R
LEP
2.1.02
Laws of lenses and optical instruments
Related topics
Law of lenses, magnification, focal length, object distance, telescope, microscope, path of a ray, convex lens, concave lens,
real image, virtual image.
Principle and task
The focal lengths of unknown lenses are determined by measuring the distances of image and object and by Bessel’s method. Simple optical instruments are then constructed with
these lenses.
Equipment
Lens, mounted, f + 20 mm
Lens, mounted, f + 50 mm
Lens, mounted, f +100 mm
Lens, mounted, f +300 mm
Lens, mounted, f – 50 mm
Lens, mounted, f –200 mm
Screen, translucent, 2503250 mm
Screen, with arrow slit
Ground glass screen, 5035032 mm
Double condenser, f 60 mm
Stage micrometer, 1 mm - 100 div.
Dog flea, Ctenocephalus, mip
Slide -Emperor MaximilianOptical profile-bench, l 1000 mm
Base f. opt.profile-bench, adjust.
Slide mount f. opt.pr.-bench, h 30 mm
Slide mount f. opt.pr.-bench, h 80 mm
Diaphragm holder
08018.01
08020.01
08021.01
08023.01
08026.01
08028.01
08064.00
08133.01
08136.01
08137.00
62046.00
87337.10
82140.00
08282.00
08284.00
08286.01
08286.02
08040.00
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
5
1
2
Lens holder
Condenser holder
Swinging arm
Experiment lamp 5, with stem
Power supply 0-12V DC/6V, 12V AC
Connecting cord, 500 mm, blue
Rule, plastic, 200 mm
08012.00
08015.00
08256.00
11601.10
13505.93
07361.04
09937.01
2
1
1
1
1
2
1
Problems
1. To determine the focal length of two unknown convex lenses by measuring the distances of image and object.
2. To determine the focal length of a convex lens and of a
combination of a convex and a concave lens using Bessel’s
method.
3. To construct the following optical instruments:
1. Slide projector; image scale to be determined
2. Microscope; magnification to be determined
3. Kepler-type telescope
4. Galileo’s telescope (opera glasses).
Set-up and procedure
The experiment is set up as shown in Fig. 1. A parallel light
beam is produced with the lamp and the double condenser.
1. The object (screen with arrow slit) is directly behind the condenser, and a clear image is projected on to the screen with a
lens. The distances of image and object from the lens are
measured (assume that the lenses are thin).
Fig. 1: Experimental set-up (microscope).
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R
LEP
2.1.02
Laws of lenses and optical instruments
The measurement of distances of image and object is repeated, using both lenses and with the lens and the screen in different positions.
2. If, at a fixed distance d between object and image (case I),
we alter the position of the lens so that the image and object
distances are transposed (case II), we still obtain a clear
image of the object. In case I the image is magnified, in case
II it is reduced (Fig. 2).
3.2 Microscope
A magnifield image of a small object (stage micrometer and
micro-slide of a dog flea) is produced with a lens L1 of short
focal length ƒ1 = +20 mm. The real intermediate image is
observed through an eyepiece L2 (ƒ2 = +50 mm) (Fig. 4). The
ground glass and the object holder with the object are fixed in
the swinging arm. L1 is brought as close to the object as possible. The object is illuminated through a ground glass screen.
The size of the image and thence the overall magnification are
roughly determined by comparing it with a scale at the least
distance of distinct vision (approximately 25 cm). To do this
we look through the microscope with the right eye and at the
scale with the left. With practice the two images can be superimposed.
Fig. 2: Determination of focal length after Bessel.
Using a convex lens of focal length +100 mm, for instance,
measure the distance e at which a sharp image is obtained for
both possible lens positions (repeat the measurement and calculate the average value e–. Now take a measurement in the
same way but using the convex lens from the first measurement and a concave lens (- 200 mm for example). Make the
distance d as large as possible, and measure at least four
times the combined forcal length.
3.1. Slide projector
Place the slide – Emperor Maximilian – immediately behind the
condenser and project an image on the screen with the lens L2
(ƒ2 = +100 mm).
Fig. 4: Path of a ray in the microscope.
3.3 Telescope after Kepler
A convex lens of long focal length ƒ1 (+300 mm, for example),
and one of short focal length ƒ2 (e. g. +50 mm) are secured to
the optical bench at a distance of ƒ1 + ƒ2 (Fig. 5).
To obtain the best image illumination set the condenser so
that the image of the lamp coil is in the plane of objective lens
L2 (Fig. 3).
Determine the magnification V of the image
V=
B
G
Fig. 5: Path of a ray in a Kepler telescope.
If we look through the lens of short focal length, we can see
an inverted, magnified image of a distant object.
3.4 Galileo telescope (opera glasses)
A convex lens of long focal length ƒ1 (+300 mm, for example)
and a concave lens of short focal length ƒ2 (e. g. - 50 mm) are
set up at a distance of ƒ1 – | ƒ2 -| (Fig. 6).
Fig. 3: Path of a ray in a slide projector.
2
22102
Through the concave lens we can see distant object magnified
and the right way up.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
LEP
2.1.02
Laws of lenses and optical instruments
Substituting into the lens formula gives
d2 – e2
4d
ƒ=
The focal length of the convex lens can therefore be determined from the measured values of d and e.
Fig. 6: Path of a ray in Galileo telescope.
Theory and evaluation
The relationship between the focal length ƒ of a lens, the
object distance g and the image distance b is obtained from
geometrical optics. Three particular rays, the focal ray, the
parallel ray and the central ray, are used to construct the
image (Fig. 7).
If we now use a lens system of focal length ƒcomb. consisting
of the convex lens already measured (focal length ƒs ) and a
concave lens, and carry out the measurement in the same
way, we obtain the following for the focal length of the concave lens ƒz :
1
1
1
ƒ
·ƒ
5
– or ƒz 5 comb. s
ƒz ƒcomb. ƒs
ƒs – ƒcomb.
Here we assume that
1
|ƒs|
>
1
|ƒz|
as otherwise no real images would be produced. ƒ2 was
99.7 mm for the convex lens (+100 mm), ƒcomb. was 180 mm
for the combination of two lenses (+100 mm/- 200 mm) so that
ƒz = - 223 mm represents the focal length of the concave lens.
(The combination of two lenses involves a systematic error as
the distance between the principal planes is disregarded).
3.1 The magnification is obtained from the relationship
between object size and image size
Fig. 7: Image construction with three principal rays.
V5
b–ƒ
B
5
G
ƒ
When the image distance b is 700 mm and the focal length
ƒ = 100 mm, then V = 6.
From the laws of similar triangles,
B
b
G
ƒ
5 and 5
G g
B
b–ƒ
where B is the image size and G is the object size.
By transforming we obtain the lens formula
1
1
1
b·g
5 1 or ƒ 5
ƒ b g
b1g
1. From the values of b and g measured in Problem 1 we calculate ƒ, the average value of ƒ and its standard deviation. For
the first lens (100 mm) ƒ1 was 100.2 mm with a standard deviation sf1 of 0.6 mm; for the second (50 mm), ƒ2 was 53.1 mm
with a standard deviation sf2 of 0.9 mm. (The focal lengths
marked on the lenses have a tolerance of ± 5 %.)
2. Since gI = bII (the object distance in case I = image distance
in case II) and since bI = gII ,
gI + bI = d
gI + bI = e
(see Fig. 2).
If we solve the equations for gI and bI we obtain
1
(e + d)
2
1
bI = (d – e)
2
gI =
3.2 The overall magnification is obtained by multiplying the
magnification due to the objective (Fig. 4),
bobjective =
Y’ a’ a’
5 5 –1
Y
g
ƒ1
by the angular magnification of the eyepice
GL =
250 mm
ƒ2
With the lenses used we obtain an overall magnification V = 60.
3.3 The objective L1 provides a real, inverted image of size Y’1
of a very distant object, and this image is observed through
the eyepiece L2. The angular magnification (for small angles) is
GL =
«9 Y91 /ƒ2 ƒ1
5 9
5
«
Y1 /ƒ1 ƒ2
3.4 A concave lens is placed in the path of the ray in front of
the real first image produced by objective L1 so that the focal
points F19 and F2 coincide. The eye then sees a virtual, upright
image. The magnification is once again
GL =
ƒ1
|ƒ2|
Note
The markings on the lenses used to measure focal length
should be removed and replaced by a code, e. g. coloured
dots or etched lines, known only to the instructor.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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R
LEP
2.1.01
Measuring the velocity of light
Related topics
Refractive index, wavelength, frequency, phase, modulation,
electric field constant, magnetic field constant.
Principle and task
The intensity of the light is modulated and the phase relationship of the transmitter and receiver signal compared. The
velocity of light is calculated from the relationship between the
changes in the phase and the light path.
Equipment
Light velocity measuring app.
Screened cable, BNC, l 1500 mm
Oscilloscope, 20 MHz, 2 channels
Block, synthetic resin
11224.93
07542.12
11454.93
06870.00
1
2
1
1
Problems
1. To determine the velocity of light in air.
2. To determine the velocity of light in water and synthetic
resin and to calculate the refractive indices.
Set-up and procedure
The deviating mirror and the lenses are set up in such a way
that the incident and emergent light rays are parallel to the
base plate (Fig. 1) and a maximum signal reaches the receiving diode (detailed directions can be found in the operating
instructions).
The modulation frequency of 50.1 MHz (quartz stabilised) is
reduced, to the approximately 50 kHz so that the transmitter
and receiver signals can be displayed on the oscilloscope.
1. First of all, the mirror is placed as close to the operating unit
as possible (zero point on the scale). A Lissajous figure appears
on the oscilloscope (XY-operation) and is transformed into a
straight line using the ‘phase’ knob on the operating unit. The
mirror is then slid along the graduated scale until the phase has
changed by p, i.e. until a straight line sloping in the opposite
direction is obtained.
The mirror displacement Dx is measured; the measurement
should be repeated several times.
2. The water-filled tube or the synthetic resin block is placed in
the path of the ray so that its end faces are perpendicular to the
optic axis; the mirror is placed directly behind them (top of
Fig. 3). A supporting block can be used with the resin block so
that the light passes through it in both directions.
A straight line is obtained on the oscilloscope again with the
‘phase’ knob. The medium is then taken out of the path of the
Fig. 1: Experimental set-up for measuring the velocity of light in synthetic resin.
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2.1.01
Measuring the velocity of light
Fig. 2: Diagram of the experimental set-up for measuring the velocity of light in air.
rays and the mirror moved until the Lissajous figure agains
shows the same phase difference.
The mirror displacement Dx is measured several times.
m0 = 1.257 · 10–6 H
m
is the magnetic field constant,
e the relative permittivity of the medium and m its permeability.
Theory and evaluation
The velocity of light is obtained as follows from Maxwell’s
equations:
c =
1
(1)
The refractive index of a medium is the quotient of the light
velocity in a vacuum and in the medium.
n = AFFFF
e·m
(2)
m = 1 for most transpartent substances.
where e0 = 8.854 · 10–12 F
m
Relative permittivity and refractive index are dependent of frequency (dispersion) because of the natural vibration of atoms
and molecules. Red light (LED) is used in the experiment. The
phase relationship between transmitter and receiver signal is
represented by a Lissajous figure on the oscilloscope. If it is a
straight line, the phase difference is 0 in the case of a positive
slope and p in the case of a negative one.
is the electric field constant,
1. In order to measure the velocity of light in air, the light path
is extended by
D l = 2 · Dx
(Fig. 2), to produce a phase change of p: i.e. to travel this
distance the light requires a time
Dt = 1
2ƒ
where ƒ = 50.1 MHz, the modulation frequency.
The velocity of light in air is thus expressed by
cL = D l = 4ƒ · D x
Dt
(3)
Fig. 3: Measuring the velocity of light in other media.
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R
LEP
2.1.01
Measuring the velocity of light
In water, the distance measured lm = 1 m, so that the term
The average of 10 measurements was:
cL = (2.98 ± 0.01) · 108
m
s
k ·
cL
.6·k
ƒ · lm
Value taken from literature:
cL = 2.998 · 108
In synthetic resin, for a distance m of 30 cm, the term
m
s
k ·
2. The velocity of light in water or synthetic resin, cM, is measured by comparing it with the velocity of light in air cL (Fig. 3).
In the first measurement (with the medium), the light travels a
distance l1 in time t1.
l1 = 2x1
t1
cL
. 20 · k
ƒ · lm
From the expected magnitude for the refractive index we can
deduce that
k = 0, therefore t1 = t2
(5)
The measurements in water give
1
1
=
(l – l ) +
l
cL 1 m
cM m
nH2O = 1.335 ± 0.002
In the second measurement (no medium), the light travels a
distance
cH2O = (2.23 ± 0.01) · 108 m
s
Values from literature:
l2 = l1 + 2Dx
nH2O = 1.333
in time
t1 =
1
( l + 2Dx)
cL 1
cH2O = 2.248 · 108 m
s
The phase relationship between transmitter and receiver signal is the same in both cases, so that
t1 = t2 +
n synthetic = 1.597 ± 0.003
k
; k = 0, 1, 2…
ƒ
resin
c synthetic = (1.87 ± 0.01) · 108 m
s
resin
We thus obtain the refractive index
n =
cL
k · cL
= 2 · Dx + 1 + +
cM
ƒ · lm
lm
For the synthetic resin block we obtain the following:
(4)
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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LEP
2.2.01
Interference of light
Related topics
Wavelength, phase, Fresnel biprism, Fresnel mirror, virtual
light source.
Principle and task
By dividing up the wave-front of a beam of light at the Fresnel
mirror and the Fresnel biprism, interference is produced. The
wavelength is determined from the interference patterns.
Equipment
Fresnel biprism
Prism table with holder
Fresnel mirror
Lens, mounted, f + 20 mm
Lens, mounted, f +300 mm, achrom.
Lens holder
Swinging arm
Slide mount f. opt. pr.-bench, h 30 mm
Slide mount f. opt. pr.-bench, h 80 mm
Optical profile-bench, l 1000 mm
Base f. opt. profile-bench, adjust.
Laser, He-Ne 1.0 mw, 220 V AC
Measuring tape, l = 2 m
08556.00
08254.00
08560.00
08018.01
08025.01
08012.00
08256.00
08286.01
08286.02
08282.00
08284.00
08181.93
09936.00
Problems
Determination of the wavelength of light by interference
1. with Fresnel mirror,
2. with Fresnel biprism.
1
1
1
1
1
2
1
2
2
1
2
1
1
Set-up and procedure
The experimental set up for producing interference with the
Fresnel mirror is as shown in Fig. 1. The laser (2 cm), the lens
holder and lens of focal length ƒ = 20 mm (23.3 cm) and a
mount with Fresnel mirror (43.2 cm) are mounted on the optical bench. A light surface at a distance of about 2 to 5 m is
used as a screen.
Before starting the experiment, the movable part of the
Fresnel mirror is adjusted so that the two halves of the mirror
are approximately parallel.
The mirror surface is now aligned parallel to the optical bench.
The laser is adjusted that the enlarged beam of rays strikes
both halves of the mirror equally.
Two light spots, separated by a dark zone, should now be visible on the screen. By turning the adjusting screws of the
Fresnel mirror the movable part of the mirror is tilted until
these zones overlap. The visible interference pattern and its
relationship to the angle of inclination of the mirrors are observed on the screen. The pattern should look like that given in
Fig. 4.
The experimental set up with the biprism is similar as shown
in Fig. 1 (right). The optical bench carries, in addition to the
laser and the first lens, a slide mount with a prism table and
the biprism (45 cm), and a lens mount with a lens of focal
length 300 mm (approx. 60 cm). The widened beam strikes the
central edge of the biprism. With the aid of the lens at 60 cm,
Caution: Never look directly into a non attenuated laser beam
Fig. 1: Experimental set-up for producing interference with the Fresnel mirror.
PHYWE series of publications • Laboratroy Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22201
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2.2.01
Interference of light
the two virtual light sources project an image on to a bright
surface about 3 m away. The distances between the two
points of light, the image-forming lens and the image, and the
object distance – lens 1 to lens 2 minus the focal length of lens
1 – are measured. If lens 2 is removed, and interference pattern is observed. The distance between m succesive interference bands is measured.
Theory and evaluation
If light of wavelength l from two lumilous points whose phase
difference is constant (coherence) falls on a point P, then the
two beams of light interfere.
If the two vector amplitudes for propagation in the x direction
are represented by:
and
d = 2 r sin a.
If the distance a between the screen and the mirrors is large
compared with the distance between two adjacent interfer
ence maxima, the following applies approximately:
r2 = r1 = a
pd
r2 – r1 =
a
since
(r2 – r1) (r2 + r1) = 2pd .
The phase difference d is thus
si = ai e i(Z/ l–di ) ,
where di represents the phases, the individual intensities being
given by
d = 2p
r2 – r1
=
l
2p pd
.
la
According to equation (1), maxima occur on the screen for
distances p equal to:
Ii = si · s*i ,
the superimposition gives
p = n·
I = I1 +I2 + 2 EIBB
1 I2 cos d
la
d
, n = 0, 1, 2, …
(2)
(1)
and minima for
where
p =
d = d1 – d2 .
According to equation (1), I possesses maxima and minima as
a function of the phase difference d. In the case of the Fresnel
mirror a wave from the light source Q falls on to two mirrors
inclined at an angle a. The interference pattern is observed on
the screen S. The mirror with light source can be replaced by
two coherent light sources Q1 and Q2, separated by a distance d.
If r is the distance between Q and the point A at which the mirrors are touching, then from Fig. 2:
AQ1 = AQ2 = r
1n + 2 2
1
·
la
d
, n = 0, 1, 2, …
(3)
The distance d between the two virtual light sources is determined by projecting a sharp image of them on the screen,
using a lens of focal length ƒ and measuring the size of the
image B:
1
1
1
+
=
g
b
ƒ
g
d
=
b
B
where g and b represent the object-to-lens and the image-tolens distance respectively.
d =
B·ƒ
b–ƒ
(4)
(See experiment 2.1.2 “Law of lenses and optical instruments”)
Fig. 2: Geometrical arrangement, using the Fresnel mirror.
2
22201
Fig. 3: Geometrical set up, using the Fresnel biprism.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
LEP
2.2.01
Interference of light
Fig. 4: Interference pattern of the Fresnel mirror.
In the case of the Fresnel biprism the distance d is determined
exactly as for Fresnel mirror, using equation (4).
Equation (3), similarly, applies for the distance p between the
interference bands if the effect of the refractive index and the
thickness of the prism are neglected. Using equations (4), (3)
and (2), the value for l was determined as
l = 624.0 nm.
Literature value: 632.8 nm
From this, and from equation (2) or (3), l was determined as
the mean of various measurements, using different angles of
inclination of the mirror.
n = 1 and formula (2)
p =
la
d
l =
dp
a
d =
B·ƒ
b–ƒ
or
with
a is the distance between two neighbouring maxima.
l = 626.5 nm .
PHYWE series of publications • Laboratroy Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22201
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R
LEP
2.2.02
Newton’s rings
Related topics
Coherent light, phase relationship, path difference, interference in thin films, Newton’s ring apparatus.
Principle and task
In a Newton’s rings apparatus, monochromatic light interferes
in the thin film of air between the slightly convex lens and a
plane glass plate. The wavelengths are determined from the
radii of the interference rings.
Equipment
Newton rings apparatus
Lens, mounted, f +50 mm
Interference filters, set of 3
Screen, translucent, 2503250 mm
Lamp, f. 50 W Hg high press. lamp
Power supply for Hg CS/50 W lamp
Double condenser, f 60 mm
Lens holder
Slide mount f. opt. pr.-bench, h 30 mm
Slide mount f. opt. pr.-bench, h 80 mm
Optical profile-bench, l 1000 mm
Base f. opt. profile-bench, adjust.
Rule, plastic, 200 mm
08550.00
08020.01
08461.00
08064.00
08144.00
13661.93
08137.00
08012.00
08286.01
08286.02
08282.00
08284.00
09937.01
1
1
1
1
1
1
1
2
4
1
1
2
1
Problems
Using the Newton’s rings apparatus, to measure the diameter
of the rings at different wavelengths and:
1. to determine the wavelengths for a given radius of curvature
of the lens
2. to determine the radius of curvature at given wavelengths.
Set-up and procedure
The Newton’s rings experiment is set up as shown in Fig. 1.
The mercury vapour high-pressure lamp with the double condensator (focal length 60 mm) fitted, the lens holder with the
interference filter, the Newton’s rings apparatus, the lens holder with the lens of focal length 50 mm and a transparent
screen about 40 cm away from the lens are all set up on the
optical bench. At the beginning of the experiment the path of
the rays is adjusted, first without colour filters, until interference rings can be observed on the screen. Then the yellow filter
is inserted in the lens holder and the room darkened. By turning the three adjusting screws on the Newton’s rings apparatus to and fro, the plano-convex lens is set on the plane-parallel glass plate so that the bright centre of the interference
rings is in the mid-point of the millimetre scale projected on
the screen. When making this adjustment, ensure that the lens
Fig. 1: Experimental set-up for determining wavelength using the Newton’s apparatus.
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2.2.02
Newton’s rings
Fig. 2: Creating Newton’s rings.
Fig. 3: Geometry used to determine the thickness d.
and the glass plate only just touch. This is achieved when no
more rings emerge from the ring centre when the adjusting
screws are tightened up.
The radii rn of the interference rings are measured for the
various interference filters at the appropriate ordinals.
For the interference rings of maximum cancellation,
Theory and evaluation
If two phase-locked wave trains of the same frequency and
plane of polarisation (coherent light) overlap, after travelling
different paths, interference occurs. In a limited aperture ange
g, the light of wavelength l leaving a surface of diameter a fulfils the coherency condition
a · sin g << l/2 .
“Newton’s rings” occur through monochromatic light interfering in the thin intermediate film between a convex lens and a
plane glass plate. Ray 1 reflected at the underside of the lens
thus interferes with ray 2 reflected at the top of the glass plate
(Fig. 2).
The film of air at a distance r from the point of contact between the lens and the glass plate has a thickness D = d ± do.
As ideal contact is not present, we must take do into account.
do is positive when, for example, there are particles of dust
between the lens and the glass plate, but is can so also be
negative when the pressure is greater. The geometric path difference d’ of the interfering rays is therefore:
In addition, the ray reflected from the plane glass surface
experiences a phase shift p at the transition from the optically thinner to the optically denser medium. The effect of this
corresponds to a distance travelled of length l/2. In all, therefore, there is an apparent path difference
22202
2 (d ± do) = ln.
(2)
In accordance with Fig. 3, there is a relation
(3)
between the radius rn of the nth dark ring, the thickness d and
the radius of curvature R of the plano-convex lens (in the ideal
case do = 0).
In case of slightly convex lenses, d << R, so that for the dark
rings, using (2) and (3), we have:
–
rn2 = nRl + 2do R
(4)
For the evaluation, rn2 is plotted against n (Fig. 4). At the given
radius of curvature, R = 12.141 m, the wavelength l of the
transmitted light is obtained from the slope of the straight line:
b = R·l
(5)
lyellow = 582 ± 4 nm
lgreen = 545 ± 4 nm
d’ = 2 (d ± do).
2
or
d · (2R – d ) = rn2.
Interference figures whose brightness can differ in places, can
occur.
d = 2 (d ± do) + l/2
d = 2 (d ± do) + l/2 = (n + 1/2) l
(1)
lblue
= 431 ± 4 nm
At a given wavelength l, the value of R obtained from (5) is the
average value of the radius of curvature of the plano-convex
lens:
R = 12.13 m
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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2.2.02
Newton’s rings
Fig. 4: Radius of the interference rings as a function of the
order number for various wavelengths.
Note
In the set-up described, the Newton’s rings are observed in
transmitted light. The interference rings are complementary to
those in reflected light. In the latter case, therefore, the light
rings are counted and not the dark ones.
PHYWE series of publications • Laboratroy Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22202
3
R
LEP
2.2.04
Fresnel’s zone construction / Zone plate
Related topics
Huygens-Fresnel principle, Fresnel and Fraunhofer diffraction,
interference, coherence, Fresnel zone construction, zone plates.
2. The focal points of several orders of the zone plate are projected on a ground glass screen. The focal lengths to be
determined are plotted against the reciprocal value of their
order.
3. The radii of the zone plate are calculated.
Principle and task
A zone plate is illuminated with parallel laser light. The focal
points of several orders of the zone plate are projected on a
ground glass screen.
Equipment
Laser, He-Ne 1.0 mW, 220 V AC
Zone plate, after fresnel
Lens holder
Lens, mounted, f + 20 mm
Lens, mounted, f + 50 mm
Lens, mounted, f +100 mm
Lens, mounted, f – 50 mm
Object holder, 535 cm
Ground glass screen, 5035032 mm
Polarising filter, 50 3 50 mm
Optical profile-bench, l 1000 mm
Base f. opt. profile-bench, adjust.
Slide mount f. opt. pr.-bench, h 30 mm
08181.93
08577.03
08012.00
08018.01
08020.01
08021.01
08026.01
08041.00
08136.01
08613.00
08282.00
08284.00
08286.01
1
1
4
1
1
1
1
2
1
1
1
2
7
Problems
1. The laser beam must be widened so that the zone plate is
well illuminated. It must be assured that the laser light beam
runs parallel over several meters.
Set-up and procedure
Fig.1 shows the complete experimental set-up. The slide
mount for the laser is placed at the head of the optical bench.
To start with, the laser beam is widened with lenses L1, L2 and
L3 to a diameter of approx. 5 mm (cf. Fig. 2). Careful shifting
of lenses L2 and L3 allows to make the laser beam parallel over
a length of several meters (maximum 10 m). The correct
values for the different focal lengths of the zone plate can only
be obtained under this condition. For this purpose, a piece of
black cardboard, into which a hole is punched with a desk
punch and which is used as a test diaphragm, proves useful.
The other components should then be mounted, making sure
the zone plate is well illuminated. The image of the zone plate
is observed on the ground glass plate, which is located nearly at the end of the optical bench at the beginning, with
magnifying lens L4. Moving the ground glass screen and L4 in
the direction of the zone plate simultaneously, the different
focal points of the zone plate are searched for and the corresponding focal lengths are determined. The polarising filter,
which is used to reduce the brightness of the image, is set
together with the ground glass screen in the same mounting
frame.
Caution: Never look directly into a non attenuated laser beam
Fig. 1: Experimental set-up to determine the different focal lengths of a zone plate.
PHYWE series of publications • Laboraroty Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22204
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LEP
2.2.04
Fresnel’s zone construction / Zone plate
Fig. 2: Position of
f (L1) = +
f (L2) = –
f (L3) = +
f (L4) = +
the optical components.
20mm
ZP = zone plate
50mm
S = ground glass screen
100mm
P = Polarisation filter
50mm
Assuming the distance between P and the centre of the zone
plate to be f1, in case of constructive interference at P, the following holds for radii rn (n=1,2,3 ...):
2
2
r2n= 1f + n l2 – f 2 = f 2 + nf l + n2 l = nf l;
2
4
2
2 l
with nf l >> n
4
(2)
For the radii rn of the zone plate and the focal length f we thus
have:
1
(3)
rn = (nfl)1/2; f = r2n ·
nl
If the point of observation P is shifted along OP towards the
zone plate, alternating brightness and darkness are observed,
which means that the zone plate has several focal points.
fm = f1/m (m = 1, 3, 5, 7, ...)
Theory and evaluation
According to Fresnel, interference of waves diffracted by
obstacles may be treated simply by splitting the primary wave
front into so called zones. The optical path difference from the
common boundaries of a zone pair up to a point of observation P is always l/2. Secondary waves originating from neighbouring zones impinge in P with opposed phases, thus extinguishing each other except for a part coming from the first
zone.
Using a so called zone plate, which consists of alternating
transparent and opaque circles, it is possible to let either the
odd or the even zones exert an influence at a point of observation P.
If the number of zones is 2k, the amplitude A at point P is
(under the justified assumption that the secondary waves
have the same amplitude, due to the fact that the areas of the
single zones are equal):
A = A1 + A3 + A5 + ... + A2k-1; A ' kA1
(1)
At the point of observation P, the amplitude A without zone
plate is 1/2 A1 (contribution of half of the first zone). Using a
zone plate, it is thus possible to increase light intensity at P by
a factor of 4k2. This means that the zone plate acts as a focusing lens.
In Fig.3, the first rings of a zone plate illuminated by a plane
wave (parallel beam) are shown.
Fig. 3: Geometry of the zone plate.
2
22204
(4)
The existence of these focal points of higher order is due to
the difference in the optical path of the zone rays of 3/2l,
5/2l, 7/2l, 9/2l ...
The zone plate used for the experiment has 20 zones, the
radius of the first dark central circle is r1 = 0.6 mm. The following radii are found to be:
rn = n1/2 · 0.6 mm
(5)
In Table 1, the averaged experimental values are compared to
the values calculated according to (3), (4) and (5) and with l =
632.8 nm. Fig 4 shows the empirical focal lengths as a function of the inverse value of the order of the focal points.
m
f(theor.)/cm
f(exp.)/cm
n
r(theor.)/mm r(exp.)/mm
1
56.9
57.8
1
0.60
0.61
3
19.0
19.3
2
0.85
0.86
5
11.4
11.7
3
1.04
1.05
7
8.1
8.5
4
1.20
1.21
9
6.3
6.6
5
1.34
1.35
Fig. 4: Focal length of first and higher order of the zone plate
as a function of the reciprocal value of the order.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
LEP
2.3.01
Diffraction at a slit and Heisenberg’s uncertainty principle
Related topics
Diffraction, diffraction uncertainty, Kirchhoff’s diffraction formula, measurement accuracy, uncertainty of location, uncertainty
of momentum, wave-particle dualism, de Broglie equation.
Principle and task
The distribution of intensity in the Fraunhofer diffraction pattern of a slit is measured. The results are evaluated both from
the wave pattern viewpoint, by comparison with Kirchhoff’s
diffraction formula, and from the quantum mechanics standpoint to confirm Heisenberg’s uncertainty principle.
Equipment
Laser, He-Ne 1.0 mW, 220 V AC
Diaphragm, 3 single slits
Diaphragm, 4 double slits
Diaphragm, 4 multiple slits
Diaphragm holder
Photoelement f. opt. base plt.
Slide mount, lateral. adjust., cal.
PEK carbon resistor 1 W 5 % 2.2 kOhm
Multi-range meter A
Universal measuring amplifier
Optical profile-bench, l 1500 mm
Base f. opt. profile-bench, adjust.
Slide mount f. opt. pr.-bench, h 80 mm
Connecting cord, 500 mm, red
Connecting cord, 500 mm, blue
08181.93
08522.00
08523.00
08526.00
08040.00
08734.00
08082.03
39104.23
07028.01
13626.93
08281.00
08284.00
08286.02
07361.01
07361.04
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
Problems
1. To measure the intensity distribution of the Fraunhofer diffraction pattern of a single slit (e. g. 0.1 mm).
The heights of the maxima and the positions of the maxima
and minima are calculated according to Kirchhoff’s diffraction formula and compared with the measured values.
2. To calculate the uncertainty of momentum from the diffraction patterns of single slits of differing widths and to confirm
Heisenberg’s uncertainty principle.
Set-up and procedure
Different screens with slits (0.1 mm, 0.2 mm and 0.05 mm) are
placed in the laser beam one after the other. The distribution
of the intesnity in the diffraction pattern is measured with the
photo-cell as far behind the slit as possible. A slit (0.3 mm
wide) is fitted in front of the photocell. The voltage drop at the
resistor attached parallel to the imput of the universal measuring amplifier is measured and is approximately proportional to
the intensity of the incident light.
Important: In order to ensure that the intensity of the light
from the laser is constant, the laser should be switched on
about half an hour before the experiment is due to start. The
measurements should be taken in a darkened room or in constant natural light. If this is not possible, a longish tube about
4 cm in diameter and blackened on the inside (such as a cardboard tube used to protect postal packages) can be placed in
front of the photcell.
Caution: Never look directly into a non attenuated laser beam
Fig. 1: Experimental set-up for measuring the distribution of intensity in diffraction patterns.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22301
1
R
LEP
2.3.01
Diffraction at a slit and Heisenberg’s uncertainty principle
Fig. 2: Diffraction (Fraunhofer) at great distance (Sp = aperture
or slit, S = screen).
Fig. 3: Intensity in the diffraction pattern of a 0.1 mm wide slit
at a distance of 1140 mm. The photocurrent is plotted
as a function of the position.
The principal maximum, and the first secondary maximum on
one side, of the symmetrical diffraction pattern of a slit 0.1 mm
wide (for example) are recorded. For the other slits, it is sufficient to record the two minima to the right and left of the principal maximum, in order to determine a (Fig. 2).
Theory and evaluation
1. Ovservation from the wave pattern viewpoint
When a parallel, monochromatic and coherent light beam of
wave-length l passes through a single slit of width d, a diffraction pattern with a principal maximum and several secondary
maxima appears on the screen (Fig. 2).
The intensity, as a function of the angle of deviation a, in
accordance with Kirchhoff’s diffraction formula, is
I (a) = I (0) ·
(sinb b)
2
(1)
where
b=
pd
· sin a
l
The intensity minima are at
an = arc sin n ·
l
d
where n = 1, 2, 3 …
The angle for the intensity maxima are
a’0 = 0
l
a’1 = arc sin 1.430 ·
d
l
a’2 = arc sin 2.459 ·
d
The measured values (Fig. 3) are compared with those calculated.
Minima
Measurement
Calculation
a1 = 0.36°
a1 = 0.36°
a2 = 0.72°
a2 = 0.72°
a3 = 1.04°
a3 = 1.07°
Maxima
a1 = 0.52°
a1 = 0.51°
a2 = 0.88°
a2 = 0.88°
I (a1)
= 0.044;
I (0)
I (a1)
= 0.047
I (0)
I (a2)
= 0.014;
I (0)
I (a2)
= 0.017
I (0)
The relative heights of the secondary maxima are:
I (a1) = 0.0472 · I (0)
I (a2) = 0.0165 · I (0)
2
22301
Kirchhoff’s diffraction formula is thus confirmed within the limits of error.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
LEP
2.3.01
Diffraction at a slit and Heisenberg’s uncertainty principle
2. Quantum mechanics treatment
The Heisenberg uncertainty principle states that two canonically conjugate quantities such as position and momentum
cannot be determined accurately at the same time.
