4.5 Circular apertures and resolution  (Page 2/8)

The first minimum is at an angle of $\theta =1.22\text{λ}\text{/}D$ , so that two point objects are just resolvable if they are separated by the angle

where $\text{λ}$ is the wavelength of light (or other electromagnetic radiation) and D is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression, $\theta$ has units of radians. This angle is also commonly known as the diffraction limit.

All attempts to observe the size and shape of objects are limited by the wavelength of the probe. Even the small wavelength of light prohibits exact precision. When extremely small wavelength probes are used, as with an electron microscope, the system is disturbed, still limiting our knowledge. Heisenberg’s uncertainty principle asserts that this limit is fundamental and inescapable, as we shall see in the chapter on quantum mechanics.

Calculating diffraction limits of the hubble space telescope

Strategy

Solution

1. The Rayleigh criterion for the minimum resolvable angle is
   
   $\theta =1.22\frac{\text{λ}}{D}.$
   
   Entering known values gives
   
   $\theta =1.22\frac{550\times {10}^{\text{−}9}\text{m}}{2.40\text{m}}=2.80\times {10}^{\text{−}7}\text{rad}.$
2. The distance s between two objects a distance r away and separated by an angle $\theta$ is $s=r\theta .$
   Substituting known values gives
   
   $s=\left(2.0\times {10}^6\text{ly}\right)\left(2.80\times {10}^{\text{−}7}\text{rad}\right)=0.56\text{ly}.$

Significance