Figure 11-1 Vibrational tolerances
Vibrations introduce a change of the optical path of the interference
beams that results in a variation of the phase difference between
the object and reference waves and consequently a shift of the
interference fringes over the hologram. The complete analysis
of the effect requires the examination of the particular geometry
of the optical arrangement.
We have from figure 11-1
(11.1)
h = 
(11.2)
(11.3)
The transverse tolerance for the object is the order of
(11.4)
We have also from figure 11-1
(11.5)
(11.6)
(11.7)
(11.8)
and the longitudinal tolerance is given by
(11.9)
Calculation of the objects transverse tolerance when 
= 1/10 and
= 30o :
We have from expression (11.4)
This means that the transverse tolerance for the object is one
tenth of an micrometer.
Calculation of the objects longitudinal tolerance when = 1/10
and = 30o :
We have from expression (11.9)
From these calculations can we see that vibrations of the object,
in this case the LCD, of more than 0.11 
m will make reduction
of the holograms quality.
Analogously, the vibrational tolerances for the holographic film
plate can be found from
(11.10)
and
(11.11)
Calculations of the film plates transverse tolerance when = 1/10
, 
= 0.63 m, and
1 = 2 = 30o.
We have from expression (11.10)
Calculations of the film plates longitudinal tolerance when =
1/10 , = 0.63 m, and
1 = 2 = 30o.
We have from expression (11.11)
From these calculations can we see that vibrations of the film
plate or the holographic printer, for more than 0.05m will make
reduction in the holograms quality.
This kind of analysis extends to the stability of all the components of the optical arrangement with adequate adaptation. It is seen that to use a design angle (see section 5.1.2) much in excess of the resolving power required unnecessarily reduces these tolerances.
These tolerances ensure no loss of the diffraction pattern. However, there is a further aspect of vibrational stability related to the loss of resolving power which is, in principle, comparable to the vibration amplitude. The achievement of high resolution imposes, then, very tight control vibrational stability, in particular, the avoidance of microphonisms on the different components of the experimental arrangements.
In conclusion, a geometry is recommended that meets the following
criteria:
- Records in the near field of the diffraction from the object.
- Uses a plane reference wave.
- The reference and the object beam strike the holographic recording layer before going through the holographic plate base.
- The object is placed as near as possible to the holographic film plate.
- The reference beam has the wider angle of incidence
- The holographic film plate normal approximately bisects the angle between the reference and object beams.
The optical set-up for measuring of vibration stability is shown in figure 11-2.
Figure 11-2 Michelson interferometer used in vibration stability measurement.
In this experiment there have been performed several different
measurements with different sampling rates over various period.36
To alternate the sampling rate and the period, it is possible
to find the data that can give the result which are best suited
for a presentation. There have been made 3 different measurements
of stability shown in the list on the following page.
1: Vibration stability measurement of the equipment shown in figure
11-2. This is done
when all the optical equipment is mounted to the optical
table and the printer and
shutter are not moving.
2: Vibration stability measurement when the glass cage is
mounted on the table. The printer
and shutter are not moving.
3: Vibration stability measurement when the glass cage is mounted on the optical table. The shutter and the printer is activated. This is the same procedure as used during the recording of holographic stereograms.
All of the measured data presented in the plots, is sampled every
500 ms in 250 seconds. In the front of the interference fringes
on the optical set-up, shown in Figure 11-2, there is mounted
a stable slit of 0.5 mm width and 20 mm length. This slit covers
the Laser power meters detector. The slit's extension (length)
must have the same direction as the interference fringes. This
method is a well known method to measure movement of interference
fringes, i.e. vibrations.
Figure 11-3 Measure of vibrations, with and without glass cage.
Figure 11-3 shows the measurement of vibrations of optical equipment with and without the glass cage. The amplitudes of the peak measurement made without glass cage are bigger than the peaks without glass cage. The stability for the measurements made with the glass cage mounted on the optical table is also better. From other measurements this picture is repeated, and the plot above gives a representative picture of the vibration stability. The purpose of the glass cage is to reduce air vibrations produced by the ventilator and air vibration from talking and doors slamming from other rooms in the building.
An experiment has also be carried out with slamming the door and talking in the laboratory, when the interference fringes have been visually observed. The movements of the interference
fringes have a considerably higher amplitude without, than with the glass cage placed on the optical table.
A test of the printers stability during the recording process was done, the set-up shown in figure 11-2 was used. To make holographic multi-stereograms of good quality, the printer (film holder) must be stable during the exposure.
To make the measurement as real as possible the data logging was executed during the recording process.
Figure 11-4 Measurement of vibrations under holographic recording process.
The plot, figure 11-4, shows the vibrations measured during a
recording process, with the glass cage mounted on the optical
table. The measurement time is 250 seconds, where the data logging
starts at t=0, when nothing is in movement on the table.
After 30 seconds the printer motor starts and move the film to
the first exposure position of the multi-stereogram. When the
printer has reach the position, the printer stops and the equipment
is fixed for 30 seconds. At t=110 seconds the shutter opens
to expose the film for 40 seconds, and the shutter then closes.
The equipment is fixed to t=190 seconds before the printer's
motor starts to move the film to the next position. At t=220
seconds the film plate has reached the new position and the motor
stops. Then the above procedure is repeated.
When the data logging started the slit and laser power meters
detector were placed in the area where the light from the laser
has a constructive interference fringe (maximum light power).
When the printer motor starts the interference fringes are moving
all the time. There is a great deal of vibration from the printer,
which can also be seen visually. When the printer reaches the
new position and stops, the interference fringes become stable
for a short time (1-2 seconds). The vibrations made by the printer
quickly die out. This means that the exposure can start 2 seconds
after the printer had stopped and the film plate reaches the new
position, without any vibrations created by the printer's motor.