The angle a1 of the first minimum is thus
sin a1 =
l
d
(8)
according to (1).
Let us consider, for example, a totality of photons whose residence probability is described by the function ƒy and whose
momentum by the function ƒp. The uncertainty of location y
and of momentum p are defined by the standard deviations as
follows:
Dy · Dp ^
h
4p
(2)
where h = 6.6262 × 10-34 Js, Planck’s constant (“constant of
action”), the equals sign applying to variables with a Gaussian
distribution.
For a photon train passing through a slit of width d, the
expression is
Dy = d
(3)
Whereas the photons in front of the slit move only in the direction perpendicular to the plane of the slit (x-direction), after
passing through the slit they have also a component in the ydirection.
The probability density for the velocity component ny is given
by the intensity distribution in the diffraction pattern. We use
the first minimum to define the uncertainty of velocity (Figs. 2
and 4).
Dny = c · sin a1
(4)
If we substitute (8) in (7) and (3) we obtain the uncertainty relationship
Dy = Dpy = h
(9)
If the slit width Dy is smaller, the first minimum of the diffraction pattern occurs at larger angles a1.
In our experiment the angle a1 is obtained from the position of
the first minimum (Fig. 4a):
tan a1 =
a
b
(10)
If we substitute (10) in (7) we obtain
h
a
sin arc tan
l
b
(
Dpy =
)
Substituting (3) and (11) in (9) gives
d
a
sin arc tan
=1
l
b
(
)
Dpy = m · c · sin a1
(5)
where m is the mass of the photon and c is the velocity of light.
(12)
after dividing by h.
The results of the measurements confirm (12) within the limits
of error.
where a1 = angle of the first minimum.
The uncertainty of momentum is therefore
(11)
Width of slit*
d/mm
0.101
0.202
0.051
First minim
a/mm
b/mm
7.25
1140
3.25
1031
10.8
830
a
d
sin arc tan
b
l
(
)
1.01
1.01
1.05
* The widths of the slits were measured under the microscope.
The momentum and wavelength of a particle are linked
through the de Broglie relationship:
h
=p=m·c
l
(6)
Thus,
Dpy =
h
sin a1
l
(7)
Fig. 4: Geometry of diffraction at a single slit
a) path covered
b) velocity compoment of a photon
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22301
3
R
LEP
2.3.02
Diffraction of light at a slit and an edge
Related topics
Intensity, Fresnel integrals, Fraunhofer diffraction.
Principle and task
Monochromatic light is incident on a slit or an edge. The intensity distribution of the diffraction pattern is determined.
Equipment
Laser, He-Ne 1.0 mw, 220 V AC
Photocell, selenium, on stem
Multirange meter with amplifier
Dry cell, 1.5 V
Lens holder
Lens, mounted, f -50 mm
Slit, adjustable
Screen, metal, 3003300 mm
Diaphragm with slit
Tripod base -PASSBarrel base -PASSMeter scale, demo. l = 1000 mm
Bench clamp, -PASSMeasuring tape, l = 2 m
Connecting cord, 750 mm, red
Connecting cord, 750 mm, blue
08181.93
06750.00
07034.00
11620.34
08012.00
08026.01
08049.00
08062.00
09816.02
02002.55
02006.55
03001.00
02010.00
09936.00
07362.01
07362.04
1
1
1
6
1
1
1
1
1
1
3
1
1
1
1
1
Problems
1. Measurement of the width of a given slit.
2. Measurement of the intensity distribution of the diffraction
pattern of the slit and
3. of the edge.
Set-up and procedure
The experimental set up is as shown in Fig. 1. The divergent
lens of focal length – 50 mm is placed in front of the laser to
expand the beam. An inner edge of the slit which is fully open
serves as the edge. The distance between lens and slit is
75 mm. The laser power is adjusted to 1 mW.
For diffraction at the slit the laser beam is directed symmetrically onto the vertical closed slit edges. The metal screen with
the tape scale stuck to the middle, is set up at a certain distance (e. g. 3 m).
The slit is opened and the slit width is calculated from
b=
2m 1 1
·l
2 · sin am
where
sin am =
xm
! x2m 1 r2
b = slit width
m = serial order of the maximum from the centre
outwards
xm = distance of the mth maximum
r = distance between the slit and the screen
l = wavelength of the laser light
To ensure glare-free reading at the screen it is necessary to
cover up the intensely bright centre of the pattern (e. g. with a
pencil in a barrel base).
Caution: Never look directly into a non attenuated laser beam
Fig. 1: Experimental set-up for the diffraction of light at a slit and an edge.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22302
1
R
LEP
2.3.02
Diffraction of light at a slit and an edge
For diffraction at the edge, the screen with a single slit (vertical) is stuck to the photocell with adhesive tape. The metre
scale, on which a barrel base with the photocell can be moved
at right angles to the laser beam, is secured at a certain distance (e. g. 3 m). The photocell is connected to the multirangemeter with amplifier (mV = nA).
Fig. 3: Intensity distribution on diffraction at the slit, as a function of the position along a straight line parallel to the
plane of the slit, standardised on the intensity without
the slit.
First of all, the intensity I0 is measured without the edge –
initially without the laser (dark value) and then with it (light
value). These values must be taken into account in the evaluation.
The edge (an edge of the slit) is moved into the laser beam so
that half of it is masked. This requires some care. In some circumstances, an intensity measurement can be carried out
more rapidly with the slit screen lying horizontally. In this case
the edge is moved into the beam until only half the voltage is
recorded.
Theory and evaluation
If light of wavelength l falls onto a slit of width b, each point
of the slit acts as the starting point of a new spherical wave.
The diffraction pattern is formed on a screen behind the slit as
a result of the interference of these new waves.
If this diffraction is treated according to the Fraunhofer
approximation, the intensity at point P on a screen parallel to
the slit, using the symbols of Fig. 2, is:
I=c·
(
2
pb
sin u
l
pb
sin u
l
sin
)
x=n·
al
b
(1)
c is a constant which depends on the wavelength and the
geometry. Intensity maxima occur for
tan
where a >> x, the minima are approximately equidistand, and
pb
pb
sin u =
sin u .
l
l
The first maximum is thus obtained for u = 0. The following
maxima occur if the argument of the tangent assumes the values:
If light falls on to a slit formed by a straight edge (parallel to
the y axis), it is diffracted. If the origin of coordinates is placed
at the intersection of the connecting line PQ between the light
source and the point of incidence with the plane of the diffraction screen, the intensity distribution of the diffraction pattern
behind the diffracting edge is
I5
I0
2
2
2
( (U(v) 1 21 ) 1 (V(v) 1 21 ) )
(2)
1.43 p, 2.459 p, 3.47 p, 4.479 p, …
Intensity minima occur when
pb
sin u = n p; n = 1, 2, …
l
Fig. 2: Diffraction at the slit.
2
22302
Fig. 4: Diffraction at the edge.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
LEP
2.3.02
Diffraction of light at a slit and an edge
Using the symbols of Fig. 4 we have
I0 =
1
(R0 1 R)2
v = x · cos d
(3)
!2l (R1 1 R1 )
Fig. 5: Intensity distribution on diffraction at the edge, as a
function of the position on a straight line at right angles
to the line connecting the light source and the edge,
standardised on the intensity without the edge.
(4)
0
U and V are the Fresnel integrals, defined as follows:
v
p 2
cos
n dn.
U(v) =
2
E
( )
E
p
sin n2 dn.
cos
2
0
v
V(v) =
0
( )
The intensity on the shadow side decreases regularly. On the
light side the intensity exhibits maxima and minima, while the
total intensity according to (3) decreases quadratically with the
distance between the light source and the point of incidence.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22302
3
R
LEP
2.6.03
Recording and reconstruction of holograms
Related topics
Dispersion, reflection, object beam, reference beam, real and virtual image, volume hologram, Lippmann-Bragg hologram,
Bragg reflection.
Principle and task
The laser beam is divided into an object beam (illuminating
beam) and a reference beam by using a beam splitter. The
expanded illuminating beam is diffusely reflected by the surface of the object and then interferes with the reference beam
in the plane of the hologram.
A laser light hologram is recorded and reconstructed by using
the same laser light as for recording. In the second part of the
experiment the object beam and the reference beam strike the
hologram plate from opposite sides. This results in an interference pattern which is structured in the depth of the lightsensitive emulsion and forms semi-transparent silver layers or
layers of different refractive index when the hologram is processed. For the purposes of reconstruction, the hologram can
be illuminated with white light and viewed in reflection. The
virtual image is due to Bragg reflection on the layers, and is
monochromatic.
A white light reflection hologram is recorded by using laser
light and reconstructed by using a point white light source at
distance.
Equipment
Optical base plate w. rubber ft.
Laser, He-Ne 0.2/1.0 mW, 220 V AC *
Magnetic foot f. opt.base plt.
Sliding device, horizontal
08700.00
08180.93
08710.00
08713.00
1
1
6
1
xy shifting device
Adapter ring device
Objective 253 N.A. 0.45
Pin hole 30 micron
Adjusting support 35335 mm
Surface mirror, large, d = 80 mm
Surface mirror 30330 mm
Beam splitter 1/1, non polarizing
Holder f. diaphr./beam splitter
Object for holography
Darkroom equipment for holography
Set of photographic chemicals
Holographic plates, 20 pieces **
Potassium dichromate
250 g
Sulphuric acid, 95-98%
500 ml
08714.00
08714.01
62470.00
08743.00
08711.00
08712.00
08711.01
08741.00
08719.00
08749.00
08747.88
08746.88
08746.00
30102.25
30219.50
2
1
1
1
2
1
2
1
2
1
1
1
1
1
1
* Alternative to 1 mW Laser:
He/Ne Laser, 5mW with holder
Power supply f. laser head 5 mW
08701.00
08702.93
1
1
** Alternative to holographic plates:
Holographic sheet film
Glass plate, 120x120x2 mm
08746.01
64819.00
1
2
Problems
1. Record a laser light hologram and process it to get a phase
hologram. Reconstruct it by verifying the virtual and the real
image.
2. Record a white light reflection hologram and process it to
get a phase hologram. Laminate it for reconstruction by a
white light source.
Fig. 1: Experimental set-up for recording a transmission hologram.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22603
1
R
LEP
2.6.03
Recording and reconstruction of holograms
Transmission hologram
– Perform the experimental set-up according to Fig. 7 The
beam path height is 13 cm. If the 1-mW laser is used, the
SH switch [3,8] (magnetic catch, shutter) cannot be employed.
– Since this set-up is a two beam arrangement in which the
reference and the object beams follow different paths after
passing through the beam splitter BS [1,2], particular care
must be taken to ensure the set-up’s mechanical stability.
– Allow the laser to warm up for approximately one hour before beginning the experiment in order to avoid oscillations
in the wavelength.
– The laser beam (initially without the E25x expansion system
[1,4] is adjusted with the mirrors M1 [1,8] and M2 [1,1] in
such a manner that the object O [10,5] is well illuminated.
Then insert the beam splitter BS [1,2] (metallized side
toward mirror M1), which splits the laser beam into the reference (R) and object beams (O). Half of the object beam,
which has already been adjusted, passes through the splitter; whereas the other fraction is deflected to the large mirror M3 [4.5, 6.5]. This large mirror is now adjusted in such a
manner that the beam strikes the centre of the hologram
plate H [10,3] (or the holographic sheet film between the
glass plates) at the height of the beam path.
