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Description  |
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This invention relates to a method for producing a synthetic hologram from
a wavefront being defined by geometrical points in space and by the phase
and amplitude of the points.
BACKGROUND OF THE INVENTION
Since holography was first described by D. Gabor--A New Microscope
Principle, Nature, V.161 (1948), p. 777--the development within this field
has proceeded quickly, in particular following the introduction of the
laser for reconstruction, see E. N. Leith and J. Upatnieks: Reconstructed
Wavefronts and Communication Theory, J. Optical Society of America, V. 53
(1963), p. 1377. This development from what may be called classical
holography to digital holography for producing synthetic holograms--T. S.
Huang: Digital Holography, Proc. of the IEEE, V. 59 (1971) no. 9--has made
it possible to look at three-dimensional representations of objects which
have been described mathematically, but which do not exist.
Within classical holography a wave coming from a real object is combined
with a reference wave, and the sum of these waves is recorded on a
modulator for a reconstruction wave. By directing the reconstruction wave
against the hologram thus produced, the object is reconstructed. In the
computer generation of holograms the combination of an imaginary wave from
the mathematically described object and an imaginary reference wave is
calculated mathematically, such that the imaginary total wavefront is
calculated in quantized areas in the plane in which the hologram is
located during reconstruction. This involves the calculation of amplitude
and phase for the total wave field. In optical holography a calculated
wave information is usually recorded by plotting the interference pattern
between the two light waves at a practical scale as an artwork which is
then scaled down photographically. The present invention comprises the
direct generation of quantized hologram areas, for instance by means of a
scanning electron microscope.
Several methods have been developed for recording amplitude and phase
information for a light wave front. In the Lohmann's technique the
hologram generated is binary, thus it consists of opaque and transparent
windows. The size of a transparent window is then proportional to the
desired amplitude, and its position is related to the desired phase. In
Lee's technique the complex wave information is decomposed into four real
parts displaced from each other, so that both the real and the imaginary
parts of the desired information are recorded. In both these techniques
the object points considered to be emitting light against the hologram
plane, must be on a plane or a collection of planes located in the
Fraunhofer region, so that the fast fourier transform (FFT) can be used to
calculate the light wave amplitude and phase at quantized apertures in the
hologram plane. In Waters' technique the individual object points as well
as the hologram due to each of these points are considered separately. In
other words the zone plate pattern is recorded. In such case the object
points can be in the Fresnel region. In the kinoform technique the
amplitude of the light wave is assumed to be constant, and only the phase
is recorded, which will be approximately correct for objects giving
diffuse reflection. The phase information can be recorded by means of
binary selection, such that when the phase is between 0 and .pi. radians,
a window is made on the hologram, and when the phase is between .pi. and
2.pi. radians, no record is made.
Generally one of the big problems in connection with computer generation of
holograms is that the time for computing the interference pattern between
the object wave and the reference wave may be so long that the generation
excludes itself. Even the simplified methods used are complicated and time
consuming. With respect to the kinoform technique in particular there must
be used a gray scale in order to simulate the actual phase angle. In a
sampled representation of the object and binary quantization also
conjugate and higher order images are created.
SUMMARY OF THE INVENTION
The purpose of the present invention is to provide a simpler method for
producing synthetic holograms.
This has been made possible among others by employing a scanning electron
microscope, a controlled laser beam or other equipment which can plot with
sufficient accuracy for exposing a modulator. In this connection it has
been observed that the hologram areas essentially are sources of point
diffraction. Moreover, it has been observed that bombardment of the
modulator causes phase alteration of an incoming light field.
During operation of the above equipment it has been determined that the
amplitude of the light field at any point on an object or in a light
wave-front during diffraction is proportional to the number of hologram
areas being exposed, and the size thereof. Moreover, it has been found
that the phase of the light field from the object point is determined by
the distance between the object point and the center of the exposed area
in the photosensitive material. These observations have general validity
to the extent it is known that holography can be employed for other waves
than light.
On this background this invention has provided a simple method for
producing synthetic holograms, this method being characterized in that the
hologram areas of each point are considered in relation to the defined
phase and amplitude of the point, determining for the phase the positions
of the hologram areas of the point by adjusting the length of the radius
vector from the point to each associated hologram area by using the phase
in this point and the phase of a reference wave on the hologram area, and
determining for the amplitude a number of areas for the point such that
this number is proportional to the amplitude in the point.
By varying the number of hologram areas employed, amplitude modulation thus
can be obtained very accurately. The determination of any phase is made by
adjusting the length of the radius vector from the object point to the
center of the exposed hologram area. This is done by selecting initially a
hologram area position, which is preferably done in an arbitrary way since
any correlation between the hologram areas which are not used for
reconstructing a point, leads to noise and/or undesired images. Then the
length of the radius vector between the point and the hologram areas is
calculated. The positions of these areas are then somewhat altered in
horizontal or vertical direction, or in a specified direction, so as to
adjust the length of the radius vector accurately to give the desired
phase.
This method has made it possible to substantially simplify the mathematical
calculation of the wavefront. When the necessary means are available, the
method besides will make possible the production of synthetic volume
holograms, since an arbitrary choice of hologram area position and
alteration thereof in a horizontal, a vertical or a specified direction
permits hologram point generation in three-dimensional modulators.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention shall now be explained more closely by means of an example.
Reference is made to the drawing in which:
FIG. 1 is a previously known illustration of the principle for generating
synthetic holograms by means of a scanning electron microscope which is
used for carrying out this invention,
FIG. 2 shows a geometrical configuration illustrating the invention, and
which is also used as a starting point for the mathematical consideration,
FIG. 3 shows a two-dimensional object which is reconstructed with a
hologram produced with the method according to this invention, and
FIG. 4 shows a three-dimensional object which is reconstructed with a
hologram produced with this method.
DETAILED DESCRIPTION OF THE INVENTION
In FIG. 1 reference numeral 11 designates geometrical data which
mathematically and in numerical form describe the coordinates of a number
of object points. These numerical data are stored in the memory of a large
computer 12 which carries out the calculation of the wavefront from the
object according to some suitable, but often complicated mathematical
technique, as well as the interference pattern between the calculated
object wavefront and a reference wave. The result is supplied to a
magnetic tape station 13 for a small computer 14 which controls a scanning
electron microscope 15 by controlling the line shift 16, the deflection 17
and the intensity 18 of the electron beam 19, which thereby exposes a
modulator 20 in accordance with the calculated result. After exposure and
possible subsequent treatment the modulator constitutes the hologram. The
object can then be viewed by illuminating the hologram with a laser. The
computer 12 and the magnetic tape station 13 in the present case can be
eliminated by using a computer 14 which also carries out the calculations.
In FIG. 2 there is shown an object 21 and a number of geometrically defined
object points 22 which describe the object 21 mathematically for the
computer 12 in FIG. 1. The object 21 can also be a sampled wavefront. The
positions of the object points 22 in space are chosen as desired provided
that they are not outside the diffraction limited zone. One of these
points 22 has been given the coordinates (x.sub.o, y.sub.o, z.sub.o). The
hologram to be produced is here an optical planar hologram considered to
be located in the plane xy, and the extent of the hologram is indicated by
rectangle 23 which may have a size of 2 .times. 2 mm. Within this
rectangle there may be provided as much as 4096 .times. 4096 areas or
apertures 24 having a transverse dimension down to about 1 micron.
Although the areas are shown quadratic here, they may also be circular for
example. The positions of these areas in the hologram plane are calculated
by means of the mathematical equations being shown below. The number of
areas for each object point is chosen to be proportional to the light wave
amplitude which is desired in this object point, as mentioned above. As an
example can be mentioned that with 10 object points having amplitudes in
the ratio of 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, the number of areas for each
point is chosen to be 100 000, 90 000, 80 000, 70 000, 60 000, 50 000, 40
000, 30 000, 20 000 and 10 000, respectively. Moreover, the phase is
determined by locating the areas arbitrarily and then afterwards shifting
them and thus adjusting the distances as described above. One of the areas
has been given the coordinates (x.sub.i, y.sub.i, z.sub.i) and has a
radius vector length of r.sub.oi from the point (x.sub.o, y.sub.o,
z.sub.o). Letter .delta. indicates the angle between the vector r.sub.oi
from the center of the area (x.sub.i, y.sub.i, z.sub.i) to the object
point (x.sub.o, y.sub.o, z.sub.o) and the normal to the area (x.sub.i,
y.sub.i, z.sub.i). The arrow 25 indicates an incident laser beam of wave
length .lambda..