The plot, figure 11-4, shows that the interference fringes do not stop at the same position as before the printer was started. This has no practical importance to the quality of the recorded hologram.
The vibrations from the shutter's movement (open/close) was impossible to measure. It was not possible to see any movement of the fringes visually either. So the shutter causes no destructive vibrations during the exposure.
A optical set-up with the use of printer, shutter and glass cage should not cause any vibration problems for the recording of holographic multi-stereograms.
The light output from the laser used in this thesis is linearly
polarised. Therefore, the reference beam is always linearly polarised.
But polarisation of the object beam will depend on the nature
of the object surface.
Depolarisation can occur as a result of reflections from metallic and dielectric surfaces. It is also caused by scattering from particles like fog and dust when the laser light goes through them.
The particles of irregular shape and complex refractive index
strongly scatter the light beam in all directions. The result
is a significant depolarisation of the laser beam.
An LCD is used in the holographic recording process in this thesis.
The liquid crystals in the LCD changes the polarising direction.
Therefore, most LCD's have polarizers on both sides of the liquid
crystal panel. The result is that the light passing through such
an LCD is polarised. In holography it is necessary to spread the
light from the LCD. In this thesis ground glass is used to spread
the light from the LCD. The ground glass depolarises the light
with about 20% of the transmitted light (see section 8.1).
The effect of depolarisation is to degrade the reconstructed image's quality with the possibility of a complete loss of information of those parts of the object from where the scattered beam is depolarised.
We let the object beam amplitude be Oe(iøo) and that due to the reference beam be Re(iør), where O and R are corresponding absolute amplitudes. In conventional holographic recording the irradiance at the recording medium is
I = | Oe(iøo) + Re(iør) | 2 (11.12)
= O2 + R2 +2OR cos`
Where ` = o - r is the phase difference
between object and reference beams assuming that both beams are
linearly polarised and parallel to each other. If the object beam
is depolarised after scattering from the object, the equation
over would not be a true representation of the irradiance at the
recording plane. If we describe the scattered object beam as O
= Op + On,
where Op denotes that component of the beam which is
polarised parallel to the reference beam polarisation, and On
denotes the orthogonal component of O.
The degree of polarisation P is defined as
P = Ip /( Ip + In) = 1 / (1
+ 
) (11.13)
Where
= In / Ip (11.14)
The reconstruction efficiency is given by
(11.15)
Where K is a constant and depends on the transfer characteristics
of the recording medium and 
is the reference-to-object beam intensity
ratio.
The dependence of (
/ K) on the degree of polarisation P is shown
in figure 11-6 for different values of . It is observed that the
efficiency of the hologram reduces drastically with a slight decrease
in the value of P.
Figure 11-6 Degree of polarisation
There is a 43 % decrease in when P decreases from 1 to 0.9. When
the degree of polarisation becomes zero, i.e. there is no parallel
polarised component in the object beam, the efficiency falls to
zero.
This spatial filtering is achieved by blocking the higher frequencies associated with a pinhole placed at the focus of a microscope objective so that only the desired smooth intensity profile is transmitted.
Figure 11-7 The principle of a spatial filter.
An objective should be chosen according to the amount of beam expansion required. A pinhole can then be selected to provide the necessary frequency cut off for a given beam diameter. The pinhole size can be determined using this simplified expression:
(11.16)
where
P = Pinhole diameter.
= The lasers wavelength.
f = Focal length of objective lens.
d = Diameter of the laser beam.
Microscope objective, magnification 63 X :
Microscope objective, magnification 40 X :
Some time before the recording process of a hologram the filmplate must be taken out of the refrigerator, and be tempered in room temperature.
When the filmplate is heated, the part of the glass plate where the emulsion is bound will expand. When the glass plate expands the grains in the emulsion will also be drawn in the same directions that the glass plate expands. If this happens during the exposure, the image that created in the hologram may lose information.
Figure 11-8 Expansion of the filmplate
The expansion of the filmplate can only takes place as shown in figure 11-8.
Technical specifications
Lx = 127 mm
Ly = 102 mm
= 8E-6 oK-1
= 0.633 E-6 m
where
Lx = The filmplates length
Ly = The filmplates height
= Coefficient of thermal expansion
= The lasers wavelength
Lx = Thermal expansion in x-direction
Ly = Thermal expansion in y-direction
T = Temperature difference in oK
The formula for thermal expansion is generally given by
L = *Lo*T (11.17a)
The thermal expansion of the glass plate in x and y
direction can be written
Lx = *Lx*T (11.17b)
and
Ly = *Ly*T (11.17c)
We know that vibrations and movement of the filmplate, in the range of more than 10 % of the wavelength will create a reduction of the quality of the hologram image. This means that a movement of the filmplate for more than 0.06310-6 meter will disturb the hologram.
If we illuminate the entire holographic filmplate and look for maximal brightness of the image, there is no possibility of high temperature differences during the exposure.
Calculation of temperature limit for thermal expansion in x and y directions.
Lx = 8E-6 oK-1 . 0.127m . T
0.06310-6 m = 1.01610-6 moK-1 . T
T = 0.06 oK
Lx = 8E-6 oK-1 . 0.102m . T
0.06310-6 m = 0.81610-6 moK-1 . T
T = 0.08 oK
From the calculations above we can see that a temperature variation
over 0.06 oK reduce the brightness of the hologram.
During the production of multi-stereograms with horizontal parallax, there is no great problem of thermal expansion in x-directions. This is because the light wave illuminates only
1-2 mm width of the hologram. Even if the slit is only 1-2 mm in width, the slit's height is the same as the filmplate's, so the temperature variation should not be higher than 0.08 oK during the exposure when we use the holographic printer.
