Download Physics 316 B2 1 Revised 3/7/08 Experiment B2: Monochromatic

Physics 316
B2
Experiment B2: Monochromatic Fringes with the Fresnel Biprism
Reference: Hecht (4th ed.)
p. 398
The wave nature of light was established in the early part of the 19th century, although many
interference and diffraction phenomena had long been known. Fresnel's experiment with the
biprism was one of the earliest experiments to yield values for the wavelength of light. As you will
find, accurate values do not usually emerge, but the experiment does provide some useful
experiences in careful alignment and fine adjustment.
The basic concept is simple: monochromatic light from a source passes through a slit S. Light
passing through the two halves of the biprism is deviated so that it seems as though there are two
(‘virtual') slits, S’ and S", and the resulting interference pattern is the same as for two slits.
path difference
s'
!
a
"
r2
s
"
s"
!
r1
r0
interference
fringes
x = ave.
spacing
r
Figure 1
The wavelength can be calculated from the average fringe separation (x), in terms of the various
distances (ro and r):
2r ( n " 1)# x
[1]
! = 0
n = refractive index of prism
(r + r0 )
Apparatus:
Biprism (make a note of its reference number)
Slit
Laser (He-Ne)
Measuring microscope (optional)
Beam expander lens (Addendum)
Condensing lens (Addendum)
Mercury lamp with green filter (wavelength 5461 Å, Addendum)
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Procedure
Set up the biprism as shown in the diagram (Fig. A1). The biprism is to be mounted on the special
holder that permits rotation of the biprism in its own plane. Accurate alignment of the edge of the
biprism with the slit (S) is critical to obtain the best fringes.
Align the laser, filter, slit (S) and biprism and look, by eye, for fringes. You should be able to see
them, and if you use a simple magnifying lens, they should be easily seen. Arrange the separation
of the components so that you obtain a large number of fringes when you view them from across
the room.
Start with the observing position no more than 100 cm from the prism. When you have adjusted to
get sharp fringes, increase the prism-observing distance to about 2m, making whatever adjustments
are needed to maintain fringe quality.
Measurement of the wavelength requires knowledge of the biprisim angle. This is not easy to
measure accurately because it is so small; several methods are outlined in different optics books,
but we have developed our method which is easy to use and gives accurate results.
To measure the distance between components several meters apart, ask the T.A. for a steel tape
measure.
!
am am
e
r be
lase ected b
fl
e
r
and
"
reflected,
refracted
beam
Figure 2
Measurement of the biprism angles, using a He-Ne laser beam
The biprisms are not usually symmetrical, so you will need to measure both angles and use the
average in the calculation of the wavelength.
For a beam of light at normal incidence (Fig. 2), one reflected beam will return to the laser, but the
other reflected beam (from the back surface) will be refracted and will emerge at angle β = 2nα
(since α, β are small angles).
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Mount the biprism in a lens-holder, so that you can rotate it and vary the angle of incidence easily.
Reflected spots can be located on the wall on the far side of the room, and the angle then
determined by measuring the distances between the reflected spots and the distance apart of the
components. Using the laser, the angle of minimum deviation can be found:
Figure 3
! + D$
sin"#
2 % and with α, D small angles, we have
With n =
!
sin" $
# 2%
" ! + D$
# 2 %
D
n=
& 1+
"! $
!
# 2%
We then have two equations: n = 1 +
[2]
D
and ! = 2n" .
!
Estimate the standard errors in n and α.
(NOTE: angles α, β, D are in radians in this calculation.) Solve for n and α using the
measured values of β and D. Separation of the virtual slits is a = 2r0 (n - 1)α.
Note on approximations:
Hecht (p. 439 problem 9.15), asks you to investigate the conditions under which the approximation,
path difference = r1 - r2 ! asin " , is valid, in the derivation of the expression for the average fringe
spacing. This approximation requires that r >> a 2 ! where r = the distance between biprism
and fringes, and a = the separation between virtual slits. Typically, we find a ! 0.5cm , and with
! " 5 # 10 $5 cm , we find that we need r ! 5000cm , which can not be met, unless a is very much
reduced.
Closer examination shows that this requirement comes from calculation of the path difference
between rays to a single fringe, from the two virtual slits, and this yields
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B2
a2
!r = asin" #
cos2 "
2r
" = mean angle to point of observation
[3]
Failure to meet the condition r >> a 2 ! means that !r turns out negative!
Further examination, however, shows that this is not the way to consider the problem, but that the
approximation is indeed valid:
"
There will be a dark fringe when !r = n
(n odd)
2
So
!
"2
2
[4]
n = " sin# $
cos #
2
2r
whence
'
dn 2 $
#2
[5]
= &# cos ! +
cos! sin ! )
d! " %
r
(
since θ is small we put cos θ =1
'
dn 2 $
#2
= &# +
sin! )
d! " %
r
(
[6]
thus
$
'
)
" &
1
d! =
& #
) dn
2# & 1+ sin! )
%
r
(
The angle between adjacent dark fringes (dθ) is given by dn = 2:
"
#
d! = %& 1 $ sin ! '(
#
r
whence
"
! = " d# $% 1 + sin# &'
r
! 0.5
!
Typically
~
~ 10 "3 so ignore the sin " term
r 500
r
! " # d$
# x &
= [2r0 (n ! 1)" ]%
( as before.
$ r + r0 '
[7]
[8]
[9]
[10]
[11]
Calculations:
Once you have the optical components aligned and have fringes ready to measure, make a rough
measurement, to check that you are obtaining reasonable values.
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B2
Also, make a rough estimate of the percentage error that is attainable in the various measurements.
This will guide you to the precision needed in measuring the large distance, r. From a
consideration of the error propagation, you can see where it is necessary to be very careful, and
where apparently inaccurate measurements can still be tolerated.
In obtaining the best estimate of the wavelength, you will need to calculate the average fringe
spacing. Do this by plotting a graph of fringe position vs. fringe number, and fitting a least squares
line. Why is this better than taking a straight average of the measured spacings?
Qualitative demonstration:
The Fresnel biprism lends itself well to a demonstration that interference fringes cannot be obtained
when the two beams of light do not have the same polarization.
As you have carried out the experiment, the two beams (coming through the two halves of the
biprism) have random polarization since they both originate in the mercury lamp. You are
provided with two pieces of Polaroid. Attach a piece of masking tape to each as a marker, and by
rotating one Polaroid relative to the other, find out which relative orientation produces total
extinction. You can then label the polaroids to indicate the directions of their optic axes.
Using tape and some cardboard, mount the two polaroids as shown in the diagram and then hold
them over the biprism, so that one of the emerging beams will be polarized in one plane, and the
other beam has its polarization axis at right angles to the first. The fringe pattern should
disappear—you should see a generally illuminated field of view. Remove the polaroids, and the
fringes should reappear.
Figure 4
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