# Calculating Fresnel Diffraction Pattern from Circular Apertures

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• Douglas-Stinson
In summary: Your Name]In summary, the speaker is attempting to calculate the Fresnel diffraction pattern using circular apertures and following the procedure outlined in a paper by Klaus D. Mielenz. However, they have encountered discrepancies in their results when using the Lommel function V. They are seeking help from others who may have attempted these calculations and are open to suggestions and advice to troubleshoot the issue.
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.

#### Attachments

• Fresnel Pinhole Mielenz-Lommel rtf.pdf
534.4 KB · Views: 156

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.

## 1. What is Fresnel diffraction?

Fresnel diffraction is a phenomenon that occurs when light waves pass through a small aperture or around an obstacle, causing the wavefronts to bend and create a diffraction pattern.

## 2. How is Fresnel diffraction different from Fraunhofer diffraction?

The main difference between Fresnel and Fraunhofer diffraction is the distance between the aperture or obstacle and the observation point. In Fresnel diffraction, the observation point is close to the aperture, while in Fraunhofer diffraction, the observation point is far from the aperture.

## 3. What are the factors that affect the Fresnel diffraction pattern?

The factors that affect the Fresnel diffraction pattern include the size and shape of the aperture, the wavelength of the light, and the distance between the aperture and the observation point.

## 4. How do you calculate the Fresnel diffraction pattern from circular apertures?

To calculate the Fresnel diffraction pattern from circular apertures, you can use the Fresnel-Kirchhoff diffraction integral. This involves integrating the complex amplitude of the incident wave over the aperture and then propagating the resulting wavefront to the observation point.

## 5. What are some practical applications of Fresnel diffraction?

Fresnel diffraction has many practical applications, including in the design of optical systems, such as lenses and telescopes. It is also used in the study of wave phenomena and in various imaging techniques, such as holography and microscopy.

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