Lecture 37: Diffraction on circular apertures

Diffraction on circular aperture

The case of circular aperture is very important in optical devices. Microscopes, telescopes, cameras, anything utilizes spherical lenses or mirrors are subject to diffraction due to finite size of the aperture through which the light passes. Diffraction place fundamental limit on angular resolution of such devices

• The interference pattern on the screen behind the circular aperture of a finite size is created similarly to the slit - the waves from different secondary sources in the apperture (Hyugens principle) interfere. The resulting pattern is a set of co-centric circular bright and dark bands with the highest peak in the center
  
• It is just a notch more difficult to derive the intensity shape of the image, because the integrals cannot be done in elementary functions (they lead to special functions, in particular Bessel functions . However the radial plot of the profile is ( k = 2 π/λ, α=D is the diameter of the aperture)
  
• The positions of the minima are at sin&theta_min = 1.22 λ/D, 2.23 λ/D, 3.24 λ/D
• The positions of the maxima, beside the central peak, are at sin&theta_max = 1.63 λ/D, 2.68 λ/D, 3.70 λ/D
• This pattern was investigated by George Airy and the central bright spot is known as Airy disk

• All the diffraction rules apply
• Larger the aperture or shorter the wave length, closer the right
• When aperture diameter becomes as small as one waveleght, the central peak spreads through the whole image
• The central peak is approximately twice wider, 2.44 λ/D , that subsequent bright rings which have widths ≈ &lambda/D

• Although peak intensity falls of quickly in the secondary rings, there is still notable total power coming from second and third ring, because due to circular nature they occupy larger and larger areas
  
• 83.8% of the power is contained within the first dark ring (i.e in a central peak), 91% within the second and 93.8% within the third (so 6% of the power is beyond the first three bright bands)

Circular appertures and angular resolution of optical devices

• Diffraction limits the resolution of devices that create images - lenses and mirrors

• The image of a source due to diffraction is not a point, but a diffraction pattern, in particular the lens does not focus the parallel rays strictly into focus point. Image is smeared
• Most importan is the finite size of the Airy disk, over which the point image is mostly spread.
• If one has two sources (or two parts, head and tail, of one source), they form two diffraction patterns in the image plane.
  
• If the Airy disks do not overlap, we can easily distinguish the two images
• But if they overlap, ultimately we'll not see that there are two images, but one
• The rule-of-thumb Rayleigh's criterion when we can still distinguish two images is when the central peak of one image is in the first minima of the other image . I.e, angular separation (assuming small angle, as always the case with lenses/mirrors) between two images is Δθ = 1.22 λ/D
• Although θ is the angluar direction from aperture to the image, for the distant sources under our approximations it is also angular direction to the source. Thus, we can say that if two sources are separated by less than Δθ = 1.22 λ/D , they will produce blurred single image, and will not be resolved. This angular separation is thus diffration limit for angular resolution of optical instrument
• Example - optical telescope with diameter 1 m will not be able to resolve two stars separated by less than 6 × 10^-7 radians observing at wavelength of 500 nm. (this is 0.13 arcsec )
• Diffraction from the main mirror is a fundamental limit on how good the resolution can be. In practice, due to other effects, it can be only worse. In ground-based telescopes, the main limit on the angular resolution usually comes from turbulent motions of air in atmosphere, rather than from the diffraction on main mirror.
• That's why the shapest images are made from the sky ( Hubble Space telescope has 2.4 m mirror - what is its diffraction limit ? )