Resolving power question

Question

Imagine you have a flat piece of glass as a lens so the focal length is infinity since in the formula$$ \frac{1}{f}~=~\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) \,,$$where $R_1 \to \infty$ and $R_2 \to \infty$.

The limit of resolution is given by$$ d \, = \, 1.22 \frac{f \, \lambda}{D} \,.$$So if I understand correctly since $f$ approaches infinity, then the limit of resolution should approach infinity. But how could that be?

Since at home if I look outside through the window of my home (a flat plane of glass) at night at two different lights outside I can resolve them easily. So what am I missing here?

Answer 1

What you are missing is that the formula for resolution describes the radius of the Airy disk (which is the spot size, for all intents and purposes) which was produced by the lens from a collimated (parallel) ray bundle.

It follows that if your focal length is infinity, then a parallel ray bundle going through a flat window will converge at infinity, or in other words, stay parallel.

From here it should be intuitive to understand that converging\diverging rays should continue to converge or diverge after passing through the window, so no problem with seeing through a glass window.

What actually happens when you see something through the window in your home is that rays from distant objects come towards you relatively parallel, pass through the window (remaining relatively parallel), and then are focused by the lens of your eyes onto your retina.