Closeup of the Off-Axis Focus of a Parabolic Mirror

  One can get more detail on the off-axis focus of the parabolic mirror found in the last blog. We find that the curves of equal action are caustics with a cusp on the left. Here the step in path length has been reduced to Δℓ=0.002 and the path length for the curve with cusp at z=1 is ℓ=2. The curve moves upwards as the path length increases.

Notice that the density of points increase near the cusp. The corresponding points on the curves are on the rays of the "system." A peculiarity of Excel is that points in a plot are not centered on the data point but displaced somewhat. The upper left corners of the square point markers are closest to the data points. The same is true for the focal point. These curves are the parts of the surfaces of constant action in the plane of the plot. To get the caustic for the focal plane one has to project the constant action surface onto this plane. The "density of points" determines the relative brightness of the caustic.