I'm trying to implement a model for diffraction-limited imaging, following "Microlithography" by Sheats and Smith. You can skip to the bottom for my question, but I'll explain the setup here.
The basic idea is that I have a black and white image $m(x, y)$ (measuring the electric field, with 0 being dark and 1 fully illuminated). The case I'm especially interested in is where $m$ is an array of repeating spaces and lines, but my question applies to general images.
The book says to take the Fourier transform and then multiply by a pupil function, modeling the lens, before taking the inverse Fourier transform to get the final aerial image that I should expect as a result of diffraction and a given numerical aperture (NA) for my lens.
If $M(f_x, f_y) = \mathcal{F}(m(x, y))$ represents the Fourier transform, then the final image should be $\mathcal{F}^{-1}(M(f_x, f_y) P(f_x, f_y))$, where $f_i$ is the spatial frequency and the pupil function $P$ is defined below.
$$P(f_x, f_y) = \begin{cases} 1 & \text{if } \sqrt{f_x^2 + f_y^2} \le \text{NA}/\lambda \\ 0 & \text{if } \sqrt{f_x^2 + f_y^2} > \text{NA}/\lambda \\ \end{cases} $$
My question is about this pupil function. When I coded it up, I had an intuition that the pupil window could be made arbitrarily big by just increasing the numerical aperture. But it seems like $\text{NA} = \sin \alpha = 1$ is the maximum possible numerical aperture (assuming the medium is air with an index of refraction around 1 and $\alpha$ is the maximum half-angle of the diffracted light that still enters the lens), which creates a window of radius $1/\lambda$ in the Fourier domain.
But the book says the following:
Does an ideal lens produce a perfect image? No. Because of the finite size of the numerical aperture, only a portion of the diffraction pattern enters the lens. Thus, even an ideal lens cannot produce a perfect image unless the lens is infinitely big. Since in the case of an ideal lens the image is limited only by the diffracted light that does not make it through the lens, we call such an ideal system diffraction limited.
I'm struggling to reconcile these two previous paragraphs. For an infinitely big lens, we should have $\text{NA} = \sin \alpha = 1,$ so the pupil function would have a limited radius of $1/\lambda$, while the quoted passage seems to suggest that the image should be perfect, so the pupil function would be identically 1 everywhere. Which one of these is true? Is the pupil function always limited by $n/\lambda$ ($n$ the index of refraction) or can a large enough lens in theory give a perfect reconstruction of the original image, regardless of the wavelength $\lambda$ of light used?
Any help/clarity you could give would be much appreciated! Thanks for reading.
