U.S. patent application number 11/081214 was filed with the patent office on 2005-07-21 for shear inducing beamsplitter for interferometric image processing.
This patent application is currently assigned to Lenslet Ltd.. Invention is credited to Bergstein, Leonard, Eisenbach, Shlomo, Garcia, Javier, Glushko, Boris, Goldenberg, Efraim, Mendlovic, David, Miron, Yehuda, Sariel, Aviram, Shabtay, Gal.
Application Number | 20050157313 11/081214 |
Document ID | / |
Family ID | 27271928 |
Filed Date | 2005-07-21 |
United States Patent
Application |
20050157313 |
Kind Code |
A1 |
Mendlovic, David ; et
al. |
July 21, 2005 |
Shear inducing beamsplitter for interferometric image
processing
Abstract
A shearing generator comprising: an input light source; an image
generator that generates two images of the input light source at an
output plane, the image generator comprising a beam splitter that
splits light from the input source into at least one pair of
interfering light waves at an output thereof, and defines different
optical paths for the light propagation of the light waves, said
optical paths including at least one phase shifting element that
provides for a different phase shift for the two paths.
Inventors: |
Mendlovic, David; (Tel-Aviv,
IL) ; Glushko, Boris; (Ashdod, IL) ;
Goldenberg, Efraim; (Ashdod, IL) ; Shabtay, Gal;
(Petach-Tikva, IL) ; Garcia, Javier; (Valencia,
ES) ; Bergstein, Leonard; (Teaneck, NJ) ;
Eisenbach, Shlomo; (Kfar-Pines, IL) ; Miron,
Yehuda; (Tel-Aviv, IL) ; Sariel, Aviram;
(Ramot-Hashavim, IL) |
Correspondence
Address: |
WOLF, BLOCK, SCHORR & SOLIS-COHEN LLP
250 PARK AVENUE
NEW YORK
NY
10177
US
|
Assignee: |
Lenslet Ltd.
Herzliya Pituach
IL
46733
|
Family ID: |
27271928 |
Appl. No.: |
11/081214 |
Filed: |
March 16, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11081214 |
Mar 16, 2005 |
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10257425 |
Feb 24, 2003 |
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6879427 |
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10257425 |
Feb 24, 2003 |
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PCT/IL01/00334 |
Apr 10, 2001 |
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Current U.S.
Class: |
356/520 |
Current CPC
Class: |
G06E 3/00 20130101; G01B
9/02098 20130101; G06E 3/001 20130101; G02B 26/06 20130101 |
Class at
Publication: |
356/520 |
International
Class: |
G01B 009/02 |
Foreign Application Data
Date |
Code |
Application Number |
Apr 10, 2000 |
IL |
135576 |
Jan 23, 2001 |
IL |
141041 |
Mar 7, 2001 |
IL |
141856 |
Claims
1. A shearing generator comprising: an input light source; an image
generator that generates two images of the input light source at an
output plane, the image generator comprising a beam splitter that
splits light from the input source into at least one pair of
interfering light waves at an output thereof, and defines different
optical paths for the light propagation of the light waves, said
optical paths including at least one phase shifting element that
provides for a different phase shift for the two paths.
2. A shearing generator according to claim 1 wherein the phase
shifting element comprises a coating at said beam splitter.
3. A shearing generator according to claim 2 wherein the phase
shifting element comprises a phase shifting element in one of the
paths.
4. A shearing generator according to claim 3 wherein the phase
shifting element is a coating on an external surface of an optical
block containing the beam splitter.
5. A shearing generator according to claim 1, wherein the light
waves form coplanar images at said output, said waves propagating
normally to said output, the images being provided without
reflection except from said beam splitter.
6. A mirror-less shearing generator comprising: an input source; an
image generator that generates two images of the input light source
at an output plane, the image generator comprising a beam splitter
that splits light from the input source into at least one pair of
interfering light waves at an output thereof, and defines different
optical paths for the light propagation of the light waves, neither
of said optical paths including a reflecting surface, except for
said beam splitter.
7. A shearing generator according to claim 6, wherein the beam
splitter provides two non-parallel waves and including at least one
refractive element that refracts at least one of the waves, so that
the waves are parallel.
8. A shearing generator according to claim 7 wherein the refractive
elements are interfaces at outer walls of the beam splitter.
Description
RELATED APPLICATIONS
[0001] The present application is a divisional application of U.S.
application Ser. No. 10/257,425 filed on Feb. 24, 2003, which is a
U.S. national application of PCT Application No. PCT/IL01/00334,
published as WO 01/78012, filed on Apr. 10, 2001.
FIELD OF THE INVENTION
[0002] This invention is generally in the field of signal transform
techniques, and relates to an optical discrete transform method and
system.
BACKGROUND OF THE INVENTION
[0003] Transform techniques have played an important role in signal
processing for many years. The need for transform techniques is
associated with the fact that the amount of generated data (an
input signal) may be so great that it results in impractical
storage, processing and communication requirements. In such cases,
representations beyond the simple sampling and quantization are
needed. In addition, many kinds of frequency domain processing are
known. When the input data is not in the frequency domain, a
transformation of the data is generally required to apply such
processing.
[0004] Image compression addresses the problem of reducing the
amount of data required to represent a digital image. Various
communication techniques recently developed, such as Asymmetric
Digital Subscriber Line (ADSL), deal with the transmission of a
great amount of data to a subscriber premise. ADSL can
substantially transform an existing public information network from
one limited to voice, text and low resolution graphics, to a
powerful, ubiquitous system capable to bringing multimedia
(including full motion video) to every home.
[0005] In order to process an input signal (e.g., an image of an
object scene) using conventional ADSL techniques, the input signal
must first be received by a transducer and converted into an
appropriate form. The choice of a particular transform in a given
application depends on the amount of reconstruction error that can
be tolerated and the computational resources available. Fourier
transform is very popular because of its wide range of
applications. Fourier transformations can be performed
electronically using a suitable computer and software. For
electronic processing, the input data presented in a time domain is
converted into a frequency domain and vice versa, and for coherent
optical processing the input data is converted into amplitude
transmittance variations.
[0006] Processing by a computer is usually serial in nature and the
processing speed is very limited. The use of an array processor
increases the amount of parallelism, as well as the processing
speed. True real time (speed of light) processing, however, is
still not possible with this approach.
[0007] Coherent optical processors can perform the Fourier
transformation in real time. However, spatially coherent
illumination suffers from phenomenon such as a speckle effect,
i.e., the appearance of bright dots (interference phenomena) in the
output correlation plane. When dealing with pattern recognition,
which is part of an image processing technique, this effect is
generally undesirable, since speckle can cause a false alarm in
identification of the object.
[0008] The use of incoherent light, avoids the speckle effect
(reducing signal noise) and may increase the dynamic range of the
resultant transform. Since optical processing is frequently
performed over the intensity rather than the field amplitude, an
incoherent system has a superior dynamic range over a coherent
system. Incoherent light based systems are generally less sensitive
to component deformation (e.g., flatness of a spatial light
modulator), thus reducing the severity of component specifications,
as compared to those of the coherent light based system.
[0009] Techniques aimed at performing the Fourier and related
transforms utilizing spatially incoherent light have been
developed. A shearing interferometer based technique appears to be
an attractive technique of the kind specified. This technique is
also useful for measuring wavefront parameters. The constructional
and operational principles of the shearing interferometer are
disclosed in the following publications:
[0010] (1) "OTF Measurements With a White Light Source: an
Interferometric Technique", J. C. Wyant, Applied Optics 14, 1619
(1975);
[0011] (2) "Use of an AC Heterodyne Lateral Shear Interferometer
With Real-Time Wavefront Correction Systems", J. C. Wyant, Applied
Optics. 14, page 2622, (1975).
[0012] A joint transform correlator (JTC), based on the shearing
interferometer, is disclosed in the following publication:
[0013] (3) "Joint Transform Correlator With Incoherent Output", D.
Mendlovic et al., JOSA A11, 3201-3205 (1994).
[0014] Generally speaking, the shearing interferometer based
technique deals with input objects that are (quasi-) monochromatic,
but spatially incoherent. In general, this means that points in the
input signal are (locally) temporally coherent but spatially
incoherent (with the other points). Herein, such objects are termed
"locally temporally coherent".
[0015] The principle of shearing interferometer based techniques
can be thought of as the creation of an interference pattern in an
output plane. This interference pattern can be formed by two light
sources each corresponding to an input signal. For example, one of
the sources may be a vector or an array of locally spatially
coherent light sources. Each source is the inverse image (either
real or virtual) of another source, and each position in each of
the sources is coherent only with its image source. To this end, a
common shearing interferometer optical setup is provided with two
signals indicative of the object, wherein each of these two signals
is obtained by imaging each position on the object (which is
spatially coherent with itself), thus enabling the interference
between these two signals.
[0016] FIGS. 1A and 1B illustrate known optical setups 1A and 1B
of, respectively, a shearing interferometer and a spatially
incoherent JTC utilizing the same. To facilitate understanding, the
same reference numbers are used for identifying those components,
which are common in the setups 1A and 1B.
[0017] The shearing interferometer setup 1A is composed of an
incoherent light source assembly 2 composed of spatially
continuous, spatially incoherent and locally temporally coherent
light, a beam splitter 3, a regular mirror 4 that creates a virtual
coherent image of each point source, and a comer prism 5. There are
two optical paths: one with via regular mirror 4, and the other via
corner prism 5. The path of the regular mirror 4 reflects the input
spatially incoherent image as is, and the path with the corner
prism 5 provides a reflected mirror image. Thus, the wavefronts
emanating from both optical paths interfere. This interference can
be detected in an output plane OP (located downstream of the beam
splitter), where an output acquisition device, e.g., CCD, is placed
(not shown). Interference based on source-and mirror-images is
known in the art, e.g., the Lloyd's mirror (Born & Wolf,
Principles of optics, 1980, p. 262).
[0018] In the incoherent JTC setup 1B, the shearing interferometer
of FIG. 1A is associated with a conventional coherent system 6 for
performing a Fourier transform. The system 6 comprises a coherent
light source 26 producing an input object, a lens 7, and an
optional filter (not shown). Mounted in the optical path of light
ensuing from the system 6 and propagating towards the shearing
interferometer 1A, is a rotating diffuser 9.
[0019] Referring additionally to FIG. 1A, due to the light passage
through the beam splitter 3, each point on the input image is
doubled. The fact that each point in the input image is temporally
coherent only with one single point in its mirrored image provides
a separate interference pattern due to each point of the image. Due
to the wave nature of light, the free space propagation of two
coherent points is an interference pattern with a frequency
proportional to the distance between these points. All other parts
of the image are fully incoherent with the two points, thus the
intensity follows the cosine transform. This is described in the
above publication (3).
[0020] The mathematical analysis in (3) shows that the amplitude
impulse response of the system is as follows:
.delta.(x-x.sub.0) cos(kx.sub.0.nu.) (1)
[0021] Here, .delta. is the delta functional; x.sub.0 is the
shifted center of the input signal (information to be transmitted);
k is a constant associated with the geometry of the shearing
interferometer, e.g., k=2.pi./.lambda.z corresponds to the
coordinate of the output plane.
[0022] Since for an impulse at the input, the output is purely
coherent, the output function is bipolar (includes positive and
negative values). However, available detectors are sensitive to
intensities, rather than fields. Thus, the intensity of the impulse
response must be considered:
I(x-x.sub.0) cos.sup.2(kx.sub.0.nu.)=0.5(1+cos(2kx.sub.0.nu.))
(2)
[0023] It should be noted that the sum of many incoherent signals
obtained, for example, from many discrete point sources of the
input object is represented by the sum of their intensities, rather
than their fields.
[0024] The intensity also acts as a cosine transform, but with a
certain bias. The mathematical analysis of the same are given in
the above publication (3). Evidently, this optical setup needs no
lens for performing the Cosine transform.
SUMMARY OF THE INVENTION
[0025] As indicated above, the prior art describes a method for
producing cosine transforms of an object utilizing a spatially
continuous light source. In general, sine transformations are
desirable or needed to fully define an input image. Furthermore,
many optical processing systems utilize a modulated light source
producing input light from an array of light emitting elements, or
a combination of a coherent light emitting element and a Spatial
Light Modulator (SLM). The modulated light beam is processed by
various optical elements, and the processed light is detected by a
detector such as a CCD. Both the modulated light source and the
detector, are discrete optical elements, while the nature of light
processing based on the known transformation techniques is
continuous. This introduces a mismatch between the input signal to
be processed (i.e., modulated light) and the detected output
signal.
[0026] An aspect of some embodiments of the invention, is concerned
with the provision of sine transforms for extended optical light
sources. In some embodiments of the invention both sine and cosine
transforms are provided. In some embodiments of the invention, the
source may be discrete or continuous.
[0027] In exemplary embodiments of the invention a shearing
generator is used in the production of the transform. In some
embodiments, two or more displaced and optionally inverted replicas
of an extended input optical signal are produced by the shearing
generator. The input signal modulated, locally temporally coherent
and spatially incoherent. Each point in the replicas has a
corresponding point in the other replica (which may be the input
signal) with which it is temporally coherent. However, it is
incoherent with all other points in the same and other replica. The
replicas are capable of interfering with each other in an output
plane. The so obtained interference pattern presents the transform
of the input signal.
[0028] For some embodiments of the invention, the replicas have the
same phase an output of the generator. These embodiments can be
used to produce cosine transforms. In other embodiments, the phase
difference between the replicas is .pi./2. These embodiments can be
used to produce sine transforms. Other transforms, such as Fourier
and Hartly transforms and two dimensional transforms can also be
produced utilizing the apparatus and methods disclosed herein.
[0029] An aspect, of some embodiments of the invention, is
concerned with the provision of transforms of discrete light
sources.
[0030] Some embodiments of the invention obtain an output signal,
indicative of the discrete transform (cosine and/or sine), of an
input signal (which may be either continuous or discrete), in an
output plane (where a detection means are placed), by means of an
optical system. In an exemplary embodiment of the invention, the
optical system is based on the principle of shearing
interferometry, and has a predetermined geometry constrained by a
matching condition between the input and output signals.
[0031] An aspect of some embodiments of the invention is concerned
with the provision of input/output matching in shearing
interferometers.
[0032] The inventors have found that one-dimensional and
two-dimensional discrete transform (cosine or sine) can be obtained
with a shearing interferometer configuration, provided it satisfies
a matching condition. Furthermore, the inventors have developed new
configurations of the shearing interferometer. Some of these
designs are less complex and/or more compact as compared to that of
the conventional shearing interferometer based systems.
[0033] The matching condition defines a predetermined distance
between the corresponding points in the input and output signals
(i.e., the center of a pixel of the input signal and the center of
a corresponding pixel of the output signal). Such matching is
termed herein position matching. Optionally, the size and or
intensity/sensitivity of the sources/detectors may be varied to
optimally match the source and detector. Such matching is described
below and in PCT application PCT/IL00/00282, filed May 19, 2000 and
published as WO 00/72105, the disclosure of which is incorporated
by reference.
[0034] An aspect of some embodiments of the invention relates to
optically generating a discrete transform of a complex input
signal.
[0035] Some embodiments of the invention deal with one-dimensional
objects, some with two-dimensional objects and some with both. In
the case of a one-dimensional object, the optical system provides
one pair of interfering signals, namely the original signal and one
additional signal, as described above. In the case of a
two-dimensional object, two mutually perpendicular pairs of the
interfering signals are provided (either simultaneously or
sequentially), the two signals of each pair interfering with each
other.
[0036] The term "locally temporally coherent" as used herein
signifies that each point in the input signal is only coherent with
itself and incoherent with the other points. It should be
understood that such coherence need not be perfect and that the
invention is also applicable to quasi-monochromatic signals for
which the wavelength variation during the difference in time of
flight of two interfering signals is negligible in comparison with
the mean wavelength of the radiation. This implies that generation
of the interfering images can be obtained from a single point
source and that dispersion effects can be neglected.
[0037] To provide the spatially modulated input signal, the light
source may comprise one or more light emitting elements (e.g., a
vector or an array of such elements or an expanded light source),
and an SLM, for example, an acousto-optic modulator. Alternatively,
the light source may comprise an array of vertical cavity surface
emitting lasers (VCSELs), thereby eliminating the need for any
additional SLM. Thus, the input signal is created either by
spatially modulating the light source or by changing the output of
each point-like light emitting element of the light source in
accordance with the information to be transformed. Alternatively,
acousto-optic modulators can be used as well to represent the input
signal.
[0038] In order to obtain N-elements discrete input signal, light
emitting elements of the light source assembly may be arranged in a
one- or two-dimensional array, based on the required
application.
[0039] To provide the at least one pair of interfering signals,
splitting of the input signal may be required, and possibly also
the appropriate rotation of the split-images, when needed. The
splitting can be achieved by prisms or by locating a suitable
splitting means, such as diffractive optical element(s), splitting
the energy in the optical path of light emitted by the light
source. Generally speaking such a splitting means creates two
images of an original input, such that each of the created images
has preferably half of the energy of the original input. When the
emitting divergence angle of the light source is not sufficiently
large to enable sufficient interference area in the output plane,
the splitting means is of a kind providing diffraction angles
higher than the divergence angle of the light source.
[0040] The splitting of the input signal can be achieved without
any specific splitting means, by using the light source
characterized by sufficiently wide divergence (e.g., suitable LEDs
or VCSELs). In this case, the two signals are obtained by using a
different part of the opening for each signal.
[0041] As for the rotation of the input signal or at least one of
the split signals, in some embodiments of the invention, it is
achieved as a by-product at the splitting stage, by using a mirror
as the splitting means. In other words, when dealing with one pair
of interfering signals (i.e., one-dimensional object), the input
signal can be directed onto a mirror normally (with zero incident
angle), thereby producing an inverted image of the input signal.
The two interfering signals, i.e., the input signal and its
inverted image, will produce an interference pattern in the output
plane. When dealing with two pairs of interfering signals (i.e.,
two-dimensional objects), a right-angle mirror may be used
producing three images of the input signal, thereby resulting in
two pairs of interfering signals, such that the signals in one pair
are parallel to each other, and the signals of different pairs are
perpendicular to each other.
[0042] In another embodiments of the invention, the mirror is
replaced by lens and/or prisms setups. The at least one pair of
interfering signals produced by splitting the input signal
propagates through the corresponding number of the lens and/or
prisms setups, which create the output signal in the form of the
interference pattern in the output plane.
[0043] In yet another embodiments of the invention, the optical
setup is designed like a planar sheet containing a beam splitter
and mirrors formed at some edges of the beam splitter.
[0044] According to yet another embodiments of the invention, the
optical assembly is based on a cubic beam splitter and a
phase-shifting means, so as to obtain simultaneously cosine and
sine discrete transform of the input signal.
[0045] An aspect of some embodiments of the invention is concerned
with the provision of practical two dimensional inputs for one
dimensional shearing generators. Where large amounts of data are to
be processed, it may be impractical to generate a single array of
discrete points to produce the input signal required for the
shearing generator. In exemplary embodiments of the invention,
where N elements are required, the elements are divided into m
pieces each having p elements, such that m*p=N; and the m pieces
are configured to be offset in a direction perpendicular to a data
axis of the shearing generator.
[0046] An aspect of some embodiments of the invention is concerned
with the provision of shearing generators of novel design and/or
performance characteristics.
[0047] In some embodiments of the invention the shearing generator
splits the image at an input plane sending them down different
paths to form two images at an output plane. These image are the
same except for an optional rotation and for a possible phase
shift. The phase shift may be zero. In some embodiments it is
.pi./2. In some embodiments, the difference in phase is produced by
providing a phase shifting element in one or both paths to
introduce a differential phase shift between the images at the
output plane. The phase shifting element can be a phase shifter
placed in one of the paths. In other embodiments, the phase
shifting element is part of a beam splitter that splits the input
image.
[0048] In some embodiments of the invention, the generation of two
images in a plane is produced without any reflective surfaces,
except for a partial reflection at a beam splitter. The beams are
guided to parallelism utilizing refractive surfaces of the beam
splitter itself or utilizing a separate prism or prisms.
[0049] There is thus provided in accordance with an exemplary
embodiment of the invention, optical apparatus for obtaining a
discrete transform of an input signal in an output plane, the
system comprising:
[0050] a light source positioned at an input surface, the source
comprising an array of N spaced temporally coherent, spatially
incoherent light elements representative of the signal, spaced with
a first spacing;
[0051] a detector positioned at an output surface, the detector
comprising an array of N spaced detectors, spaced with a second
spacing, such that it N samples light at the output surface;
and
[0052] an optical transformer that collects light from the light
source and transforms it into a pattern at the output surface, said
optical transformer being constructed such, such that the N
sampling of the output signal results in the discrete transform of
the input signal,
[0053] wherein said optical transformer includes a shearing
generator that provides two images of the light source that are
inverted forms of each other, interference between said images
providing a continuous interference pattern that is sampled by said
detector. Optionally, the shearing generator introduces a phase
difference between corresponding points on said two images such
that the transform is a sine transform. Alternatively or
additionally, the shearing processor provides a phase difference
between corresponding points on said two images such that the
transform is a cosine transform.
[0054] There is also provided in accordance with an exemplary
embodiment of the invention, optical apparatus for obtaining a sine
transform of an input signal in an output plane, the system
comprising:
[0055] a light source positioned at an input surface, said source
being a temporally coherent, spatially incoherent light
distribution representative of the signal;
[0056] a detector positioned at an output surface, that detects
light at the output surface; and
[0057] an optical transformer that collects light from the light
source and sine transforms it into a pattern at the output surface,
said output pattern representing a sine transform of the input
signal,
[0058] wherein said optical transformer includes a shearing
generator that provides two images of the light source that are
inverted forms of each other and have a phase shift between them,
such that interference between said images provides a continuous
interference pattern that is sampled by said detector. Optionally,
the input source is a discrete source and the detector samples the
output to provide a discrete sine transform of the input
source.
[0059] In an exemplary embodiment of the invention, said optical
apparatus provides both sine and cosine transforms or at least one
combination of sine and cosine transforms.
[0060] In an exemplary embodiment of the invention, said transform
is a two dimensional transform. Optionally, the optical assembly
comprises anamorphic optics accommodated in the optical path of the
two interfering signals, the system thereby providing for obtaining
a two-dimensional transform of a two-dimensional input signal.
[0061] In an exemplary embodiment of the invention, the transform
is a JPEG transform. Alternatively, the transform is a Fourier
transform. Alternatively, the transform is a Hartly transform.
[0062] In an exemplary embodiment of the invention, the signal
represents an image.
[0063] In an exemplary embodiment of the invention, the optical
transformer is constructed and configured to match the input source
and detector geometry, and said matching condition is at least
partly determined by a distance between the input and output
signals. Optionally, said matching condition comprises matching
position of detector elements to the geometry of elements of the
source. Alternatively or additionally, said matching condition
comprises matching one or more of sensitivity, gains, intensity and
sizes of elements of the source and detector and the optical
geometry between the input and output.
[0064] In an exemplary embodiment of the invention, the light
source includes a highly coherent or partially coherent light
emitting element, and a spatial coherence removing element.
[0065] In an exemplary embodiment of the invention, the light
source includes an array of temporally coherent, but mutually
spatially incoherent light emitting elements. Optionally, said
light source includes vertical cavity surface emitting lasers
(VCSEL).
[0066] In an exemplary embodiment of the invention, the light
source comprises a one-dimensional array of N light emitting
elements, the total length of said array being not less than N*a,
wherein a the pitch of the array.
[0067] Alternatively, the light source comprises a two-dimensional
array of N light emitting elements.
[0068] Alternatively, the N light emitting elements are aligned
along a shearing symmetry X-axis of the system, said
two-dimensional array being formed by a plurality of m parallel
portions extending along the X-axis and aligned in a spaced-apart
relationship along a Y-axis, each of the portions containing n
light emitting elements aligned in a spaced-apart relationship
along the X-axis with a pitch a, the entire number N of the light
emitting elements of the light source being n*m.
[0069] In an exemplary embodiment of the invention, the N light
emitting elements are aligned along a shearing symmetry X-axis of
the system, said two-dimensional array being formed by a plurality
of m parallel sections, each extending along the X-axis and aligned
in a spaced-apart relationship along a Y-axis, each of the sections
containing n light emitting elements aligned in a spaced-apart,
overlapping relationship along the X-axis, such that the each light
emitting element in each section is separated by the distance a
from a nearest light emitting element in another, locally adjacent
section and separated by a distance m*a from an adjacent element on
the same section.
[0070] In an exemplary embodiment of the invention, the light
source comprises a two dimensional rectangular array ofN light
emitting elements is arranged in an mxn matrix, wherein N=m*n, the
m and n elements being aligned along the X- and Y-axes with pitches
a and b, respectively, and said input signal is created by tilting
the matrix by an angle .beta., satisfying the condition of
a*cos.beta.=n*b*sin.beta..
[0071] In an exemplary embodiment of the invention, the shearing
generator comprises a splitting sub-assembly that splits light from
the input source into at least one pair of interfering light waves,
and defines different optical paths for the light propagation of
the light waves. Optionally, said splitting sub-assembly comprises
a diffractive optical element. Alternatively or additionally, said
splitting sub-assembly comprises a beam splitter. Optionally, the
beam splitter is a cubic beam splitter.
[0072] In an exemplary embodiment of the invention, the splitting
sub-assembly comprises a mirror. Alternatively, said splitting
sub-assembly comprises mirrors accommodated with respect to the
light source assembly so as to provide three images of the input
signal, the input signal and the three images forming two pairs of
the interfering signals, the system to provide a two-dimensional
transform of a two-dimensional input signal.
[0073] In an exemplary embodiment of the invention, the appartus
includes a phase shifting element formed by a coating on an inner
diagonal surface of the beam splitter so as to provide a desired
phase shift between the different parts of the input signal
transmitted through and reflected from two parts of said surface at
two opposite sides of a bisection line of the beam splitter,
respectively.
[0074] In an exemplary embodiment of the invention, said splitting
sub-assembly comprises a grating assembly splitting the input
signal, the optical assembly further comprising a rotating
sub-assembly in the optical path of the splitting sub-assembly for
creating the two pairs of interfering signals.
[0075] In an exemplary embodiment of the invention, the optical
assembly defines a substantially equal length of optical paths for
the two images, thereby providing a cosine transform of the input
signal in the output plane.
[0076] In an exemplary embodiment of the invention, the optical
assembly comprises a phase shifting element in one of the two
optical paths, so as to provide a .pi./2 phase difference in the
length of optical paths for the two images, thereby providing a
sine transform of the input signal in the output plane. Optionally,
the phase shifting element comprises a coating on a reflecting or
refractive surface of the splitting sub-assembly.
[0077] In an exemplary embodiment of the invention, the input light
is a conjugate symmetric signal, the light source assembly
comprising two separate arrays of light emitting elements for
producing real and imaginary parts of the input signal.
[0078] In an exemplary embodiment of the invention, the input light
is a conjugate symmetric signal, the light source assembly
comprising two separate arrays of light emitting elements for
producing real and imaginary parts of the input signal and wherein
the optical transformer includes a cubic beam splitter accommodated
with respect to the arrays of light emitting elements so as to
transmit the real and imaginary parts of the input signal through
an inner diagonal surface of the beam splitter from opposite sides
of said surface.
[0079] In an exemplary embodiment of the invention, said optical
assembly comprises at least one pair of separate setups for
creating said at least one pair of interfering images in the form
of, respectively, an inverted image and non-inverted image of the
input signal. Optionally, the optical assembly has a planar
structure, two optical paths defined by the two separate setups
being placed beside each other.
[0080] There is also provided in accordance with an exemplary
embodiment of the invention, a shearing generator comprising;
[0081] an input light source;
[0082] an image generator that generates two images of the input
light source at an output, the image generator comprising a beam
splitter that splits light from the input source into at least one
pair of interfering light waves at an output thereof, and defines
different optical paths for the light propagation of the light
waves, said optical paths having a difference in length of .pi./2.
Optionally, the optical paths including at least one phase shifting
element that provides for a different phase shift for the two
paths.
[0083] There is also provided in accordance with an exemplary
embodiment of the invention, a shearing generator comprising:
[0084] an input light source;
[0085] an image generator that generates two images of the input
light source at an output plane, the image generator comprising a
beam splitter that splits light from the input source into at least
one pair of interfering light waves at an output thereof, and
defines different optical paths for the light propagation of the
light waves, said optical paths including at least one phase
shifting element that provides for a different phase shift for the
two paths.
[0086] Optionally, the phase shifting element comprises a coating
at said beam splitter. Alternatively or additionally, the phase
shifting element comprises a phase shifting element in one of the
paths. Optionally, the phase shifting element is a coating on an
external surface of an optical block containing the beam
splitter.
[0087] In an exemplary embodiment of the invention, the light waves
form coplanar images at said output, said waves propagating
normally to said output, the images being provided without
reflection except from said beam splitter.
[0088] There is also provided in accordance with an exemplary
embodiment of the invention, a mirror-less shearing generator
comprising:
[0089] an input source;
[0090] an image generator that generates two images of the input
light source at an output plane, the image generator comprising a
beam splitter that splits light from the input source into at least
one pair of interfering light waves at an output thereof, and
defines different optical paths for the light propagation of the
light waves, neither of said optical paths including a reflecting
surface, except for said beam splitter.
[0091] In an exemplary embodiment of the invention, the beam
splitter provides two nonparallel waves and including at least one
refractive element that refracts at least one of the waves, so that
the waves are parallel. Optionally, the refractive elements are
separate from the beam splitter. Alternatively, the refractive
elements are interfaces at outer walls of the beam splitter.
[0092] There is also provided in accordance with an exemplary
embodiment of the invention, a method of constructing an discrete
input having N elements, for a one dimensional shearing generator
having a data axis, the method comprising:
[0093] providing m sections each having p elements, such that
m*p=N; and
[0094] configuring said m pieces offset in a direction
perpendicular to the data axis.
[0095] Optionally, the elements in each section are separated in
the data direction by a distance .alpha., adjoining sections being
offset in the data direction such that a first element of one
section is offset by .alpha. distance a from a last element of an
adjoining section, such that the N elements are spaced by .alpha.
in the data direction. Alternatively, the elements in each section
are separated by a distance m*.alpha. in the data direction,
adjoining sections being offset in the data direction such that a
first element of one section is offset by a distance a from a last
element of an adjoining section, such the N elements are spaced by
.alpha. in the data direction. Alternatively, the elements in the
sections form a rectangular matrix of evenly spaced elements and
wherein the elements are spaced with respect to a first axis of the
matrix by a spacing .alpha./cos.beta. and rotating the rectangular
matrix with respect to the data axis such that the first matrix
axis makes an angle of .beta. with the data axis.
BRIEF DESCRIPTION OF THE DRAWINGS
[0096] In order to understand the invention and to see how it may
be carried out in practice, a preferred embodiment will now be
described, by way of non-limiting example only, with reference to
the accompanying drawings, in which:
[0097] FIGS. 1A and 1B are prior art shearing-interferometer-based
systems;
[0098] FIG. 2 is a schematic block diagram of an optical system
according to an embodiment of the invention;
[0099] FIGS. 3A to 3E illustrate several possible examples for a
light emitting elements arrangement suitable for use in a light
source of the system of FIG. 2, accordance with embodiments of the
invention;
[0100] FIG. 4 illustrates an example of a cubic beam splitter based
optical assembly suitable to be used in the system of FIG. 2, in
accordance with an embodiment of the invention;
[0101] FIG. 5 illustrates another example of a cubic beam splitter
based optical assembly suitable to be used in the system of FIG. 2,
in accordance with an embodiment of the invention;
[0102] FIGS. 6A and 6B illustrate two more examples of a cubic beam
splitter based optical assembly suitable to be used in the system
of FIG. 2, in accordance with embodiments of the invention;
[0103] FIG. 7 more specifically illustrates an example of the
system of FIG. 2 designed for directly obtaining DCT and DST of a
conjugate symmetric input signal in an output plane, in accordance
with an embodiment of the invention;
[0104] FIGS. 8A and 8B illustrate two more examples, respectively,
of the system according to an embodiment of the invention,
utilizing a cubic beam splitter based shearing interferometer;
[0105] FIGS. 9A to 9C illustrate three mirror-based setups,
respectively, suitable to be used in the optical assembly of the
system of FIG. 2, according to embodiments of the invention;
[0106] FIGS. 10A and 10B show two possible examples of the
implementation of the system of FIG. 2 suitable for processing
one-dimensional objects, in accordance with embodiments of the
invention;
[0107] FIGS. 11A and 11B show other examples of apparatus according
to embodiments of the invention suitable for processing
one-dimensional signals, with more compact (e.g., shorter)
configurations, as compared to those of the examples of FIGS. 10A
and 10B;
[0108] FIG. 12 illustrates a system according to an embodiment of
the invention utilizing a multi-channel setup, each constructed in
a manner similar to the system of FIG. 2;
[0109] FIG. 13 illustrates one possible example of a system
according to an embodiment of the invention utilizing planar
optics, suitable for processing one-dimensional signals, wherein
two optical paths are placed one over the other;
[0110] FIGS. 14A and 14B illustrate the top and side views,
respectively, of a system according to another possible example of
the planar optics based system suitable for processing
one-dimensional signals, in accordance with an embodiment of the
invention;
[0111] FIG. 15 illustrate a system according to an embodiment of
the invention, suitable for performing one-dimensional cosine
transform, in which a mirror is used as a splitting
sub-assembly;
[0112] FIG. 16 illustrates an example of a system according to an
embodiment of the invention for carrying out a two-dimensional JPEG
transform;
[0113] FIG. 17 schematically illustrates the operational principles
of a system, according to an embodiment of the invention, for
generating 2-D shearing interference; and
[0114] FIG. 18 illustrates another possible example of a system,
according to an embodiment of the invention, carrying out the
kernel transform.
DETAILED DESCRIPTION OF SOME EMBODIMENTS OF THE INVENTION
[0115] FIGS. 1A and 1B illustrate the conventional configurations
of the shearing interferometer based systems that are known as
capable of performing the Cosine transform of an input signal.
[0116] As indicated above, for most practical cases of signal
processing, a discrete format of transformation is needed. It will
be shown in the description below that a discrete cosine and sine
transforms can be obtained with configurations similar to or, in
some cases the same as the conventional shearing interferometer
configurations.
[0117] Reference is made to FIG. 2 illustrating a schematic block
diagram of an optical system 10 according to the invention. The
system 10 comprises a light source 12 that produces a spatially
modulated spatially incoherent locally temporally coherent input
signal IS (corresponding to an input object or other image, such a
computer generated image, which is not specifically shown, or a
non-image signal), which is to be transformed in an output plane
OP, and an optical assembly 14 that produces at least one pair of
interfering signals producing an interference pattern in output
plane OP.
[0118] In the output plane, although not specifically shown, a
detector array is generally placed. The manner in which, among
other things, this array should be placed and detectors be
configured with finite size sources is defined by the matching
condition described below.
[0119] As indicated above, the light source 12 is locally
temporally coherent (and, as defined above, spatially incoherent).
Such a light source may include a highly coherent light emitting
element (e.g., diode laser) or a partially coherent light emitting
element (such as LED), and a suitable coherence breaking means
(e.g., diffuser). Alternatively, an array of coherent light
emitting elements such as vertical cavity surface emitting lasers
(VCSEL) can be used. The main features to be considered when
choosing a light source, as well as the advantages of using VCSELs,
will be described more specifically below.
[0120] The optical assembly 14 operates like a shearing
interferometer, namely, provides a pair of images of the input
signal IS, and defines the optical length for their propagation to
the output plane OP, so as to enable interference between these
images.
[0121] Consider the input signal transformation to be carried out
by optical assembly 14 for obtaining an output signal in the form
of a discrete cosine transform (DCT) in the output plane. It should
be noted that, in the following disclosure, various mathematical
versions of the DCT were used to derive the equations in each of
the specific cases. Each of the aforesaid equations, which were
derived using a specific version, can be derived using the other
versions in a similar way.
[0122] For a given input signal .function.(x), the shearing
interferometer provides the following output: 1 F ( v ) = 0 D f ( x
) [ 1 2 * ( 1 + cos ( 2 k x v ) ) ] x ( 3 )
[0123] wherein D is the dimension of the input signal.
[0124] The input signal .function.(x) is locally temporally
coherent, but is spatially incoherent, such that:
(.function.(x.sub.1).multidot..function.(x.sub.2)=.delta.(x.sub.1-x.sub.2)
(4)
[0125] wherein the angle brackets stand for temporal averaging.
[0126] Considering discrete signal processing, the input
information contains N elements (i.e., N light emitting elements of
a light source or modulated light elements). Assume that the output
information is sampled Ntimes. The samples in the output plane
(i.e., detected output signal) are thus: 2 F ( k ) = n = 0 N - 1 f
( n ) cos ( n k N ) ( 5 )
[0127] Here, the constant bias has been omitted. It is assumed here
that a matching condition between the input and output signals is
satisfied (as will be described further below).
[0128] The above equation (5) presents one possible definition of a
DCT (DCT-I). It should be understood that another definitions of
DCT could be obtained, defined by appropriate sampling locations at
the output plane.
[0129] FIGS. 3A-3E, illustrate several possible examples for
arranging light emitting elements for use in light source 12 of an
system 10 for obtaining N-elements discrete input signal IS, in
accordance with various embodiments of the invention.
[0130] A simple example is shown in FIG. 3A. The input signal IS is
formed by a one-dimensional array 112A of N light emitting
elements, generally indicated as LE. Such an array may be an array
of VCSELs or a laser diode array. The minimal size of array ll2A is
determined by a pitch size a, i.e., the space between the centers
of two adjacent elements LE. Hence, the total length of the array
112A cannot be less than the product N.multidot.a.
[0131] In some applications, where tens or even hundreds of light
emitting elements are required to form an input signal, such
one-dimensional array 112A will significantly increase the
dimensions of the entire system, and will be very complicated to
manufacture. Manufacturing limitations may include the maximum
number of light emitting elements that can be produced into one
array, and/or the minimal distance (pitch) between each two
adjacent light sources in this array.
[0132] FIGS. 3B-3E exemplify how the above limitations of a single
one-dimensional array can be overcome. The method is based on the
fact that a lensless 1-D shearing interferometer produces a
one-dimensional output, with no explicit dependence on the other
dimension. Generally, here, N input units (N elements of the input
signal) are formed by m parallel arrays (pieces) each containing n
light emitting elements, the entire number N of input units being
thereby n m. The number n of the elements in the pieces is defined
by manufacturing capabilities and system dimensions. The n light
emitting elements in each piece A.sub.i (i=1, . . . , m) are
linearly aligned in a spaced-apart relationship defining an x-axis,
and the m pieces A.sub.i are aligned with offsets in one or both of
the x- and y-axes.
[0133] FIG. 3B illustrates one of the m pieces A.sub.i composed of
n light emitting elements LE. FIGS. 3C-3E illustrate three
different configurations, respectively, utilizing m arrays A.sub.i,
wherein the configuration of FIG. 3C overcomes the limitation
associated with the light source array size, and the configurations
of FIGS. 3D and 3E can generally overcome both manufacturing and
space limitations.
[0134] In a light source 112C of FIG. 3C, the pieces A.sub.i (only
three pieces A.sub.1, A.sub.2 and A.sub.3 being shown in the
figure) are shifted with respect to each other along the x-axis
with a certain overlapping distance between each two locally
adjacent pieces. This overlapping distance is determined such that
the n.sup.th light emitting element LE.sub.n in the piece A.sub.i
is separated by a distance a from the first light emitting element
LE.sub.1 in the piece A.sub.i+1. It should be noted that for a
one-dimensional shearing setup, only the distances along the x-axis
are relevant (assuming the x-axis is the shearing symmetry axis of
the setup).
[0135] In a light source 112D of FIG. 3D, the pieces Ai (only
pieces A.sub.1-A.sub.4 being shown in the figure) are shifted with
respect to each other along the x-axis a distance of a/m. As a
result, (m-1) light emitting elements of (m-1) successive pieces
A.sub.(2), A.sub.(3), . . . A.sub.(m), respectively, are aligned
along the x-axis within the space between two locally adjacent
light emitting elements of the piece A.sub.(1). As clearly seen in
the figure, two locally adjacent light emitting elements
LE.sup.(1).sub.1 and LE.sup.(1).sub.2 of the piece A.sub.1 are
aligned along the x-axis with the pitch a, and light emitting
elements LE.sup.(2).sub.1, LE.sup.(3).sub.1 and LE.sup.(4), of
pieces A.sub.2, A.sub.3 and A.sub.4, respectively, are aligned
along the x-axis within this space a, being spaced from each other
the distance a/m; light emitting elements LE.sup.(2).sub.2,
LE.sup.(3).sub.2 and LE.sup.(4).sub.2 are located within the space
between the elements LE.sup.(1).sub.2 and LE.sup.(1).sub.3;
etc.
[0136] FIG. 3E illustrates a light source 112E in the form of
m.times.n matrix (block) of light emitting elements LE.sub.ij (i=j,
. . . , m; j=1, . . . , n). In the present example: m=6 and n=7.
The light emitting elements LE.sub.ij are aligned in a spaced-apart
relationship along the x- and y-axes with pitches a and b,
respectively.
[0137] It is understood that, when the block 112E is tilted by an
angle .beta., pitches between two adjacent light emitting elements
LE.sub.i1 and LE.sub.(i+1).sub..sup.2 along the x- and y-axes are,
respectively, a-cos.beta. and b-sin.beta., such that all of the
light emitting elements (i.e., N=nm) are spaced along the x-axis
with a pitch (a.multidot.cos.beta.)/n between each two adjacent
light emitting elements. To this end, it is helpful if the angle
.beta. satisfies the following condition:
a.multidot.cos.beta.=n.multidot.b.multidot.sin.beta.- . It should
be noted that this configuration has an advantage for a
two-dimensional shearing setup. As far as there is flexibility with
respect to the numbers n and m, this configuration allows for the
pitches between the light emitting elements along the x- and
y-axes. If, for example, the matrix is symmetric (m=n) and
.beta.=45.degree., then the spacing along the x- and y-axes are
equal.
[0138] The DFT, {.sub.k}, of an N-sample sequence, {u.sub.m}, is
defined as follows, for the discrete Fourier Transform: 3 u ~ k = m
= 0 N - 1 u m exp ( - j 2 N m k ) with k = 0 , 1 , , N - 1 ( 6
)
[0139] The output signal can be split into its real and imaginary
parts: 4 u ~ k = m = 0 N - 1 [ Re { u m } cos ( 2 N m k ) + Im { u
m } sin ( 2 N m k ) ] + ( 6 b ) i m = 0 N - 1 [ Im { u m } cos ( 2
N m k ) - Re { u m } sin ( 2 N m k ) ]
[0140] It is clear that the output is a combination of type I
cosine and sine transforms of the real and the imaginary parts of
the input, provided that only even samples are considered. Although
equation (6b) is not the only possible decomposition, it shows that
the capability to perform type I cosine and sine transforms is
sufficient for obtaining a DFT of an arbitrary complex function. It
is understood that since the real and imaginary parts of the output
are space coincident, the real and imaginary parts are not
separable.
[0141] A case of particular interest is when the output of the DFT
is real. In order to obtain a real output of the DFT, the input
sequence {u.sub.m} must be conjugate symmetric as in a case of
complex valued {u.sub.m}, i.e., u.sub.m=u*.sub.-m. As particular
cases, the DFT of a symmetric real input or an antisymmetric
imaginary input are also real. A conjugate symmetric vector can be
constructed from an arbitrary complex valued vector {u.sub.m}, by
defining a 2N-sequence, namely {.nu..sub.m}, obtained by adding a
mirror image to the real part of {.sub.k} and a sign reverted
mirror image (anti-symmetric) to its imaginary part, namely, as
follows:
{.nu..sub.m}={0, u.sub.1, u.sub.2, . . . , u.sub.N-1, 0,
u*.sub.N-1, u*.sub.N-2, . . . , u*.sub.1} (7)
[0142] The output of such a 2N-DFT is as follows: 5 v ~ k = m = 0 2
N - 1 v m exp ( - j N m k ) = 2 [ m = 0 N - 1 Re { u m } cos ( 2 N
m k ) + m = 0 N - 1 Im { u m } sin ( 2 N m k ) ] ( 8 )
[0143] The above expression (8) presents cosine and sine transforms
of, respectively, the real and imaginary part of the input
{u.sub.m}.
[0144] For the optical implementation, a source located at position
x.sub.0 in the (x,y)-plane produces a complex amplitude at the
({tilde over (x)}, {tilde over (y)})-plane separated a distance z
from the source. Assuming Fresnel approximation, this complex
amplitude is as follows: 6 u z ( x ~ ) = exp ( - j z ( x ~ - x 0 )
2 ) ( 9 )
[0145] In the shearing interferometer setup, for each point in the
input separated a distance x.sub.0 from the origin of coordinates
of the system, a mirror image is obtained at a distance (-x.sub.0).
The mirror source will, in general, have a different phasing than
the original source, due to the reflection or the different paths
between both sources and the output plane. If .phi. is the phase
added to the mirror image, the total field of a set of N sources
(light emitting elements) with the spacing .DELTA.x and amplitudes
{u.sub.m} is determined by the following equation (10): 7 u z ( x ~
) = m = 0 N - 1 u m exp ( - j z ( x ~ - m x ) 2 ) + m = 0 N - 1 u m
exp ( - j z ( x ~ + m x ) 2 + j ) = m = 0 N - 1 u m exp ( - j z ( x
~ 2 + m 2 x ) 2 ) [ exp ( j 2 z m x ~ x ) + exp ( j ) exp ( - j 2 z
m x ~ x ) ] = m = 0 N - 1 u m exp ( - j z ( x ~ 2 + m 2 x 2 ) ) exp
( j 2 ) [ exp ( j ( 2 z m x ~ x - 2 ) ) + exp ( - j ( 2 z m x ~ x -
2 ) ) ] = 2 m = 0 N - 1 u m exp ( - j z ( x ~ 2 + m 2 x 2 ) ) exp (
j 2 ) cos ( 2 z m x ~ x - 2 )
[0146] For input sources that are mutually incoherent, when the
output is sensed in a square law detector, the detected intensity
is: 8 I z ( x ~ ) = 2 m = 0 N - 1 u m 2 cos 2 ( 2 z m x ~ x - 2 ) =
m = 0 N - 1 u m 2 [ 1 + cos ( 4 z m x ~ x - ) ] = m = 0 N - 1 u m 2
+ m = 0 N - 1 u m 2 cos ( 4 z m x ~ x - ) ( 11 )
[0147] Except for an additive factor, the output is a harmonic
transformation of the intensities in the input sources. Sampling
the output with an interval .DELTA.{tilde over (x)}, the following
final result is obtained: 9 I z ( x ~ = k x ~ ) = I k = m = 0 N - 1
u m 2 + m = 0 N - 1 u m 2 cos ( 4 z m k x ~ x - ) ( 12 )
[0148] The results depend on the phase .phi., as follows:
[0149] if .phi.=0, the output of the interferometer is a cosine
transform;
[0150] if .phi.=.pi., the output of the interferometer is a minus
cosine transform;
[0151] if .phi.=.pi./2: the output of the interferometer is a sine
transform.
[0152] It is this evident that the relation between the result in
(12) and the output for the DFT of a conjugate symmetric function
(equation (8) above) is straightforward since .phi. determines
whether symmetric or anti-symmetric input is used for the
interference and subject to application of a matching condition. It
should be noted that, in order to obtain the mathematical transform
(8) by the optical transform (12), the following matching condition
should be satisfied: 10 4 z m k x ~ x = N m k or 4 N x ~ x = z ( 13
)
[0153] Here, .DELTA.x is the location of a specific pixels in the
input signal, .DELTA.{tilde over (x)} is the location of the
corresponding pixels in the output signal, m and k are numbers of
the pixels in the input and output signals, respectively, and z is
the distance between the input and output planes. Such a matching
results with DCT-I and DST-I transforms of the real and the
imaginary part, respectively, of the original input {.sub.k}.
[0154] It should be understood that in order to practically
optically realize DFT of Conjugate symmetric input (i.e., equation
(7) above), as required in an OFDM (Orthogonal Frequency Division
Multiplex), by performing DCT and DST of the original complex, the
input is split into real and imaginary parts and each part is
treated separately in the optical system. The sampling and the
matching conditions can be adjusted to obtain other discrete
transform definitions, as described below.
[0155] In order to optically perform a DFT transform of a conjugate
symmetric input, two separated optical systems with different
optical path difference are desirable. This does not substantially
change the geometry of the setup, and parallel fringes are still
obtained in the output plane. Since a fast phase change is
generally difficult to achieve, optical assembly 14 (FIG. 2) is
generally composed of two separate setups, one for the DCT and the
other for the DST.
[0156] FIG. 4 illustrates an example of such an optical assembly
114, in which a cubic beam splitter 103 is utilized such that
incident light Linc (generated by a light source, which is not
specifically shown) propagates towards beam splitter 103 parallel
to its diagonal side 104, which acts as an interface. While
propagating through the beam splitter and impinging onto the
interface 104, the incident light beam is split into two parts
(i.e., light components reflected and transmitted through the
interface 104, respectively). Thus, two output beams L.sup.(1) and
L.sup.(2) are produced, which are capable of interfering with each
other. Beams L.sup.(1) and L.sup.(2) further propagate along
spatially separated optical paths (presenting two setups,
respectively). In the present example, a desired phase difference
between the output beams is achieved by providing one of the
setups, e.g., that associated with the propagation of beam
L.sup.(1), with a retarder RT (e.g., phase plate). It should,
however, be noted that the same result could be achieved by
fabricating the diagonal side 104 in the form of a multilayer
structure that adjusts one or both of the phase of one of the beams
for either cosine or sine transform.
[0157] According to another example shown in FIG. 5, the design of
a cube beam splitter based optical assembly 214 one can parallelize
for DFT of a conjugate symmetric input, as required for example in
ADSL communication systems. Here, two separate setups (optical
paths) of light propagation are created by dividing a beam splitter
203 into two halves, an upper part 203A and a lower part 203B,
located at opposite sides of a plane 205. Upper and lower parts
204A and 204B of a diagonal side 204 are fabricated so as to
provide appropriate phase difference between transmitted and
reflected light components. To this end, the upper part 204A has a
coating of a kind producing no phase shift between transmitted and
reflected beams, while the lower part 204B produces a .pi./2 phase
shift. Separate light source arrays are provided for producing,
respectively, the real and imaginary parts of an input signal and
directing them into the optical assembly 214. A detector array 215
is appropriately mounted in the output plane, and is preferably
oriented such that its axis intersects with the bisection line 205.
The detector 215 senses the addition of the cosine and sine
transforms of the light source arrays, respectively. Alternatively,
by operating the input sources in two different cycles, DCT and DST
transforms can be obtained sequentially. Alternatively, multiple
sets of detectors can be provided. In this an other embodiments, in
which a coating is used to introduce a phase shift at the beam
splitter, a phase shifter (such as a coating at an output surface
of the structure or a separate phase shifter in one of the paths)
may be used in some embodiments.
[0158] Turning now to FIGS. 6A and 6B, there is illustrated an
optical assembly 314 utilizing a cube beam splitter 303 having its
diagonal side 304 formed with a multilayer coating designed such
that the relative phase between transmitted and reflected beams
either L.sup.(1) or L.sup.(2) is different, depending on the
direction of input beam propagation towards the interface 304
(diagonal side). If the input beam L.sub.0 impinges onto the
diagonal side at its surface 304A (FIG. 6A), the resultant phase
difference is 0, and if it impinges onto the opposite surface 304B
(FIG. 6B), the resultant phase difference is .pi./2.
[0159] FIG. 7 (which is structurally the same as FIG. 6)
illustrates an example of a system 400 designed for directly
obtaining DCT and DST of the conjugate symmetric input signal in
the output plane. As shown, a light source 412 is a two-part
source, the source parts 412A and 412B generating, respectively,
real and imaginary parts I.sub.r and I.sub.im of the input signal.
An optical assembly 414 includes a cube beam splitter 403,
configured with respect to light source parts 412A and 412B such
that the light components I.sub.r and I.sub.im impinge onto the
opposite surfaces 404A and 404B, respectively, of the diagonal side
404. Output beams L.sup.(1) and L.sup.(2) propagate towards a
detector 415 located in the output plane.
[0160] FIGS. 8A and 8B illustrate two more examples, respectively,
of embodiments of the invention, utilizing a cube beam splitter
based shearing interferometer. The cube beam splitter is
accommodated such that its inner reflecting surface extends along
the optical axis of light propagation through the system.
[0161] In a system 500A (FIG. 8A), a VCSEL array (light source)
512A presenting an input signal is located at one side of a cube
beam splitter 503A. Light propagation through the beam splitter
results in the creation of a first image I.sub.1 and a second,
inverted image I.sub.2 of the VCSEL array 512A. A prism 505A is
accommodated in the optical path of images I.sub.1 and I.sub.2 so
as to ensure the parallel propagation of these light components
towards the output plane OP in which a sensing surface of a
detector (e.g., CCD) is located. Further provided in system 500A is
an imaging lens L, which is installed in the optical path of the
light components exiting from the prism and concentrates the fringe
pattern from all the VCSEL elements at the center.
[0162] In a system 500B (FIG. 8B), a VCSEL array 512B is situated
at an angle to a cube beam splitter 503B, such that light
components exiting from the beam splitter and representing two
images I.sub.1 and I.sub.2 of the VCSEL array, are spatially
separated and propagate parallel to the optical axis. Similarly,
the imaging lens L concentrates the fringe pattern from all the
VCSEL elements at the center. This optical setup provides
propagation of both images without any additional optical
elements.
[0163] Reference is now made to FIGS. 9A-9C, illustrating
mirror-based setups suitable to be used in the optical assembly. As
shown in FIG. 9A, the shearing can be obtained with a single mirror
614A. The selection of the type of mirror plays a fundamental role.
For practical considerations (e.g., setup and detector dimensions,
resolution), the shearing requires a substantially grazing angle of
incidence (e.g., close to .pi./2).
[0164] A DCT transform can be obtained using a metallic mirror
(single reflection in a metal). The use of such a mirror will
result in a phase difference between incident and reflected beams
of nearly n radians for grazing incidence. This generates a pair of
images (one real and one imaginary) suitable for providing a DCT.
It should be noted that the phase and amplitude changes with the
angle of incidence faster when the polarization of incident light
is parallel to the plane of incidence. Therefore, perpendicular
polarization is preferably used. By coating the surface with a
phase retarding material, the phase of the reflected wave can be
reversed, so that the two images are suitable for a DST.
[0165] A dielectric interface (single reflection in the interface
between two dielectric media) may be used. Alternatively or
additionally, internal and/or external reflection is used. In the
case of internal reflection, the refraction index of that side of
the interface where the emitters and detectors are places is larger
than that of its opposite side. At grazing incidence, although the
modulus of the reflection coefficient is unity for both
polarization states, the phase varies as a function of the angle of
incidence. Therefore, at grazing incidence, the DCT can be
obtained, while at the angle of incidence providing .pi./2 phase
difference with a single polarization state, the setup may be used
to obtain the DST. In the case of external reflection, the
refraction index of that side of the interface where the emitters
and detectors are placed is smaller than that of the opposite side
of the interface. The phase is exactly .pi. for a wide range of
incidence angle for the two polarization states, thereby enabling
to carry out the DCT. The amplitude varies faster in the case of
parallel polarization. If either parallel or perpendicular
polarization are used, the only effect will be a different
weighting factor for every frequency in the DCT. This error is
correctable, for example, by pre-weighting the intensity of the
sources.
[0166] An interface coating can be used, namely, the mirror's
surface, metallic or dielectric, can be formed with single or
multilayer coating. The design of the number of layers, refractive
indices and thickness can provide any desired phase difference
between incident and reflected beams for a given polarization.
Hence, it is possible to obtain either a DCT or a DST
transform.
[0167] FIG. 9B exemplifies a combination of mirrors suitable for
simultaneously obtaining DCT and DST of an input function by
combining .pi. (or 0) and .pi./2 phase differences between incident
and reflected beams. Here, a mirror 614 is divided into two parts
614A and 614B, the upper mirror-part 614A being a metallic or
dielectric mirror providing a .pi. phase shift, and the lower
mirror-part 614B being a multilayer mirror, providing .pi./2 phase.
That light component of input light which, whilst propagating from
an input source 612 towards an upper array of detectors 615A
impinges onto the mirror part 614A, is reflected therefrom without
phase shift. The DCT is thereby obtained on the array 615A. As for
the light component of the input light interacting with the lower
mirror part 614B, it undergoes a .pi./2 phase shift. Therefore the
lower photodetector array 615B will sense the DST.
[0168] The optical setup would need two cycles for a DFT
transformation of a hermitic vector. In the first cycle, the real
part is presented in the array, and the DCT is sensed in the upper
detectors 615A (corresponding to the metallic mirror). In a second
cycle, the imaginary part is displayed, and the DST is sensed in
the lower detectors 615B (corresponding to the dielectric
reflection).
[0169] As shown in the example of FIG. 9C, by slightly modifying
the above setup, the DFT of a hernitic vector can be obtained in
one cycle. To this end, two separated arrays of sources 612A and
612B, e.g., for the real and imaginary parts, respectively, are
used. Source 612A corresponding to the real part is confronted with
the mirror-part 614A introducing no phase change, while the
imaginary part associated source 612B faces the mirror-part 614B
with a .pi./2 phase shift. The DFT of the input is sensed in a
detector array 615.
[0170] In the above description, special emphasis has been made on
the calculation of DFT of a conjugate symmetric vector. As
particular cases, DCT and DST of real vectors can be obtained. The
general case of DFT, as defined by equations (6) and (6b) above,
requires four real transformations (DST or DCT), that can be
achieved in a similar fashion to the above descriptions or by using
two cycles in those systems, each with a different input.
[0171] It should be noted that DST and DCT could be obtained with
the same shearing interferometer but with the input signal having
3N elements, where N of them are zeros. The first element of the
object is shifted by N with respect to an axis of light propagation
through the interferometer, and for the input signal .function.'(n)
we have: 11 f ' ( n ) = { f ( n - N ) N n 3 N 0 otherwise ( 14
)
[0172] In this case, for the same size of the input object, 50%
more length units at the input plane are required, as compared to
the previous case with the "nonshifted object". For the output
signal F'(k), we have: 12 F ' ( k ) = n = N 3 N - 1 f ' ( n ) cos (
n k 2 N ) = m = 0 2 N - 1 f ( m ) cos ( k ( m + N ) 2 N ) = m = 0 2
N - 1 f ( m ) { cos ( k m 2 N ) cos ( k 2 ) - sin ( k m 2 N ) sin (
k 2 ) } ( 15 )
[0173] wherein m is the coordinate describing the location of the
input elements in the input plane of the original non-shifted
object, and k is the coordinate describing the locations of the
discrete output elements of the transform. It should be noted that
in these notations, n, m and k are dimensionless. The following
change of variable was conducted:
m=n-N (16)
[0174] Since only half of the .function.(m) elements differ from
zero, equation (15) can be rewritten: 13 m ' = 0 N - 1 f ( m ) {
cos ( k ' m ' N ) cos ( k ' ) - sin ( k ' m ' N ) sin ( k ' ) } (
17 )
[0175] wherein additional variable changes are used: 14 m ' m 2 , m
' = 0 , 1 , N - 1 and k ' k 2 , k ' = 0 , 1 2 , 1 , 3 2 N - 1.
[0176] The behavior of the function F'(k) is as follows:
[0177] For samples at k=0, 2, 4, . . . , exactly the cosine
transform, C(k), is obtained;--For samples at k=1, 3, 5, . . . ,
the minus cosine, -C(k), is obtained which is the DCT-I;
[0178] In order to describe the Sine transform, the variable k'
will be changed to j such that 15 j = k ' - 1 2 , k ' = 1 2 , 3 2 N
- 1 2 .
[0179] For the term of the sine only it is found that: 16 m ' = 0 N
- 1 f ( m ) { - sin ( ( j + 1 2 ) m ' N ) sin ( ( j + 1 2 ) }
[0180] For samples at j=0, 2, . . . , a sine transform, S(k), is
obtained; and
[0181] For the last set of samples, 1, 3, 5, . . . , a minus sine
transform, -S(k), is obtained which is the DST-II transform.
[0182] Turning now to FIGS. 10A and 10B, there are illustrated two
possible examples of the implementation of an optical system
according to embodiments of the invention.
[0183] In a system 700A, shown in FIG. 10A, a light source 712A has
relatively narrow beam divergence, as compared to a light source
712B of a system 700B (FIG. 10B). Generally, the light source 712A
does not have sufficient divergence to illuminate the region of
interest in output plane by all the points in the modulated input
signal. Consequently, system 700A also comprises a splitting
sub-assembly 716 in the form of a diffractive optical element (DOE)
that splits the input image IS into two light components, which
then propagate along two optical paths (channels), respectively,
defined by optical setups 718A and 718B. The splitting sub-assembly
716 provides diffraction angles greater than the divergence of the
light source 712A, but is limited in angle to avoid any overlap
between the two channels (paths) of light propagation. The design
of such DOE 716 can be based on the channels' dimensions. As for
the system 700B (FIG. 10B), it utilizes a light source comprising
suitable LEDs or VCSELs which are characterized by sufficiently
wide divergence, thereby eliminating the need for a splitting
means.
[0184] The imaging setups 718A and 718B are designed to create,
respectively, an inverted image 719A and a non-inverted image 719B
of the input signal IS. In this specific example, the setup 718A
comprises two identical lenses L.sup.(A).sub.1 and L.sup.(A).sub.2,
wherein lens L.sup.(A).sub.1 is situated such that DOE 716 is
located in its back focal plane. Lenses L.sup.(A).sub.1 and
L.sup.(A).sub.2 are spaced from each other a distance 2f along an
optical axis of imaging setup 718A, whereinf is the focal length of
the lens L.sup.(A).sub.1 and L.sup.(A).sub.2. Setup 718B is
composed of two identical lenses L.sup.(B).sub.1 and
L.sup.(B).sub.2 each having the focal length .function./2. Lenses
L.sup.(B).sub.1 and L.sup.(B).sub.2 are positioned similar to
lenses L.sup.(A).sub.1 and L.sup.(A).sub.2, namely lens
L.sup.(B).sub.1 is spaced from the input image IS by a distancef
and from the lens L.sup.(B).sub.2, by a distance 2f. In order to
obtain a DST transform, a .pi./2 phase plate 715 is inserted in
either one of the setups 718A and 718B--in the setup 718A in the
present example. The phase plate 715 is located upstream of the
lens L.sup.(A).sub.1 (or L.sup.(B).sub.1). This results in a .pi./2
phase shift between the two images (i.e., interfering signals),
thereby providing a DST transform.
[0185] As further shown in the figures, each setup also comprises a
prism P accommodated downstream of the lens L.sup.(A).sub.1 (and
lens L.sup.(B).sub.1) proximate thereto. The provision of the prism
P is optional and is aimed at appropriately deflecting light beams
impinging thereon, so as to provide beams parallel to the optical
axis of light propagation. Further optionally provided in the two
setups, are field lenses and additional lenses L.sub.4. The field
lens is typically used for improving the efficiency of the optical
system by ensuring that the light passes by one lens is directed
into the subsequent lenses. As for the additional lenses, they aim
to correct the spherical curvature of the images 719A and 719B.
[0186] Thus, setups 718A and 718B define two symmetrical optical
paths with respect to an optical axis OA of light propagation
through the system, for the propagation of corresponding light
components impinging on lenses L.sup.(A).sub.1 and L.sup.(B).sub.1,
respectively. Since each point in the image 719A is coherent with
the corresponding point in image 719B, the two images interfere
with each other. This construction is less complex and more
compact, as compared to that of the conventional shearing
interferometer shown in FIGS. 1A and 1B.
[0187] In the above examples of FIGS. 10A and 10B, two images of
the input signal IS are provided (either by the use of a splitting
sub-assembly or by the use of an appropriate light source), and one
of the replicated images is appropriately inverted while both
images propagate through the optical setups. System 700A (or 700B)
defines a free space d for the light free space propagation aimed
at producing the interference between the two images 719A and 719B
in the output plane, thereby obtaining the DCT of the input signal.
It should be noted that all the elements in FIGS. 10A and 10B
including lenses and prisms could be implemented as diffractive
optical elements.
[0188] In the above examples, each of the two imaging paths creates
an output image with magnification equal to 1. The total length of
the optical path is:
L=4f+d (20)
[0189] Considering the divergence angle .alpha. of the light source
(or of the replication means, as the case may be) and the dimension
.DELTA.x of the input object, in order to produce the necessary
interference, the following condition should be satisfied: 17 d 2 =
2 x . ( 21 )
[0190] Thus, for .DELTA.x=16 mm, .alpha.=0.15 radians and f/2=8 mm,
the total desired length L of the optical path is: L=490.7 mm
[0191] Assuming that the lens parameter .function..sub.0 defining
the amount of light collected by the lens (i.e., the so-called
"lens speed" .function..sub.0), which is determined as the ratio
between the focal length and the diameter of the lens) is 2.5.
Assuming that the light source used is VCSEL with a 250 .mu.m
pitch, then the maximal volume of the setup is determined as
follows: 18 V = ( 2 250 m ) max { f f 0 , x } L = 3925.3 [ mm 3 ] (
22 )
[0192] wherein the coefficient 2 is associated with the two optical
paths of the light propagation.
[0193] It can be noted that the performance of the entire system is
determined inter alia by such main parameters as the size and
coherence length of the light source. The coherent length is an
extremely important parameter, since the formation of the
interference pattern by all the pixels of an input image is
achievable only if the coherence length of the source permits it
The following table summarizes these parameters for three main
options for a light source.
1 LED Diode Laser VCSELs Size(.mu.m) 5-10 1-5 1-3 Coherence length
(mm) 0.01-0.03 0.1-2 5-15
[0194] The coherence length .DELTA. of the light source must be
larger than the maximal light path differences .DELTA..sub.h, which
can be approximated as: 19 h = L 2 + ( 2 x ) 2 - L 2 x 2 L ( 23
)
[0195] The coherence length .DELTA. is determined by the properties
of the light source resonator, that is:
.DELTA.=.multidot.h (24)
[0196] wherein is the finesse number and h is the length of the
resonator.
[0197] The relation between the coherent length .DELTA. and the
maximal light path differences .DELTA..sub.h should satisfy the
following condition:
.DELTA.>.DELTA..sub.h (25)
[0198] Assuming the above values for .DELTA.x and L, the required
coherence length should be: 20 h = 2 16 2 426 = 1.2 mm ( 26 )
[0199] As indicated above with regard to the optical length to be
defined by the system so as to meet the requirements of the
shearing interferometry, this length is governed by two conditions:
the divergence of the light source that should ensure that each
point source illuminates the whole output region of interest, and
the matching condition between the input and output signals. The
optical setups of FIGS. 10A and 10B are rather long, since they
require the inner imaging length.
[0200] The above examples of FIGS. 10A and 10B are suitable for
processing one-dimensional signals. The same processing can be done
with a system of shorter configuration. This is exemplified in
FIGS. 11A and 11B. Here, the two symmetric imaging channels of
FIGS. 10A and 10B are replaced by asymmetric configurations with
some demagnification.
[0201] In a system 800A of FIG. 11A, an imaging setup 818A
comprises a phase plate 815, a lens L.sup.(A).sub.1 with the focal
length .function., and a lens L.sup.(A).sub.2 with the focal length
.function.' such that .function.'<.function.. The
demagnification factor of this channel is .function.'/.function..
An imaging setup 818B is composed of a lens L.sup.(B).sub.1 with
the focal length .function./2, and a lens L.sup.(B).sub.2 with the
focal length .function.'/2. The setups 818A and 818B provide,
respectively, the inverted and non-inverted (direct) images 819A
and 819B, and are shorter than the setups 718A and 718B of FIGS.
10A and 10B. This is due to the following reasons:
[0202] Shorter focal lengths of the lenses result in a shorter
setup; and
[0203] Owing to the demagnification factor, the angular divergence
is .function./.function.' higher, and consequently, the required
distance for the shearing interference is shortened by the same
factor.
[0204] For simplicity, the focal lengths of the upper part lenses
L.sup.(A).sub.1 and L.sup.(A).sub.2 in the above example will be
denoted by .function..sub.1 and .function..sub.2, and the focal
lengths of the lower part lenses L.sup.(B).sub.1 and
L.sup.(B).sub.2 will be denoted by .function..sub.1' and
.function..sub.2'.
[0205] The demagnification of the upper part (setup 818A) is: 21 M
= f 2 f 1 ( 27 )
[0206] The demagnification of the lower part (setup 818B) is
determined as follows: 22 1 f 1 + 1 d i = 1 f 1 ' 1 f 1 + f 2 - d i
+ 1 f 2 = 1 f 2 ' M = d i f 1 f 2 f 1 + f 2 - d i ( 28 )
[0207] wherein d.sub.i is the position downstream of the first lens
L.sup.(B).sub.1, where the image of the input signal is created. In
order to obtain equal magnifications in both channels 818A and
818B, this parameter d.sub.i should, in this specific example,
satisfy the following condition: 23 d i = f 1 + f 2 2 ( 29 )
[0208] If d.sub.i satisfies the above condition, than for the focal
lengths of the lower part lenses L.sup.(B).sub.1 and
L.sup.(B).sub.2, we have the following relations: 24 f 1 ' = f 1 [
f 1 + f 2 3 f 1 + f 2 ] f 2 ' = f 2 [ f 1 + f 2 f 1 + 3 f 2 ] ( 30
)
[0209] Thus, the total length L of the demagnification setup of
FIG. 11A and the space d needed to be provided between the last
lens in the setup and the output plane, are as follows: 25 L = 2 f
1 + 2 f 2 + d d = ( f 2 f 1 ) 2 4 x ( 31 )
[0210] Assuming again that f.sub.2=16 mm, we have: 26 L f 1 = 0 2 -
2 4 x f 2 f 1 3 = 0 f 1 = ( 4 x f 2 ) 1 / 3 ( 32 )
[0211] This means that f.sub.1=19 mm, f.sub.1'=9.1 mm and
f.sub.2'=8.4 mm. The total length L is thus 372 mm, and the total
volume V is 2980 mm.sup.3.
[0212] In the example of FIG. 11B, negative focal lenses
L.sup.(A).sub.2 and L.sup.(B).sub.2 are used for the further
reduction of the length and volume of the system. This
implementation also utilizes a demagnification factor that enables
a reduced shearing interference length. In this case, if the lenses
are still positioned in the same locations, the focusing is
obtained at a distance of (-.function..sub.2) relatively to the
last lens. Thus, the total length L will be:
L=2f.sub.1+d (33)
[0213] Here, d is greater than .eta..sub.2.
[0214] In order to maintain the elements of the lower path in their
positions, the following condition should be satisfied: 27 1 f 1 +
1 d i = 1 f 1 ' 1 f 1 + f 2 - d i - 1 f 2 = - 1 f 2 ' ( 34 )
[0215] For the same demagnification M, we have: 28 f 1 ' = f 1 [ f
1 + f 2 3 f 1 + f 2 ] f 2 ' = f 2 [ f 1 + f 2 f 1 - f 2 ] ( 35
)
[0216] Thus, in the example of FIG. 11B, similar to that of FIG.
11A, the focal lengths of the lenses should satisfy the following
condition: .function..sub.1>.function..sub.2. Assuming that
f.sub.2=16 mm, f.sub.1=19 mm, f.sub.1'=9.1 mm, we have:
f.sub.2'=186.7 mm, d=302.5 mm, and L=340.5 mm.
[0217] Utilizing the above equation (22), the volume is V=2720
mm.sup.3, if .function..sub.0 is greater than 11. Hence, the input
plane, and not the numerical apertures of the lens, is the
restricting dimension. Such a value of .function..sub.0 (i.e.,
f.sub.o=11) is not high enough to damage the resolution since the
pitch is only 250 .mu.m. Indeed:
2.44.lambda..function..sub.0=2.44-0.86.mu..multidot.11<250
.mu.m
[0218] As indicated above, in order to produce spatially modulated
light (i.e., input signal) indicative of an input object, the light
source is typically equipped with a spatial light modulator (SLM).
By using VCSELs, however, the need for such an SLM can be
eliminated. This is due to the fact that VCSELs are characterized
by ultra fast modulated components (up to 3 GHz) and high light
intensity (up to 1 mwatt per single VCSEL source). Thus, when using
an 8.times.8 VCSEL array at the input plane in a system according
to an embodiment of the invention, there is no need for any spatial
light modulator (subject to the availability of a fast enough 8 by
8 Photo Diode Array (PDA)). VCSELs can be modulated to present the
block images and at ihe output, the DCT will be optically achieved.
For instance, if a PDA of 500 MHz is used, then:
500.multidot.10.sup.6.multidot.64=32 GHz/S
[0219] In other words, when a 8.times.8 VCSEL block is used (i.e.,
64 channels operating in parallel), the total transmission speed
(from all the channels together) is about 32 GHz/s Referring to
FIG. 12, there is illustrated a system 900, utilizing a
multi-channel setup--three channels 910A-910D in the present
example, wherein each of these channels is constructed similarly to
the above described system 700. Namely, each of the channels
910A-910D is composed of two sub-channels 718A and 718B constructed
as described above. It should be understood that the sub-channels
could be constructed as those in the example of either of FIGS. 11A
and 11B.
[0220] Yet another example of the transformation of one-dimensional
signals utilizes the so-called "planar optics implementation", and
provides a compact architecture based on the fact that the
inversion of one of the signals is carried out in a separated
optical path. According to this technique, the two optical paths
defined by two setups may be placed one over the other, resulting
in a much more compact package. The setups may be constructed as
described above with reference to FIGS. 10A, 10B, 11A, 11B and
12.
[0221] One possible planar optics implementation is schematically
illustrated in FIG. 13, showing a system 1000, in which the
one-dimensional input signal IS is placed horizontally (into the
figure), and is replicated along its narrow dimension (up or down)
into two light components propagating along two paths. The two
paths are treated separately in two one-dimensional planar packages
(setups) 718A and 718B as described above. The optical elements
(for the replication action and within the planar package) might be
refractive or diffractive optical elements. FIG. 13 is similar in
function to FIG. 10, however, the elements are flat rather than
round.
[0222] Another possible planar optics implementation for processing
one-dimensional signals is illustrated in FIGS. 14A and 14B,
showing, respectively, the top and side views of a system 1100. The
system 1100 comprises a shearing interferometer constructed
generally similar to that of FIG. 1A, namely, comprising a beam
splitter 1103 and a mirror 1104, but utilizing an angular mirror
1105 instead of the Dove prism typically used. The interferometer
is designed like a planar sheet 1114, whose thickness is very
small, such that light may be guided along the narrow dimension of
the sheet. Thus, the beam splitter 1103 is accommodated within the
sheet, and the mirrors 1104 and 1105 are attached to some of the
sheet edges. This embodiment is functionally similar to that of
FIG. 1A, except that it is flat.
[0223] If the sheet is of 10 mm.times.10 mm dimensions, the
divergence of the light source is 15.degree. in each direction, and
the light source is of 5 mm size, then the total propagation length
L should be at least: 29 5 sin ( 15 ) 20 mm
[0224] This implementation distinguishes from the above-described
examples in that the light, while propagating between the input and
output planes, passes in two opposite directions, while in the
previously described examples, light passes in one direction only
from the input to the output plane.
[0225] Turning now to FIG. 15, there is illustrated a system 1200
capable of performing one-dimensional cosine transform, in which a
mirror 1216 is used as the splitting sub-assembly. In some cases,
this design will lead to a more compact system and sometimes to
even a less complex optical configuration. As shown, the input
signal IS (indicative of the image of an input object) is placed at
a predetermined distance from the mirror 1216, which generates a
virtual image IS'. The distance between any point on the input
signal IS and the mirror 1216 determines the frequencies that will
be obtained by the signal transform. Both images generate the
shearing interference at the output plane OP. It should be noted
that the overlapping area between the wavefronts from each couple
of a point source (e.g., a light emitting element of the light
source) and its image decreases as moving along the input signal IS
away from the mirror. Therefore, the maximal distance determines
the overlapping area in which the transform can be obtained.
[0226] In system 1200, where the split signals are obtained due to
mirrors, the total length L is determined as follows: 30 L = d = 4
x ( 36 )
[0227] For the data used before, L=426.7 mm and V=250
.mu.m.multidot..DELTA.x.multidot.L=1706 mm.sup.3. It is thus
evident that such a system is half in its volume, as compared to
the previously described examples. This can be understood by the
fact that one of the images and paths is virtual.
[0228] When dealing with a two-dimensional input object, two
variants of transformation are carried out. The first is a kernel
one that includes spatial frequencies along one direction only. For
example, a discrete Fourier transform of the input signal
.function.(m,n) is 31 m , n = 0 N - 1 f ( m , n ) exp ( N ( m k + n
l ) ) ( 37 )
[0229] This example may lead also to the following two-dimensional
cosine transform: 32 F ( k , l ) = m , n = 0 N - 1 f ( m , n ) cos
( N ( m k + n l ) ) ( 38 )
[0230] This transform can be achieved using the above-described
examples, but with a two-dimensional input object replacing a
one-dimensional object. The latter should contain a grid of point
sources that are spatially incoherent, for example, by using a
two-dimensional VCSEL array or LED array. The reason for this is
associated with the fact that the impulse response of a system is
still a cosine function with one-dimensional symmetry.
[0231] For some applications, a kernel that includes spatial
frequencies along two orthogonal directions is required. For
example, the two dimensional JPEG standard requires the following
discrete Cosine transform kernel: 33 F ( k , l ) = m , n = 0 N - 1
f ( m , n ) cos ( N m k ) cos ( N n l ) ( 39 )
[0232] FIG. 16 illustrates a system 1300 for carrying out a
two-dimensional JPEG transform by utilizing the same optical setup
as above with two images. The system 1300 is capable of obtaining
two images of the object in two successive cycles. To this end,
this system is constructed generally similar to the example of FIG.
12, but also comprises anamorphic optics, for example, in the form
of a cylindrical lens L.sub.5, which provides imaging of an object
along one axis only (with interference taking place along the other
axis.
[0233] The cylindrical lens should be positioned such that its
focal plane is located at the distance .function./2 along the axis
perpendicular to the plane of the figure. The focal plane is at the
distance .function./2, however, the image in this specific setup is
obtained at a distance .function. from the lens, i.e., not at the
focal plane.
[0234] Thus, in the direction along this axis, both setups 718A and
718B perform imaging. However, in the perpendicular plane (i.e.,
the plane of the figure), interference takes place. In other words,
the interference pattern in the output plane is formed by
interfering signals (images) indicative of only one dimension of
the input signal. To provide interference of signals indicative of
the two-dimensional input object along two mutually perpendicular
axes, the system 1300 operates with two cycles: the object
.function.(n,m) is first "interfered" along one axis in the output
plane, thereby obtaining an intermediate interference pattern,
F.sub.int(n,k), formed by the interference between the input signal
along its one dimension with the image of the input signal along
its second dimension. This intermediate pattern is then rotated at
90.degree. (for example using a Dove prism with its hypotenuse
plane tilted by -45.degree.) resulting in a signal
F.sup.T.sub.int(n,k), which is fed back into the system as an input
signal.
[0235] Thus, in the first cycle, the system 1300 creates the
intermediate interference pattern F.sub.int(n,k) indicative of the
input object .function.(n,m), described as follows: 34 F i n t ( n
, k ) = m = 0 N - 1 f ( m , n ) cos ( N m k ) ( 40 )
[0236] In the second cycle, the same optical system 1300 is fed by
the rotated signal F.sup.T.sub.int(n,k), that is:
F.sub.int.sup.T(n,k)=F.sub.int(k,n) (41)
[0237] The so-obtained output signal F(k,l) is determined as
follows: 35 F ( k , l ) = n = 0 N - 1 F i n t ( k , n ) cos ( N n l
) ( 42 )
[0238] Another possible way of performing a two-dimensional cosine
transform is based on the use of a doubled sheared optical setup.
This implementation requires a more complicated setup, aimed at
generating an impulse response of: 36 cos ( N m k ) cos ( N n l ) (
43 )
[0239] FIG. 17 schematically illustrates the operational principles
of such a doubled sheared optical setup based system denoted 1400.
Each point of the input signal IS should be split four times. This
may be achieved by the above described setup of FIG. 10A or FIG.
10B, but with two-dimensional diffraction grating and prisms. As
shown in FIG. 17, four images I.sub.1-I.sub.4 are obtained after
the light passage through the diffraction grating assembly, and are
rotated by the lenses and prisms, thereby resulting in four output
images I'.sub.1-I'.sub.4.
[0240] More specifically, the system 1400 starts with a
self-illuminated image IS which is spatially incoherent. Then, a
rectangular grid diffraction grating (not shown here) splits the
image into four images. After the splitting, the propagation
direction of the light components indicative of these images should
be corrected. This can be done using four prisms (not shown) in a
similar way used for the two-dimensional case as described above.
After obtaining four identical images I.sub.1-I.sub.4 placed side
by side, three of them are appropriately rotated: image I.sub.1 by
180.degree., image I.sub.2 by 90.degree., and image I.sub.3 by
-90.degree.. The light components indicative of image I.sub.4
propagate towards the output plane without any rotation.
[0241] An optical assembly used for rotating the images can be
composed of four Dove prisms aligned at 90.degree., 45.degree.,
-45.degree. and 0.degree. angular orientations. Thus, the split
images will be rotated (clockwise with respect to the vertical
axis) in the following manner: the upper image I.sub.1 will be
rotated by 180.degree. by a Dove prism with its hypotenuse plane
titled 90.degree.; the left image I.sub.2 will rotated by
90.degree. by a Dove prism with its hypotenuse plane titled by
-45.degree.; and the right image I.sub.3 will be rotated by
-90.degree. degrees by a Dove prism with its hypotenuse plane
titled by 45.degree..
[0242] FIG. 18 illustrates a system 1500 presenting yet another
possible example for the implementation of the kernel transform
with the impulse response (43) above. Here, two mirrors 1516A and
1516B constitutes the splitting sub-assembly, which provide three
virtual images I.sub.1, I.sub.2 and I.sub.3 of an input signal IS.
The so obtained four signals IS, I.sub.1, I.sub.2 and I.sub.3
generate interference patterns. For the 64 VCSELs input,
.DELTA.x=16 mm. For the 512 VCSELs input, we have .DELTA.x=64 mm
and V=6824 mm.sup.3.
[0243] In order to obtain the required kernel transform (expression
(43) above), the following processing should be applied:
[0244] The following expressions can be derived from the shearing
kernel: 37 [ cos ( k m N / 2 ) cos ( n l N / 2 ) ] 2 = 1 4 [ 1 +
cos ( k m N ) ] [ 1 + cos ( n l N ) ] ( 44 ) = 1 4 + 1 4 cos ( k m
N ) + 1 4 cos ( l n N ) + 1 4 cos ( k m N ) cos ( l n N ) F ( k , l
) = D C + k m I ( m , n ) cos ( k m N ) ( 45 ) cos ( l n N ) + k m
I ( m , n ) cos ( k m N ) + k m I ( m , n ) cos ( l n N ) or F ( k
, l ) = D C + I ^ ( k , l ) + I ( k ) + I ( l ) ( 46 )
F(k,l)=DC+(k,l)+{circumflex over (I)}(k)+{haeck over (I)}(l)
(46)
[0245] For the purposes of the present invention, the function
I(k,l), which is the required transform, can be found using
equations (45) and (46). It should be noted that in the output
plane along the line (k,0) one obtains:
F(k)=DC.sub.1+2(k)+DC.sub.2 (47).
[0246] DC.sub.1 and DC.sub.2 can be determined separately, and
therefore (k)can be extracted from the equation (47). In a similar
way, {haeck over (I)} (k)can be obtained, and, finally, (k,l) can
be determined using equation (46).
[0247] As indicated above, the optical length required for
performing the shearing interferometer itself is governed by the
divergence of the light source (ensuring that each point source
illuminates the whole output region of interest), and by the
matching condition. The above description is mainly associated with
the continuous cosine transform and a sampled version thereof. The
shearing interferometer can be used for realizing the DCT, and,
under certain conditions presented below, any discrete linear
transformation.
[0248] The following is a derivation for matching conditions for
the- shearing interferometers described above, for generating an
arbitrary discrete transform. It should be understood that the
systems described above can generate practically any transform
between spatial intensity ard spatial frequency domains, provided
the proper matching condition is met.
[0249] The matching conditions comprise two portions, although both
portions are generally determined and met in the same structure.
One of these is spatial matching in which the sampling points on
the detection side are chosen to match the detector elements used
so that the output is a discrete transformation of the discrete
input. Various geometric parameters must be satisfied, for example
these may include the distance between the input and output planes.
The second condition is the provision of "apertures" as defined
below, which are directed to achieving/allowing a particular
transform at the expense of others and to compensate for
non-optimal sampling size. As used herein, apertures may include
finite sized holes and/or a mask having an absorption varying with
position. Such mask may be present at the input (source) or output
(detector) side of the transformer or apertures may be present at
both sides.
[0250] The following discussion provides a general methodology for
providing such matching and specifics for application to apparatus
containing shearing generators. Position matching is described
first.
[0251] The output intensity distribution provided by the shearing
interferometer is determined as follows: 38 F ( x ) = 0 .infin. f (
x _ ) cos ( 2 x x _ D ) x _ + E f ( 48 )
[0252] Here, .function. ({overscore (x)}) and F(x) are the input
and output intensity distributions respectively, .lambda. is the
wavelength of the illuminating source, D is connected with the
geometry of the interferometer (for example, the distance between
the first pixel of the input signal relative to the optical axis
defined by the light propagation in the interferometer), and
E.sub.f is the energy of the input signal.
[0253] The DCT to be realized is given by: 39 F ( k ) = n = 0 N - 1
f ( n ) cos [ k 2 N ( 2 n + 1 ) ] ( 49 )
[0254] wherein k is the discrete coordinate.
[0255] Assuming an ideal input pixels' shape (in the form of
Dirac's delta functional) and an ideal sampling in the output
plane, wherein N input pixels and N output pixels are located,
respectively, at the following coordinates n.DELTA.{overscore
(x)}+{overscore (x)}.sub.0 and k.DELTA.x, and utilizing the above
equation (48) connecting the input and output intensity
distributions, the following result is obtained: 40 f ( x _ ) = n =
0 N - 1 f ( n ) ( x _ - n x _ - x _ 0 ) ( 50 ) and F ( k ) = n = 0
N - 1 f ( n ) ( x _ - n x _ - x _ 0 ) cos [ 2 D k x ( n x _ + x _ 0
) ] x _ + E f ( 51 )
[0256] The above equations (50) and (51) describe, respectively,
the discrete input of the interferometer, and the analog output
thereof.
[0257] Simplifying the above equation (51), we have: 41 F ( k ) = n
= 0 N - 1 f ( n ) ( x _ - n x _ - x _ 0 ) cos [ 2 D k x ( n x _ + x
_ 0 ) ] x _ + E f ( 52 )
[0258] The comparison between the last equation (52) and the DCT
(equation (49) above) provides the following reciprocity
relations:
.DELTA.{overscore (x)}=2{overscore (x)}.sub.0 (53)
[0259] 42 x x _ = D 2 N ( 54 )
[0260] These relations indicate the exact connection between the
locations of the various samples in the input and output planes.
Hence, the output signal (52) obtained with the shearing
interferometer is the DCT. Using the above analysis and optionally
a simulation, one can determine the largest pinhole size that will
yield a required accuracy.
[0261] It is often the case that the pixels of input and output
devices cannot be approximated as delta functions. This is
especially true of detector elements, since making them too small
will result in unacceptable power loss. In this case, the pixels'
response and the output sampling process should be specifically
considered. The following is a relatively general analysis of this
situation.
[0262] Assuming that the response of each of the input pixels can
be modeled by an arbitrary function l(x) (which can be determined,
for example, by imaging a turned-on pixel along the input source),
the input may be presented as 43 f ( x _ ) = l ( x _ ) n = 0 N - 1
f ( n ) ( x _ - n x _ - x _ 0 ) ( 55 )
[0263] wherein {circle over (x)} corresponds to the convolution
operation.
[0264] Assuming that f(x<0)=0, the output intensity distribution
is as follows: 44 F ( x ) = 1 2 - .infin. .infin. f ( x _ ) exp (
i2 x x _ D ) x _ + 1 2 - .infin. .infin. f ( x _ ) exp ( - i2 x x _
D ) x _ + E r ( 56 )
[0265] For the sake of simplicity, E.sub.f will be ignored
temporarily. It is considered below. The above equation (56) can
thus be rewritten as: 45 F ( x ) = 1 2 f ~ ( - x D ) + 1 2 f ~ ( x
D ) = 1 2 L ( - x D ) n = 0 N - 1 f ( n ) exp [ i 2 x _ D ( n + 1 2
) x ] ++ 1 2 L ( x D ) n = 0 N - 1 f ( n ) exp [ - i 2 x _ D ( n +
1 2 ) x ] ( 57 )
[0266] wherein L and {tilde over (f)} are the Fourier transforms of
l and .function., respectively.
[0267] Assuming that the input pixels response l (and therefore L)
is a symmetric function, for the output intensity distribution F(x)
we obtain: 46 F ( x ) = L ( x D ) n = 0 N - 1 f ( n ) cos [ 2 x _ D
( n + 1 2 ) x ] ( 58 )
[0268] Considering now a non-ideal sampling process in the output
plane, we have: 47 F ( k ) = 1 x k x - x 2 k x + x 2 F ( x ) W k (
x ) x ( 59 )
[0269] wherein W.sub.k(x) is the weighting function of the k.sup.th
output pixel.
[0270] Introducing the following definition: 48 R ( x ; k ) = L ( x
D ) W k ( x ) ( 60 )
[0271] we obtain: 49 F ( k ) = 1 x n = 0 N - 1 f ( n ) k x x 2 k x
+ x 2 R ( x ; k ) cos [ x _ D ( 2 n + 1 ) x ] x ( 61 )
[0272] Comparing the latter result to the DCT (equation (52)
above), we obtain a set of N*N equations in the form: 50 cos [ k 2
N ( 2 n + 1 ) ] = 1 x k x - x 2 k x + x 2 R ( x ; k ) cos [ x _ D (
2 n + 1 ) x ] x ( 62 )
[0273] Considering a different reciprocity relation (as compared to
the above equations (53) and (54)), that is: 51 x x _ D = 1 ( 63
)
[0274] we obtain R(x;k) in the form: 52 R ( x ; k ) = 2 n = 0 N - 1
cos [ k 2 N ( 2 n + 1 ) ] cos [ x ( 2 n + 1 ) x ] ( 64 )
[0275] It should be noted that the use of the above technique can
facilitate the realization of an arbitrary discrete linear
transformation of the form: 53 F ( k ) = n = 0 N - 1 f ( n ) C ( k
, n ) ( 65 )
[0276] In this case, R(x;k) becomes: 54 R ( x ; k ) = 2 n = 0 N - 1
C ( k , n ) cos [ x ( 2 n + 1 ) x ] ( 66 )
[0277] The solution for R(x;k) is indicative of the matching
condition to be satisfied by the relation between the input and
output pixels (i.e., the centers of these pixels). It should be
noted that E.sub.f, which was omitted in the previous
considerations, will appear in the output plane, and will result in
a different value for each one of the output pixels. However, its
contribution can be calculated in advance and may be
compensated.
[0278] In some cases a vector to be processed by a shearing
processor, in accordance with an embodiment of the present
invention, may have a number of components that is larger than the
number of light sources in an array of light sources used to
represent vectors in the shearing processor. The vector cannot
therefore be completely represented by the light source array and
the shearing processor cannot produce a processed result for the
vector in a single step. In such cases, in accordance with an
embodiment of the present invention, the vector may be partitioned
into a plurality of shorter "partial" vectors each having a number
of components that is equal to or less than the number of light
sources in the light source array. Each of the partial vectors is
then processed by the shearing processor. Results from the
processing of all the partial vectors are then combined to provide
a processed result for the vector. Partitioning "overlong" vectors
to be processed by a shearing processor, in accordance with an
embodiment of the present invention is discussed in PCT application
entitled "OFDM Apparatus and Method" filed on even date with the
present application in the Israel Patent Office. This application
is hereby incorporated herein by reference.
[0279] In addition, PCT application PCT/IL99/00479, filed 5 Sep.
1999 and published as WO 00/72267 describes various DCT
configurations. This application is incorporated herein by
reference. Combinations of some of these configurations, with a
shearing generator may be useful in the practice of the present
invention.
[0280] Those skilled in the art will readily appreciate that
various modifications and changes can be applied to the embodiments
of the invention as hereinbefore described, without departing from
its scope defined in and by the appended claims. For example, any
suitable configuration of a shearing interferometer may be used for
carrying out the discrete cosine or discrete sine transform,
provided a matching condition is appropriately considered. The
shearing interferometer configurations according to the invention,
however, provide more simple and compact designs, as compared to
the known configuration, utilizing the combination of a beam
splitter, a mirror and a dove prism, and provides for
simultaneously obtaining both cosine and sine discrete transforms.
of an input signal.
[0281] Furthermore, while a large number of examples have been
disclosed, each having numerous elements, some embodiments of the
invention do not require all of the elements shown and some
embodiments of the invention utilize elements shown in different of
the disclosed embodiments.
[0282] As used in the claims, the terms "comprise", "include", and
"have" and their conjugates mean "including, but not necessarily
limited to".
* * * * *