Diffraction and interference of microwaves
TEP
Related topics
Microwaves, electromagnetic waves, Huygens-Fresnel principle, double-slit, interference
Principle
If a double-slit is positioned in the divergent microwave beam, a characteristic intensity
profile results behind this double-slit. The periodicity of the intensity profile can be used to
determine the wavelength of the microwaves.
Note
Prior to performing this experiment, it would be helpful, though not mandatory, to perform
the experiment P2460501 "Standing waves in the microwave range" first.
Equipment
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(1)
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Microwave set 11742-93
Microwave transmitter
Microwave receiver
Microwave probe
Microwave control unit
Meter rule
Angle scale
Double-slit
Cover plate
Additional equipment
Multi-range meter, analogue
Connecting cord, 32 A, 750 mm, red
Connecting cord, 32 A, 750 mm, blue
Barrel base PHYWE
Support rod, stainless steel 18/8, l = 250 mm, d = 10 mm
Right angle clamp PHYWE
Vernier calliper, stainless steel
Adhesive tape
07028-01
07362-01
07362-04
02006-55
02031-00
02040-55
03010-00
Fig. 1: Set-up for the interference experiment
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TEP
Diffraction and interference of microwaves
Tasks
First, familiarise yourself with the phenomenon of diffraction through a single-slit and on a
small obstacle. Then, measure the intensity profile that results from a diffraction through a
double-slit and determine the wavelength λ of the electromagnetic waves based on this
intensity profile.
Theory
If a diffraction object, e.g. a slit, double-slit, or grating, is placed in the beam path of a
source of light, an intensity pattern, which is characteristic of the object that is used, can
be observed at a certain distance behind this object. This is due to the diffraction of the
light on the edges of the object. This phenomenon can be explained by way of the
Huygens-Fresnel principle according to which every point of the object edge is considered
the starting point of a new wave. When the waves interfere with each other at a distant
point, the result is an intensity profile that cannot be explained by way of geometrical
projection (shadow-casting). Therefore, proof of interference is also proof of the wave
nature of light (here: of light in the microwave range).
Fig. 2: Concerning the geometry of the set-up
As far as diffraction through a grating is concerned, the expected intensity distribution can
be stated as a function of the location. The distribution primarily depends on the number of
slits. This experiment is about the special case of a double-slit for which the following is
true (see Fig. 2):
I ( α)=I 0⋅(
sin (γ) 2
2
γ ) ⋅cos (δ)
(1)
with
k
γ= ⋅b⋅sin(α)
2
(2)
k
δ= ⋅a⋅sin( α)
2
(3)
and
These two substitutions are used in order to make equation 1 clearer. The parameter k, the
so-called wave number, is defined as follows:
k=
2
2π
λ
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P2460901
Diffraction and interference of microwaves
TEP
The angle α is given by the geometry of the experiment (see Fig. 2):
x
α=arctan( )
d
(5)
In addition, the following relationship can be stated for the periodicity Δp of the intensity
distribution:
d
Δ p=λ⋅
a
(6)
The experimental determination of the periodicity can be used, for example, in order to
determine the wavelength λ.
However, the relationship described above applies only if the so-called far-field
approximation is used: Only if the distance between the aperture and the location of the
intensity measurement (here: between the double-slit and probe) is sufficiently long can
the diffraction effects on the slit on which the interference is based be sufficiently
developed.
In order to estimate as to whether far-field approximation can be applied to an experiment
set-up, the so-called Fresnel number F is defined:
F=
b2
d⋅λ
(7)
Here, b is a characteristic size of the aperture (here: width of the slit b) and d is the
distance between the aperture and the location of the intensity measurement. The Fresnel
number is a dimensionless number. Far-field approximation is fulfilled if:
F≪1
(8)
This is why it must be ensured that the distance d of the probe from the double-slit is not
too small, since it is incorporated into the Fresnel number in a reciprocal manner.
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TEP
Diffraction and interference of microwaves
Set-up and procedure
First preliminary experiment: Diffraction through a slit
Set the experiment up as shown in Fig 3.
Fig. 3: Experiment set-up
Connect the microwave transmitter and receiver to their associated sockets of the control
unit. Connect the multi-range meter to the voltmeter output of the control unit and select
the 10 V measuring range (direct voltage). Set the amplitude controller to maximum. The
loudspeaker and internal or external modulation are not required for this part of the
experiment.
Combine the angle scale and meter rule by way of the screw on the back of the angle scale
and the recess in the meter rule. Set the mark of the scale to 180°. Turn the meter rule in
order to align the reference mark (arrow) on the angle scale with the one of the meter rule
so that they coincide (see Fig. 4).
