Answer to Question #201203 in Optics for Camron

Question

A modern mobile phone screen has pixels around 5.0 µm across.

We wish to make a telescope with a diffraction-limited angular resolution the same as the angular size of each pixel on your mobile phone screen when you are holding your phone 50 cm from your face. How large must the aperture of the telescope be? (to 2 s.f and in cm)

[Note: assume a wavelength of 500 nm.]

Answer 1

The Rayleigh criterion:

$$\theta=1.22\frac \lambda D.$$

Find the aperture diameter:

$$D=1.22\frac \lambda \theta.$$

Find the angle of resolution:

$$\tan\frac \theta 2=\frac{x/2}{d},\\\space\\ \theta=\arctan(\frac xd),\\\space\\ D=1.22\frac{\lambda}{\arctan(x/d)},\\\space\\ D=1.22\frac{500·10^{-9}}{\arctan(5·10^{-6}/0.5)}=1.06·10^{-3}\text m.$$