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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 tex2html_wrap_inline208 at point P in the imaging plane, the Fraunhofer diffraction integral is
where k is the wavenumber tex2html_wrap_inline210, tex2html_wrap_inline212 and tex2html_wrap_inline214 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, tex2html_wrap_inline216 = x/R and tex2html_wrap_inline218 = y/R. Since telescopes generally have circular apertures, we will let a be the radius of the aperture and write tex2html_wrap_inline212, tex2html_wrap_inline214, tex2html_wrap_inline216 and tex2html_wrap_inline218 in cylindrical coordinates where the z axis is the axis of the telescope.

In the aperture plane,

In the image plane,

If tex2html_wrap_inline228, the angle between the axis of the telescope and R, is small,

We can choose tex2html_wrap_inline230 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 tex2html_wrap_inline232 is defined as

This means equation 10 can be written as
with tex2html_wrap_inline234, where f is the focal length of the telescope.

The intensity pattern due to the constructive and destructive interference is related to tex2html_wrap_inline236 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. tex2html_wrap_inline238 of the total intensity is located within the central circle or the Airy disk. The dark destructive interference rings occur at the minimums of tex2html_wrap_inline240, where u = 3.83, 7.02 ... or tex2html_wrap_inline244 ...

Figure 1:

The Airy diffraction pattern. DAMN! It didn't work!

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 tex2html_wrap_inline246 so the radius of the central disk
(Heald and Marion, 1995; Hecht, 1987; Jackson, 1962).

Figure 2:

The Rayleigh criterion for resolution. DAMN! It didn't work!

next up previous
Next: Turbulence in the Atmosphere Up: Twinkletwinkle little star: Previous: Introduction