Homework 8:
Remember: In addition to this problem, you also have a “Mastering Physics”
assignment due outside of my office at 10:00 on Friday March 27 . There is also a
Mastering Physics Assignment due of March 7
In class when discussing the Yound double slit experiment (and diffraction gratings) we
assumed that the incident light was normal to the slits (or grating). In this problem we
work out what happens if it comes in at an angle
in
θ
as in the figure below; the dashed
line represents the direction normal to the slits . For simplicity we consider the double
slit case where the spacing between the slits is d. The arrows represent the direction of
the light; wave crests are perpendicular to it. The purpose of the problem
Because the incident wave is come in at an angle there is a phase difference between
the upper and lower slits. This phase difference alters the result from that obtained in
class.
a) As a first step show that the phase at the upper slit is ahead of the lower slit in
phase by an amount given by
λ
θπ
φ
)sin(2
in
d
=∆ . (Hint: this can be shown by the
same argument used in the derivation of Snell’s law.)
b)
Explain why the condition for a maximum to occur at some point in the light
emerging from the slits is that
(
)
m
rr
ul
πφ
λ
π
2
2+∆=
− for
L,3,2,1,0
±
±
±
=
mwhere
l
r and
u
r are distances from the point to the lower and
upper slits respectively. Recall that the maximum occurs when the light emerging
from the two slits are in phase up to
π
2 times an integer since that is when
constructive interference takes place.
c)
Show that maxima occur at angles given by
d
m
in
λ
θθ
+= )sin()sin(
for
L
,3,2,1,0
±
±
±
=
m . Hint: recall in our derivation for the normally incident
case it was shown that
(
)
)sin(
θ
drr
ul
=−
(assuming that the point is much further
than d from the slits . You can use that result here.
in
θ
θ