In the Foucault test, light from a pinhole-sized source is used to illuminate a concave mirror. When the light source is located near the center of curvature of the mirror, the reflected light is brought to a near focus at roughly the same point. A knife edge is used to cut into the converging beam to allow the user, examining the mirror from behind the knife, to ascertain the optical qualities of the mirror. A variation of this test, using a Couder mask just in front of the mirror, is depicted in figure 1. In this form of test, each zone in the mirror returns light to a single point on the axis. Measuring the axial distance of this point from the center of curvature, yields information about the conic deformation of the mirror. If the mirror is spherical, the light from the source at the center of curvature is returned exactly back to a focus at the center of curvature. If the mirror is paraboloidal and the source and knife move axially together, the axial distance from the center of curvature that a zone y returns the light is calculated by:

     y² / ( 2 * R )

where R is the radius of curvature of the mirror.

However, if the mirror to be tested is convex, the center of curvature is behind the mirror material. If a pinhole light source is placed anywhere along the axis in front of the mirror, the reflected light is divergent.

In figure 2, light from a pinhole source is sent through an auxilliary lens set to be reflected from a convex mirror before returning through the lens set to a focus near a Foucault knife edge. Suppose that the lens set itself is fully corrected for longitudinal aberration at the two conjugates. With that assumption, the setup is an accurate null test for a spherical convex mirror.

The f-number for such lens sets may make it difficult to construct one corrected at the conjugates; perhaps as difficult as constructing the secondary itself. Furthermore, such a corrected lens could not be reused for secondary mirrors with other radii of curvature.

The requirement in the test setup that the lens be accurately nulled for longitudinal aberration may, however, be removed. Figure 3 represents a calibration setup for the corresponding secondary test setup. The calibration setup removes the reflective mirror temporarily, replacing the original pinhole source with one placed at what would have been the center of curvature of the secondary mirror. This setup could be used to determine the single pass longitudinal aberration for each zone used of the final convex mirror test.

To a first approximation, in the secondary test, the zonal change for each zone is the difference between the actual zonal measurement in the secondary test and twice the zonal measurement in the calibration:

     LA_effective = LA_{secondary measurement} - 2 * LA_calibration

However, most Cassegrain secondary mirrors are not spherical, but quite hyperbolic. A test nulled against a spherical mirror will not work.

For typical secondary mirrors, the focus for the outer zones will fall short of the central focus. An estimate of this effective longitudinal aberration for the zone y_m is:

     LA_estimated = - SC * y_m² * t_sec² / R³

where:

     SC is the Schwartzschild constant for the secondary
     t_sec is the distance between the central zone knife edge and the lens set
       (This distance should be the same for both the calibration setup and the secondary test setup.)
     R is the radius of curvature of the secondary, as above

For a moving source, the longitudinal motion of the source/knife will be roughly half of the estimated longitudinal aberration.

While the estimates above do not require a strong model of the lens set itself, they are indeed only estimates.

The primary source of error is in the secondary test setup. The key, however, is still in the calibration setup. If the calibration setup uses a moving source that is coincident with the intersection of the zonal normal of what would be the secondary, then the knife crossing for that zone is coincident with the moving source location of the knife edge in the secondary test setup. Again, this is calibration is exact.

When the calibration is known for two spaced values of Schwartzschild constant, determination of intermediate values may be calculated by linear interpolation. In the case of a Cassegrain secondary, the two values of Schwartzschild constant are zero (the fixed source calibration) and the desired Schwartzschild constant for the secondary (the moving source calibration).

When this full calibration setup is used, the parallax between the zonal mask and the mirror surface must be taken into account. For the zone y_m at the zonal mask, let the zone at the secondary be y_s. Then the sagitta at that point is:

     S = C * y_s² / ( 1 + sqrt( 1 - ( SC + 1 ) * ( C * y_s )² ) )

Where C is the curvature of the secondary, or C = 1 / R. The axial distance of the normal intersection from the zone on the conic is:

     Q = sqrt( 1 - ( SC + 1 ) * ( C * y_s )² ) / C

To calibrate the zone y_m, the source must be set at a distance Q + S from the mask, measuring the distance of the knife crossing to find the difference of the knife edge in the secondary test setup. It should be noted that this test includes not just a test of the Schwartzschild constant of the conic, but, if the distance of the knife edge from the lens set is retained, the radius of curvature of the secondary. Again, a strong model of the lens set itself is not required, as the calibration subtracts out many prescription errors.

The parallax correction may be found from:

     y_s = ( Q / ( Q + S ) ) * y_m

using an initial guess of y_s = y_m / sqrt( 1 + ( y_m / R )² ), and repeating until the desired accuracy is attained.

Note: There is reason to suspect that the above does not converge well for oblate surfaces. Thus, in this tool, the recalculation of y_s is determined by Newton-Raephson iteration on the quartic:

     ( ( SC * C )² ) * y_s⁴ -
       ( 2 * ( SC + 1 ) * C² * SC * y_m ) * y_s³ +
          ( ( ( SC + 1 ) * C * y_m )² - ( SC - 1 ) ) * y_s² +
            ( 2 * SC * y_m ) * y_s -
               ( SC + 1 ) * y_m² = 0

ZONE CALCULATOR

To use this calculator, either the radius of curvature or the curvature and either the Schwartzschild constant or conic constant of the secondary need to be known. These values are entered into the appropriate box and the button under it is clicked, selecting that option. Clicking the selection automaticly calculates the non-selected option value.

The size of the secondary may be established as either the diameter of the zonal mask or the diameter of the secondary. This value is entered into the appropriate box and the button under it is clicked, selecting that option. Again, clicking the selection automaticly calculates the non-selected option value. Clicking the selection also selects the zonal type in the zone calculation below. This zonal type is either the mask or secondary.

Calculation for a specific zone may either be absolute (dimensional) or relative (fractional/root). The specification of an absolute zone is done by entering the dimensional data and clicking on the button under it. This selection also establishes the maximum zone value as half of either the mask diameter or secondary diameter, as appropriate. The specification of a relative zone is done by entering all of the zonal value, the user selected maximum zonal value, the amount of the edge of the secondary not tested, and the percent of the remainder that is not tested (in the center). The option is then selected by clicking one of the relative options. For example, setting the zone to 70 and the maximum zone to 100 will calculate the 70% zone value. Again, this will be relative to either the mask or secondary, as appropriate; the relationship is linear if the fractional option is selected and proportional to the square root if the root option is selected.

For the given zone, the values presented are the zonal dimension at the mask (y_m), the zonal dimension at the secondary (y_s), the sagitta of the secondary mirror for that zone (S), and the axial distance of the moving calibration source from the zonal intercept (Q). In addition, the sums S+Q (the axial distance of the moving calibration source from the secondary node), and S+Q-R (the zonal difference of the axial distance of the moving calibration source) are provided.