The Rayleigh Criterion (DP IB Physics: HL)

• A circular aperture allows a cone of light to enter a region behind the aperture
• Examples of circular aperture include:
  • A lens within an optical device such as a camera
  • The pupil of an eye

• A circular aperture allows light to act like a point source once passing through
• Placing two point sources near each other or viewing those sources too far away, will make them appear to be a single unresolved source of light
• Consider car headlights which are distant on a highway:
  • Initially, when the car is far away, the headlights appear as one point source
  • It is not until the car comes closer that the two individual headlights can be resolved individually 

• Light from any object passing through a circular aperture, including the human eye, will diffract and create a diffraction pattern upon the detector inside
  • In the case of the human eye, for example, the detector is the retina

• Each of these diffraction patterns need space on the detector to be resolved
  • If they are too close, then they will appear as one single source

• Resolution is defined as:

The capability of an imaging system to be able to tell if two sources are independent and produce individual images of those two sources

 

Two light sources outside a circular aperture produce diffraction patterns that have a minimum angle of resolution determined by the Rayleigh criterion

• The Rayleigh Criterion describes the limit of resolution of a system to separate two sources

Two sources that only just be resolved. The red is a single source, the blue is the other source and the purple line represents their combined intensity

• The Rayleigh Criterion states that:

Two sources are able to be just resolved if the principal maximum from one diffraction pattern is aligned with the first minimum of the other diffraction pattern

• Two sources that can be fully resolved:

Two sources that can be fully resolved. The red is a single source, the blue is the second source and the purple line represents their combined intensity

• Visually, two sources that could clearly be resolved would look like the above

• Two sources that cannot be resolved:

Two sources that cannot be resolved. The red is a single source, the blue is the second source and the purple line represents their combined intensity

• Visually, two sources that could not be resolved:

Visually, two sources that could not be resolved

• Visually, two sources that are only just resolved:

Visually, two sources that could only just be resolved

• This above example is the limiting case of the Rayleigh criterion

Visually, two sources that could clearly be resolved would look like the above

• If the two sources are separated further apart than maximum to minimum, they can be resolved
  • If the two sources are brought closer together, they cannot be resolved