424 Week 13: Interference and Diffraction
that is, it might beapproximatelyconstant over a time that is long compared to a period
of the wave, perhaps several thousands or millions of those periods, but on shorter
times it might vary within some range. This variation might be caused bye.g. thermal
fluctuations in the source, by thermal doppler shifting of a sharp natural frequency in a
gas, or by still other things (including humans, who amplitude or frequency modulate a
carrier wave to encode information).
In nature, not even quantum sources have infinitely sharp frequencies, so even “monochro-
matic” light is onlyapproximatelymonochromatic or monochromatic within some band-
width or range^111 , and the variation over longer time scales may be sufficient to cause
temporalinterference (beats) instead of thespatialinterference we will examine in this
chapter when waves that follow different paths from a common source are recombined.
The other source of incoherence is the phase angleφ(t). We recall that when we solved the
wave equation we could add an arbitrary phase constant to the argument of the harmonic
wave and we’d still have a harmonic wave. Basically, that constant simply indicated when
we “started our clock”, and we could more or less choose to use a sine wave or cosine wave
with no phase at all by starting our clock appropriately when examining or describing
the wave.
The problem is that for many sources, especiallyhotsources, this clock getsresetwhen-
ever the oscillators that are producing the wave are physically disturbed or re-energized
(the oscillation necessarily damps out over time as the energy in the oscillator is radi-
ated into the electromagnetic field). There is no reason to expect that the phase of the
oscillator producing the light will be constant over time indefinitely. Indeed, we rather
expect the opposite!
The simplest model for “hot source” incoherence is that ofphase interruption. We
imagine a sample of some element that is hot enough so that when an atom collides with
a neighbor it excites some particular oscillator state with a fixed frequency and a phase
determined by the time of the collision. It then oscillates monochromatic light with
a phase and polarization direction determined by the time and angle ofthat collision.
Eventually, however, the atom collides again, and although the sameoscillator state is
re-excited and light of the same frequency emerges, it has a (discretely) different phase
and direction of polarization!
In this (most common) case, the hot “monochromatic”^112 source is temporally phase
coherent only for the mean time between collisions, which in turn depends on things
like the density of the material and its temperature. Although our mental picture of
“collisions” is simplest to envision for a fluid like a liquid or gas, related (e.g. phonon
based) events also phase interrupt the wavetrains emitted by hotsolids, and again there
is a characteristicaverage timebetween such phase interruption events.
The effect of these phase interruptions is such that when adding the electric fields of two
completely incoherent sources, no interference or spatial diffraction is observed to occur- the intensities of the different sources simply add because the fields themselves add for
a few cycles, then cancel for a few cycles, then add, then cancel,in such a way that the
average energy transmitted smooths out and just adds. Temporal incoherence over long
time scalesdestroys spatial interference patterns and replaces them with mere
average intensity addition^113! This isvery important– it is the reason we don’t
see interference patterns all the time, e.g. why windowpanes and drinking glasses don’t
exhibitthinfilm interference like that discussed below! Whenever we add two harmonic
waves to get a harmonic wave as a result, we are implicitly assumingcoherence.
(^111) We speak of “line broadening” and the “natural width” of spectral lines to acknowledge or quantify this.
(^112) In quotes because the fourier transform of a harmonic wave with random phase interruption is no longer sharp
or monochromatic.
(^113) All of this is proven in more advanced mathematical treatments.