The parabolic mirror

5.1.1 The parabolic mirror

The fundamental requirement is to do with coherence. Imagine a wavefront arriving from a source, say a star (this is an astronomy book!) that is effectively at infi nity. In this case the wavefront will have no curvature and is called a plane wavefront. What does this mean? Simply that light everywhere across the wave- front left the star at the same time and is said to be coherent. You may have seen

a wave represented by a rotating vector which rotates once as the wave advances one wavelength through space. The angle of the vector, which conventionally rotates in an anticlockwise sense, defi nes the phase of the wave. So, if we split up our plane wavefront into little bits – we could call them wavelets – we could

represent each wavelet as a rotating vector with each wavelet having the same phase. If, through an optical system, we can bring all these wavelets together to

Introduction to Astronomy and Cosmology

Figure 5.1 Comparison of a spherical mirror and a parabolic mirror showing that, in the case of the parabolic mirror, all paraxial rays pass through one point (the focus).

one point each having travelled for the same time from a plane wavefront (and hence the source) then the vectors of the wavelets will all have the same phase and will add coherently (in a vector addition) to give a true focus. At other locations within the optical system the wavelets will add with varying phases so that the resultant will, in general, tend to zero. Figure 5.2 using a parabolic mirror will hopefully make this clear. At the focus all the vectors are in phase so the vector addition gives a large resultant but away from the focus the resultant is effectively zero. (The mirror shown has a very short focal length and is representative of the

surface of a radio telescope, but the principle is just the same for optical mirrors.)

Observing the Universe

Figure 5.3 The geometry of a parabola.

So why does a parabolic mirror give a good focus? The answer is simply because

a parabolic surface has the property that (l 1 ) the distance from any point on a plane perpendicular to the optical axis to the point on the surface below plus (l 2 ) the distance from this point on the surface to the focus is constant.

A parabola is defi ned by:

y ⫽ (1/4a) x 2

where a is the distance of the focus from the origin (Figure 5.3). The path length from the plane wavefront across the aperture fi rst to the sur-

face and then to the focus is given by l 1 2 . l 1 ⫽a⫺y

l ⫽ [x 2 2 ] 2 1/2

We can substitute for x 2 ( ⫽ 4ay) and expand the square term giving:

So the path length for all wavelets to the focus is a constant – they will add coherently.

In order to give its theoretical resolution – when the optics of a mirror or radio telescope surface are termed ‘diffraction limited’ – the mirror must conform to the

Introduction to Astronomy and Cosmology

Figure 5.4 The eff ect of an error in the surface of a parabolic surface.

precise parabolic shape to a very high degree of accuracy. Suppose, as shown in Figure 5.4, a small part of a parabolic surface lies below its correct position by a distance of 1/8th of a wavelength (this would correspond to a radio wavelength of 8 cm). The extra path length that the waves falling on this segment will travel to the focal point will be approximately twice this distance (2 cm or 1/4 of a wave- length). The phase of the waves refl ected by this segment will thus be 360/4 ⫽ 90° out of phase with the waves that have been refl ected by the surface which lies on the correct parabolic profi le. The vector summation at the focus will thus give a smaller resultant – all the waves reaching the focus are not quite coherent and, as a result, the image quality is reduced. A professional mirror or radio telescope surface will probably have errors less than 1/20th of the shortest wavelength that is to be observed.

One problem with a parabolic mirror is that, though it gives a perfect image at the focal point which lays on the axis of the parabola, away from the axis the image quality breaks down due to an optical aberration (called coma) – stars appear to look like little comets. The area of the sky that can be clearly seen is limited by this effect which is why other more complex designs have been developed as will be described later.