From: "Sidor . Kurt" 
To: 
Reply-To: "Sidor . Kurt" 


Dear List,

At my current job I have been designing optical encoders utilizing
diffractive optics principals.  I have numerous Mathcad files utilizing
Fresnel and Fraunhoffers models for simulating diffractive effects.  With
slight dimensional scaling I have run a few simulations of Raleighs and
Dawes criterions for an ATM size 6" diameter aperture case.

Lambda = 550 nM (0.00022") wavelength of light D = 6 inches, mirror diameter

Raleigh Criterion: "Stars are said to be "just resolved"
when the center of one Airy disc falls upon the first minimum of the second
Airy disc" (according to Hect, OPTICS, third edition)

For a 6" diameter 1.22*Labmda/D = 0.92 arcseconds

Dawes says stars can be resolved at 4.5/D seconds of arc, or in our case
0.77 arcseconds. (Smith, MODERN OPTICAL ENGINEERING)

I plotted some Mathcad graphs of Fraunhoffer diffraction patterns separated
at these spacings of 0.92 and 0.77 arcseconds respectively.

The results can be seen here:

http://images.andale.com/f2/115/106/3663062/1042483409819_RALEIGHDAWES1.JPG

The separate Airy disc intensities are plotted on the left with the
mathematical summations plotted to the right. The Raleigh separation shows
a "contrast" of 28% change in the intensity from the brightest
peak of one Airy disc to the dimmer "shade of gray" that resolves
it from the next one.  Due to the slope of the Airy discs central bright
core when you reduce the spacing from 0.92 arcseconds to 0.77 arcseconds
this intensity "contrast" reduces to only a 4% difference from
the brightes peak to the "shade of gray".  This result implies
that a "contrast" change as small as 4% could be detected by
Dawes defining his "limit".

I then created a new model of summing the intensities of 100 Airy discs
defining an object 5 arcseconds wide for the same Fraunhoffer pattern
created by a 6" aperture.  This was identical to the previous model
using only two to define two stars but now I am creating an
"extended" object by mathematically putting them all next to
eachother.  Then I removed one of the sections in the middle of my source
simulating a dark line 0.05 arcseconds wide.  The resulting change in the
surface intesity of my 5 arcsecond object was a 6% "contrast"
change.  This contrast change is greater then what the Dawes criterion of
4% contrast needs as a minimum.

The plot of this dark line (0.05 seconds wide) against a 5 arcsecond object
can be seen here:

 http://images.andale.com/f2/115/106/3663062/1042483417451_RALEIGHDAWES2.JPG

This does not imply that a 6" telescope can "resolve" (as
one would classically use the word) 0.05 arcseconds but it does imply that
one could detect the prescence of a dark feature that small against a
brighter background although its "gray" width would be quite
subtle.  I would guess that Martian "canals" would fall into this
category.

Before I created my model of an "extended" object 5 arcseconds
wide, I tried using smaller objects.  I tried 1.8 arseconds, twice the
Raleigh criteria. My results were poor and I could not "resolve"
dark features against it.  I believe the cause of this is the slope of the
central bright spot in the Airy disc.  This slope itself is too wide and my
"extended" object of only 1.8 arcseconds would not create a
"plateu" of uniform "flat" light intensity across its
middle width, it just looked like a wide Airy disc.  In other words the
diffractive edge effects washed out the small features because they were
too close to the edge of the object to be "resolvable".  You need
to be some finite distance in from the edge before any change in contrast
can be detected.  I then increased my width to 5 seconds of arc and got a
good looking "flat" plateu.

Any thoughts?  This is all just a mathematical model, I lack the personal
experience of being an experienced observer of things like Saturns ring
divisions or Martian canals.  I'm just an engineer who's worked with
diffraction too much in the last few years.

Regards,

Kurt Sidor
Mechanical Engineer
Dynamics Research Corp.
Encoder Division

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