RE: Talbot ?

A square wave grating when illuminated by spatially coherent light will
form an image of itself at the Talbot spacing. Thus when a readout grating
is placed at this position, it has optimum contrast.
It has been known as Talbot effect, or Talbot spacing, for a long time, but
is not widely discussed outside interferometry circles. There are articles
in Applied Optics, and probably in JOSA (journal of the optical society of
america) that hash it out.
I doubt if it is known by any other name, at least in English speaking
engineering.
There was a commercial device by the name of a Talbot interferometer about
15 years ago, but it was rather obscure, and anyone that wanted one could
make one very easily (I did).

Elliot Burke


Date: Fri, 24 Sep 1999 12:48:12 +0200
From: "Arne Chr. Jorgensen" <instel@...>
Subject: Talbot ?

Elliot:
----
I am not familiar with the "nomentclure" - what is Talbot spacing
? Is there any papers on the net, which could sheed some light
into this ?

There is something you said, that I take as:

The Talbot spacing - must be the distance between the stationary and
the moving gratings.

Could this be known under another name ? What we have is two
diffraction gratings, and just a bunch of difficult stuff appears,
interference, etc.

But you hit a "nerve" - telling me something I think I recognize. I
just can't grasp it, yet.



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Message: 4
Date: Fri, 24 Sep 1999 14:07:22 +0200
From: "Arne Chr. Jorgensen" <instel@...>
Subject: Talbot

>Elliotf wrote:

>>At the Talbot distance these orders coincide in phase,
>>reconstructing or imaging the grating.
>

>This is giving me "arne-itis" again, - this is too difficult for
>me !

>I may be wrong, but what you have written here, looks like the the
>condition for a "dark" fringe:

>sin ( theta ) = n*(lambda)/ a, ( n= +-1,+-2,+-3,....)

>? and this is for a single slit, - but we are talking about many,
>hence interference with many. That is a diffraction grating.

This is the grating equation, it gives the relationship between wavelength,
diffraction order, grating period, and diffraction angle. It assumes an
infinite number of grating lines. For a finite number of lines, the
diffraction angle has a range of values: its is convolved with the
diffraction pattern of the pupil. What this means is that it broadens a
little. For two or three lines the pattern is pretty broad and it's hard to
assign a single angle to the diffraction.

>But we have more: We have two separate diffraction gratings on top
>of each other, we must also take into the account, the source of
>light, and the sensor characteristics. Distances, and energy
>distribution.

Actually, its conceptually pretty simple. If you know the diffraction
pattern with a single point source of light, then adding another point
source will simply add intensity to the diffraction pattern. The second
pattern will be shifted with respect to the first pattern if the two point
sources are separated. Add up (integrate) the effect of all point sources
and you have the solution for extended sources. Note that intensities, not
amplitudes, are added here.

The second grating is usually just in front of the detector, so it suffices
to calculate the energy that falls between the dark lines.

>>This is a remarkable effect, and i recommend trying to view it.

>Okay, could you dream up a simple way for us to see this. Why I
>say "dream" - is that we would need an example with things we could
>use, and not some lab equipment.

If you have a transmission grating, a laser pointer, and a low power
microscope, place them all in a line: shine the laser through the grating,
and magnify the grating with the microscope. You should see an image of the
grating if you are focused on it. Now move the microscope away from the
grating, it will change in appearance in a complex way. When you get to the
Talbot spacing, the grating will again appear in perfect focus, even though
it is far from the focus of the microscope. Move that distance again, and
you'll see a second image of the grating.

>> Another step forward in understanding them requires a more
>>rigorous diffraction theory, which is a bit beyond the charter of
>>this > list.

>Well, what ever you could tell us, is okay by me - if you could
>put it in plain english :-)

Most diffraction theory is a scalar theory. This means that the e fields
are scalar, a single value for each point in space and time. This is an
approximation, as actually fields are vectors. The vectors have direction,
and even the direction changes with time. This is far harder to calculate,
and scalar theory give a pretty good result if the frequencies of the
gratings are much less than a wavelength. But there are some anomolies that
only can be explained with vector theory. Usually it is done
computationally, since most of the equations can't be solved.

>Okay, let see...hmmmm Jon Elson could make a film, this could
>be glued on top of an aluminum bar, you could shine a laser or
>something down at this bar at say, 45 degree. Have a grating at -45
>degree. You could interpolate the output, - giving much higher
>resolution than he now has. And for rapid moves, you would skip the
>interpolation. How about it ?

A properly made LED source can be used instead of a laser. I haven't
thought much about reflection gratings, but there may be a neat way to
package them.

>The thing is, - we need to understand some of the problems
>involved, but the use of it would be to make some stuff.

Yes, it's a significant engineering problem.

>I just want to encourage you to come with what ever you have to
>say, even if it is pure math. The way I said things, could very
>well been understood as "we should only focus on the practical
>stuff". This would be wrong, - even if just a few would have any
>interest in it, - it would still be very valuable. Many in this
>group are new to electronics, but they listen and the learn. And
>soon they put together some drives and wire it up. But in the
>beginning, a lot is new and difficult. To set up Linux the first
>time is difficult. What I am saying - even if some of the subjects
>tend to be difficult, - we need that part too.

>You have my attention, and I have been concerned that some of the
>things I may have said, could be misunderstood. We need it all,
>from the most brilliant input - to the stupid ones. And thank you
>all for making this interesting.

>//ARNE

Thanks for the encouragement. Gratings are a deep and mysterious subject
even when one is used. when they are used in combination things get very
complex. I will try to document some experiments and put it on my web site.


Elliot Burke