Solving Diffraction Patterns: Investigating Central Spot Intensity/Radius

[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 into 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/~dzierba/P360n/KPAD/Exps/Poisson/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]

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.

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).