– Now move the E25x expansion system [1,4] without the
objective and the pinhole diaphragm, but with the adjusting
diaphragms, into position. Align it in such a manner that the
beam passes unimpeded through the adjusting diaphragms. Now replace these diaphragms with the objective
and the pinhole diaphragm. Move the pinhole diaphragm
toward the focal point of the objective. lnitially ensure that a
maximum of diffuse light is incident on the apertured diaphragm and subsequently, concider the expanded beam.
Successively shift the positions of the objective and the pinhole diaphragm laterally while approaching the focal point in
order that an expanded beam without diffraction phenomena is subsequently provided.
– If sheet film is used, the film is tightly pressed between two
glass plates in the plate holder. To avoid undesirable interference phenomena and multiple reflection between the
glass plates, it is advisable to press the upper edges of the
glass plates together with an additional clamp or clip.
Before exposing the film, wait approximately 1 to 2 minutes
until the pressure and temperature equilibration between
the sheet film and the glass plates has taken place.
– The photosensitive layer faces the object during the imagecapture process.
– lmmediately after removing a piece of film or a plate from
the storage box, reclose the box.
– The exposure time for a 1-mW laser is 20 to 30 seconds.
With a 5-mW laser this is reduced to approximately 5
seconds.
– The exposure is performed as follows:
With the 5-mW laser the time for the magnetic shutter SH
can be set manually on the power supply.
The light beam of the 1-mW laser must be blocked off with
a piece of (blackened) card-board. At the beginning of the
exposure period this piece of cardboard is removed and
replaced at the end of it without jarring the laser (the time is
measured with a stopwatch). Since the exposure time is
very long, the incidence of diffuse light from the laser on the
hologram should be avoided.
2
22603
– The development and bleaching of the phase hologram is
performed according to the procedure given below.
– The hologram can be reconstructed according to Fig. 7 if
the object beam is blocked directly behind the mirror M2
with blackened cardboard.
Attention
Never look directly into the laser beam! The He/Ne laser has a
power of 5 mW (or 1 mW) and can cause permanent damage
to the retina. When tracing the path of the beam, only use
absorbing or strongly diffusing materials.
Development and bleaching
Only phase holograms are dealt with in the following, because
of their better appearance in reconstruction. Therefore the
following agents have to be prepared before to process the
photographic plates after exposure.
1) Developer
Mix 100 ml of holographic developer with 400 ml of deionized water and keep the mixutre ready in one of the plastic
dishes.
2) Stop-bath
Mix 12 ml of stop-bath solution with 468 ml of deionized
water and keep the mixture ready in a second plastic dish.
3) Bleaching
Dissolve 5 g of potassium dichromate in 1000 ml of deionized water. Add 5 ml of concentrated sulphuric acid. Keep
the solution ready in a third plastic dish.
Caution: Never pour water into a vessel containing sulphuric acid.
4) Rinsing
A fourth plastic dish is filled with deionized water to rinse
the bleached holograms.
The photographic plate is processed by developing it for
two minutes. After a stop-bath of 30 seconds the plate is
bleached for two minutes. Finally it is rinsed for about five
minutes and dried.
Avoid all skin contact while working with the developing and
bleaching agents. Always wear the recommended gloves.
Before recording holograms, clean all the optical components
by using lens-cleaning paper and acetone. Make sure that the
base plate is in a rather vibration-free position.
Only work in green light in the laboratory.
Theory and evaluation
In normal photography only the two-dimensional projection of
a recorded three-dimensional object is obtained.
Holography supplies a truly three-dimensional image of the
object when reconstruction is carried out. This can be done by
storing the three-dimensional wave field emitted by the object
using a coherent reference wave (coherent background) (Fig. 2).
The object wave and reference wave produce an interference
pattern at the position of the hologram which is stored as optical density (amplitude hologram) or as a change in refractive
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Recording and reconstruction of holograms
Fig. 2: The principle of holography:
interference of object wave O with coherent reference
wave R. H is the hologram plate.
a) Recording
b) Reconstruction
Fig. 4: Reconstruction of the hologram recorded in accordance
with Fig. 3.
Intensity I is obtained by time averaging (denoted by < >) and
apart from a constant factor is:
index (phase hologram). When the developed hologram is illuminated by the same reference wave, the reconstructed
object wave as was previously emitted from the actual object
appears behind the hologram. The observer therefore sees the
image where the object was previously situated. In qualitative
terms this phenomenon can be explained as follows: the
spherical wave emanating from a point on the object interferes with the (in the simplest case plane) reference wave (Fig. 3).
The recorded interference pattern consists of concentric
circles (Fresnel zone plate).
I = < E · E* >
(2)
A number of simplifying assumptions are made for the following calculations in order to reduce the calculating effort required and to emphasize the results which are of essential
importance for holography. The following initial assumptions
are firstly made:
– The time-dependent factor eivt is equal for all waves and is
therefore omitted, and it is also left out of the calculation of
intensity as a result of averaging (v = 2 pf, f = frequency).
When reconstruction is carried out (Fig. 4), a plane wave is diffracted at the Fresnel zone plate. This creates, in addition to
the light which passes through (0 = order diffraction), a divergent spherical wave (1st order diffraction) and a convergent
spherical wave (-1st order diffraction). Besides the virtual
image at the original position of the object, there is also a real
image on the other side of the hologram.
– The hologram is regarded as an “area hologram” (i. e.
thickness of photographic layer << length of light wave). It
is regarded as being in the plane z = 0.
For the quantitative treatment of holography, the light source
is described by a complex function.
Recording of holograms
Object wave O and reference wave R overlap in the plane of
the hologram (z = 0) and supply a position-dependent interference pattern with intensity I.
E (x, y, z, t) = E0 (x, y, z)ei w(x,y,z,t)
(1)
The real part of this complex function is the electrical vector of
the light wave.
A far more complicated calculation for “volume holograms”
leads to similar results.
O = OO(x, y) eic (x,y)
(3)
R = RO(x, y) ei w (x,y)
(4)
I = (O + R) · (O + R)*
= OO* + RR* + OR* + RO*
= IO + IR + OOROei (c– w) + OOROei (-c + w)
(5)
The developed hologram then has a (complex) positiondependent transmittance:
t (x,y)
= T(x,y) ei w (x,y)
(6)
If the phase w (x,y) = const., we speak of an amplitude hologram (optical density vixed after developing). On the other
hand, if T(x,y) = const. (by bleaching the hologram) we have a
phase hologram.
Fig. 3: Overlapping of a spherical wave emanating from O and
a plane reference wave R.
The transmittance is dependent on energy density W, the product of light intensity I and exposure time tB (Fig. 5).
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Recording and reconstruction of holograms
Fig. 5: Amplitude and phase transmittance in relation to energy density W for amplitude and phase holograms.
Fig. 6: Position of image points with oblique reference wave.
The exposure time and also the ratio of the intensities of the
object and reference waves should be chosen so that the transmittance is in the linear range of the characteristic shown in Fig.
5, e.g. between W1 and W2 respectively between W3 and W4.
The first term essentially reproduces the reference wave,
slightly modified by IO.
The second term describes the object wave, i. e. it appears to
the observer as if the object were still at the same point as for
the recording. IR is constant and the image is therefore undistorted provided – R a plane wave.
The third term supplies a real image, known as a cunjugate
image, because O* has the negative phase -w of O.
A divergent light beam becomes convergent.
We therefore assume for the transmission of the holograms:
– the intensity of reference wave IR and the exposure time are
chosen so that the transmittance is in the linear range of the
characteristic.
– IO << IR for the intensity of object wave IO, so that intensity
modulation remains in the linear range of the characteristic.
Under these simplifying conditions we obtains for an amplitude hologram (w = const.):
t (x,y) = a TO + b I (x,y)
a, b = const.
(7)
and for phase hologram (T = const.):
t (x,y)
a, b
= a · ei w (x,y)
< a ( 1 + iwO + i b I (x,y))
= const.
(8)
The real image
The position of the real image is studied in greater detail in the
following example. Let the reference wave be a plane wave
which hits the hologram at an angle b The object is at angle a
when the recording is made (Fig. 6).
For small angles a and b (a, b << 90 °) the virtual image is in
mirror symmetry to the real image in relation to the dash-dot
line which is vertical to the direction of the reference wave.
The following relationship is obtained by calculation:
Reference wave in the hologram plane (z = 0)
R = RO ei k x sin b
the series
(10)
Object wave in the hologram plane
ei w =
o
n
in n
w
n!
O = O' e
for IO << IR being interrupted after the first term. The relationship between t und I is also approximately linear for phase
holograms. The factor i means that the reconstruction wave
additionally undergoes a constant phase shift on passing
through the hologram.
i k x sin a
(11)
where O' = object wave at angle a = 0.
In reconstruction we obtain for the conjugate image
R2 · O* = R2O · O'* · e
i k x (2 sin b – sin a)
(12)
it therefore appears at angle g with
Reconstruction of a hologram
For the purpose of reconstruction the hologram is generally
illuminated again by reference wave R. The wave front appearing behind the hologram (according to (7) and (5)) contains the
following:
H=
t·R
(9)
= (aTO + bIO + bIR) · R
+ b IR · O
reference wave
sin g = 2 sin b – sin a
(13)
No real image exists at particular angles a and b. If
2 sin b – sin a > 1
(14)
there is no solution for g. This is already the case at a = 0° for
angles of the reference beam of b > 30 °.
object wave (virtual image)
+ b R2 · O*conjugate object wave (real image)
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Recording and reconstruction of holograms
Fig. 7: Setup for recording and reconstruction of a transmission hologram (* only required for 5 mW laser).
Fig. 8: Setup for recording a white light reflection hologram
(* only required for 5 mW laser).
Reconstruction of hologram with R*
If it is the real image which is of particular interest in reconstruction, e. g. in producing hologram copies of a master hologram, the hologram is reconstructed with R*, i. e. the hologram
plate is illuminated from precisely the opposite direction.
Instead of (9) we then obtain:
Recording a white light reflection hologram
The experiment is setup as shown in Fig. 8. The photographic
plate (or film) is fixed on the holder and put into the position
shown in Fig. 8 [10, 3]. The object is positioned directly behind
the photographic plate. The beam expanding system E25x is
introduced and adjusted as discribed before. The exposure
time is 10 to 20 seconds for the 1 mW laser and approximately 2 seconds if the 5 mW laser is used.
After processing and drying a dark laminate is applied to the
emulsion layer of the photographic plate with the squeezing
roller.
For the reconstruction of the hologram it is sufficient to have
a point white light source emitting a parallel beam, e. g. a halogen spot light at 1 m distance or more or even sunlight. The
best image quality is achieved if the hologram is illuminated
from the same side and at the same angle as the reference
beam during recording. From the continious white light spectrum only one wavelength (here the red light of the He/Ne
laser) is filtered out for the reconstruction while all the others
are removed. If the emulsion layer has shrunk, the colour of
the reconstructed image will shift towards shorter wavelengths while expansion will cause a shift towards longer
wavelengths.
H=
t R*
(15)
= (aTO + bIO + bIR) R*
+ b R2 O
+ bIR O*
The observer looks in the direction in which the virtual image
appears in reconstruction with R. The real image is at the corresponding position in front of the hologram plate.
For reconstruction the processed photographic plate is
brought back into the same position as for recording and illuminated with the same laser light. The object is eliminated and
the object beam is blocked directly behind the mirror M2 with
blackened cardboard. For the observer in front of the photographic plate a clear virtual image is created in the position
where the object has proviously been. Turning the photographic plate by 180 ° allows the observer to see the real image
in front of the plate (see theory for reconstruction with R*).
