms for the longitudinal aberration

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 in mm (s is in mm as well) and infinity focal ratio F (in units of 550nm wavelength, ω=2.9s2/DF3; as little as 20mm source-to-focus separation with 200mm f/4 mirror induces 0.09 wave RMS error of lower-order astigmatism). For s and D in inches, ω=74s2/DF3.

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.

ASTIGMATISM IN THE FOUCAULT TEST

Foucault test is known as insufficient at detecting astigmatic deformation of the test surface, but it does have the capability to detect relatively large errors, under certain conditions. Following two examples should illustrate more specifically its limitations in this respect. The first is a smaller, medium fast mirror, with which detection limit falls generally above the level of spherical aberration of the mirror at the r.o.c. The sensitivity is the lowest when astigmatic axis is aligned with the KE, and highest when it is at a 45-degree angle.

The other one is a large, fast mirror, with which the detection limit is, roughly, at half its spherical aberration at r.o.c. However, since it is several times higher than in the first example, the detection threshold is actually about twice larger astigmatic deformation. If compared to the Ronchi test, it is significantly less sensitive in this respect.

◄ 4.8.1. Testing optical quality    ▐    4.8.3. Ronchi test ►

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