Diffraction and Limits of Resolution

Diffraction through a circular aperture of diameter D

IntensityDiameterD

Image onScreen

θ = 1.22 λ /D

Because of diffraction, images formed by “perfect” optics are fuzzy.

For a circular aperture:

The angle from the centre to first dark ring (“angular radius” of central spot) is:

θ=1.22 λ/D radians

This is useful in calculating the limiting resolution of optical instruments (microscopes or telescopes).

The central spot is sometimes called the Airy disk

Quiz:

What would the central spot look like if white light were used for the beam?

A) Blue in the centre and red around the edge

B) Red in the centre and blue around the edge

C) White in the centre and red around the edge

Example: Sodium light (589nm) is used to view objects undera microscope. If the aperture of the objective has a diameter of 0.9 cm,

a) find the limiting angle of resolution (max resolution)

b) if the space between the object and the objective lens is filled with water (n=1.3), how is the resolution affected?

Example: If the eye pupil is dilated to a diameter of 5mm, what the minimum distance between two point sources that the eye can distinguish at a distance of 1km?

Image(perfect focus)

Two stars(point sources at infinity)

A “perfect” telescope only magnifies the diffraction pattern of the circular “hole” it looks through.

angle 1.22 λ /D

Resolution of two sources by a Resolution of two sources by a TelescopeTelescope

*

*D

Telescope lens/mirror

Rayleigh Rayleigh CriterionCriterion

NotResolved

Images are just resolved when the centre of one pattern overlaps the first dark line of the other pattern.

RayleighCriterion

Resolved

“Just resolved”(or “just not resolved”)

x

Resolved if 1.22 D

x

When it comes to optics, size does matter !!!

Fleas, s = 2mm apart,treated as point sources of light

R = 200 m

Find: Minimum lens diameter to resolve fleas. (λ=500nm = 1/2000 mm)

Example: The moon is 386,000 km away.

a) What is the smallest feature that you can resolve with a pair of 80mm binoculars?

b) What is the smallest feature that you can resolve with a 10m telescope?

Assume λ=500nm

Quiz:

What other factors can affect the resolution of a telescope?

A) atmospheric turbulenceB) steadiness of telescope/mountC) smoothness of mirror/lensD) all of the above

Quiz:

Which telescope will have the best resolution given the same size of optics (aperture size), and neglecting the effect of the atmosphere?A) Radio telescopeB) Optical telescopeC) X-ray telescope

The Sun

Radio

Visible

x-ray

How to increase telescope resolution?

2) We can use more than one mirror in a telescope

How to increase telescope resolution?

1) Build a larger telescope telescope

How to increase telescope resolution?

3) Put the telescope in space