Fig. 4: Set-up and alignment of the angle scale and meter rule
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Diffraction and interference of microwaves
TEP
Fig. 5: Single-slit in the microwave beam
Install the double-slit in the centre of rotation of the angle scale so that one of the two slits
is centred, and use the cover plate to cover the other slit. Position the transmitter on the
angle scale at 200 mm and the receiver on the meter rule at approximately 500 mm (see
Fig. 5). Switch the microwave transmitter on by connecting the control unit to the mains
power supply. Turn the meter rule by 45° (Fig. 6).
Fig. 6: Diffraction through a slit
Remove the double-slit from the beam path and, while doing so, observe the reaction of
the voltmeter. Note down your observation.
Second preliminary experiment: Diffraction on an obstacle
Connect the microwave transmitter and probe to their associated sockets of the control
unit. Connect the multi-range meter to the voltmeter output of the control unit and select
the 10 V measuring range (direct voltage). Set the amplitude controller to maximum. The
loudspeaker and internal or external modulation are not required for this part of the
experiment.
Fasten the probe to the support rod in the barrel base by way of the right-angle clamp.
Install the cover plate in the centre of rotation of the angle scale and position the probe
approximately 10 cm behind the plate (see Fig. 7). Switch the microwave transmitter on by
connecting the control unit to the mains power supply.
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TEP
Diffraction and interference of microwaves
Fig. 7: Diffraction on an obstacle
Move the probe perpendicularly to the direction of the propagation of the radiation and,
while doing so, observe the reaction of the voltmeter. Note down your observation.
Experiment concerning the interference of microwaves
Connect the microwave transmitter and probe to their associated sockets of the control
unit. Connect the multi-range meter to the voltmeter output of the control unit and select
the 3 V measuring range (direct voltage). The loudspeaker and internal or external
modulation are not required for this part of the experiment.
Fig. 8: Set-up for the interference experiment
Then, set the experiment up as shown in Fig. 8 and 9. To do so, install the double-slit in
the centre of rotation of the angle scale and position the transmitter at 400 mm on the
angle scale.
Fasten the probe to the support rod in the barrel base by way of the right-angle clamp.
Position the probe and meter rule behind the double-slit so that the probe is moved
perpendicularly to the direction of propagation of the microwaves (see Fig. 9). Position the
probe centrally behind the double-slit and select a distance between the double-slit and
probe of at least 10 cm (far-field approximation, see above).
6
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Diffraction and interference of microwaves
TEP
Fig. 9: Measurement of the intensity profile
Switch the microwave transmitter on by connecting the control unit to the mains power
supply. Move the probe along the meter rule in order to find the global maximum of the
intensity distribution behind the double-slit. Then, adjust the amplitude by way of the
amplitude controller so that the full measuring range of the multi-range instrument is used.
Vary the position r of the probe and note the positions of the intensity maxima and minima.
Use a step width of 1 cm. When reading the positions off the meter rule, ensure to avoid a
possible parallax. In order to prevent the meter rule from being displaced by an accident,
we recommend securing it on the support surface by way of some adhesive tape or similar.
Fig. 10: Reading the meter rule (for example the position r = 440 mm)
Measure also the distance d between the double-slit and meter rule, the width of slit b, and
the distance between the slit centres a by way of the calliper gauge or use the values that
are given in the evaluation section.
In addition, the internal modulation and the internal loudspeaker of the control unit can be
used in order to demonstrate the intensity variation behind the double-slit.
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TEP
Diffraction and interference of microwaves
Evaluation and result
First, check whether the condition for far-field approximation is fulfilled.
Then, determine the periodicity of the intensity profile based on the relative positions x of
the maxima and minima. Use them in order to determine the wavelength.
