Fourier optics with concave (diverging) lenses

[Qiao]

Hey,

I was wondering, since for a convex lens the Fourier transform of a fields is in their real focus plane. Is it for a concave lens that the Fourier transform of a field is in the virtual focus plane?

I can't find any book or paper that talks about how concave lenses work in terms of Fourier optics.

 

[Andy Resnick]

There's no conceptual difficulty- the only difference is using '-f' instead of 'f' in the diffraction integrals. The practical difficulty is that the planes of interest are virtual, rather than real.

 

[Qiao]

Thanks. So then I would assume that if the object is at focus distance "f", the image plane with a perfect Fourier transform of the object will be at "f/2". I concluded this by drawing a quick ray diagram.

 

[Qiao]

actually, I've thought about this some more with the following thought experiment. Take a concave lens, put an object on the left focal plane, next place a convex lens so that it's focus plane is at the virtual image plane of the concave lens. So now the second lens should in principle do another Fourier transform giving back you original field function, right? (this is based on the 4F system in the link without the transmission mask http://upload.wikimedia.org/wikiped...4F_Correlator.svg/430px-4F_Correlator.svg.png)

But if you draw a ray diagram, it would tell another story, for a collimated beam everything goes as expected they enter collimated and exit collimated.
But for a point source, when it passes the concave lens it diverges even more and all the information will never be refocused back to a point. This means that it is impossible to get your original function back, right?

This is confusing to me, this experiment contradicts the idea that the Fourier transform lies in the virtual plane

 

[Andy Resnick]

Seems to me, all you are describing is a telephoto lens which can be used forwards or backwards to magnify or demagnify.

 

[Qiao]

It is sort of a basic telescope setup. Except I don't expect go to get my function back when if I use a concave lens + convex lens.

 

[NaKuBa]

I have been thinking of this problem for a while now and like you guys, i have not found much online on this topic. From ray diagrams, all I can make out is that we do not have access to the intermediate Fourier plane in a 4f system built using a convex and a concave lens. So effectively it is magnifying or diminishing lens combo. You already know this. I tried to experimentally observe this once but I was not successful. I must admit it was not a sincere effort. I will try again to do this experiment and let you guys know.