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CO2-laser
Related topics
Molecular vibration, exitation of molecular vibration, electric
discharge, spontaneous emission, vibration niveau, rotation
niveau, inversion, induced emission, spectrum of emission,
polarization, Brewster angle, optical resonator.
Equipment
CO2-laser tube, detachable, 8 Watt
Modul-box for CO2-laser tube
Set of laser mirrors, ZnSe and Si
Opt. bench on steel rail l 1.3 m
HV-power supply \5 kV/50 mA DC
Ballast resistor unit
Cooling water unit, portable
Vacuum pump, two-stage
08596.00
08597.00
08598.00
08599.00
08600.93
08601.00
08602.93
02751.93
1
1
1
1
1
1
1
1
Gas filter/buffer unit
He/Ne-laser/adjusting device
Screen/diaphragm f. adj. CO2-laser
Powermeter 30 mW/10 Watt
Support for power probe
Protecting glasses, 10.6 micro-m
Cleaning set for laser
ZnSe biconvex lens, d 24 mm, f150 mm
Digit.thermom., NiCr-Ni
HV-isolated temperature probe
Control panel w. support, 1 gas*
Pressure contr. valve 200/3 bar*
Laser gas in bottle, 50 l/200 bar*
08605.00
08607.93
08608.00
08579.93
08580.00
08581.00
08582.00
08609.00
08583.00
08584.00
08606.00
08604.00
08603.00
1
1
2
1
1
1
1
1
1
1
1
1
1
* Alternative to:
Laser gas mixing unit, 3 gases
08606.88
1
Fig. 1: Experimental set-up of the CO2-laser system with laser gas mixing unit (Output power: max. 8 W CW).
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CO2-laser
Principle and task
Among molecular lasers, the CO2-laser is of greatest practical
importance. The high level of efficiency with which laser radiation
can be generated in continuous wave (cw) and pulse operation
is its most fascinating feature.
Fig. 1 a: CO2-laser tube in tri-point laser tube holder.
If a gold coated diffraction grating (with, e.g., 700 lines per cm)
and a diaphragm are used, it is possible to select higher energy
laser lines and to use them for spectroscopic purposes. If this
diffraction grating is mounted in such a manner that it can be
moved, then a CO2-laser can be tuned to a band width of 9.2 mm
to 10.8 mm. CO2-lasers of this type are used for spectroscopic
analyses in the intermediate IR region (RAMAN and FIR spectroscopy, environmental analytics). For, e.g. measurement of
atmospheric pollutants the CO2-laser is employed on the basis
of photoacustic spectroscopiy or that of the LIDAR principle.
In the field of medicine this laser is used e.g. in surgery as a laser
scalpel, in the ablative treatment of skin areas and as a source
of light for the heat treatment of deep-lying tissue layers (e.g.
neurostimulation).
As a result of the good thermal coupling of CO2-laser radiation
in high-temperature plasmas, this laser is used for cutting, welding and surface-hardening processes in the metal technology
(power range: 100 W to 50 kW).
The experimental equipment set is an open CO2-didactic laser
system of max. 8 W power output. Since it is an “open” system,
all components of the system can be handled individually and
the influence of each procedure on the output power can be
studied. One very primary and essential target in learning is the
alignment of the CO2-laser by means of a He/Ne-laser.
Problems
1. Align the CO2-laser and optimize its power output.
2. Check the influence of the Brewster windows position on
the power output.
3. Determine the power output as a function of the electric
power input and gasflow.
4. Evaluate the efficiency as a function of the electric power
input and gasflow.
5. If the gas-mixing unit is supplied the influence of the different components of the laser gas (CO2, He, N2) to the output efficiency of the CO2-laser are analyzed.
6. Measurement of temperatures differences for the laser gas
(imput / output) for study of conversion efficiency.
Fig. 2: The CO2-laser system, schematic.
1
2
3
4
5
6
7
2
Laser tube
Cooling jacket
Lasing medium
Cooling water
Brewster window
Brewster window mount
Si-plano mirror
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9
10
11
12
13
14
Zn/Se plano/concave mirror
Electrodes, positive
Electrode, mass
HV-DC power supply
Ballast resistor
Laser gas in bottle
Flow regulating valve
15
16
17
18
19
20
21
Flowmeter
Water inlet
Water outlet
Four-digit vacuum gauge
Oil-vapor trap
Woulff’s bottle
Vacuum pump
22
One-way sto-cock
23
Oil-vapor trap
24/25 Power meter
26
Gas supply (1 or 3 gases)
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CO2-laser
Set-up and procedure
The experimental set-up is shown in Fig. 1. To assemble the
complete functional CO2-laser system, the following components are needed (see Fig. 1 a and Fig. 2).
a) The laser tube
– It channels the lasering media, e. g. the CO2-laser gas,
along the laser light propagation. A typical CO2-laser gas
mxiture consists of 4.5 % CO2, 13.5 % N2 and 82 % He.
– It provides three electrodes for DC pumping of the laser gas
from a high voltage source. The source is both current and
voltage-stabilized.
– Its waer jacket allows the two electrodes to cool (for safe
operation) and the gas to be excited.
– The inner tube that carries the gas is extended out at either
side of the outer jacket to facilitate the mounting of
Brewster windows to linearly polarize the output beam.
b) Hardware
– Tri-point laser tube holder. This device holds the glass laser
tube firmly and is surrounded by an acrylene-covered steelhousing mounted on an optical bench by means of two
80 mm wide slide mounts. The optical bench is fixed to the
rigid surface of a U-shaped steel rail. The steel housing
bears the fittings for the inlet and outlet of water and gas as
well as safety sockets for the HV-supply.
– Adjustable mirror mounts. Two are needed. The mounts
provide micrometer screws for fine adjustment on two axes
and can be fixed to the optical bench by slide mounts of a
width of 100 mm. The stability of the mirror mounts will
directly affect the stability of the systems alignment.
– Brewster window mounts. Brewster windows polarize the
output of the laser linearly. Reflection losses caused by
windows placed on the tube ends at an angle of incidence
other than the Brewster angle would prevent lasering.
– Optical bench. This bench is designed to carry the laser
tube in its tube holder, the two mirror mounts and whatever components necessary external to the laser cavity, such
as power meter, He/Ne-laser for alignment etc. The optical
bench is mounted to a U-shaped heavy steel rail to ensure
sufficient rigidity and stability.
c) Optics
The laser cavity contains one Si-plano mirror and one ZnSeplano/Concave partial reflector, relfecting 95 % of the incoming light at 10.6 mm. The Si-plano mirror has an enhanced silver and dielectric coating whose flatness is 1/10 l for
10.6 mm. Mirror diameter = 25.4 mm, parall. = 3 arc sec.
The ZnSe-plano/concave mirror has the same flatness and
diameter as the Si-plano mirror. Its radius of curvature is 1 m
or 10 m. The coating is dielectric on both sides. Mirror diameter = 25.4 mm, thickness 3 mm, AR coated.
For the Brewster windows ZnSe is the preferred material
because of its high transmission at 10.6 mm. The Brewster
window mounts include sealed ZnSe windows of 2 mm thickness. The angle of inclination of the Brewster windows with
respect to the laser tube axis is 23.6 °. Flatness = l/10, parall. = 1 arc sec.
d) Electronics
A high voltage power supply is required to “pump” the laser tube
by discharging. The current and voltage electronically highly stabilized DC power unit has a nominal output of 50 mA and 5 kV.
Two ballast resistors absorb about 50% of the total supply output power. Pumping under optimal conditions (maximum laser
output), a current of approx. 18 mA at 3.01 kV is observed. The
discharging process within the tube takes place from the outside
electrodes to the central point (mass) along two equal distances.
The power supply unit has a visible power-on indicator.
The ballast resistors serve two functions. When configured as
shown in the system diagram, they split the output of the high
voltage supply and compensate for the negative plasma resistance of the tube during discharge. The ballast resistors
which are incorporated in the supply unit consist of two
100 kΩ resistors with a 150 Watt rating.
e) Vacuum
The typical operating pressure for the CO2 mixture is from 30
to 40 mbar. Therefore, a vacuum pump is required. A rotary
vane pump capable of approximately 67 litres/min which
establishes a maximum final pressure of about 2 mbar suffices and is included in the equipment set. Diaphragm pumps
are not recommended, due to their pulsing flow. Pressure fluctuation will result in a power instability of the laser output.
To prevent oil-droplets or oil vapor from entering the laser
tube, Woulff-bottle and a micro-filter are inserted into the circuit between the vacuum pump and the laser tube. A high precision four-digit vacuum gauge at the outlet of the laser tube
provides precise information about the quality of the vacuum.
Initially, before introducing the laser gas, a vacuum with a final
pressure of equal to or less than 2 mbar has to be established
in the tube. Needless to say, all vacuum fittings and connections have to be established with utmost care using vacuum
grease and stiff plastic tubing.
f) Gas
The CO2-laser gas mixture is supplied in a quantity of 50 l at a
pressure of 200 atm. Consequently, a pressure regulator is
needed to reduce the outlet pressure to one atmosphere
above normal. Before entering the laser tube, the gas passes
a flowmeter with a measuring range up to 1.5 l/min and flowregulating needle valve. The laser works under continuous
flow conditions to match the unavoidable CO2 dissociation.
Alternatively, it is possible to mix an individually variable gas
mixture from the required laser gas components (CO2, N2, He).
An adjustable dose and mix system (No. 08606.01) allows the
production of a laser gas from commercial gases with a technical purity grade.
g) Cooling circuit
The laser tube requires permanent cooling to reach higher
inversion densities. This is ensured by a closed cooling circuit
which consists essentially of a circulating pump, and a 10 litre
tank with dest. water from the cooling jacket. Since relatively
little heat dissipates from the laser tube, the 10 litres of water
do not need to be replaced or artificially cooled down unless
the system is operated for a long period. The water flow rate
approx. (0.5 l/min) is controlled by a flow meter.
h) Power meter
The power output of the laser, e. g. the intensity of the infrared
beam is measured using a power meter whose most sensible
range allows the instantanous detection of a change in power
of 1 mW. The maximum measurable power is 10 Watt with
mounted heat sink.
During operation, the power meter is put on the optical bench
and placed next to the plano/concave ZnSe-partial reflector
outside of the cavity. In this way it acts simultanuously as a
perfect absorber for any outcoming infrared radiation thus fullfilling an important safety requirement.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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CO2-laser
The CO2-laser system should only be assembled and
operated by experienced persons. The operating voltages
are lethal, and its output can burn or cause eye damage in
a fraction of a second. Consequently, the precautions and
safety measures must be observed to ensure safe operation of this device.
After assembling the CO2-laser system as described and after
alignment (see chapter below) you can proceed to fire the
laser.
Firing procedure
1. Turn on coolant water and verify its proper flow
(approx. 0.5 l/min)
2. Close the needle valve on the gas flowmeter and the valve
of the gas filter / buffer unit.
3. Turn on the vacuum pump and allow the system to pump
down to a pressure of approx. 2 mbar.
4. Open the main valve of the gas supply.
5. Adjust the regulator on the tank to 1 bar outlet pressure.
6. Gradually raise the output voltage until both sides of the
tube have fired. This should occur between 3 kV and 4 kV.
Adjust the voltage to the maximum (5 kV) and set the current to 20 mA.