Thus when (x.sub.o, y.sub.o, z.sub.o) is considered as the observation
point, and (x.sub.i, y.sub.i, z.sub.i) is the position of a phase-shifting
area or aperture, the mathematical development of the present method in
the general case can be based upon the Huygens-Fresnel principle for a
collection of N apertures on a plane z = o (see Introduction to Fourier
Optics, T. W. Goodman, McGraw Hill, San Fransisco, 1968). This leads to:
##EQU1##
in which U(x,y,z) is the light field and
##EQU2##
in which k is the wave number, k = 2.pi./.lambda., and .lambda. is the
wave length.
Since the hologram dimensions concerned are small compared to the distance
from the object, .delta. can be assumed to be constant. If the area or
aperture dimensions are also very small compared to the variations of the
incoming wave, equation (1) can be written with approximation as:
##EQU3##
where .theta. is the phase shift due to the area. This equation is
rigorous if the incoming wave during reconstruction is planar and
perpendicular.
Since each aperture or area is made up of a number of point diffracting
sources and here has been chosen to be rectangular in the x-y plane with
dimensions d.sub.x and d.sub.y, and since the area has a central point
(x.sub.ci, y.sub.ci, o) whose radial distance from the observation point
is r.sub.oi, given by
kr.sub.oi = 2.pi.n + .phi..sub.i where n is an integer (4)
equation (3) by using Fraunhofer approximation can be written as:
##EQU4##
where X.sub.i = x.sub.o - x.sub.ci
Y.sub.i = y.sub.o - y.sub.ci
R = the average value of r.sub.oi.
If all phase shifts .phi..sub.i are made equal, and x.sub.i, y.sub.i <
r.sub.oi, the sinc functions can be replaced by 1, so that:
##EQU5##
Thus the amplitude of the field will be proportional to d.sub.x d.sub.y N,
and its phase will be .phi.. If there are groups of such areas that
satisfy equation (7) at different points in space, there is created a
sampled wavefront with a certain amplitude and phase at each point. Note
that the modulation of equation (7) is very simple, as d.sub.x and/or
d.sub.y and/or N can be varied for the amplitude and .phi. for the phase.
The fact that N is normally a large number, means that the number of areas
can be varied almost continuously so that amplitude modulation can be
achieved very accurately.
If the apertures or areas are circular, the sinc functions are replaced by
a first order Bessel function, but equation (7) essentially remains the
same.
If the amplitude of all the object points to be generated in space is
constant, and their phases are zero, the conditions for generation simply
become:
r.sub.oi = n.lambda. where n is an integer. (8)
If the phase at an object point is .phi. radians, equation (8) is
transformed to:
r.sub.oi = n.lambda. + .phi..lambda./2.pi.. (9)
If the amplitudes of the object points vary, the number of areas for each
object point has to be proportional to the amplitude of the object point.
As mentioned above each area in the hologram plane is, when producing a
planar hologram, chosen randomly and is then moved slightly in the x-
and/or y-direction so that its center coordinates satisfy equation (8) or
(9). If overlapping of areas is considered negligible, there is no need
for memory. The hologram points generated in this fashion by means of a
digital computer can be stored in a magnetic tape and can then be used to
drive for example a scanning electron microscope by means of a small
computer as described with reference to FIG. 1, in order to expose the
modulator in the calculated positions, for instance a modulator of the
type described by O. Ersoy: A Study of Electron Beam Exposure of Positive
Resists, Optik, January 1975, p. 479.
According to what is said above no use has been made of a reference beam
because the wave to the hologram was assumed to be plane perpendicular.
However, any type of reference wave can easily be incorporated if desired.