r in mm
U in V
x in mm
α
sin(α)
γ
δ
Isim
320
1.100
-150
-0.896
-0.781
-1.943
-4.097
0.186
330
1.275
-140
-0.862
-0.759
-1.890
-3.983
0.272
340
0.750
-130
-0.825
-0.735
-1.829
-3.855
0.388
350
0.425
-120
-0.785
-0.707
-1.760
-3.710
0.537
360
0.375
-110
-0.742
-0.676
-1.682
-3.545
0.716
370
0.925
-100
-0.695
-0.640
-1.593
-3.359
0.910
380
1.400
-90
-0.644
-0.600
-1.493
-3.148
1.081
390
1.45
-80
-0.588
-0.555
-1.381
-2.910
1.162
400
1.300
-70
-0.528
-0.504
-1.254
-2.644
1.075
410
1.250
-60
-0.464
-0.447
-1.113
-2.346
0.772
420
1.150
-50
-0.395
-0.385
-0.957
-2.018
0.331
430
0.700
-40
-0.322
-0.316
-0.787
-1.659
0.015
440
0.3
-30
-0.245
-0.243
-0.604
-1.272
0.185
450
1.950
-20
-0.165
-0.164
-0.409
-0.863
0.970
460
2.000
-10
-0.083
-0.083
-0.207
-0.436
1.965
470
2.425
0
0.000
0.000
0.000
0.000
2.425
480
2.400
10
0.083
0.083
0.207
0.436
1.965
490
2.100
20
0.165
0.164
0.409
0.863
0.970
500
1.500
30
0.245
0.243
0.604
1.272
0.185
510
0.750
40
0.322
0.316
0.787
1.659
0.015
520
0.15
50
0.395
0.385
0.957
2.018
0.331
530
0.350
60
0.464
0.447
1.113
2.346
0.772
540
0.700
70
0.528
0.504
1.254
2.644
1.075
550
1.050
80
0.588
0.555
1.381
2.910
1.162
560
1.300
90
0.644
0.600
1.493
3.148
1.081
570
1.65
100
0.695
0.640
1.593
3.359
0.910
580
1.550
110
0.742
0.676
1.682
3.545
0.716
590
0.750
120
0.785
0.707
1.760
3.710
0.537
600
0.1
130
0.825
0.735
1.829
3.855
0.388
610
0.150
140
0.862
0.759
1.890
3.983
0.272
620
0.900
150
0.896
0.781
1.943
4.097
0.186
Table 1: Example data with a theoretical prediction Isim of the intensity profile
8
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Diffraction and interference of microwaves
TEP
Fig. 11: Comparison of measured and simulated values for the intensity profile
With the values b = 2.5 cm, d = 12 cm, and λ = 3.158 cm (see the experiment P2460501
"Standing waves in the microwave range"), the Fresnel number F can be calculated as
follows:
F=
b2
(2.5 cm)2
=
≈0.165≪1
d⋅λ (12cm⋅3.158 cm)
This means that the condition for far-field approximation is fulfilled in an approximative
manner.
If the wavelength λ is to be determined based on the measurement, the values of the
measurement example lead to a value of
a
5.27 cm
λ=Δ p⋅ =80 mm⋅
=35.1 mm
d
12 cm
In fact, the microwave transmitter is operated with a frequency of 9.5 GHz, i.e. with a
wavelength of λ = 3.158 mm (see above).
The theory predicts the existence of three maxima for the measuring range that is used for
this experiment (see Fig. 11). This has been confirmed by the experiment. Here, the
maximum intensity I0 is the measured value for normalising the simulated intensity profile.
The deviations of the measurements from the theoretical prediction can be explained by a
possible parallax when reading the probe positions off the meter rule and by the rather
large step width (1 cm) when screening the intensity profile. Please note that the intensity
in the boundary areas is superimposed by a weak, undiffracted interference signal of the
microwave transmitter, since the expansion of the double-slit is limited and scattered
radiation may pass the slit on the sides.
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TEP
Diffraction and interference of microwaves
Interpretation
Interference and diffraction are phenomena that can only be explained by describing light
(here: light in the microwave range) as waves.
During the diffraction through a (single) slit in the first preliminary experiment, for
example, the microwaves are diffracted into an angle (or angular range) so that, when the
slit is removed from the beam path, the intensity under the same angle is lower than
before.
During the diffraction on an obstacle (second preliminary experiment), a limited intensity
can be measured behind the cover plate, although the plate is made of reflecting metal.
This limited intensity is due to the fact the microwaves are diffracted on the edges of the
plate and that elementary waves propagate into the (alleged) shadow space.
These two experiments cannot be explained without consideration of the concept of
diffraction of waves, i.e. they cannot be explained by geometrical projection (casting of
shadow).
For instance, the diffraction patterns of two complementary objects, e.g. of a slit and an
obstacle of the same width, or of a circular disc and a circular hole of the same diameter,
cannot be distinguished from one another. This fact is known as Babinet's principle and is
true for all diffraction effects.
In the case of interference behind the double-slit, the diffracted waves of two slits interfere
with one another so that a characteristic intensity profile results. This is also a wave
phenomenon that cannot be explained by the radiation or particle theory.
10
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