Introduction:

Foucault testing of telescope mirrors using a Couder screen is well described in Jean Texereau: How to Make a Telescope (Willmann-Bell), the standard handbook of mirror making. If you do not have this book available, here is a very brief description: A Couder screen has several openings in pairs, each displaying a part of an annular zone of the mirror under test. The Foucault tester has a narrow light source and an occluding "knife edge". This straight edge is movable along the optical axis as well as across it. You hold your eye a little behind the edge, looking past it at the screen, to compare the brightness of the holes. For each hole pair in turn, you determine the position along the optical axis where the holes are equally dimmed by the knife edge as it is moved sideways to occlude part of the returned light.

For each zone, the knife edge position determines the average slope, and you can use the technique described by Texereau, or easier, an evaluation program such as Tex or Sixtests (see below) to estimate the figure. If the mirror surface is very smooth, and also rotationally symmetric, this test can give a very good estimate. However, with a finite width of the samples, you cannot measure deviations or irregularities that are smaller than the width of a zone, or not rotationally symmetric.

You can go to the S*T*A*R page of links and find many links about Foucault testing (as well as other ATM subjects), and you can download Tex or Sixtests there.

This simulated image shows a screen with 5 hole pairs, and the knife edge is set for the middle pair to match.

This image was generated with the wave optics simulation program DIFFRACT.EXE by Jim Burrows, showing a 240 mm f/5 mirror - the center zone is larger than I would recommend, see later. The image should best be displayed on a monitor with a smooth greyscale.

In chapter II.31 there is an example of a Couder screen for the "standard" mirror of 8" f/6, using 4 zones, 3 of equal area and a center zone of about half this area. This screen turns out to be well suitable for its intended mirror, but the dimensioning in general is discussed only in the vaguest of terms - "the width of the outside zone is determined by practical considerations: if it is too narrow, good measurements are difficult to make; if too wide, the zone will not appear sufficiently uniform in brightness". While the zone width is not highly critical, I believe that a poor choice of screen openings may lie behind the commonly held (but groundless, I think) belief that testing with a Couder mask is inferior to other related tests such as the Everest pin-stick testing, particularly for "fast" mirrors of small focal ratio.

On this web page, I present a general approach to the suitable dimensioning of the zones and screen openings. If you feel the mathematics here are beyond your interest, you can skip the theory and jump directly to the Zone Number Calculator or the Zone Radii Calculator

Theory:

We will assume a perfect paraboloid mirror with an aperture diameter of D, and a paraxial radius of curvature (ROC) of R. The focal length is R/2, and the focal ratio f=R/2D. A zone on the mirror has a center or mean radius z and a half-width of h - the inner radius is (z-h), the outer radius is (z+h) and the area is 4*pi*zh .

We can calculate the optical path lengths from a point on the optical axis at a distance R + b from the mirror, to the points on the mirror at radius z and z+x, respectively. The difference in path lengths (from R+b to the mirror and back!) is (approx):
d=(x⁴/4 + zx³ + 1.5z²x² - x²bR + xz³ - 2xzbR)/R³

For b=z²/2R, the first order x term is cancelled and d=(x⁴/4 + zx³ + z²x²)/R³. The normal to the mirror at z intersects the optical axis at a distance R+b from the mirror where b is the relative knife edge offset (using moving source).

The terms in d of different power have different effects on the image of the zone openings as seen past the knife edge.
The second order x² term causes a brightness gradient across each opening. That is, one opening of the pair is brighter near its outer edge, the other near its inner edge. It seems the brightness difference across the opening depends on the maximum value of the term z²x²/R³. To get a reasonably even brightness, the zonal half-width h should be chosen to limit this term to a suitable value k=z²h²/R³, that means h=(kR³)^½/z. A large k will make the zone appear quite dark at one side and fully bright near the other, and comparing the sides is difficult (see lower pair). The contrast between the holes is proportional to their width, so if k is too small, the sensitivity of the test is lowered. In Texereau's example, k = 37 nm (1.45*10^-6inches), or approx. 1/15 wavelength of 550 nm light (see upper pair - for the lower pair, k=180 nm). This seems to be a good compromise.

The total area of the annular zone (see above) is 4*pi*(zh), or 4*pi*(kR³)^½ - this is independent of zonal radius. Making the zones of equal area is thus a direct consequence of making the zones equally well readable.

