# 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?

The simplest model of a finite-aperture lens is... just the aperture. This is the limit of a thin lens as its optical power approaches zero. Such a "lens" is used in pinhole cameras, and it's a useful theoretical model sometimes used in computer graphics.

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.

As your lens becomes more complicated, i.e. getting some optical power, it gets other factors that limit its field of view. Consider a glass biconvex lens. It transforms a parallel beam into a converging one by the mechanism of light refraction. As you increase incidence angle of the beam, more light is reflected and less refracted until at some critical angle you get no refracted light at all: total internal reflection occurs. Looking through the lens you'll see other parts of the scene, not the ones you'd expect from a perfect lens.

A lens can become even more complicated, becoming so called lens assembly, where several simple lenses work together to reduce aberrations in the image. In this case light rays have a complicated path, and some of them just get lost geometrically inside the assembly. See e.g. this ray tracing of the Canon EF-S 24mm f/2.8 lens: