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BACKGROUND OF THE INVENTION
The present invention relates generally to generation of holograms, and
more particularly to the generation of x-ray holograms of biological
specimens.
Recent advances in coherent x-ray source technology are making
diffraction-limited holograms of microscopic structures, with
corresponding high spatial resolution, a reality. A useful application of
snapshot x-ray holography is the study of microscopic biological
structures in the living state. X-rays offer high resolution and high
contrast ratios for the important structures within living organisms,
thereby rendering the staining of specimens, essential for optical and
electron microscopy unnecessary if the wavelength is properly chosen.
Picosecond time resolution, which would eliminate blurring due to thermal
heating from deposition of incident energy and due to normal biological
activity of the sample is also possible. Finally, with sufficiently high
photon fluxes, such as those available from x-ray lasers, the x-ray
snapshot can be accomplished with a single pulse yielding complete
three-dimensional information of a sample having normal biological
integrity at the moment of the snapshot.
A description of holographic techniques for imaging microscopic structures
with a short-pulse, high intensity, high-quantum-energy laser is set forth
in "Holography At X-Ray Wavelengths," by J. C. Solem, G. C. Baldwin, and
G. F. Chapline, Proc. Int'l. Conf. on Lasers, pp. 293-305 (1981). Several
important points therein will be summarized. First, Fresnel holography has
the simplifying aspect of requiring but one laser beam. The subject
specimen is placed in the laser beam itself, which beam also provides the
reference. This technique, however, requires very high resolution
recording media. That is, the minimum spacing which can be resolved is
greater than twice the grain spacing of the medium. This result is
independent of the wavelength of the incident radiation as long as the
angles are small. At large diffraction angles and short wavelengths, the
surface smoothness of the medium becomes important as well as its
intrinsic graininess.
Fourier transform holography, by contrast, requires a reference source
which emits spherical or convex curved waves, which interfere with the
waves from the subject specimen at a recording surface. The specimen is
illuminated by a plane wave source. The procedure is called Fourier
transform holography because every distance from the reference source maps
to unique spatial frequency at the recording surface. The maximum spatial
frequency of the interference pattern can be adjusted arbitrarily by
placing the object at various distances from the reference source. A
shortcoming of the Fourier holography method described, supra, is that a
spherical recording surface is required in order to obtain a complete
cycle of intensity fringes for closely spaced features in the specimen.
However, if the point spacing is less than the wavelength, a full cycle is
never obtained. The physical spacing of the fringes can be made
arbitrarily large by expanding the radius of the sphere. Therefore,
ordinary film of arbitrarily large grain size could be employed as long as
the trade-off between sensitivity and resolution was favorable.
In order to obtain the spherical reference wave for Fourier holography one
must have a lens that focuses to a pinhole in the shadow plane. In the
x-ray region of the electromagnetic spectrum a Fresnel zone plate is used
to accomplish this. However, the hologram resolution is limited to finest
spacing on such a plate, currently about 10 nm. An alternative would be to
use a coherent scattering backward reflector to generate the spherical
reference waves. In FIG. 6 of the Solem reference, supra, the authors show
a parabolic reflector enclosed in a spherical shell recording surface. For
best contrast ratio, the paraboloid would have to be approximately the
same size as the object. The reference scatterer need not be a paraboloid.
In principle, the hologram could be unfolded for any convex reference
scatterer as long as the shape and dimensions thereof were known to within
a fraction of a wavelength.
In "X-Ray Biomicroholography," by Johndale C. Solem and George F. Chapline,
Opt. Eng. 23, 203 (1984), the authors state that most of the information
about the fine details of the specimen appears at large scattering angles
and can be degraded by recording surface roughness. The problem is
mitigated by using a spherical recording surface. However, the reference
scatterer will have a low scattering efficiency, as will the specimen, and
the intensity of reference- and specimen-scattered waves will
approximately match for highest contrast. The authors also discuss briefly
the use of an x-ray photocathode and microchannel plates. However, the
authors state that such devices saturate easily, have a small dynamic
range, and are available only in fairly small sizes.
Accordingly, it is one object of the present invention to provide an
apparatus for recording high resolution x-ray holograms.
Another object of our invention is to provide an apparatus for electrically
recording high resolution x-ray holograms of biological samples using
currently available electronics technology.
Yet another object of the present invention is to provide a method for
obtaining a faithful reproduction of objects from detected holograms
thereof.
Additional objects, advantages and novel features of the invention will be
set forth in part in the description which follows, and in part will
become apparent to those skilled in the art upon examination of the
following or may be learned by practice of the invention. The objects and
advantages of the invention may be realized and attained by means of the
instrumentalities and combinations particularly pointed out in the
appended claims.
SUMMARY OF THE INVENTION
To achieve the foregoing and other objects, and in accordance with the
purposes of the present invention, as embodied and broadly described
herein, the apparatus of this invention includes a spherical reference
scatterer located in the vicinity of the sample under investigation, means
for generating substantially monochromatic and substantially coherent
x-radiation having sufficient size and intensity to simultaneously
illuminate the sample and the spherical reference scatterer, means for
detecting and recording spatial frequencies in the forward direction
relative to the incident x-radiation resulting from the interference of
x-radiation scattered by the sample and by the spherical reference
scatterer, and means for reconstructing the image of the sample.
Preferably, the spherical reference scatterer has approximately the same
cross section to the incident x-radiation as the cross section of the
sample. It is also preferred that the means for generating x-radiation
includes a pulsed x-ray laser operating at a wavelength between 0.1 and 10
nm which includes the "water window", but also shorter wavelengths which
correspond to the absorption edge of other biologically important elements
such as phosphorus and calcium to name two. Preferably also the spherical
reference scatterer is fabricated from nickel, rhenium, iridium, or
osmium, or combinations thereof depending on the wavelength employed since
the maximum angle of significant specular reflection is wavelength
dependent.
The present invention may also include means for flowing the spherical
reference scatterer and the sample through the x-radiation.
In a further aspect of the present invention, in accordance with its
objects and purposes, the method of reconstructing an image from a
detected hologram hereof includes calculating basis functions which are
the set of all holograms of the individual points which comprise the
object, and projecting the hologram function measured by the detector
array onto the basis functions thereby yielding the amplitude,
.vertline.R.sub.i .vertline., of the points in the reconstructed object.
Benefits and advantages of our invention includes the ability to perform
3-dimensional x-ray microholography of living hydrated biological
materials using either synchrotron or laser sources, only one optical
element, a low resolution, two dimensional detector, and analyzing the
detector output by employing a suitable algorithm and computer hardware.
BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are incorporated in and form a part of the
specification, illustrate two embodiments of the present invention and,
together with the description, serve to explain the principles of the
invention. In the drawings:
FIG. 1 is a schematic representation of one embodiment of the apparatus of
the present invention showing the incident x-radiation from a source
thereof, the specimen under investigation and the spherical scatterer
(both fixed in position on a substrate), the x-ray charge-coupled device
array for viewing the scattered electromagnetic radiation in the forward
direction, and the computer system for data processing.
FIG. 2 is a schematic representation of a supporting foil and grid for
fixing the spherical scatterer and the sample under investigation in the
path of the x-radiation.
FIG. 3 is a schematic representation of a second embodiment of the
apparatus of our invention showing the same elements shown in FIG. 1
hereof except that the spherical scatterer and the specimen are flowed
through the region of x-radiation and are detected by a second laser
device utilized for activating the x-ray source.
FIG. 4 is a schematic representation of the image reconstruction system
showing the hologram detector, the video digitizer, the array processor
the display controller, the host computer and the display.
FIG. 5 is a schematic representation of the details of the geometric
relationships among the scattering bodies and the charge-coupled device.
In particular, the location on the charge-coupled device array of an
incident x-ray reflected at an angle .theta. by the reference scatterer is
shown, as are the pixel spacing d and the scattering system-charge-coupled
device separation 1.
FIG. 6 is a flow diagram describing the process for generation of basis
functions necessary for the reconstruction of the image from a detected
hologram.
FIG. 7 is a flow diagram describing the process for reconstructing an image
from the basis functions generated according to the flow diagram of FIG. 6
hereof and the intensities detected on a hologram recording device.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE INVENTION
Briefly, the present invention includes a Fourier-transform holographic
microscope applicable for use with both x-ray lasers and synchrotron
sources. The Fourier configuration is particularly suitable for
high-brightness sources. Presently envisaged laboratory x-ray lasers will
produce high intensity beams with narrow divergence. Self-channeling
lasers may achieve diffraction limited divergence with high spatial and
temporal coherence. The narrow divergence, as well as the necessity for
high resolution recording media, makes Fourier-transform holography
preferable to Fresnel transform holography for investigating the
ultrastructure of biological and other specimens, such as semiconductor
devices. To perform Fresnel transform holography a large area of the
recording surface must be illuminated with a reference beam, a procedure
which would require an impracticable distance between the laser and the
recording surface. Moreover, a high-resolution recording medium such as a
photoresist would be required, adding significant complexity and time to
the process of data acquisition and interpretation. According to the
teachings of the present invention, an x-ray beam simultaneously
illuminates both a reference scatterer and the specimen under
investigation. The resulting scattered electromagnetic radiation is
detected in the forward direction relative to the incident x-radiation by
a charge-coupled device, since high resolution in the interference pattern
recording is not required. The output from such a detector array is
digitized and linked directly to a computer for reconstruction of the
physical details of the specimen using a suitable algorithm which will be
described hereinbelow.
The essential distinguishing features of the present invention over that
described in the two Solem et al. references, supra, is the use of a
spherical forward scatterer, not a backward reflector, and the positioning
of a substantially planar detector array in the forward scattering
direction, as opposed to a spherical detection system which observes
backward scattered electromagnetic radiation. In fact, there are no single
materials which have significant scattering efficiency at large angles. In
order to achieve large angle reflectivity, one must utilize multilayer
materials which are useful only over a very narrow range of wavelengths.
For such materials it is also difficult to precisely define the shape.
This latter problem is a serious one since the generation of accurate
physical characteristics of the specimen under investigation is dependent
on the details of the shape of the reference scatterer.
Reference will now be made in detail to the present preferred embodiments
of the invention, examples of which are illustrated in the accompanying
drawings. All similar or identical structure will be identified using the
same callout numbers. Turning now to FIG. 1, there is schematically
illustrated one embodiment of the apparatus of the present invention.
Radiation 10 from x-ray source 12 is directed onto scatterers 14 and 16
which include a sample to be investigated 14 and a reference scatterer 16.
Electromagnetic radiation 18 scattered in the forward direction from
scatterers 14,16 is detected on planar charge-coupled device 20. The
output from detector 20 is recorded by data processing system 22, and
displayed by graphic display terminal 24. The apparatus of our invention
is compatible with both x-ray laser sources and synchrotrons. A laser
source is preferable, since, as a consequence of the substantially greater
brightness, a sufficient exposure rate will be present to record the
holographic data in a single pulse. This feature eliminates blurring due
to incident energy being deposited in the specimen. The wavelength region
of particular interest for hydrated biological samples is in the region of
the water window (2.36-4.47 nm), since this range of wavelengths gives the
greatest contrast for biological structures in the presence of water.
However, in general, x-radiation in the region from 0.1-10 nm will be
useful to contrast elements such as calcium, phosphorus, sulfur, sodium
and potassium.
Spherical reference scatterer 16 is located in proximity to specimen 14 and
furnishes the reference beam required for Fourier transform holography. If
the scatterer were a perfect reflector, the reference illumination would
be isotropic. However, since specular reflectivity is not strong in the
soft x-ray region of the electromagnetic spectrum, and changes rapidly
with grazing angle, forward scattering was observed in order to maximize
the signal-to-noise ratio of the apparatus. A survey of the periodic table
for elements having optimum specular reflectivity to serve as reference
scatterers for Fourier transform holography in the spectral region of the
water window has yielded that nickel is optimum at the water window
threshold (4.47 nm) and osmium is optimum at the wavelength of maximum
contrast (3.16 nm), nitrogen K.sub..alpha. -edge. Both elements are
sufficiently reflective at moderately large angles (20.degree.-30.degree.)
to render the present apparatus useful for microholography. In order to
use spheres of these elements as reference scatterers, it is necessary to
fabricate such spheres in the appropriate sizes (approximately having the
same cross section to the incident radiation as the specimen under
investigation) and having sufficient sphericity and surface smoothness.
Calculations suggest that nickel spheres in the 2-10 .mu.m range can be
fabricated in laser produced plasma sprays to a high degree of surface
smoothness and roundness. Smoothness of the reference scatterer is
important to avoid speckle as will be discussed further hereinbelow.
Extant data on droplet formation in a plasma jet for nickel-base brazing
alloy and copper support the conclusion that plasma sprays can generate
appropriate reference spheres. See, e.g., "Fine Powders Produced by Plasma
Processing," by W. A. Johnson, N. E. Kopatz, and E. B. Yoder, in Progress
in Powder Metallurgy, Vol. 42 (Netal Powder Industries Federation, 1986)
p. 775. Although investigation of the spheres generated from such sprays
indicates evidence of imperfections arising from particle-particle
collisions occurring in the spray, it is believed by the coinventors that
a reduction of particle density can eliminate this effect. It should be
mentioned that any shape scatterer would be appropriate for
Fourier-transform microholography as long as the exact shape and
orientation are known. Microspheres are therefore most appropriate for use
since their orientation is irrelevant.
An evaluation of potential biological samples has been made and the
apparatus of the present invention may be used to investigate numerous and
varied specimens. Clearly, according to the teachings of our invention,
samples other than biological specimens can be studied. For example, the
physical characteristics of semiconductor chips might be evaluated. Given
the opacity of water in the soft x-ray region, a useful and practical
biological sample thickness would be approximately 1 .mu.m. As a matter of
practicability, it is desirable to have a statically mounted scattering
system with both the reference scatterer and the sample placed on a thin
foil. In such a configuration, it is necessary to choose a foil having low
scattering power so that it negligibly influences the transmission of the
soft x-rays. Moreover, it is important that the structure not degrade the
spatial coherence of within a fraction of a wavelength over the useful
portion of the beam. Therefore, the foil must have a substantially uniform
thickness over the region exposed. Suitable materials for selection are
thin sections of low-Z material such as beryllium and carbon. Commercially
available support grids for films include structures made of beryllium,
carbon composite, nylon mesh coated with carbon, copper, and gold. Such
grids are available in a range of meshes, as well as honeycomb, with round
and slit hole designs, and are routinely used for electron microscopy. The
coinventors believe that carbon and beryllium films having a thickness
between 100 and 200 .ANG. can be used.
FIG. 2 shows a schematic representation of the spherical reference
scatterer 16 and the biological specimen 14 under investigation mounted on
a supporting foil 26 which is located on a supporting grid structure 28.
Since carbon and beryllium films have a low affinity for biological
materials, a thin layer of poly-L-lysine 30, which has a high charge
affinity for many biological specimens, should adequately anchor the
samples. The carbon foil 26 would first be suspended over the open regions
of the grid. The Ni/Os reference sphere would then be placed on the foil
and encapsulated by a thin evaporated carbon layer 32, thereby fixing its
position. At this stage, the system is a blank on which a suitable
biological specimen can be located. This last step in the preparation of
the scattering system, the placement of the biological material, can be
performed under circumstances suitable for the handling of a hydrated
sample. Micromanipulation of the scattering system must be used to place
the target properly in the path of the x-ray beam.
A second embodiment of the apparatus of the present invention is shown in
FIG. 3. Scattering systems would be injected into the region of the x-ray
beam using a microflow device 34, such as a flow cytometer. Radiation 36
from laser 38 is scattered into detector 40 by the spherical reference
scatterer and the sample under investigation. The output from detector 40
is amplified by amplifier 42 which is directed to the source of
x-radiation 12 for the purpose of triggering this source when the
scatterers are in the appropriate position. This procedure eliminates any
degradation of the scattering image through imperfections in the foil
required in the static system shown in FIG. 2 hereof.
FIG. 4 is a schematic representation of the interrelationship among the
electronic components of the image reconstruction system of the present
apparatus. Shown are hologram detector 20, video digitizer 44, which
processes the output from detector 20, array processor 46, display
controller 48, and graphics display 24 which receives the image
reconstructed from the hologram detected on hologram detector 20. Video
digitizer 44, array processor 46, and display controller 48 interact with
computer 50 in the image reconstruction process.
FIG. 5 shows the geometric relationship among the scatterers and the
detection device in two-dimensions. In actuality, a three-dimensional
image would be reconstructed from a two-dimensional hologram.
Charge-coupled devices are currently being developed for x-ray astronomy.
Such devices can be used for imaging over the range between 0.1 and 1000
nm with a greater than 80% efficiency in the soft x-ray region. Moreover,
these direct electronic readout devices have a noise level of about 1
e.sup.- /pixel/s and have a sufficiently high pixel density to provide
spatial resolution adequate for Fourier transform holography at reasonable
scatterer-detector distances.
It is desirable to avoid a large disparity between the reference-wave
intensity and the specimen-scattered-wave intensity. The approximate
mismatch which can be tolerated depends on the noise and dynamic range of
the charge-coupled device. Since specular reflection is a rapidly
decreasing function of scattering angle in the soft x-ray region of the
electromagnetic spectrum, there is some angle .theta. beyond which the
mismatch becomes too great and information cannot be extracted. Estimates
derived from scattering data indicate that .theta.=30.degree. for a nickel
sphere at 4.47 nm and .theta.=22.degree. for an osmium sphere at 3.16 nm.
These estimated scattering angles are sufficiently large to give
wavelength scale resolution in both the transverse and the longitudinal
directions. Therefore, the acceptance angle imposed by the reference
scatterer does not limit resolution of the subject apparatus.
Fourier transform holography basically maps distances in the specimen into
spatial frequencies at the recording surface. The highest spatial
frequency derives from the largest distance in the specimen. Therefore,
the charge-coupled detector pixel size will then determine the greatest
distance to be resolved which is the gross diameter of the specimen. A
pixel spacing of about 15 .mu.m is reasonable and with a mosaic of
3.times.10.sup.3 .times.3.times.10.sup.3, which could be built with
presently available technology, a scattering system would subtend an angle
of 30.degree. at a distance of about 7.4 cm from the charge-coupled
device.
In the process of imaging a biological specimen in the x-ray region, there
is little reflection or refraction in the specimen that can cause the
random phase shifts which give rise to speckle. The optical element which
may give rise to speckle is the spherical reference scatterer which
creates the reference waves by specular reflection. Roughness on this
surface can produce speckle and careful fabrication of the sphere is
mandated, but is within currently available technology.
Finally, an estimate of the required x-ray pulse energy must be made.
Simple calculation shows that this quantity is less than 10 .mu.J, which
can be achieved by currently available x-ray lasers (See, e.g.,
"Demonstration of Soft-X-Ray Amplification in Nickel-like Ions," by B. J.
MacGowan, S. Naxon, P. L. Hagelstein, C. J. Keane, R. A. London, D. L.
Matthews, M. D. Rosen, J. H. Scofield, and D. A. Whelan, Phys. Rev.
Letters, 59, 2157 (1987).). It should be mentioned that cw x-ray sources
may be employed if the sample is immobile. That is, for a biological
sample, one might cool the sample to reduce motion in order for the sample
to receive sufficient irradiation to provide reliable data.
The theoretical approach to the reconstruction process is based on
projecting the magnitude of each of a set of basis functions which map to
individual points in the reconstruction space. A more detailed description
of the analytic process is described in "A Description Of The Theory And
Apparatus For Digital Reconstruction Of Fourier Transform Holograms," by
W. S. Haddad, J. C. Solem, D. Cullen, K. Boyer, and C. K. Rhodes, summary
submitted on Sep. 12, 1987 for Electronics Imaging '87, held on Nov. 2-5,
1987 in Boston, Mass., the disclosure of which is hereby incorporated by
reference herein. The object is described as a collection of point sources
of scattered radiation. In the limit of large distances from the object
and reference scatterers, the reconstruction reduces to a Fourier
transform. However, this limits the analysis to a two-dimensional object,
and a small aperture for the detector. In order to have good longitudinal
and transverse resolution, a large detector aperture is necessary and the
Fourier transform is no longer suitable as the reconstruction algorithm. A
solution is found by approaching the problem as represented by the Fourier
transform, but with the use of a different set of basis functions, not
necessarily trigonometric. These functions can be thought of as the set of
all holograms of the individual points which comprise the object. The
projection or inner product of the hologram function measured by the
detector array and the basis functions yields the amplitude,
.vertline.R.sub.i .vertline., of the points in the reconstructed object
(.vertline.R.sub.i .vertline..sup.2 =brightness). In principle, it is
desirable that the B.sub.i s be orthogonal. However, the lack of complete
orthogonality due to the detector specifications and the system geometry
and will always be present in any finite system, and is the principle
source of distortion in the reconstruction process.
Since a hologram is intrinsically a phase-sensitive recording, it is
expected that unwanted phase shifts will degrade the reconstructed image.
This sensitivity is related to the fact that a phase error affects all of
the information in the hologram. For example, when the basis set is
calculated, the geometrical parameters of the holographic system must be
entered as basic data. One such datum is the position of the reference
scatterer which will always contain some uncertainty in its position
relative to the detector. A computational procedure for enabling the
measured holographic data to be used to correct the alignment of the basis
set has been developed by the present inventors. If this correction is
performed prior to the reconstruction of the image, full compensation can
be achieved.
It is also possible to use digital processing to improve image quality and
performance of the system. That is, the "confusing wave" effects can be
removed by applying a high pass frequency filter to the hologram function
before performing the reconstruction. The "confusing wave" arises from the
interference among the various scattering bodies within the object under
investigation. The hologram detected from this interference has low
spatial frequencies since these bodies are close together. What is of
interest, however, is the interference pattern of the scattering bodies
within the object and the reference scatterer. If the reference scatterer
is sufficiently far from the object, the important information can be
observed at high spatial frequencies. Therefore, a high pass filter
applied to the detected data will simplify the reconstruction process.
Fast Fourier transforms can be used in a manner enabling efficient and
rapid computation for this procedure. A Fourier-transform is obtained and
the zero- and low-frequency terms made zero before the reverse-transform
is calculated.
An algorithm has been developed and tested in two dimensions for
reconstruction of the physical characteristics of samples from measured
spatial frequencies. Geometrical optics and plane-wave irradiation were
assumed. It was also assumed that the spherical reference scatterer is
perfectly spherical. An expansion of the procedure to three-dimensions is
now outlined. Referring now to FIG. 5 hereof, the two-dimensional
description of the system to be analyzed, and generalizing to
three-dimensions, if one defines x.sub.o, y.sub.o, and z.sub.o as the
coordinates of pixels in the reconstruction volume, x.sub.r, y.sub.r, and
z.sub.r as the coordinates of the center of the reference sphere, x.sub.h
and y.sub.h as the coordinates of pixels on the surface of the hologram
detector, N as the number of points in the reconstruction volume,
H(x.sub.h,y.sub.h) as the hologram function detected by the charge-coupled
detector array, Bi(x.sub.h,y.sub.h) as the basis functions used for the
reconstruction, R.sub.i as the amplitude coefficients of pixels in the
reconstruction volume, .theta.(x.sub.h,y.sub.h) as the grazing angle of
reflection of the x-rays from the reference scattering sphere,
A(x.sub.h,y.sub.h) as the reflectivity function for waves scattered by the
sphere, s.sub.x and s.sub.y as the error in the actual transverse
position of the reference sphere, n as the "stretch" applied to the
hologram to correct for error in the longitudinal position of the
reference sphere, C(s.sub.x,s.sub.y,n) as a sharply peaked function from
which s.sub.xp, y.sub.xp and n.sub.p may be determined, s.sub.xp,
y.sub.xp, and n.sub.p being the coordinates where this function is
maximized, and M is a constant background value to be subtracted from the
detected hologram, the reconstruction process proceeds as follows.
FIGS. 6 and 7 are flow charts for performing the reconstruction process.
FIG. 6 describes the calculation of the basis functions, B.sub.i
(x.sub.h,y.sub.h), necessary for reconstruction of the scattering specimen
from the detected hologram. One begins by inputting the geometrical
parameters, x.sub.o, y.sub.o, z.sub.o, x.sub.r, y.sub.r, z.sub.r, x.sub.h,
and y.sub.h, the coordinates of the pixels, of the center of the reference
sphere, and of the pixels on the hologram detector surface, respectively,
as defined hereinabove and described in FIG. 5 hereof. Additionally, the
sphere radius, r, and the table of reflectivities as a function of grazing
angle are supplied. The phase correction introduced by
.theta.(x.sub.h,y.sub.h), the grazing scattering angle for incident x-rays
from the reference scattering sphere is then calculated. This is a purely
geometrical calculation and is necessary since the incident radiation is
scattered from the surface of the sphere having a radius, r>0 rather than
from a point source. Since the incident radiation reflected by the
reference sphere is planar, a particular point (x.sub.h,y.sub.h) on the
detector surface is reached by reflected light having a unique angle
.theta.. The basis vectors, B.sub.i represent interferences between a
reference point (x.sub.r,y.sub.r) and a point (x.sub.o,y.sub.o) in the
object. Therefore, phase corrections in the reference wave will generate
corresponding corrections in these basis functions. A(x.sub.h,y.sub.h),
the reflectivity function for scattered radiation by the reference
scattering sphere, which is a function of the material of the sphere and
the wavelength of the incident radiation, is evaluated for each value of
.theta. in order to determine the intensity of the scattered wave.
Basically, the amplitude of the reflected wave diminishes for increasing
angles of reflection. This causes the basis functions to decrease in
amplitude at locations on the recording surface corresponding to large
angle scattering, which will affect both the resolution and contrast of
the reconstruction. Selective processing the holographic information prior
to the reconstruction process to correct for the degradation of image has
been found to improve the images obtained. One can then calculate and
store the basis functions, B.sub.i (x.sub.h,y.sub.h). There is one B for
each point in the reconstruction volume.
FIG. 7 schematically shows how one corrects the detected signal H for
various physical factors, to be explained hereinbelow, and utilizes the
values of the Bs just obtained to reconstruct the details of the specimen
under investigation. Consider a misalignment in one of the transverse
dimensions x or y. Such a misalignment translates directly into an error
in the phase of the basis functions B.sub.i. A misalignment in the
longitudinal coordinate, z (i.e., the distance from the detector),
corresponds to a compression or extension of the basis functions with
respect to the hologram. As defined hereinabove, the compression/extension
factor n, and the corresponding corrections to the position in the x and y
coordinates, s.sub.x, and s.sub.y, respectively, are made the variables in
a newly-defined correlation function
##EQU1##
Given the set of C.sub.i corresponding to the basis vectors B.sub.i, a
function, D.sub.i .ident.C.sub.i
(s.sub.xp,s.sub.yp,n.sub.p).delta.(s.sub.xp,s.sub.yp,n.sub.p) is
constructed, in which (s.sub.xp,s.sub.yp,n.sub.p) are the coordinates for
which C.sub.i is a maximum, and where .delta.(s.sub.xp,s.sub.yp,n.sub.p)
1, .delta. being equal to zero for all other coordinates. One then defines
a new function, C(s.sub.x, s.sub.y, n).ident..SIGMA.D.sub.i which has been
found to exhibit a sharp peak at the point (s.sub.xp,s.sub.yp,n.sub.p) for
any hologram. This function permits the quantities s.sub.xp, s.sub.yp, and
n.sub.p to be determined for any holographic exposure, and the appropriate
correction to the B.sub.is made. The measure of s.sub.xp will be in units
of x.sub.h, the s.sub.x and s.sub.y being integers because of the descrete
pixel grid. Model calculations have confirmed the efficacy and
practicality of this procedure. That is, phase errors are introduced into
a hologram and reconstruction of the "damaged" hologram is performed. A
phase correction of only 30 pixels will completely destroy the image of
the object. If the correctional procedure outlined hereinabove is employed
before reconstruction, the faithful reproduction is fully restored.
Uncontrolled variations in the amplitude of the hologram can degrade the
quality of the image. Such variations can occur from either variations in
the intensity of the reference wave, or from modulations in the
sensitivity of the recording charged coupled device array. Fortunately,
the sensitivity of the reconstructed image to variations in the amplitude
of the hologram has been examined and found to be very low. High quality
images can be reconstructed from holograms grossly altered by the loss of
amplitude information over substantial regions of the exposure.
The "confusing wave" is now removed as described hereinabove. Finally, M, a
constant background value is subtracted from the detected hologram in
order to produce a purely oscillatory detected signal with no offset
before the .vertline.R.sub.i .vertline. are calculated.
The foregoing description of two preferred embodiments of the invention has
been presented for purposes of illustration and description. It is not
intended to be exhaustive or to limit the invention to the precise form
disclosed, and obviously many modifications and variations are possible in
light of the above teaching. The embodiments were chosen and described in
order to best explain the principles of the invention and its practical
application to thereby enable others skilled in the art to best utilize
the invention in various embodiments and with various modifications as are
suited to the particular use contemplated. It is intended that the scope
of the invention be defined by the claims appended hereto.
* * * * *
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