Theory and formulas developed byNils Olof Carlin |
Web page calculator developed byKen Slater and Nils Olof Carlin |

Copyright 2000 by Nils Olof Carlin/Ken Slater, Non-commercial use only |

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

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=(x ^{4}/4 + zx³ + 1.5z²x² - x²bR + xz³ - 2xzbR)/R³
**

For **b=z²/2R**, the first order **x** term is cancelled and
**d=(x ^{4}/4 + zx³ + z²x²)/R³**.
The normal to the mirror at

The terms in

The second order

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.

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**.

The upper image is for

The middle image is for

The lower image is for

Thus, for best accuracy, the corrected zonal radius to use in calculations is:**(z+d)=(z² + h²/2) ^{½}**;

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.

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

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

Use this calculator to find the best number of zones to use for your mirror.

In the table below are the calculated optimal number of zones for your mirror,

using the formula **n = (1 - 0.01e)D²/16(R³k) ^{½ }**as described in the theory section above.

Use this calculator to compute the optimal zone radii for your mirror. Measure the free diameter of the mirror face, 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.

For each zone, the inner and outer radii are calculated,

and the corrected zonal center radius, as described in the theory section above.

*To calculate zone percentages, set Mirror Diameter to 2, resulting in a Unit Radius.*

*This page was last revised on 2000-Jan-29.*