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PROJECT FOR SECTION 15.4
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Fraunhofer Diffraction by a
Circular Aperture
Y
M
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DISTRIBUTION
S
X
ρ
θ
P
ψ
w
L
O
O
Lens
© Jones & Bartlett Learning, LLC
Anton M. Jopko, Ph.D.
Figure 2
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Department of Physics
and FOR
Astronomy,
McMaster University
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lens, and its origin is where all the light from the star
would appear in the absence of diffraction. Because of
diffraction, however, some light also appears at P. Point
The stars©inJones
the sky &
areBartlett
an enormous
distance
from
us,
P is a general
point but
very closeLearning,
to O, being only
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LLCarc
so they can be considered point sources of light. If you
seconds
away.
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look at such a star through a telescope, you might expect
In Figure 2, we have merged the aperture and the
to see just another point of light, albeit a much brighter
lens, because in practice the edge of the lens also defines
one. However, this is not the case. Because it is a wave,
the aperture. Because of the circular symmetry of the
light is diffracted as it passes through the circular aperlens and the diffraction pattern, it is very desirable to
ture&ofBartlett
the telescope
so that theLLC
light is spread out over a
convert
to polar coordinates.
a wave be emitted at a
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small fuzzy region that we call the diffraction pattern.
point
S
in
the
lens
with
coordinates
(X, Y) or 1r, u2 and
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This project will investigate the shape of the diffraction
arrive at P with coordinates (L, M) or angular coordipattern for light passing through a circular aperture of ranates 1w, c2 . Then X r cos u, Y r sin u, and
dius R.
L w cos c, and M w sin c. Here r is the radial disFor simplicity, we assume the light is of one wavetance from the center of the lens to the source S of the
length l, or color. This light
has the form
of a spherical
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emitted wave and u is its ©
polar
angle; &
w Bartlett
is the angular
wave front near the star, but by the time it reaches us the
c
radius
of
P
and
is
its
polar
angle.
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wave front forms a plane wave. All points on the wave
The waves emitted at the aperture are in phase and
front have the same phase. We now point the telescope
have the same amplitude, but they all travel different
with its circular aperture and lens directly at the star so
distances to point P so they become out of phase there.
that the plane wave fronts are incident from the left, as
The intensity of light at P will be proportional to the
in Figure©1.Jones & Bartlett Learning, LLC
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square of©the
resultant
amplitude Learning,
of all waves arriving
there.
We
now
need
to
calculate
this
resultant
amplitude
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Y
by taking into account the waves’ phase differences.
M
We define the wave number of the incident and emitX
Lens
P
ted waves to be k 2p>l. Then according to Principles
of Optics, seventh edition, by Born and Wolf, the resultL
ant &
amplitude
P from all the
emitted waves in the
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Bartlettat Learning,
LLC
O Learning, LLC
aperture
is
just
the
Fourier
transform
of the aperture:
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U1P2 C
Aperture
radius R
ik1LXMY2
e
dXdY
aperture
Figure 1 Diffraction of light
where C is a constant, proportional in part to the bright© Jones & Bartlett Learning, LLC
© Jones
Bartlett
LLC
ness of the star. The intensity
at P will&then
be givenLearning,
by
From Huygen’s principle,
each
pointSALE
in the opening
NOTpattern
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ORasDISTRIBUTION
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FOR
OR DISTRIBUTION
U1P22. This is the diffraction
the star
a
of the circular aperture emits a wave in all directions.
function of the angular radius w.
Fraunhofer diffraction requires that the waves leave the
aperture in a nearly parallel bundle traveling toward a
Related Problems
very distant point P. The only purpose of the lens is to
1. Show©
that
the resultant
amplitude
at P usingLLC
the two
form a ©
point
image&ofBartlett
this parallel
bundle at LLC
a much
Jones
Learning,
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& Bartlett
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systems
of
polar
coordinates
can
be
written
closer distance
to
the
aperture.
Diffraction
would
still
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occur even without the lens. The dotted line joining the
R 2p
two origins is also the axis of the aperture and lens. The
eikrw cos 1uc2rdudr
U1P2 C
LM system of coordinates is in the focal plane of the
0 0
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PROJECT FOR SECTION 15.4 Fraunhofer Diffraction by a Circular Aperture
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4591X_PROJ_Zill.qxd
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2. Using the identity
7. What is the value of the smallest nonzero root of J1?
Using l 550 nm, R = 10 cm, and the smallest root
just
found, Learning,
calculate the angular
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LLC radius w (in arc sec© Jones & Bartlett
onds)
of
the
central
diffraction
disk.
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2 J1 1kRw2
where Jn is the Bessel function of the first kind, show
8. Draw a plot of
as a function of kRw as well
kRw
that the resultant amplitude reduces to
as the intensity, its square. The diffraction pattern of
R
central disk surrounded by
U1P2 ©2pC
dr Learning, LLC the star consists of a©bright
JonesJ0 1krw2r
& Bartlett
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LLC
several thin, faint concentric
rings.
This disk isLearning,
named
0
NOT
FOR
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OR
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NOT FOR SALE OR DISTRIBUTION the Airy disk in honor
of G. B. Airy, who was the first
for any c. We choose c 0. (This expression is also
to calculate the diffraction pattern of a circular aperture
known as a Hankel transform of a circular aperture.)
in 1826.
3. Using the recurrence relation
9. What happens to the angular width of the diffraction
pattern
if the radius
R of the aperture
is doubled?
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Bartlett Learning, LLC
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3u Jn1 1u2 4 un1 Jn 1u2,
10. What
happens
the angular
of the diffraction
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NOT
FORtoSALE
OR width
DISTRIBUTION
pattern if the wavelength of the light is doubled?
show that
11. What happens to the angular width of the diffraction
x
pattern if the focal length of the lens is doubled?
u J0 1u2 du x J1 1x2
12. Suppose that a circular aperture has the shape of an an0
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with inner
radius a and
outer radius b. Find U(P).
2J
1kRw2
1
NOT FOR4. SALE
ORU1P2
DISTRIBUTION
ORisDISTRIBUTION
(This result
of practical importance because reflectShow that
. Therefore the in-NOT FOR SALE
CsR2
kRw
ing telescopes almost always have an obstruction in the
tensity is given by
central part of the aperture.)
13. Suppose that the annulus in Problem 12 is very narrow
2 J11kRw 2
small butLearning,
not in¢a being
U 1P2©2 c
d I0
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©, with
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kRw
finitesimal. Show then that the approximate resultant
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by U1P2 C12pa¢a2J0 1kwa2 .
2 J1 1kRw2
[Hint: Interpret the result U(P) from Problem 12 as an
5. What is lim
?
wS0
kRw
d1u J1 1u2 2
approximation for
u J0 1u2 with u kwa.]
6. What is the physical significance of I0?
du
in
a ina
Learning,
eix cos LLC
e da Jn 1x2,
2p 0
2p
v
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© Jones & Bartlett Learning, LLC
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© Jones & Bartlett Learning, LLC
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© Jones & Bartlett Learning, LLC
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PROJECT FOR SECTION 15.4 Fraunhofer Diffraction by a Circular Aperture
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xxxi