When producing volume holograms according to the above described method,
the reference wave must of course be incorporated. In the general case
equation (9) is modified to:
r.sub.oi = n.lambda. + (.phi. - .phi..sub.R) .lambda./2.pi.(10)
where .phi..sub.R is the phase of the reference wave in the center of each
hologram area.
By means of the method described there is produced optical planar holograms
with a 30kV JEOL scanning electron microscope controlled by a Kongsberg SM
402-S mini-computer. The working area for continuous exposure was 2
.times. 2 mm. Although the size of this area can be increased to 7.5
.times. 7.5 cm by means of stepping motors, this is not recommendable for
a single hologram because of an uncertainty of .+-. 5.mu.m in positioning.
The larger size area can, however, advantageously be used for reproducing
so that the viewing window and intensity of the image are larger. The
number of areas with the smallest possible transverse dimension of about 1
.mu.m was 4096 .times. 4096. The size of each area can be increased by
exposing adjacent areas which overlap from center to center. In the
experiments performed there was used an electron sensitive material as
described by O. Ersoy, B. Spjelkavik, K. Lovass, Applied Optics, January
1975.
In the following examples the mathematical calculations were carried out on
the basis of a He-Ne laser having a wave length .lambda. = 0.6328 .mu.m.
In connection with FIG. 3, 11 points were chosen on a line being 3 cm long,
so that x = 4 cm, z = 60 cm and o < y < 3 cm and with the origin of the
coordinate axis in the upper left corner of the hologram. The number of
areas used was 120 000, and each area had a size of 8 .times. 8 adjacent
areas. The object points were made of equal light intensity.
The picture shown was taken at approximately 60 cm from the hologram,
namely in the focal plane. The main light beam was blocked in order not to
overexpose the film. Both the real and the conjugate images can be seen,
as well as the blocking of laser light. The pictures are less than 3 cm
long because of the reduction with the Polaroid camera used. When
measuring the length of the line, the distance from the hologram plane to
the image could easily be found.
FIG. 4 has been included to show the three-dimensional effect. Each of the
four letters in the word LOVE was chosen on a different plane. The
distances of each of the planes from the hologram plane were 60 cm, 70 cm,
80 cm and 90 cm, respectively. If all letters were on a single plane, the
distance between them would be 1 cm. In the picture it is seen that this
distance is decreasing from the first to the last letter because of the
depth effect. The picture was taken at approximately 90 cm from the
hologram, namely in the focal plane of the letter E, which is the reason
why E is most bright, and L is the least bright.
The number of areas used was 100 000, and each area had a size of 4 .times.
4 areas. The object points were made of equal intensity. The hologram was
duplicated on a matrix with 4 .times. 4 holograms and enlarged 16 times so
that it could easily be viewed by means of a laser or a mercury arc lamp.
The images of the letters were clearly observed in space in their
respective places.
The method described has both advantages and disadvantages when compared
with known techniques, as will appear from the following evaluation.
Since each object point is considered independent of the others, the
hologram will saturate after a certain number of image points due to the
finite size of the hologram. Studies have indicated, however, that this
number is several thousands with the equipment described and a hologram
area of 2 .times. 2 mm. If a composite hologram is made by using stepping
motors, the number of image points can be increased by a factor of 1380 to
create a rather complicated image. In such case, however, the light coming
from each individual hologram would be highly directed so that an
additional system would be necessary to combine all the information coming
from the different holograms.
The main advantages of the method are its simplicity, ease of choosing the
image points in space arbitrarily rather than on a plane as in the case of
Fourier techniques. Moreover, it is quite simple to perform the amplitude
modulation. Besides, this method results in a high signal/noise ratio,
which makes it ideal in applications that do not require many image
points. Thus, for example it can replace the so-called step-and-repeat
cameras being used in connection with integrated electronics. The method
can also be used for mixing laser beams having different wave length, at
an exact point in space, for example a 10.6 .mu.m laser beam from a
CO.sub.2 laser and a 0.6328 .mu.m laser beam from a He-Ne laser can be
mixed.
Another interesting possibility afforded by this method, is, however, that
object points with desired phase and amplitude can be created. This can be
very important in connection with optical filtering and information
processing where it is desired to re-combine a certain number of object
points in one part of space to give some desired image in another part of
space.
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