Niels
Noordhoek produced the
Bahtinov
Grabber, which uses line recognition algorithms to detect the diffraction
spikes produced by the Bahtinov mask. This takes much of the guesswork out of
trying to decide if the central spike is really centred.
Grabber analyses the three lines and
calculates the displacement between the central line and the intersection of
the other two lines.

This example shows a
simulation using another of Niels' software marvels 
Maskulator.
The focus error was 300 microns, and the distance between the 'X'
intersection and the centre line was 7.63pixels.
When approaching the region of
'critical focus' the distance can be less than 1 pixel and so is difficult to
measure with high accuracy.





With
this in mind, I tried a few different designs that would produce a greater
displacement for the same focus error. In theory, if the distance being
measured is greater, and there is no loss in accuracy of detecting the lines,
then there should be an improvement in accuracy. 



Assume that the angle between the two sloping lines is
2theta, and each line is displaced a distance
x. Some simple geometry shows that the intersection point will move a
distance 2xcostheta/cos(902theta)
The distance x
is determined by the change in focus position, so to ensure a larger
displacement of the intersection point, we must decrease theta.










The standard
Bahtinov mask has the angled slits inclined at 40 degrees to each other. Pavel
Bahtinov chose this angle because it results in a diffraction pattern that
makes it relatively easy to judge by eye if the middle spike is truly central.
This results in theta in the above diagram being
70 degrees. (Note that the diffraction spikes are perpendicular to the
slits). 




I constructed this mask, which
has the angled spikes inclined at 170 degrees to each other. This makes
theta equal to 5 degrees. The resulting
diffraction pattern is markedly different. 




This
'85 Bahtinov' mask is of no use whatsoever for visual work, because it is now
very difficult to judge when the middle spike is central. However, the Bahtinov
Grabber software has no such problems, and can locate the spikes with subpixel
accuracy.
With 85
degree angles, mathematics predicts that the displacement between the central
spike and the 'X' intersection should be 6.04 times greater than with
the traditional Bahtinov mask.
In the simulation a shift of 7.63 pixels has been increased to
46.5 pixels. This is a ratio of 6.09 which agrees closely with the
maths.
The diagram
to the right is to the same scale as the first diagram on this
page.



To
test the new mask I made a series of simulations of diffraction patterns using
'Maskulator'. I initially chose a range from 100 microns focus error to +100
microns, with images made every 10 microns.
A
similar series was made using the traditional design.
Niels Noordhoek obligingly rewrote
the grabber program so that it could analyse the new mask patterns, and he also
made a modification to display the measured focus error to 2 decimal
places. 

Initial results showed that there was indeed an increase in
accuracy. The 20 degree mask at 100 microns focus error showed a displacement
of 1.99 pixels. The 85 degree mask showed a displacement of 12.68
pixels for the same focus error. This is 6.37 times
greater. 

This graph shows the
difference between the focus error as measured by Grabber, and the generated
focus error in the simulations.
As can be seen, the 85 Bahtinov was within 2 microns
across the entire range.
(The repetitive undulations in the graphs is
due to the discrete size of the pixels in the images).



This animation shows the Bahtinov Grabber
in action as it analyses each
image.



This graph shows the
displacement between the vertical line and the intersection of the sloping
lines for a diffraction spike shift of 1 pixel.
An
89 degree Bahtinov would show a displacement of nearly 60 pixels, but the
diffraction spikes would be only 2 degrees apart, and a small error in locating
the spike positions would result in a large error in determining the
intersection point. It is also likely that Bahtinov Grabber would stand more
chance of getting confused if the lines were that close
together.



An
experimental 87 degree mask showed errors of over 2 microns over the
100 to +100 micron range.
The worst error with the 85 degree mask was 1.66 microns. It is
likely that 85 degrees is close to the optimum
value.



Preliminary conclusions are that
the 85 degree variant of the Bahtinov mask will
provide greater accuracy than the 40 degree mask when used in conjunction with
software such as the 'Bahtinov Grabber'.


See this
YouTube video which shows how
the diffraction spikes move for the 85 degree mask.
Note
that when purely visual inspection is used to determine optimum focus,
then the traditional Bahtinov should be used, or
another mask design, the Carey
Mask.


Star tests (15th May
2010)

Initial tests on the sky show that
the 85 degree Bahtinov works in principle, but the error in precisely locating
the diffraction spikes is significantly greater than with the 20 degree mask.
This was tested by fixing the telescope focus and then taking 10 exposures with
each type of mask. The 20 degree mask gave values
of defocus of 23.23, 25.10, 29.13, 26.29, 14.73, 36.88, 31.52, 24.43, 21.39,
29.17 microns The 85 degree mask gave
6.42, 4.00, 0.70, 9.98, 11.67, 8.82, 1.77, 7.03, 28.12, 0.88
microns 

The
spread of values for the 85 degree mask was broader than for the 20 degree
mask. Although both masks will allow critical focus to be reached it was
disappointing that the 85 degree mask did not live up to
expectations.



21 May
Another test. Fixed focus.
Quite close agreement between the
two masks this time, but still greater spread with the 85 degree
mask.





The diagram above illustrates
the problem. Examples are for 20, 45, 70 and 85 degree masks. Dark blue lines
represent the diffraction spikes. Paler lines show the error uncertainty in
measuring the true line positions. Assuming that the error in finding the
intersection of the sloping lines lies to the left, then the orange shaded
areas show all possible positions that the measurement could give. The numbers
given show relative values. Clearly the 85 degree mask could have a maximum
error 6.07 times greater than the 20 degree mask, neatly cancelling out any
advantage!
If time
permits, I will try 45 and 70 degree masks.
Regrettably the
hoped for increase in sensitivity of 6 has not been realised.


Yet another mask
design...
It is possible to get even
greater displacements between diffraction spikes. This variant has two pairs of spikes. The intersections of the
spikes move perpendicular to each other.
The
slits are arranged so as to generate lines that move in directions that will
produce the maximum displacement. 



In this
simulated image the one intersection has moved to the left, the other
downwards. This example is also for a defocus of 300 microns.
Line detection software that
measured the distance between the two intersections would find a 70.8 pixel
displacement (compare with 46.5 and 7.63 for the 85 and 20 degree
Bahtinovs).
See
this YouTube video








