A Fourier optics model of a 4f system

[Skaiserollz89]

TL;DR Summary

I am attempting to model a 4f system in matlab. However, before I do I want to ensure I am understanding the fourier optics involved in doing so.

In my system I am trying to represent two lenses. L1 with focal length f1=910mm and the other lens, L2 with focal length f2=40mm. These lenses are space such that there is a distance of f1+f2 between the lenses. I have a unit amplitude plane wave incident on L1. My goal is to find the resulting irradiance pattern after passage through both lenses.

I think I only need to perform a couple fourier transforms. For L1, I will use the Fraunhofer diffraction integral on the incoming field u_in(x,y) with z=f1. This results in the transform field U_in(fx,fy) at f1. From here I'm not sure what to do. For lens L2 do I need to do an inverse version of the Fraunhofer diffraction integral to get out of frequency space while simultaneously accounting for the focal length of the 2nd lens f2? Any advice would be much appreciated!

 

[member38644]

The Fourier optics model of a 4f system is a powerful tool for analyzing optical systems with multiple lenses. In your system, you have two lenses, L1 and L2, with focal lengths of 910mm and 40mm respectively. These lenses are spaced such that there is a distance of f1+f2 between them. To find the resulting irradiance pattern after passage through both lenses, you are correct in saying that you will need to perform a couple of Fourier transforms.

First, for L1, you will need to use the Fraunhofer diffraction integral on the incoming field u_in(x,y) with z=f1. This will result in the transformed field U_in(fx,fy) at f1. This transformed field will then act as the input for L2. However, in order to properly account for the focal length of L2, you will need to use the inverse version of the Fraunhofer diffraction integral. This will allow you to transform the field back into real space and account for the focal length of L2 at the same time.

Once you have performed the inverse transform, you will have the field at the focal plane of L2. From here, you can use the standard thin lens equation to determine the field at the final image plane. This will give you the resulting irradiance pattern after passage through both lenses.

In summary, to find the resulting irradiance pattern in your system, you will need to perform two Fourier transforms and use the inverse version of the Fraunhofer diffraction integral to properly account for the focal length of L2. This will allow you to accurately analyze the behavior of your 4f system and determine the resulting irradiance pattern.