LEP
1.5.18
-00
Diffraction of ultrasound at a Fresnel zone plate /
Fresnel’s zone construction
Related topics
Longitudinal waves, Huygens’ principle, Interference,
Fraunhofer and Fresnel diffraction, Fresnel’s zone construction, zone plates.
Principle
An ultrasonic plane wave strikes a Fresnel zone plate. The
ultrasonic intensity is determined as a function of the distance
behind the plate, using an ultrasonic detector that can be
moved in the direction of the zone plate axis.
Equipment
Ultrasonic unit
Power supply f. ultrasonic unit, 5 VDC, 12 W
Ultrasonic transmitter on stem
Ultrasonic receiver on stem
Fresnel zone plates for ultrasonic, 1 pair
Digital multimeter
Optical profile bench, l = 150 cm
Base f. optical profile bench
Slide mount f. optic. profile bench, h = 80 mm
Stand tube
13900.00
13900.99
13901.00
13902.00
13907.00
07134.00
08281.00
08284.00
08286.00
02060.00
1
1
1
1
1
1
1
2
3
3
Plate holder
Connecting cord, l = 50 cm, red
Connecting cord, l = 50 cm, blue
02062.00
07361.01
07361.04
1
1
1
Tasks
1. Determine and plot graphs of the intensity of the ultrasonic behind different Fresnel zone plates as a function of the
distance behind the plates.
2. Carry out the same measurement series without a plate.
3. Determine the image width at each distance of the transmitter from the zone plate and compare the values
obtained with those theoretically expected.
Set-up and procedure
Set up the experiment as shown in Fig. 1.
Fix the zone plate on the optical bench at the 50 cm mark (it
is to be kept at this position throughout the experiment) and
set the transmitter and the receiver to the same height as the
central axis of the zone plate.
Connect the transmitter to the TR1 diode socket of the ultrasonic unit and operate it in continuous mode “Con“. Connect
the receiver to the left BNC socket (prior to the amplifier).
Fig.1: Experimental set-up
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
21518-00
1
LEP
1.5.18
-00
Diffraction of ultrasound at a Fresnel zone plate /
Fresnel’s zone construction
Connect the signal received to the analog output of the digital
multimeter to have it displayed subsequent to amplification
and rectification. To ensure proportionality between the input
signal and the analog output signal, avoid operating the amplifier in the saturation range. Should such a case occur and the
“OVL“ diode light up, reduce either the transmitter amplitude
or the input amplification.
To start with, position the transmitter at the end of the optical
bench so that it is at a distance of approx. 95 cm from the
zone plate. To determine the ultrasonic intensity behind the
zone plate, move the receiver away from it in steps of
0.5-1.0 cm and measure the receiver voltage U at each step.
Repeat this measuring procedure for different distances
between the zone plate and the transmitter (see Fig. 3).
Subsequently carry out measurements as above but without
the zone plate (comparison measurements, see Fig. 4) and
then with the negative zone plate.
Note:
To avoid measurement field interference, the person carrying
out the experiment should not stand too close to the measurement area when taking readings.
Theory and Evaluation
A plane wave of wavelength l falls vertically on a plane S (see
Fig. 2). According to Huygens’ principle, each point on the
plane can be a source of a new spherical wave. A Fresnel zone
plate is situated in the plane S. This plate is divided into ring
segments which act alternatively as gaps (transmit) or obstacles (impervious), each of which have not only the same area
but also boundary radii which are so designed that marginal
rays on their way to point F have a path length difference of
l/2 .
Because of the equal areas of the Fresnel zones, each wave
has an n’th zone wave from the neighbouring (n ± 1)-zone with
a path length difference of l/2. Such waves cancel each other
out.
A zone plate stops the destructively interfering wave range
and allows only waves having a phase difference of n · 2p
through, so that constructive interference is given at point F,
and an increase in intensity can be observed. A Fresnel zone
plate acts as a converging lens for a certain wavelength l.
From equation (1), the focal length of it is given by:
1 2
l
4
n·l
r2n n2
f
(2)
In this experiment, however, the zone plate is not illuminated
by a plane wave, but by a spherical wave whose source is at
a distance d from the zone plate.
On using the image equation here, analogously to in geometric optics,
1
1
1
b
g
f
(3)
which gives for the image width b:
1 2
l b
4
b
1
n · l · g r2n n2 · l2
4
g a r2n n2 ·
(4)
The zone radii (starting with r1 = 3 cm to r10 = 10.3 cm; see
equation 1) are so designed that the zone plates have a focal
point at f = 10 cm for a wavelength of l = 0.88 cm. As the
transmitter transmits at a frequency of f = 40 kHz, it follows in
the present case that the length of the ultrasonic wave is: l =
0.86 cm (c = l · f mit c = 343.4 ms-1 bei T = 20°C).
For the wavelength l = 0.86 cm and the given radii rn of the
zone plates, these have a focal length of f 10.2 cm.
In Fig. 3, the receiver voltage U is plotted as a function of the
distance d behind the zone plate for various distances of the
ultrasonic transmitter from the positive zone plate (object distances). For a positive zone plate, the central region is
opaque.
Table 1 lists image widths b for different object distances g
that were calculated using equation (4) and determined from
Fig. 3.
Table 1: Calculated and experimentally found image widths.
Fig. 2: Diagram of Fresnel zone beams
g / cm
The following is valid for the radii of the 1st and n’th zone:
r21 a f l 2
l2
b f2 f · l 2
4
(1)
r2n a f n
2
21518-00
l 2
l2
b f2 n · f · l n2
2
4
b
theo.
/ cm
b
exp.
/ cm
95
11.5
13
50
12.9
16
30
15.6
21
Fig. 3 shows that the zone plate focusses. This finding can be
further illustrated by the comparison measurement (Fig. 4) in
which the intensity was measured in the same d range as in
Fig. 3, but without the zone plate.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
1.5.18
-00
Diffraction of ultrasound at a Fresnel zone plate /
Fresnel’s zone construction
3a
Fig. 4: Ultrasonic intensity in the d region without zone plate
3b
For large distances g of the transmitter from the zone plate,
the experimentally image widths b are in approximate agreement with the calculated values. When the object distance is
reduced, however, then lens errors come into effect which
lead, for example, to a broadening of the focussing curve.
Fig. 5 shows the focussing action of the negative zone plate,
the central region of which is transparent.
A comparison with Fig. 3a shows that the negative and positive zone plates with identical radii have the same imaging
effects.
3c
Fig. 3: Ultrasonic intensity as a function of the distance d
behind a positive Fresnel zone plate (plate centre with
transmission T = 0). Transmitter-zone plate distance =
g = object distance: 3a: g = 95 cm; 3b: g = 50 cm; 3c:
g = 30 cm.
Fig. 5: Ultrasonic intensity as a function of the distance d
behind a negative Fresnel zone plate (plate centre with
transmission T = 1). Transmitter-zone plate distance:
g = 95 cm.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
21518-00
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1.5.18
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Diffraction of ultrasound at a Fresnel zone plate /
Fresnel’s zone construction
21518-00
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen