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# Diffraction Limited Resolution

All telescopes have an inherent limitation to their angular resolution due to the diffraction of light at the telescope's aperture. When a continuum of wave components pass through an aperture, the superposition of these components result in a pattern of constructive and destructive interference. For astronomical instruments, the incoming light is approximately a plane wave since the source of the light is so far away. In this far-field limit, Fraunhofer diffraction occurs and the pattern projected onto the focal plane of the telescope will have little resemblance to the aperture. (Hecht, 1987; Heald & Marion, 1995).

What does the diffraction pattern of a star look like and what is the limit of a telescope's resolution? For wavefunction at point P in the imaging plane, the Fraunhofer diffraction integral is

where k is the wavenumber , and are the aperture's width and height, R is the distance from the aperture's center to P, x and y define the position of P on the image plane, = x/R and = y/R. Since telescopes generally have circular apertures, we will let a be the radius of the aperture and write , , and in cylindrical coordinates where the z axis is the axis of the telescope.

In the aperture plane,

In the image plane,

If , the angle between the axis of the telescope and R, is small,

and

We can choose because of the cylindrical symmetry of the telescope. Therefore, equation 1 becomes

We are only concerned with the real part of this integral so

The Bessel function is defined as

and

This means equation 10 can be written as

with , where f is the focal length of the telescope.

The intensity pattern due to the constructive and destructive interference is related to by

This intensity pattern of constructive and destructive interference rings is known as the Airy diffraction pattern (Figure 1), named after Sir George Airy, the Astronomer Royal of England who first derived it. of the total intensity is located within the central circle or the Airy disk. The dark destructive interference rings occur at the minimums of , where u = 3.83, 7.02 ... or ...

### Figure 1:

The Airy diffraction pattern.

The limit for the telescope's resolution is set by the diffraction at the aperture of the telescope. For a point source, like a star, the resulting image is a Airy pattern. The Rayleigh criterion for resolution of two point sources is that the central maximum of one images lies at the first minimum of the second image (Figure 2). Thus the limit of the angular resolution is

The Airy disk has angular radius so the radius of the central disk

(Heald and Marion, 1995; Hecht, 1987; Jackson, 1962).

### Figure 2:

The Rayleigh criterion for resolution.

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