Ellipses

When discussing lenses, we claimed that an ellipse of appropriate eccentricity satisfies the equation \[ n_1d_1 + n_2d_2 = \text{const}, \tag{1}\] where \(d_1\) is the distance of a point on the ellipse to a vertical line and \(d_2\) is the distance from the point to one of the foci. Here we will check that this is the case.

At the left of the diagram is a vertical black line, representing a wavefront. A single horizontal ray leaves this line, heading to the right. The right of the diagram, and a ellipsoidal bulge are highlighted in blue to indicate the presence of glass. The light ray travels a distance of d one before striking the upper distance of d two to the focal point.
Figure 1: Figure from chapter 5.

Two definitions

There are at least four common definitions of an ellipse — we will need two.

Sum of distances: Given two fixed points (the foci) and a constant \(2a\), find the set of points such that the sum of distances to each focus equals \(2a\). These points form an ellipse.1

1 Here \(a\) is the length of the semi-major axis — the line from the centre of the ellipse to its furthest point.

Figure 2: Ellipse defined in terms of the sum of the distances to the two foci.

Directrix and focus: Given a straight line (called the directrix) and a number \(e\) (the eccentricity) between zero and one, find the set of points such that the ratio of the distance to the focus and the distance to the directrix equals \(e\). These points form an ellipse.

Figure 3: Ellipse defined in terms of distances to one focus and to the directrix.

Deriving a condition on \(d\) and \(f_2\)

How are \(d\) and \(f_2\) related? Taking the two definitions, we can eliminate \(f_1\) to get \[ de + f_2 = 2a. \] To relate this to the discussion on lenses, multiply by \(n_1 / e\) to get \[ n_1d + \frac{n_1}{e}f_2 = \frac{2n_1a}{e}. \] Furthermore, to relate to figure 1, let \(d = d_1\) and \(f_2 = d_2\), to get \[ n_1d_1 + \frac{n_1}{e}d_2 = \frac{2n_1a}{e}. \] If this is to match equation 1, we require2 \(e = n_1 / n_2\), which produces \[ n_1d_1 + n_2d_2 = 2n_2a. \] The left hand side now matches that of equation 1. The right hand side is a constant, but perhaps not the same constant as in equation 1. This can be fixed by simply moving the vertical line.

2 Note that this rules out a circular/spherical surface, since the eccentricity of a circle is zero.

We have therefore discovered that an ellipse (or ellipsoid in three dimensions) of eccentricity \(e = n_1 / n_2\) is the correct shape for the interface between air and glass to focus parallel light rays to a single point. Notice that larger ellipses, of the same eccentricity, produce longer focal lengths. It may not be necessary to use the entire ellipse — if we wish to focus a narrow beam of rays to a distant focal point, only the central portion of the ellipse (or ellipsoid) is required.