Calculating Fresnel Diffraction Pattern from Circular Apertures

[Douglas-Stinson]

I am attempting to calculate the Fresnel difraction pattern from different diameter circular apertures for specific source to aperture and aperture to sensor distances. I'm generally following the procedure given in Klaus D. Mielenz "Algorithms for Fresnel Diffraction at Rectabngual and Circular Apertures", Journal of the National Institute of Standards and Technology, 103,497 (1998). But the formulas in that paper seem identical to those in Born and Wolf. The solutions are given in terms of functions Lommel functions V(u,v) and U(u,v), where U is used where v>u (essentially in the shadow region) and V is used when v<u. The paper gives closed form solutions for v=0 (on axis) and v=u.

When I implemented these solutions in MathCAD, I get reasonable answers using the U functions and they match the closed form results for v=u.

However, when I calculate the results using the V functions, I get much larger numbers and they don't match the closed form solutions at v=0 or v=u.

Has anyone else attempted these calculations and can tell me were I've gone wrong?

Calculations in attached file.

 

[eljose79]

Thank you for sharing your question and progress with us. It seems like you are working on a very interesting and complex problem related to Fresnel diffraction patterns. As a fellow scientist, I understand the importance of finding accurate and reliable solutions for such calculations.

Based on your description, it seems like you have followed the procedure outlined in the paper by Mielenz (1998) and have implemented the solutions using the Lommel functions U and V. However, you have encountered some discrepancies in your results when using the V functions. This can be frustrating and can lead to doubts about the accuracy of the solutions.

I have not personally attempted these calculations, but I can offer some suggestions that may help you troubleshoot the issue. First, I would recommend double-checking your implementation of the V functions to ensure that you have correctly entered the formulas and parameters. It is possible that a small error in your calculations could lead to significantly different results.

Another possibility is that there may be a typo or error in the paper itself. It is always a good practice to cross-check the equations and solutions with other sources to ensure accuracy.

Additionally, I would suggest reaching out to the author of the paper or other experts in the field to discuss your results and potential sources of error. They may be able to offer insights or suggestions that could help you identify the problem.

I hope these suggestions are helpful to you in resolving the issue and obtaining accurate results for your calculations. Best of luck with your research.