Parabola

Mirror is any surface reflecting light. As light is an electromagnetic radiation in the visible range of waves, the conclusions derived from analyses of parabolic mirror are true for communication antennas as well. Only the size of an antenna must be tailored to wavelength of the radiation.

The white lines in Fig. 17 illustrate the rays of the light. The parabolic mirror is illuminated by a plane wave and the reflected rays concentrate in a point on the axis of the mirror. This point is labelled as focus. The reflection obeys the low of geometric optics: the angle of incidence equals to the angle of reflection. As all the rays meet in one point on the mirror axis, the focal length is independent on the aperture of the mirror.

Geometric optic does not solve the distribution of the light along the axis in a vicinity of the focus. Distribution of light along the axis is solved by the QED sum-over-paths method. The paths of photon are shown in Fig. 17 as black lines only for segments parallel to the mirror axis. The paths from parabola to points on axis can be easily imagined. The parabolic mirror focus is the point on the optical axis where the probability of hitting the axis by photon acquires the maximum.

Fig. 17 Probability pattern for low (red), middle (green) and high (blue) frequency (parabolic-mirror.exe)

There are shown three probability patterns for lights of red, green and blue frequencies in Fig. 17. As the photon travels over paths always in the same medium, no dispersion effect takes place and the focuses are not separated in contrast with the focuses separation by lens. Nevertheless, the width of the probability pattern is getting larger with decreasing photon frequency. A focus calculated by the sum-over-pats method is consistent with the focus computed according to the reflection postulate of geometric optic.

Parabola is defined by the equation x = Ky². The parameter K is optional in the program parabolic-mirror.exe and defines not only the curvature of the parabola but the focal distances as well. With enlarging K the focal distance is shortening and the probability pattern is getting narrower.