We reviewed the concept of two-slit diffraction – that the two slits act like new
sources (that are coherent, i.e. in-phase). Before reaching a screen from which the
light is observed, the two rays travel different distances and thus have a path difference
which leads to a phase difference. Hence a diffraction pattern arises –
bright spots (constructive interference) interspersed with dark regions (destructive
interference).
After reviewing two-slit diffraction, we started to consider single-slit diffraction. We
now take into account the thickness of the slit which we will designate as
a. In single-slit diffraction we focus on the "dark side." It is easy to argue when
each ray is cancelled out by some partner. We divide the slit into an even number of segments
(two segments, four segments, six segments, etc.) We look at rays coming from the top of
the first and second segments. If these two rays cancel (have a path difference that is
half a wavelength), then so too will the rays from the bottoms of the two segments. In fact,
each ray from Segment 1 will cancel with a ray from Segment 2. Hence we will have complete
cancellation.
The path difference is a sin(θ) /2 where θ is the angle between the
rays (they are small and close enough together to consider them to be equal). As usual when the
path difference is half of a wavelength, we have destructive intereference. This leads to
a sin (θ) / (2n) = λ /2
where we broke the segment into 2n segments. Using a small angle approximation
(sin (θ) ≈ tan(θ))
a sin (θ) = a y / L = n λ
One of the pitfalls of these problems is the disparate lengths that are used: millimeters (mm)
for the slit width a, meters (m) for the slit-to-screen distance L, and
nanometers for the wavelengths λ
