Engineering with trip times

Bending mirrors to make all path times the same

Angling two mirrors can focus two beams. You can predict where it will be brighter by drawing rays from two sources, and using the rules about reflection. That gives a rule-based approach to accounting for the behaviour. But earlier we started to suggest that least-time paths were all important in predicting beams – that is they predicted the rays. Now you've got a slightly different challenge in that the effect of two rays is being predicted (it could easily be more, as we might have a multi-segmented mirror, perhaps even an infinite number of segments, to give a smooth parabola, or any other shape that we choose, by altering the angle between the segments). Earlier, multiple paths were considered as possibilities, leading to a unifying principle: rays are to be drawn along the line where the path trip time is least.

This diversion is something of a preview of episode 03, where multiple paths contribute, and also a bit of underpinning for the rules introduced for drawing ray diagrams for mirrors (here) and lenses (to follow).

Let's assume that the vibrators are in step at the sources (that's a good plan, because we could just be modelling a spread out beam, by selecting the extremes of the beam). How can we guarantee that the beams will add up to a large contribution at the detector? Simple – arrange the paths so that the delay injected between source and detector is the same or nearly the same. And how do you do that? Make sure the trip times are the same. The interactive allows you to do just that, and see a parabolic mirror emerge as if by magic. Only it's not magic: it's only working with the essence of waves – as do like me, but later – and geometry, which sets the trip times.

Shaping a glass block to make all path times the same

Lenses are ground, and in a particular way, so that the trip times from source to detector are constant, if the detector is placed at the focus. That is, lenses are ground so that the contributions from each path are all in step. This happens when the trip times from source to detector are equal, or very nearly so.

If we had a slightly imperfect lens, or one that exhibited chromatic aberration, then one frequency of the radiations might be in step, and another not quite. But – near enough, perhaps, for the eye to be fooled. These kinds of judgements must be made in the craft of making high-quality lenses that have to work over a range of frequencies.

Here you can model the construction of a simple lens, for only a single frequency. The apparent speed of light in the glass varies with frequency, and this gives rise to differing trip times for the different frequencies, for a given curvature of the lens. That variation in speed cannot be modelled here, but you can magically assemble a lens by fixing the thicknesses of the different prisms that can be thought of as making up the lens.

Apart from the pleasure of seeing new connections – exploiting the essence of waves (do like me, but later) and geometry to set the trip times – there is another important message here. Where the differences between trip times are small, there interesting things happen. Hold that thought for episode 03, where contributions from different paths are essential to a discussion of superposition, held by many to be the real signature effect of waves.