Week 13: Interference and Diffraction 439
maximum you are observing and the number of slits that are coherently illuminated by
the beam which contribute to it. AsN goes up, the first minima squeeze ever more
tightly around the principle maxima and the resolving power improves.However, asm
increasesallof the angles increase, as well as all of the separations of the angles. Since
the width of the principle maxima doesnot vary withm, higher order maxima have
better resolution, all things being equal. If we want to know if we canresolve two lines
with separation ∆λ(both very nearλ), we can merely evaluate:∆λmin=λ
mN(1065)
for the order considered and if the two lines are separated by morethan this spread, they
will be resolved.
There are other places in our daily lives where “diffraction gratings” can be observed.
CD or DVD ROMs, for example, consist of many “tracks” carved intoa shiny reflective
plater and pitted by means of a laser to encode information. The reflective grooves
behave just like multiple slits and split white light up into a veritable rainbow of colors
when the reflective grooved surface is viewed at various angles. There is norealcolor to
the shiny disk; all of the color arises from multiple slit interferences.
This same process worksbackwards, as well. A radio telescope is made out of a regular
array of antennae spread out in a two dimensional lattice. If we imagine all of the
antennae radiating coherently at the save frequency and wavelength, we expect the waves
they emit to only constructively interfere and hence radiate most of their energy along
certain directions. If we reverse this, however, by adjusting thephase of the signalspicked
upby the antennae and combining them into one phase delayed superposition signal, we
can arrange it so that they only coherentlyreceivefrom certain directions in the sky. In
fact, by appropriately sweeping the phase delays, we can sweek the telescope across the
sky and make a highly directional map of all of the radio signals emittedby the sun, by
stars, even by remote galaxies. We even expect resolution to improve as we increase the
number of antenna, in a way that should now be intuitively familiar.
Now, let us think about multiple slits and Huygens’ Principle. Huygens’Principle states
that all of the points on a wavefront behave like coherent radiators, which sounds alot
like what multiple slits that sample just some of those radiators do. The difference is that
with a wavefront, the number of coherent radiators has to go to infinity at the same time
that the distance between radiators has to go to zero at the sametime the amplitude
emitted by each radiator (which we’ve been treating as a givenconstantfor the many
slit problems) has to also go to zero, but in such a way that the totalenergy emerging
from a piece of the wavefront is conserved!
Handling all of this correctly lets us understanddiffraction, the interference of a wave
that e.g. passes through asingleslit withitself. Understanding diffraction is absolutely
essential to the understanding of the diffraction/wave based limitations of optical in-
struments such as microscopes and telescopes. We begin by completely analyzing and
solving for the diffraction intensity produced by light passing through asingleslit of
widtha > λ, in the usual Fraunhofer approximation.13.6: Diffraction
We have seen how coherent, monochromatic light passed through multiple slits, when it
recombines after traversing different path lengths, interferes –sometimes creates a wave
with an amplitude greater than that produced by a single slit, sometimes cancelling
altogether – and that this creates a modulation of the intensity observed on a distant