Airydisk derivation

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Diffraction limits the resolution according to θ = 1.22λ/D = y/L.  Here the height of the object to be resolved is y and the distance to the object is L.  Solving for D we find D = 12.2 m.

A spy satellite travels at a distance of 50 km above Earth's surface.  How large must the lens be so that it can resolve objects with a size of 2 mm and thus read a newspaper?  Assume the light has a wavelength of 400 nm.

Airydisk formula

If you look at a far-away object, then the image of the object will form a diffraction pattern on your retina.  For two far-away objects, separated by a small angle θ, the diffraction patterns will overlap.  You are able to resolve the two objects as long as the central maxima of the two diffraction patterns do not overlap.  The two images are just resolved when one central maximum falls onto the first minimum of the other diffraction pattern.  This is known as the Rayleigh criterion.  If the two central maxima overlap the two objects look like one.

Airydisk calculator

If r0 is the distance from the point s = 0 on the optical axis to a point y on the screen, then the contribution dE to the total amplitude on the screen from the point at s = 0 is dE(y = d*tanθ) = (Asds/r0)cos(kr0 - ωt).For off-axis points for which s ≠ 0, the distance to the screen is longer or shorter than r0 by an amount Δ.

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The closer you are to two objects, the greater is the angular separation between them.  Up close, two objects are easily resolved.  As your distance from the objects increases, their images become less well resolved and eventually merge into one image.

Airydisk radius

Airy patterncalculator

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Airydisk Microscopy

To find the total amplitude E(y) we have to add up the contributions from all points on the aperture.  Because there are an infinite number of points, the sum becomes an integral.

The angle through which the light spreads is approximately θ ≈ λ/a(0).  Therefore a(z)/2 ≈ z/θ.  Because the laser beam diameter is typically much larger the wavelength of light, or a(0) >> λ, θ is quite small.  Consider a HeNe laser, for which λ = 633 nm with a beam waist of ~ 0.6 mm.  Then θ ~ 10-3 rad = 1 millirad.  The beam must propagate ~ 3 m before the diameter increases by a factor of 10.

We define sinθ = Δ/s.  Since r0 >> Δ, we approximate 1/(r0 + Δ) with 1/r0.  However we cannot drop the Δ inside the cosine function, since kΔ(s) is not necessarily much smaller than 2π.  We then have

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If all ray aberrations in an optical system can be eliminated, such that all of the rays leaving a given object point land inside of the Airy Disc associated with the corresponding image point, then we have a diffraction-limited optical system.  This is the absolute best we can do for an optical system that has lenses with finite diameters.

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Airydisc diffraction

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AiryDisk telescope

The time-averaged intensity has a peak in the center with smaller fringes on the sides.  For small angles we may approximate sinθ ~ θ.  Then the first zeros on the sides of the central peak occur when πasinθ/λ ~ πaθ/λ = π, or θ = λ/a.

The resolving power of an optical instrument is its ability to separate the images of two objects, which are close together.  Some binary stars in the sky look like one single star when viewed with the naked eye, but the images of the two stars are clearly resolved when viewed with a telescope.

The width of the central maximum in a diffraction pattern depends on the size of the aperture, (i.e. the size of the slit).  The aperture of your eye is your pupil.  A telescope has a much larger aperture, and therefore has a greater resolving power.  The minimum angular separation of two objects which can just be resolved is given by θmin=1.22λ/D, where D is the diameter of the aperture.  The factor of 1.22 applies to circular apertures like the pupil of your eye or the apertures in telescopes and cameras.

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