W9_parallel_resonance.eps

440 Week 13: Interference and Diffraction

screen, basically transforming it into a pattern of light and dark bars (or something more

complex if we have sources more complicated than “slits”).

We havealsoseen that Huygens’ Principle tells us that every point on a wavefront of an

advancing wave behaves like a “source” for the future time evolution of the wavefront.

This suggest that we don’tneedmultiple slits in order to see a wave interfere – all we

need isoneslit, but one that is wide enough that it contains “many” Huygens radiators

in the wavefronts that are incident upon it!

Calling this interference would be very confusing – one slit? two? ten?– so we introduce

a new term to describe “interference” of a wave with itself, or the interference patterns

produced byverylarge numbers of slits/sources, so many that they form a near contin-

uum. We call this kind of phenomenadiffraction, and speak of thediffractionof a wave

through a single slit, or the diffraction of a wave around an obstacle,or the diffraction

patterns produced on a screen or piece of film by light that passes through one or more

slits that are wide enough that the light that goes through them caninterfere with itself.

θ

λ P

a

θ

a/2 sin θ

Figure 184: The geometry of single slit diffraction. Waves of some wavelengthλpass through a slit
of widtha, whereais typically somewhat larger thanλ(to get an “interesting” diffraction pattern)
and fall upon a screen under Fraunhofer conditions, where the screen is distant compared toaand
λand roughly equidistant from the center of the slit

The geometry of diffraction is straightforward and is representedin figure 184. Note its

similarity toNslits – all of theNlittle round circles in the slitarepresent Huygens

radiators on the wavefront there.

As before, we’ll assume that we have Fraunhofer conditions, so that the screen is far

(compared toaandλ) from the slits, and we’ll either ignore any radial variation in the

field strength with distance or imagine that the screen bends in a halfcylinder around

the center of the slit. Note that we don’t have to do this – wecouldwork all of this

out (and in later courses physics majors very likely will) but doing so doesn’t help you

understand the basic idea of diffraction itself so we won’t bother^122.

Locating maxima and minima – especially maxima – will prove more difficult for a single

slit (of widtha) than it did for two or more very thin slits! Before we tackle actually

solving for the intensity in a formally justifiable way, let’s point out a couple of heuristic

features that will – for the most part – suffice to help us understand at least the gross

features of the diffraction pattern that results.

(^122) We’ll also (as we’ve been doing) more or less ignore the vertical dimension of the slit (the one perpendicular to
the paper) even though that is itself a “slit” and hardly seems to be as negligible as we’ve been making it out to be...