# What's limiting the field of view of a lens

An ideal lens with finite aperture leads to the formation of an image in the image plane as $$$$g(x, y) = \int \text{psf}(x/\lambda d - x', y/\lambda d -y') f(x', y') dx'dy',$$$$ where $$g(x)$$, $$f(x)$$ is the scalar light-field in the image-, object-plane and $$\text{psf}(x)$$ is the point-spread function. $$d$$ is the distance between the lens and the image plane.

The larger the aperture, the sharper $$\text{psf}(x)$$ is localized around $$0$$. Despite neglecting aberrations, the model nicely captures the finite resolution with which optical systems can image objects. Furthermore, at least conceptually, the effect of aberrations can be seen as a additional broadening of $$\text{psf}(x)$$.

However, in this model an arbitrarily large object is still imaged "perfectly" (with limited resolution of course), which is unphysical. So how does this effect of a limited field of view emerge? Is there a simple toy model that illustrates it, much like the simple ideal lens + aperture illustrates blurring due to diffraction?

For a uniform light beam incident at a zero-thickness pinhole at incidence angle $$\theta$$, the projected size of the pinhole diminishes $$\propto\cos\theta$$. This means that at e.g. $$\theta=60°$$ the image intensity (on a spherical screen) is just half that in the center of the image, at $$\theta=0$$, while at $$\theta=84°$$ there's just $$10\%$$ of center intensity. This is called vignetting. This is the first effect that limits useful field of view.