Between this and the next assignment, we want to get a better under- standing of how light interacts with the eye. Here are two questions to get us started, focused on diffraction (i. e., the spreading of light when it passes through a narrow opening). A. To regulate the intensity of light reaching our retinas, our pupils1 change diameter anywhere from 2 mm in bright light to 8 mm in dim light. Find the angular resolution of the eye for 550 nm wavelength light at those extremes. In which light can you see more sharply, dim or bright

θ₁ = 3.35 10⁻⁴ rad ,  θ₂ = 8.39 10⁻⁵ rad

Explanation:

This is a diffraction problem for a slit that is described by the expression

       sin θ = m λ

the resolution is obtained from the angle between the central maximum and the first minimum corresponding to m = 1

      sin θ = λ / a

as in these experiments the angle is very small we can approximate the sine to its angle

        θ = λ / a

In this case, the circular openings are explicit, so the system must be solved in polar coordinates, which introduces a numerical constant.

       θ = 1.22 λ / D

where D is the diameter of the opening

 let's apply this expression to our case

indicates that the wavelength is λ = 550 nm = 550 10⁻⁹ m

the case of a lot of light D = 2 mm = 2 10⁻³ m

       θ₁ = 1.22 550 10-9 / 2 10⁻³

       θ₁ = 3.35 10⁻⁴ rad

For the low light case D = 8 mm = 8 10⁻³

      θ₂ = 1.22 550 10-9 / 8 10⁻³

      θ₂ = 8.39 10⁻⁵ rad