How to Simplify the Fourier Transform of a Gaussian Times a Rectangle Function?

[yawphys]

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.

 

[Dr Transport]

You can write the solution as

$$F(e^{-x^2}) \otimes F(rect(x))$$

where the $$\otimes$$ 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.