Fourier optics for diffraction pattern

[sphys]

I need to determine the diffraction pattern of 2 different kinds of prisms.
1. For a right angled prism.
2. For a symmetric prism

For a right angled prism, the transmission function is exp[2ipixa]*rect(x/d)
where d is the base width of the prism, a is a conastant. So the diffraction pattern can be found by determining the Fourier transform of above function. I determined the Fourier transform using the convolution theorem.

For a symmetric prism, the transmission function is exp[2ipia(1-|x|)]*rect(x/d).
I am having trouble calculating the Fourier transform for this function. Can someone help me please?

 

[eljose79]

Hello there,

I can definitely help you with determining the diffraction pattern for these two types of prisms.

For the right angled prism, the transmission function can be written as exp(2iπxa) * rect(x/d), where d is the base width of the prism and a is a constant. To find the diffraction pattern, we can use the convolution theorem which states that the Fourier transform of a product of two functions is equal to the convolution of their individual Fourier transforms.

So, in this case, the Fourier transform of the transmission function will be the convolution of the Fourier transform of exp(2iπxa) and the Fourier transform of rect(x/d). The Fourier transform of exp(2iπxa) is a delta function centered at -a, and the Fourier transform of rect(x/d) is a sinc function.

Therefore, the diffraction pattern for a right angled prism will be a sinc function centered at -a and with a width of d.

Moving on to the symmetric prism, the transmission function can be written as exp(2iπa(1-|x|)) * rect(x/d). To calculate the Fourier transform of this function, we can divide it into two parts - exp(2iπa) and rect(x/d) - and use the convolution theorem.

The Fourier transform of exp(2iπa) is again a delta function centered at -a, and the Fourier transform of rect(x/d) is a sinc function. However, in this case, the delta function will be multiplied by 2 since the function is symmetric.

Hence, the diffraction pattern for a symmetric prism will be a sinc function centered at -a with a width of 2d.

I hope this helps you in determining the diffraction patterns for these two types of prisms. Let me know if you have any further questions. Happy experimenting!