Poisson's Spot

Dan Forth

Hi

hoping someone can help me with this problem. If I block plane monochromatic light (wavelength lambda) with a circular disc of radius r, and then look at the diffraction pattern on a screen placed a distance x behind the disc, I get a bright spot in the centre of the shadow. All well and good. What I want to know is how do the intensity and radius of that central spot vary as a function of x. Specifically I want to know how small I need to make x in order to effectively eliminate the bright spot. Clearly at x=0 there is no bright spot - as x increases I don't know whether the spot appears with increasing radius or increasing intensity (presumably both) but I need to be able to put some numbers in to find at what point the spot becomes intolerably bright/large. For anyone interested the reasoning behind my problem is a botched photolith job that I'm trying to do an autopsy on and avoid repeating the same mistakes!!

Thanks very much for any help anyone can give,

Dan

Doc Al

Look up a mathematical discussion of Fraunhofer diffraction in a physical optics text (or just poke around the web). That spot is also called the Airy disk. The intensity is described by a Bessel function. Here's a site that describes it a little: http://dustbunny.physics.indiana.edu...n/poisson.html

Dan Forth

Hey Doc

cheers for the reply. Yeah I've been looking through the textbooks (Fresnel diffraction rather than Fraunhoffer I think) but it's not as simple as I'd hoped. The Bessel functions from the standard analysis will never give you the dot disappearing, as the analysis is invalid for x approaching zero (which it's going to have to do.) The classic undergrad approach fails at the first hurdle. I've got a feeling it's going to involve a lot of unpleasant maths starting right from first principles of diffraction (doubtless quickly resulting in an analytically impossible integral) - just wondered/hoped whether anyone out there had already ploughed through this or had a better way in mind!

Doc Al

Oops, you're right. Obviously I haven't looked at this stuff in ages: I was thinking Airy disk (and far-field limits) while you were talking about the Poisson spot from a circular obstruction (a consequence of Fresnel diffraction).