436 Week 13: Interference and Diffraction
used to identify e.g. atoms and molecules from their “signature” optical spectra, and
are the basis for much of what we know of the Universe. For example, we know that
the physical laws governing very distant stars very far away (andhence being observed
today in their distant past due to the speed of light delay) is pretty muchidentical to
the laws we observe today!
This may sound silly, but this is anenormously important result. If things like the
gravitational constantG, the electric permittivityǫ 0 , the magnetic permeabilityμ 0 , the
speed of lightc– constants of nature, as it were – weren’tconstantover time frames of
billions of years, it woul radically alter our perceptions and understanding of the Universe
we find ourselves apparently living in. Instead we find that no matterhow far away or
how far back in time we look, the spectra of atoms in stars are pretty much the same,
something that actually testsmanyof the constants of nature all at once. The physics
governing those stars there, then, seems the same as the physics we learn and use today.
Of course spectrographs are also useful throughout science and technology in a strictly
mundane way. We havemanyoccasions to wish to identify a material, and if we heat
almost anything until it glows and then examine its light with a spectrograph, we can
instantly identify at least all of the elements in the sample and their relative abundance,
if not the molecules made up of those elements. Chemistry, engineering, and a variety
of physical sciences use this capability every day, using machines that have more or less
automated the process. It does seem wise for us to learn at least ingeneral how this
works, and what limits the resolution and accuracy of the process.
The second place understanding the interference of “many” slits will aid us is in boot-
strapping our understanding of diffraction itself. There a mix of Huygens principle and
our knowledge ofN-slit interference will let us quickly come to understand how a single
“wide” slit can produce an intensity pattern, cast on a distant screen, that is the result of
part of the light passing through the slit interfering with the rest, awave interfering with
itself. In the next two sections we will therefore apply the concepts we have learned for
2 , 3 , ..., Nslits, beginning withN-slit interference for largeNstraight up, thediffraction
grating.13.5: The Diffraction Grating – Rayleigh’s Criterion for Resolution
Consider now adiffraction grating– basically an opaque material with many transparent
narrow slits inscribed through the opacity, each separated from its neighbor by a dis-
tanced. We will imagine this grating to be normally illuminated by polychromatic light
(with many frequencies/wavelengths) in such a way thatNof them produce outgoing
waves that recombine coherently at the screen, where in application the screen is indeed
wrapped around in a cylinder at a distance that is large compared tod > λ(for anyλ
in the visible band).
As we saw in the previous section, the angles at which the primary maxima occur are
determined only by the distanceddsuch that:θmaxm = sin−^1(
mλ
d)
(1051)
independent ofN– indeed, they are at the same angles for 2 slits as they are for 2000.
What changes as we increase the number of slits is the location of theminimaand the
secondary maxima in between. Consider the two minima that “bracket” each primary