Fourier Transform for Diffraction

In summary, the conversation is about calculating the Fourier transform of a gaussian multiplied by a rectangular function. The speaker is struggling to find a solution using analytical functions and is looking for help in simplifying the result. Another person suggests using the convolution operation instead of Fresnel integrals to obtain a solution.
  • #1
yawphys
1
0
Hi all,

I'm working in an exercise of advanced optics related to diffraction, in Fraunhoffer's aproximation.

I need to calculate the FT of a gaussian multiplied by a rectangle function, i.e, FT(exp(-x^2)*rect(x/a)), and I can't obtain a result expressed using analytical common functions. I can only obtain one solution using Fresnel integrals.

I think there could be a simpler way of expressing this result. Anyone can help me?

Thanks.
 
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  • #2
You can write the solution as

[tex] F(e^{-x^2}) \otimes F(rect(x)) [/tex]

where the [tex] \otimes [/tex] is the convolution operation. No need for Fresnel integrals because you integrate the exponential over all space and the rect funtion becomes a sinc funtion upon transform.
 

Related to Fourier Transform for Diffraction

What is the Fourier Transform for Diffraction?

The Fourier Transform for Diffraction is a mathematical tool used to analyze the diffraction patterns of light. It converts a complex wave function, such as a diffraction pattern, into a sum of simpler wave functions with varying frequencies and amplitudes.

How is the Fourier Transform for Diffraction used in science?

The Fourier Transform for Diffraction is used in various fields of science, such as optics, crystallography, and spectroscopy. It allows scientists to analyze diffraction patterns and extract information about the structure and properties of materials.

What is the relationship between Fourier Transform and diffraction?

The Fourier Transform and diffraction are closely related. Diffraction patterns can be analyzed using the Fourier Transform, which decomposes the pattern into its individual frequency components. This allows for the identification of specific features and properties of the diffracting object.

What are the benefits of using the Fourier Transform for Diffraction over other methods?

The Fourier Transform for Diffraction is a powerful tool because it can provide a detailed analysis of complex diffraction patterns. It allows for the identification of specific frequencies and their amplitudes, which can provide valuable information about the structure and properties of the diffracting object.

Are there any limitations to using the Fourier Transform for Diffraction?

While the Fourier Transform for Diffraction is a useful tool, it does have some limitations. It assumes that the diffraction pattern is produced by a continuous, linearly varying function. This may not always be the case, and in some situations, other methods may be more appropriate for analyzing diffraction patterns.

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