Week 13: Interference and Diffraction 423
13.1.1: Hot Sources and Wave Coherence
Although we’ve see that Maxwell’s equations in free space become theelectromagnetic
wave equation (so that light is plausibly and electromagnetic wave) wehaven’t spent
much time considering how light arises in the first place, how charges can end upemitting
electromagnetic waves. The bulk of our understanding came from thinking about a
Lorentz model atom – an electric dipole moment that harmonically oscillates, producing
an electric field that propagates and oscillates, inducing its companion magnetic field as
it goes to produce a wave.
That’s pretty much how it (classically) goes, so this isn’t a bad thing. We also get
electromagnetic radiation (usually at radio frequencies) if we make amagnetic dipole
moment oscillate in time, for example by putting an alternating current into an antenna
consisting ofNcircular turns of wire, but radiation from atoms is predominantly elec-
tric dipole radiation. The only “catch” is that the radiation is aquantumprocess and
hence only comes out of the atoms in particular frequencies and “allat once” instead of
continuously and at varying frequencies as we might expect classically.
There are two generalkinds of sources we need to be concerned with when dealing
with electromagnetic waves and superposition leading to interference and diffraction:
CoherentandIncoherent. These are both relative terms – no causal, periodic source
of electromagnetic waves is perfectly coherent or perfectly incoherent (it would have be
periodic over an infinite amount of time to manage this, which seems infinitely unlikely in
a “messy” Universe), and ultimately source coherence is thus described by a real number
that can vary over some range.
A source is said to be coherent if:a) It is (approximately) monochromatic (or at least, a fixed mixtureof frequencies
that are independently otherwise coherent).
b) The waves emitted by these source areideally harmonic, that is, their phase tem-
porally accumulates asωtfor the fixed frequencyωand with a constant additional
phase, if any.The latter implies the former, as you can see.
Coherence, we see, is implicit in our writing down (anx-polarized harmonic wave prop-
agating in thezdirection):E~(z, t) =E 0 xxˆsin(kz−ωt) (1014)An ordinary monochromatic harmonic wave is perfectly coherent.
To understand why coherence is important to us, let us consider what a “harmonic” wave
might look like that isnotcoherent^110 :E~(z, t) =E 0 xˆxsin(kz−ω(t)t+φ(t)) (1015)In this wave I have illustrated two common sources of incoherence.One is a frequency
that isn’t really constant in time but e.g. slowly varies in such a way thatit has some
constantaveragevalue, e.g.
ωavg= limT→∞∫T
0ω(t)dt (1016)(^110) And hence, of course, not perfectly harmonic or monochromatic! Students who have taken more advanced math
can understand this in terms of theFourier transformof the wave above, which willnotbe a Dirac delta function
of any single frequency but rather will involve abandof frequencies around a peak atωavg. This in turn takes us
back to the discussion of amplitude modulated waves from theAC Circuits chapter above compared tofrequency
modulatedwaves that can also be used to carry encoded information. Deep waters underlie these simple concepts.