426 Week 13: Interference and Diffraction
question, and even white light from hot sources – as a mixture of many frequencies that
areallcoherent over similarLcoh– can be locally sufficiently coherent to support e.g.
thin film interference in all of the colors/frequencies independently.13.1.2: Combining Coherent Harmonic Waves
The unifying idea of this entire chapter is then: Monochromatic coherent light from
some source follows two (or more) different paths to reach a detector (e.g. – an eye,
a screen observed by an eye, a piece of film, a photoelectric detector). Along the way
it accumulatesphase differences between the waves due to the different path lengths
that they follow (and possibly other things such as reflection that introduce phase shifts
discretely along the way). The electric (and magnetic) fields thenrecombine, and the
intensity of the resulting electromagnetic field is registered by the detector.
Provided that the maximum path differences involved are less than the coherence length
Lcohof the light, we will then have to repeatedly evaluate below sums suchas (for a
single polarization component of the wave):Etot=E 1 sin(kx−ωt+δ 1 ) +E 2 sin(kx−ωt+δ 2 ) +E 3 sin(kx−ωt+δ 3 ) +... (1019)where the phase shiftsδiare all determined by the path differences plus discrete shifts.
It is too difficult to solve this equation generally. Instead we will make avariety of
simplifying assumptions that are all reasonably valid in the context ofthe following
specific topics. The primary ones will be that we will generally assume that all of the
field amplitudes are the same (although we could certainly deal with specific cases where
they are different in some simple way using the methodology we develop). We will usually
setoneof the phases e.g.δ 1 to be zero (setting our clock, as it were, by the first source).
The other phase differencesδiwill usually be assumed to be constant in time (the light
from all of the paths is perfectly coherent at the time of recombination).
With those assumptions, we can usually reduce thealgebraicproblem of adding the
harmonic waves to the simplergeometricproblem of adding two or more make-believe
vectors, calledphasors. Phasor addition will simplify the problem of finding the inter-
ference and diffraction patterns produced by idealized slits and apertures to where it is
straightforward, if not quite easy.
Along the way we will also endeavor to establish some very simpleheuristic rulesthat
enable one to determine where interference or diffraction patterns aremaximum^117 or
minimum.
The heuristic rules are worth stating here, although we’ll repeat them many times below.
One will generally get interferencemaxima when the waves arrive at the detectorin
phase, which in turn means that the path difference will contain aninteger number of
wavelengths(and still be less thanLcoh). One will get interferenceminimawhen the
path difference contains anodd half-integer number of wavelengths so that the waves
arrive exactlyout of phase.Tres simple, no?
Let’s start with the simplest of interference problems: Two Slit Interference.(^117) Since this is a common enough point of confusion, let me make it clear that the termmaximumin interference
or diffraction problems already refers to amaximum in intensity at the point of observation of the e.g.
interference pattern, not “maximally interfering” and hence ofminimumintensity. Similarly aminimumrefers
to the minimum (usually zero) in the interference or diffraction intensity at the receiver, on the screen, to the eye.