Interference Pattern w/ D ~ d: Qualitative Analysis

[buffordboy23]

A simplifying assumption in determining the interference pattern from a double-slit experiment is to assume that the screen distance, D, is much larger than the slit separation, d. This allows us to make the approximation that the two optical rays are parallel as they travel to the common point, P, on the viewing screen. Does anyone know of any good websites, hopefully ones with pictures, that discuss the appearance of the interference pattern when D is about the same order of magnitude as d? If possible, I'd prefer a qualitative discussion rather than a quantitative one at this moment in time. Thanks.

 

[buffordboy23]

Well, I still couldn't turn anything up, so I wrote a Mathcad simulation to model the behavior and present it for any feedback. Depending on the parameters, interesting behavior is present. I assumed the same application of physics with the new derivation (presented below) as compared to the original derivation.

New derivation: Let the central axis lie along the x-axis and be the perpendicular bisector to the line segment S1 and S2, where S1 is a point along the positive y-axis and S2 is a point along the negative y-axis; the point of intersection is taken as the origin. Then $$\overline{S1S2}$$ is the slit separation d. The screen distance D is the perpendicular distance from $$\overline{S1S2}$$ to the viewing screen. Now let point P be a point on the viewing screen along the positive y-direction and let $$R1 = \overline{S1P}$$, $$R2 = \overline{S1P}$$, $$R = \overline{OP}$$ (O is the point at the origin). For quadrant I, the path length difference is just $$\Delta L = R2 - R1$$. If we let $$\theta$$ be the angle between the central axis and R, then we can easily determine the path length difference in terms of $$\theta$$ (using the law of cosines) to be:

$$\Delta L = \sqrt{\left(\frac{D}{cos\theta}\right)^{2}+\left(\frac{1}{2}d\right)^{2}+dDtan\theta} - \sqrt{\left(\frac{D}{cos\theta}\right)^{2}+\left(\frac{1}{2}d\right)^{2}-dDtan\theta}$$

The path length difference in quadrant IV easily follows.

Link to simulation (can't upload xmcd files to thread): http://cid-810ca62460f8699f.skydrive.live.com/self.aspx/Public/Ch%2035%20-%20double-slit%20experiment.xmcd

 

[Andy Resnick]

You are asking about the complete solution to diffraction, which has not yet been written. Some of the simplifications involve scalar vs. vector representation, Kirchoff vs. Rayleigh-Sommerfeld formulations, etc. But the main approximation you are trying to overcome is finding a solution using the Fresnel approximation, rather than the more restricitve Fraunhoffer approximation.

Try looking around using any of the above search terms and see what is suitable for you.

 

[buffordboy23]

Andy Resnick,

Thanks, I will look into it. With the small dimensions, I was wondering how close this simulation would approximate reality. There may physical mechanisms unaccounted for. For D >> d, the new derivation is indistinguishable from the old derivation in terms of qualitative behavior, as it must be. Do you know any technical articles off-hand that explore this phenomena in similar conditions? If possible, I would like to compare my simulation with actual data sometime.

 

[Born2bwire]

If you want to do an accurate simulation, you can write a 2D FDTD Yee algorithm code. It's something you can easily do in Matlab or C.