7. Gradually open the needle valve of the flowmeter until the
dial gauge reads approximately 36 mbar (for a gas flow of
approx. 1l/min)
8. Verify that all persons in the area including yourself are
waring the proper safety glasses and that the power meter
is in place.
9. Using heat sensitive paper, such as that used on thermal
printers, check for the presence of the output beam in
front of the power meter. For quantitative statements, the
power meters can be used straight away in its most sensible range (0 to 30 mW).
10. If no lasering is detected, apply gentle hand pressure to
the mirror mounts in the x and y directions or make fine
adjustments on the mount itself while looking for the presence of the output beam.
11. If lasering still does not occur,check the alignment procedure.
Owing to its O-C-O structure, the CO2 molecule can oscillate
in three basic forms:
a) symmetrical vibration where the two oxygen atoms oscillate against each other:
b) flexural vibration where the carbon atom oscillates through
the oxygen atom:
c) asymmetrical vibration where the two oxygen atoms oscillate in the same direction:
Both the modes of vibration and the energy states of the
atoms are quantized. A three-digit number is used to characterize the different energy states of a molecule with three
modes of vibration:
the first digit is the quantum number for the symmetrical vibration. The second digit-represents the flexural vibration and the
third digit indicates the asymmetrical vibration.
A zero is used to indicate the ground state, ascending numbers
signifiy the higher vibrational energy levels (excited states).
Consequently, a 000 for a carbon dioxide molecule indicates
that the molecule is in the ground state. The first excited vibrational energy level with the lowest energy for CO2 is the 010
state i. e. an oscillation of the carbon atom.
The problem with molecules is that they can occur in all combinations of modes of vibration as, for example, the 111 mode
of vibration. However, for the CO2-laser it is sufficient to
The laser can be shut down in the following way:
Shut down procedure
1. Set the power supply output to zero.
2. Shut off the power and gas supply.
3. Shut off the vacuum pump and allow the tube pressure to
reach atmospheric pressure by opening the valve on the
gas filter / buffer unit.
4. Close the needle valve.
5. Shut off the coolant water.
Theory and evaluation
In atom and ion lasers, laser radiation is the result of the electron transistions close to the limit for single or double ionization i. e. far from the electron ground state. The infrared radiation of the CO2-laser, on the other hand, is the result of the
energy exchange between rotational-vibrational levels within
the electron ground level.
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Fig. 3: Energy levels for the modes of vibration of the CO2 and
N2 molecules.
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CO2-laser
Fig. 4: Energy level scheme of the CO2-laser.
understand the ground states and the transitions occuring
there. The fact that every vibrational energy level is split into
many discrete levels through the rotation of the molecule certainly increases the variety of transitions (more than 100 lines)
but is irrelevant for an understanding of how the CO2-laser
works and thus can be ignored here.
Like the CO2 molecular, the nitrogen molecle can also oscillate. However, the N2 molecule can only oscillate symmetrically, thus allowing one-digit characterization.
Fig. 3 shows the energy levels for the modes of vibration of the
CO2 and N2 molecules in part.
Fig. 4 shows the enery level scheme for the CO2-laser. The
actual laser transition occurs at the transition from the asymmetrical 001 to the symmetrical 100 mode of vibration.
The energy difference of this transition corresponds to the
wavelength of 10.6 mm.
Therefore, to attain lasering, the 001 state has to be inverted
compared to the 100 state because stimulated emissions are
only possible with inversion.
Just as with non-ionized atoms, with molecules too it is not
possible to produce inversion directly via electric discharge. A
“détour” via the N2 nitrogen molecule is necessary. As can be
seen in Fig. 4, level 1 of nitrogen and level 001 of the carbon
dioxide correspond in energy so that the energy can be transferred to the carbon dioxide by secondary collisions. Level 1 of
nitrogen is metastable (0.1 sec.) as the transition to the ground
state is forbidden. The resulting long-life of the level contributes considerably to increasing the probability of collisions with
CO2 molecules and to considerable over-population.
The over-population of the CO2 level 001 is also increased by
the transitions from level 002. The helium, which is present in
the laser tube, does not participate in the actual excitation
process; it is only required to aid in the rapid discharge of the
CO2 level 100. Consequently, discharge does not occur in
every case, only through collisions with the wall. Collisions
with omnipresent helium also occur. This makes it possible to
use a large-diameter laser tube for a CO2-laser. The output
power of the CO2-laser is greater, the greater the volume available to participate in the gain.
Due to the fact that the mirrors and the laser tube, in spite of
their rather rigid suspensions, are subject to thermal expansion, the active volume of the laser tube participating in the
gain may change. This results in a change of the laser’s working mode respectively its output power. A slight adjustment of
the mirror positions will bring the laser back into its original
mode.
1. The CO2-laser can be aligned as follows:
A He/Ne-laser of 1 mW and two 1 mm pin-hole screens in
front of the He/Ne-laser and at the end of the optical bench
are used to align the laser tube and the mirrors perfectly – a
prerequisite for oscillation (see Fig. 5).
The He/Ne-laser with the pin-hole screen in front initially
replaces the power meter on the optical bench. The pin-hole of
the pin-hole screen is 95 mm above the top of the optical
bench. The centrer of the Si-plano/concave mirros must also
Fig. 5: Alignment of the CO2-laser tube.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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CO2-laser
be 95 mm above the bench. The ZnSe-plano/concave miror,
the Si-plano mirror and the laser tube with holder and metal
housing are removed from the optical bench. The He/Ne-laser
is now adjusted in its holder till the outcoming beam passes
the pin holes of the first and second pinhole screen. The Siplano mirror is then mounted and adjusted such that the
reflected beam coincides with the incoming beam. Since that
side of the first pinhole screen directed towards the cavity is
white, it is easy to establish whether the reflected beam passes through the pin-hole again, it will definitely fall on the white
part of the screen surrounding the pin-hole. Adjusting the
micrometer screws of the Si-plano mirror holder will ensure
that the reflected beam re-enters the He/Ne-laser.
The ZnSe-plano/concave mirror with its mount is now fixed to
the optical bench directly next to the pin-hole screen. The Siplano and the ZnSe-plano/concave mirros now form a kind of
Fabry-Perrot interferometer and a pattern of dark and bright
rings can appear on the white part of the pinhole screen
around the pin-hole. The ZnSe-mirror is now adjusted with the
micrometer screws on the ZnSe-plano/concave mirror holder
until the beam reflected by its plane surface reenters the
He/Ne-laser. During this procedure the Si-plano mirror should
be covered.
The CO2-laser tube is then re-inserted into the cavity between
the two mirros and fixed on the optical bench. The Brewster
windows are taken off.
The CO2-laser tube in its tri-point holder is adjusted till the
He/Ne-laser beam passes through the center of the tube without touching the walls. Then the Brewster windows are
remounted and the laser tube in its tri-point holder is adjusted
a final time till the beam of the He/Ne-laser passes approximately through the center of the Brewster windows.
If you are satisfied with the positioning of the laser tube,
mount the transparent screen above it. With the Brewster windows positioned as shown in Fig. 5 you should catch the
reflection of the He/Ne beam on the face of the two windows.
By raising and lowering the transparent screen you can locate
the point at which these two reflections cross. Above the center of the laser tube, the two spots should overlap precisely. If
they do not, move one Brewster mount with respect to the
other until they do. If the beam really passes the tube in an
optimal way the two spots should be nearly equal in brightness.
This alignment is just as important as the mirror alignment – if
done carelessly, it can drastically reduce the power output or
prevent lasering.
After this procedure, the He/Ne-laser with the pinhole screen
in front is replaced by the power meter. After establishing the
operating gas pressure in the tube, the laser can now be
“fired” by switching on the electric power. If the alignment procedure has been carried out properly, it will burst into oscillations. It might be necessary to carry out a final adjustment of
the Si-plano mirror. Normally at the very beginning of the
lasering process the output power is still weak. It can be optimized by a final adjustment of the Si-plano mirror and a last
extremely careful adjustment of the tube within its tripoint
holder (ATTENTION: HIGH VOLTAGE!).
2. The influence of the Brewster window’s position on the
ower output can be verified as follows:
The right Brewster window is rotated or tilted around the laser
tube axis in steps of two degrees as indicated in Fig. 6 a and
Fig. 6 b.
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Fig. 6: Rotation and titing of a brewster window.
In this experiment the power supply is switched off briefly. The
power meter is put aside and the He/Ne-Laser switched on.
The reflections of the He/Ne-beam on the faces of the two
Brewster windows are observed and systematically displaced
on the transparent screen above the CO2-laser tube as shown
in Fig. 6 a and Fig. 6 b. A displacement of 1 cm corresponds to
an angle of about 2 degrees.
After each displacement the power meter is brought back into
its original position and the power supply is switched on.
The laser power output as a function of the displacement is
shown in Fig. 7 for rotation and tilting. It is evident that tilting
the Brewster window (equivalent to a wrong choice of the
Brewster angle) is much more detrimental to the power output
than a rotation.
3./4. Power output – efficiency:
In Fig. 8 the laser power output has been plotted versus the
elctric laser power input. The electric power input can be calculated from the product supply-voltage times current minus
the power dissipated by the ballast resistors. The product of
“flow x pressure” was kept constant. It can be seen that the
laser power output initially increases with the increase of the
power input. But after passing a maximum, the power output
decreases with further increase of the power input. The inversion becomes more and more disturbed by the increase in
temperature. By extrapolating curve A it can be seen that a
minimum power input of about 8 Watts (threshold power) is
needed to start the lasering process. Fig. 8 shows also the
efficiency as a function of the power output to power input.
For the working mode chosen the maximum efficiency is
2.3 % at a power output of 0.7 Watts.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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CO2-laser
Fig. 7: Laser power as a function of the angle of inclination of
the brewster window normal N.
Fig. 8: A = Laser power output as a function of laser power
input
B = Efficiency as a function of laser power input.
5. Influence of Laser Gas:
The optically active laser gas can be attained by mixing the
individual gas components (CO2, N2, He).
Advantage: Possibility of study of the reaction and optimization of the gas components on the laser process and production of the laser gas from commercial gases, some with technical purity grades. This allows the operation of a CO2 laser
also in those countries where it is not possible to obtain industrially produced laser gas mixtures.
The following gases are required to produce the laser gas mixture.
CO2, Gas, steel cylinder, 10 liters, purity grade: approx. 99%
N2, Gas, steel cylinder, 10 liters/200 bar, purity grade:
approx. 99%
He Gas, steel cylinder, 10 liters, 200 bar, purity grade:
approx. 99,9%.
The gas components are channeled with an operating pressure of 1 bar via a 2-stage pressure reducer (200 bar/max.
3 bar) to a gas control unit. Three separate flow meters with
adjustable needle vents allow an adjustment of the individual
gas volume flow, independent of the operating pressure of the
laser tube (approx. 30... 50 mbar). With the buffer with built-in
bypass-needle vent found in the filter, a further, individual
variation of the operating pressure of approx. 30 mbar can be
undertaken independent of the adjusted volumetric flow of the
gases.
This allows the determination of the essential laser parameters
depending on the gas mixture, ignition behavior, plasma stability, variation of the flow/pressure values, performance optimization and stability of the laser output power in connection
with the mechanical resonator adjustment.
The Laser Gas mixing unit, 3 gases (08606.88) includes 3 gas
bottles (CO2, N2 and He), the necessary pressure controle valves and the control panel for 3 gases.
