# Superposition of Quasi-Parallel Plane Waves

In previous chapters, we considered only individual plane-wave fields which have uniform intensity throughout space and time. Some optical fields can be well-approximated by a plane wave, but most have a more complicated structure. It turns out that any field (e.g. pulses or a focused beam), regardless of how complicated, can be described by a superposition of many plane wave fields. In this chapter, we develop the techniques for superimposing plane waves.

We begin our analysis with a discrete sum of plane wave fields and show how to calculate the intensity in this case. We will introduce the concept of group velocity, which describes the motion of interference ‘fringes’ or ‘packet’ resulting when multiple plane waves are superimposed. Group velocity is distinct from phase velocity that we encountered previously. As we saw in chapter 2, the real part of refractive index in certain situations can be less than one, indicating superluminal wave crest propagation (i.e. greater than c)! However, it is the group velocity that tracks the speed of interference fringes, which are associated with light intensity.

## Intensity of Superimposed Plane Waves

The terms involving (ωj + ωm)t oscillate rapidly and time-average to zero. By comparison, the terms involving (ωj −ωm)t oscillate slowly (especially when the ωj are all in the neighborhood of the ωm) or not at all when j = m. We retain the slower fluctuations and discard the rapid oscillations. For purposes of computing the intensity we can approximate the index as approximately constant, and write km/(ωmµ0) ≈ n²0c. With these simplifications, (7.5) becomes.

### Group vs. Phase Velocity: Sum of Two Plane Waves

The darker line in shows the intensity computed with (7.11). Keep in mind that this intensity is averaged over rapid oscillations. For comparison, the lighter line shows the Poynting flux with the rapid oscillations retained, according to (7.5). It is left as an exercise (see P7.3) to show that the rapid-oscillation peaks in Fmove with a phase velocity derived from the average k and average ω of the two plane waves

This is known as the group velocity. Essentially, vg may be thought of as the velocity for the envelope that encloses the rapid oscillations. As noted, the group velocity is often written as a derivative rather than a ratio of finite differences; the derivative will be more natural when dealing with a continuum of plane waves rather than a pair of planes.

### Frequency Spectrum of Light

Individual plane waves have infinite length and infinite duration. They do not exist in isolation except in our imagination. Moreover, a waveform constructed from a discrete sum (as in the previous two sections) must eventually repeat over and over (i.e. it is periodic).

### Last word

To create a waveform that does not repeat (e.g. a single laser pulse or, technically speaking, any waveform that exists in the physical world since no light source repeats forever) we must replace the discrete sum with an integral that combines a continuum of plane waves. Such a waveform at a point r can be expressed.