# Quantifying  Surface Features visible in a Foucault image.

By Dale Eason  5/07/2006 Copyright 2006
 Foucault Image Interferometer Interferogram

I used a Bath interferometer in an attempt to measure surface features that were also visible in a Foucault image of the same telescope mirror. The analysis technique I used is called FFT analysis and can measure detail finer than inteferogram fringe tracing methods.

A Bath interferometer is not a standard tool for making telescope mirrors nor is it needed.  Foucault testing is used by ATM's and professionals to make diffraction limited mirrors of high quality.  However Foucault testing does not give a quantitative value for surface roughness of a mirror.  The roughness can be seen in the Foucault image but not easily measured.  It has been stated in other published works that a well done Foucault image can show surface details as small as 1/100 of a wave.  Many ATM's have asked how big are those surface defects that the Foucault shadows represent. The truth is that a high quality mirror will not show any roughness in a good Foucault image. So the bottom line is, keep working on it until all roughness goes away in the Foucault image.  However, I thought it would be interesting to try to quantify what can be seen.

The technique I used was to take a mirror that showed roughness in the Foucault image and measure the same mirror with my Bath interferometer.  I used an analysis technique described by Mike Peck that allowed me to see detail in a smaller surface area than can normally be done using fringe analysis.1  A Bath interferometer is easy to build and within most ATM's budget.  However, my Bath system would not be good for analyzing how smooth a mirror is because it adds artificial surface roughness to the measurements.  I discuss this a little later.

I used the surface data obtained from the analysis to simulate the Foucault image using FFT mathematics simulation of wave theory similar to that in Diffract by Jim Burrows.  This way the real and simulated Foucault images can be compared to validate the technique.  If the images are similar then I captured and can measure real surface roughness.

The resulting images shown below are from  averaging 14 interferograms similar to this to remove air turbulence and equipment vibrations. An undesirable side effect of the FFT technique is that it can add false surface roughness to the analysis. That analysis noise must somehow be identified and ignored.   The noise can be reduced and smoothed by low pass filtering of the computed surface. After doing the smoothing I think you can see through the noise by looking for identical areas in each image.  The green image on the left is the real Foucault image.  The image on the right is the simulated one from the interferometry analysis. Any bumps present in the simulated but not in the real are the result of the noise I mentioned.

This mirror has large defects resulting from a machine figuring lap failure. But for my purposes that is even better because it give us some shadows that all can recognize and shows up well in the simulation.  The two lower images are the actual 3D surface error compared to a sphere as computed from the analysis.  The white rectangular box in the lower left is 1/8 wave high.

 I have highlighted areas in the simulated image that I think are also in the real Foucault image.  From top to bottom they are:  An angular shaped  hill inside depressed ring  A elliptical depression at the top edge of the central cone Twin peaks at 8 O'clock in the broad central raised ring. The easiest to see in the 3D images are the twin peaks.  The hardest to see in the 3D surface above is region 1. But if you look carefully you can see the raised bump in the deep valley just of the edge. There are many more bumps in the simulated image than in the real Foucault image that are caused by system noise.

### Cross sections through the computed surface

My software can measure surface heights through selected cross sections. The results are displayed below.

#### Vertical Profile

 Here is a surface profile from top to bottom through the middle of the mirror in waves of 550 nm.  This should give you some idea of the magnitude of the relative deviation.  Note: Despite what is stated on the graph the vertical scale is error on the wavefront and not at the surface.  Thus the error on the surface will by 1/2 that stated in the graph.  The numbers along the bottom are radius from the center.

#### Feature 3 height

Here is a cross section through the twin peaks of feature 3. This is wavefront error so the surface values would be 1/2 the value shown.  The horizontal numbers are not to scale but the vertical numbers are accurate.
The left peak to valley distance is about  is 1/25 wave on the surface.  We can see both peaks in the real Foucault image so that means you can see the smaller PV of 1/40 wave.

#### Feature 2 and 1 height

I measured the faint elliptical depression of feature 2 at the top of the cone to be 1/60 wave.  Feature number 1 also has a height of 1/60 wave.

#### Smallest Visible Feature Measured?

There is one region that I haven't talked about until now because I'm not sure it is real.  There may be a faint twin bump at about 10 O'clock in the central ring.  There is a similar easy to see twin peak in the simulation that may be caused by noise or it could be noise enhanced version of the real structure.  Below is a cross section through that structure.
If it is real then the smaller peak is only 1/125 wave peak to valley on the surface.  I think I can see it.  You will have to decide for yourself.

### Conclusion

The conclusion I reach is that for my setup it is pretty easy to detect features that are 1/60 wave on the surface of a mirror by examining Foucault images.  Many factors like slit width, camera and lens settings, and display type effect the visibility of Foucault shadows.  It should be noted that the best way to see fine detail in a Foucault image is when the knife is placed near the focus of that area.  That is when the area will have a gray shade that makes variations easy to see.  To see the whole mirror this can only happen when it is a sphere.  With the parabola  it becomes much harder to see the whole mirror at one time.

### Acknowledgments

Thanks to Mick Peck for describing the FFT analysis process, Jim Burrows for implementing the FFT Foucault simulation technique in Diffract and Steve Koehler for explaining how to implement both of them to me.  Special thanks goes to Dave Rowe who started the Yahoo interferometry group and championed the Bath interferometer.  Lastly, to a local ATM Mark Austin whose figuring disaster turned into a beneficial experiment.

1. Those interested in FFT interferometry analysis can find out more about it and the Bath interferometer on the Yahoo group interferometry  http://groups.yahoo.com/group/interferometry/.