# Far Field lens Imaging…Does k-wave number follow arc length equation?

If we are imaging light in the far field region. We have three situations/relations (illustrated below):

1. Arc Length: We know the distance subtended (S) by the light ray in a lens-less system will be proportional to R (distance to the screen) and theta; simple, especially if theta is very small.

2. Put a lens in the system (focused at infinity) and the distance subtended (Y) becomes a function of the focal length and theta. I confirmed this result using matrix optics!

Now, the part in question:

1. Considering the same lens system, can we say that because the imaging plane images the Far Field Distribution (k-space) and essentially angles of the incoming light can we say that the k-space subtended (dk) is proportional to k-naught and theta (see equation below)?? If so how do I go about proving this?!

Any help would be greatly appreciated!

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