# Location of a particular spatial frequency in a Fourier transform plane

I am trying to assess structure of a Fourier transform generated by a thin lens. I am using scalar diffraction theory and Fresnel approximation such that input $$U(x,y)$$ transparency located at the front focal point of a thin lens generates an FT-image of $$U(x,y) \rightarrow u(x,y)$$ in the back focal point:

$$u(x, y) = \int dx \; dy \;U(x,y) \;exp\big(-\frac{ik}{f}(xu+yv)\big)$$

My confusion is coming from trying to assess the patterns of the optical Fourier transform given a finite size of the smallest significant feature of the input transparency. It is obvious how to derive the spatial frequency as a function of an image features:

$$\nu_x = \frac{\partial \phi_x}{\partial x} = -\frac{u}{\lambda f}$$

where $$f$$ is focal distance of the lens and $$\lambda$$ is the wavelength of light. However, this suggests that $$\nu_x$$ is proportional to $$u$$ - feature size, which is in opposition to what I have previously learned, that max spatial frequency corresponds to min feature size such as $$\nu_x \sim 1/u$$.

Also, this doesn't make it clearer what would be the physical size of the Fourier domain given the min and max dimensions of the features of the input transparency.

$$u$$ is not a feature size. It is a spatial coordinate. You can map spatial frequencies of the transparency to spatial coordinates in the focal plane, as you wrote $$\nu_x = -\frac{u}{\lambda f}$$. This way, if you want to cut off a specific spatial frequency range in the transparency, you know exactly how large make a aperture to do low-pass spatial filtering.
It is hard to be precise about the min and max spatial frequencies of the transparency. It really depends on the content of the transparency itself. If it has abrupt changes in transmittance, then its spectrum will contain high spatial frequencies. However, the lens itself has a cutoff frequency, which is $$\nu_0 = \frac{D}{2 \lambda f}$$. The lens will pass frequencies up to $$2\times \nu_0$$, although they will be attenuated according to the lens's optical transfer function. There are common textbooks in Fourier optics that explain this in detail, like Goodman and Gaskill.