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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Handbook:
Physics Demonstration Experiments – Magnet Board Optics
LEP
2.7.01
Geometrical optics and theory of colours on the
magnetic board
The demonstration system presents the following advantages:
● simple handling and minimum preparation time through
components with magnets
● clear length of beams through 50 W halogen lamp with
magnet and large model objects
● clear and dust proof storage of all components in the
device shaped tray
● detailed description of experiments with figures
60 experiments covering light propagation (7), mirror (16),
diffraction (10), lenses (13), colours (6), eye (3), optical
instruments (6)
● Both sides of board can be used for mechanics and optics
● Galvanised sheet steel board in aluminium profile frame
● Mechanics side: lacquered
● Optic side: white foil with lined grid
This HANDBOOK can be purchased separately.
It contains the experiments listed below. Please ask
for a complete equipment list. Ref No 22701
Handbook • Magnet Board Optics • No. 01151.02 •
60 described Experiments
1 Propagation of light
2 Mirrors
Light path through a reversing prism (OT 3.8)
OT 1.1 (11000)
Rectilinear propagation of
light
OT 1.2 (11001)
Shadow formation by a
point light source
OT 1.3 (11002)
Umbra and penumbra with
two point light sources
OT 1.4 (11003)
Umbra and penumbra with
an extensive light source
OT 1.5 (11004)
Length of shadows
OT 1.6 (11005)
Solar and lunar eclipses with
a point light source
OT 1.7 (11006)
Solar and lunar eclipses with
an extensive light source
OT 2.1 (11007)
Reflection of light
OT 2.2 (11008)
The law of reflection
OT 2.8 (11014)
Real images with a concave
mirror
OT 2.14 (11020)
Image formation by a convex mirror
OT 2.3 (11009)
Formation of an image point
by a plane mirror
OT 2.9 (11015)
Law of imagery and magnification of a concave mirror
OT 2.15 (11021)
Law of imagery and magnification of a convex mirror
OT 2.4 (11010)
Image formation by a plane
mirror
OT 2.10 (11016)
Virtual images with a concave mirror
OT 2.16 (11022)
Reflection of light by a parabolic mirror
OT 2.5 (11011)
Applications of reflection by
plane mirrors
OT 2.11 (11017)
Aberrations with a concave
mirror
OT 2.6 (11012)
Reflection of light by a
concave mirror
OT 2.12 (11018)
Reflection of light by a convex mirror
OT 2.7 (11013)
Properties of a concave
mirror
OT 2.13 (11019)
Properties of a convex
mirror
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
3 Refraction
OT 3.1 (11023)
Refraction at the air-glass
boundary
OT 3.2 (11024)
Refraction at the air-water
boundary
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Handbook:
Physics Demonstration Experiments – Magnet Board Optics
OT 3.3 (11025)
The law of refraction
Properties of a convergent
lens
OT 4.12 (11044)´
Spherical aberration
OT 3.4 (11026)
Total reflection at the glassair boundary
OT 4.3 (11035)
Real images with a convergent lens
OT 4.13 (11045)
Chromatic aberration
OT 3.5 (11027)
Total reflection at the waterair boundary
OT 4.4 (11036)
Law of imagery and magnification of a convergent lens
OT 3.6 (11028)
Passage of light through a
planoparallel glass plate
OT 4.5 (11037)
Virtual images with a convergent lens
OT 5.1 (11046)
Colour dispersion with a
prism
OT 3.7 (11029)
Refraction by a prism
OT 4.6 (11038)
Refraction of light at a divergent lens
OT 5.2 (11047)
Non-dispersivity of spectral
colours
OT 4.7 (11039)
Properties of a divergent
lens
OT 5.3 (11048)
Reunification of spectral
colours
OT 7.1 (11055)
The magnifying glass
OT 4.8 (11040)
Image formation by a divergent lens
OT 5.4 (11049)
Complementary colours
OT 7.1 (11056)
The camera
OT 5.5 (11050)
Additive colour mixing
OT 7.3 (11057)
The astronomical telescope
OT 5.6 (11051)
Subtractive colour mixing
OT 7.4 (11058)
The Newtonian reflecting
telescope
OT 3.8 (11030)
Light path through a
reversing prism
OT 3.9 (11031)
Light path through a
deflection prism
OT 3.10 (11032)
Light transmission by total
reflection
OT 4.10 (11042)
Lens combination consisting
of two convergent lenses
4 Lenses
OT 4.1 (11033)
Refraction of light by a convergent lens
OT 4.2
2
OT 4.9 (11041)
Law of imagery and magnification of a divergent lens
(11034)
22701
OT 4.11 (11043)
Lens combination consisting
of a convergent and a divergent lens
5 Colours
6 The human eye
OT 6.1 (11052)
Structure and function of the
human eye
OT 6.2 (11053)
Short-sightedness and its
correction
OT 6.3 (11054)
Long-sightedness and its
correction
7 Optical equipment
OT 7.5 (11059)
Herschel’s reflecting telescope
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
R
Handbook:
Laser Physics I – Experiments with coherent light
LEP
2.7.02
Experimental System Advanced Optics
This experimental system allows all important experiments in
● Geometrical optics
● Wave optics
● Holography
● Interferometry
to be performed.
All experiments are supported by corresponding handbooks,
which contain detailed descriptions of experimental set-ups
and procedures, as well as results of measurements.
By use of a base plate and magnetically held adjustment devices, which can be positioned jerkfree, 1 and 2 dimensional
measuring set-ups with laser light sources can be quickly and
dependably realized. By deflecting the light path, experiments
with larger focal lengths can also be carried out on the base
plate.
The high inherent stiffness and vibration damping of the base
plate enables sensitive interferometer arrangements to be set
up.
This handbook covers the basic experiments in the field of
geometrical optics and wave optics.
This HANDBOOK can be purchased separately.
It contains the experiments listed below. Please ask
for a complete equipment list. Ref No 22702
Handbook • Laser Physics I – Experiments with coherent light • No. 01179.02 • 16 described Experiments
LP 1.1
(12166)
Diffraction of light through a
slit and at an edge.
LP 3.1
(12173)
Fresnel’s law, theory of
reflection
LP 1.2
(12167)
Diffraction through a slit and
Heisenberg’s uncertainty
principle.
LP 3.2
(12174)
Polarisation through l/4
plates
LP 1.3
(12168)
Diffraction of light through a
double slit or by a grid.
LP 1.4
(12169)
Diffraction of light through a
slit and stripes, Babinet’s
theorem
LP 2.1
(12170)
Fresnel mirror and biprism
LP 2.2
(12171)
Michelson interferometer
LP 2.3
(12172)
Newton’s rings
LP 4.3
(12180)
Determination of the index
of refraction of CO2 with
Michelson’s interferometer
LP 5.1
(12181)
Lambert’s law of radiation
LP 3.3
(12175)
Half shadow polarimeter,
rotation of polarisation
through an optically active
medium
LP 3.3
(12176)
Kerr effect
LP 3.5
(12177)
Faraday effect
LP 4.1
(12178)
Index of refraction n of a flint
glass prism
LP 4.2
(12179)
Determination of the index
of refraction of air with
Michelson’s interferometer
Experimental set-up for the qualitative verfication of Lambert’s Law of radiation
(* only required for 5 mW laser)
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
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Handbook:
Laser Physics II – Experiments with coherent light – Holography
Ralph Behrends
HANDBOOK
Laser Physics II
Holography
LEP
2.7.03
Experimental System Advanced Optics
This experimental system allows all important experiments in
● Geometrical optics
● Wave optics
● Holography
● Interferometry
to be performed.
All experiments are supported by corresponding handbooks,
which contain detailed descriptions of experimental set-ups
and procedures, as well as results of measurements.
By use of a base plate and magnetically held adjustment devices, which can be positioned jerkfree, 1 and 2 dimensional
measuring set-ups with laser light sources can be quickly and
dependably realized. By deflecting the light path, experiments
with larger focal lengths can also be carried out on the base
plate.
The high inherent stiffness and vibration damping of the base
plate enables sensitive interferometer arrangements to be set
up.
This handbook covers all experiments related to holography.
This HANDBOOK can be purchased separately.
It contains the experiments listed below. Please ask
for a complete equipment list. Ref No 22703
Handbook • Laser Physics II – Experiments with
coherent light – Holography • No. 01400.02
11 described Experiments
LH 1
(12900)
5mW Version for the experiments – Fresnel zone plate
LH 8
(12907)
Time-averaging procedure I
(with tuning fork).
LH 2
(12901)
White light hologram
LH 9
(12908)
Time-averaging procedure II
(with loudspeaker).
LH 3
(12902)
White light hologram with
expansion system
LH 4
(12903)
Transmission hologram
LH 5
(12904)
Transmission hologram with
expansion system
Transmission hologram (LH 4)
LH 10 (12909)
Real time procedure I
(bending of a plate).
LH 11 (12910)
Real time procedure II
(oscillating plate).
LH 6
(12905)
Transfer hologram from a
master hologram.
LH 7
(12906)
Double exposure procedure
Experimental set-up for real-time procedures as a holographic interferometer
for a bending plate (* only required for 5 mW laser)
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
22703
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Handbook:
Laser Physics III – Experiments with coherent light – Interferometry
Ralph Behrends
HANDBOOK
Laser Physics III
Interferometry
LEP
2.7.04
Experimental System Advanced Optics
This experimental system allows all important experiments in
● Geometrical optics
● Holography
● Wave optics
● Interferometry
to be performed.
All experiments are supported by corresponding handbooks,
which contain detailed descriptions of experimental set-ups
and procedures, as well as results of measurements.
By use of a base plate and magnetically held adjustment devices, which can be positioned jerkfree, 1 and 2 dimensional
measuring set-ups with laser light sources can be quickly and
dependably realized. By deflecting the light path, experiments
with larger focal lengths can also be carried out on the base
plate.
The high inherent stiffness and vibration damping of the base
plate enables sensitive interferometer arrangements to be set
up.
This handbook specially covers interferometer experiments.
This HANDBOOK can be purchased separately.
It contains the experiments listed below. Please ask
for a complete equipment list. Ref No 22704
01401.02
Handbook • Laser Physics III – Experiments with
coherent light – Interferometry • No. 01401.02
17 described Experiments
Michelson Interferometer – High Resolution (LPI 2)
LPI 1
(13066)
Michelson Interferometer
LPI 2
(13067)
Michelson Interferometer –
High Resolution
LPI 3
(13068)
Mach - Zehnder Interferometer
LPI 4
(13069)
Sagnac Interferometer
LPI 5
(13070)
Doppler Effect with
Michelson Interferometer
LPI 6
(13071)
Magnetostriction with
Michelson interferometer
LPI 7
(13072)
Thermal Expansion of Solids
with Michelson Interferometer
LPI 8
(13073)
Refraction Index of CO2-Gas
with Michelson Interferometer
LPI 9
(13074)
Refraction Index of Air with
Michelson Interferometer
LPI 10
(13075)
Refraction Index of Air with
Mach - Zehnder Interferometer
LPI 11
(13076)
Refraction Index of of CO2Gas with Mach Zehnder Interferometer
LPI 12
(13077)
Fabry - Perot Interferometer
– Determination of the
Wavelength of Laserlight
LPI 13
(13078)
Fabry - Perot Interferometer
– Optical Resonator Modes
LPI 14
(22611)
Fourier Optics –
2 f Arrangement
LPI 15
(22612)
Fourier Optics –
4 f Arrangement, Filtering
and Reconstruction
LPI 16
(13079)
Optical Determination of the
Velocity of Ultrasound in
PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany
Liquids – Phase Modulation
of Laserlight by Ultrasonic
Waves
LPI 17
(22605)
LDA – Laser Doppler
Anemometry
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