The third order x³ term will make the zone on one side appear a little brighter near both its edges at one side, and a little fainter near its edges at the other side, as this exaggerated example tries to illustrate (the x² term is artificially removed, the x³ term is 25 nm). If this term is large, it will make it difficult to compare sides. I believe this term should be no more than k/3, that is one third of the second order term. This means that for any zone where h>z/3, this term should limit the zonal width to for the inner zone or zones, this is the limiting constraint - here, h = R(k/3Z)^1/3 is reasonable.

In a footnote in chapter II-31 of "How to Make a Telescope", Jean Texereau discusses the center radius of the zones of Couder screens:

"...Couder...uses, instead of our arithmetic mean h(m), the quantity h(n), the 'root-mean-square' of the inner and outer radii...Theoretically, this leads to a more accurate surface figure.....The reader must let his conscience be his guide."

Near the zonal center of curvature, the image from each hole is a diffraction peak. The position of this peak is affected by the 3rd order term: the average wave front error, that is the average phase, over the zone is zero for b= (z²+h²/2)/2R. This is not an exact solution for knife-edge testing, but simulations show that zones appear better equalized for this knife edge position than for the more commonly used b= z²/2R.

In these images, the lower half of the left hole is pasted to the upper half of the right one, to enable direct comparison of the brightnesses:
The upper image is for b=z²/2R as suggested by Texereau,
The middle image is for b=(z² + h²/2)/2R as suggested here,
The lower image is for b=(z² + h²)/2R as suggested by Couder.

Thus, for best accuracy, the corrected zonal radius to use in calculations is:
(z+d)=(z² + h²/2)^½; d is approximately =h²/4z. This is, ironically, precisely midway between Couder's and Texereau's suggestions! The correction term is often small, but it is probably easier to include it than to calculate whether it is insignificant in a given case.

 

The wave front error across the zone is zh³/R³ or (kR)^3/2/z². For instance, with a 8" f/6 mirror with an inner zonal radius z=37 mm (=1.5") and k=37 nm, the error in knife-edge position is about 0.04 mm (or 0.0015"), and the wave front error is about 0.037 wavelengths.

The 4th order term is independent of z, and the smallest one for all but the center zone, where all other terms are cancelled. If this zone is large enough, this term causes a rather complex "donut" light distribution that makes side comparisons difficult. If it is small, the sensitivity is poor and the reading is of little value. At the edge of Texereau's center zone, the term is about 30 nm or 1/20 wavelength. I believe it is better to ignore the center altogether - at the very least, ignore as much of it as will be obstructed by the secondary mirror. If you decide to include it, keep the radius at about (3kR³)^1/4 to make it appear like Texereau's inner zone. The equivalent radius to use for evaluation is not obvious to me, Texereau uses half the zone radius - the left image is for zone radius=0.5*opening radius, the right image is for zone radius=0.71*opening radius. The difference in appearance is small.)

Calculating the zones:

Mirror size mm	inches	f/4	f/4.5	f/5	f/6	f/7	f/8	f/10
117	4.5	4.7	3.9	3.3	2.5	2.0	1.6	1.2
152	6	5.3	4.5	3.8	2.9	2.3	1.9	1.3
203	8	6.1	5.2	4.4	3.3	2.7	2.2	1.6
254	10	6.9	5.8	4.9	3.7	3.0	2.4	1.7
317	12.5	7.7	6.4	5.5	4.2	3.3	2.7	1.9
406	16	8.7	7.3	6.2	4.7	3.8	3.1	2.2
508	20	9.7	8.1	7.0	5.3	4.2	3.4	2.5
635	25	10.9	9.1	7.8	5.9	4.7	3.8	2.8

This table gives an overview of the number of zones needed for common mirror sizes, but you can use the calculator below to enter your mirror size. The central 5% of the area has been excluded in the table.

The screen must be made with an integral number of zones. The common way is to divide the mirror area into n areas of equal size, but if you prefer, the zones may overlap some, or be fewer than needed to cover the area. The number of zones can be calculated using the formula (as used in the calculator below) n=(1-0.01e)D²/16(R³k)^½, where e is the percentage of the area to be excluded from the center. Each zone should take 1/n of the total area (remaining). While the zonal area is not highly critical, it should not be far from the area given by the formula.

Decide the free diameter excluding the bevel - it may be wise to subtract an extra 3-5 mm (0.1-0.2") not to include a possible turned edge. Calculate the focal ratio, and look for the nearest table entry. Round the number of zones to the nearest integer (at least two).

With the zonal diameters, you can make a screen with the openings symmetric about the center. The corrected zonal center radius is given by the zonal radii calculator below, but if you prefer, you can calculate it as z+d=z+h²/2z .

You are welcome to mail your comments to Nils Olof Carlin