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Description  |
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CROSS REFERENCE TO RELATED APPLICATIONS
This application is related to U.S. patent application Ser. No. 405,476
filed Mar. 16, 1995 by H. K. Wickramasinghe and F. Zenhausern (YO995-058);
and to U.S. patent application Ser. No. 405,481 filed Mar. 16, 1995 by F.
Zenhausern and H. K. Wickramasinghe (YO995-061); and to U.S. patent
application Ser. No. 405,068 filed Mar. 16, 1995 by F. Zenhausern and H.
K. Wickramasinghe (YO995-065); and to U.S. patent application Ser. No.
405,070 filed Mar. 16, 1995 by H. K. Wickramasinghe and F. Zenhausern
(YO994-273), now U.S. Pat. No. 5,538,898; and to U.S. patent application
Ser. No. 08/511,169 filed Aug. 4, 1995 by H. K. Wickramasinghe et al
(YO995-161); and to U.S. patent application Ser. No. 08/511,166 filed Aug.
4, 1995 by H. K. Wickramasinghe et al (YO995-166); and to U.S. patent
application Ser. No. 08/511,579 filed Aug. 4, 1995 by H. K. Wickramasinghe
et al (YO995-122). The entire disclosures of those applications, all of
which are copending and commonly assigned, are incorporated by reference
herein.
FIELD OF THE INVENTION
This invention relates to a novel method and apparatus for mass data
storage processing in which access to data is preferably realizable by way
of an interferometric near-field system.
INTRODUCTION TO THE INVENTION
We are investigating a method and apparatus suitable for decoding high
density data encoded in bit patterns having at least one dimension smaller
than the focused spot diameter of an incident electromagnetic radiation,
such as for example a laser beam.
Mass data storage processing may be realized by optical memory devices such
as CD-ROM and magneto-optical disks (MO). By irradiating a focused laser
beam of about one micrometer onto these memory devices, one can read and
sometimes record stored information. Typically, optical memories today
have recording densities and performance that are higher than magnetic
memory devices.
The density of an optical memory device is mainly diffraction limited; even
with a focusing lens with high numerical aperture (NA>1), the
corresponding radius of the focused beam spot can not be smaller than half
the illumination wavelength. Current commercial devices are already close
to this resolution limit.
Several approaches have already been demonstrated to overcome this
resolution limit. First, more efficient lasers emitting shortened
wavelength may be used to record and read dam. Typically, the reduction of
wavelength by a factor two can be expected to provide about a fourfold
increase in data storage density. However, available instruments
comprising a blue-green laser diode device using multilayer compounds
grown on GaAs substrates are still at the forefront of the technology, and
manufacturing them is expensive. These devices typically afford a factor
two increase in data storage density, and further advances by reduction of
the wavelength are not in a foreseeable future.
Another approach has been reported in the literature using evanescent
waves, due to their nonisotropic properties and location with wavelengths
that are shorter than the ordinary propagating wavelength of a laser beam.
Although the application of aperture-based Near-Field Scanning Optical
Microscopy (NSOM) may potentially be fruitful for recording small bits
domains--Betzig et al. demonstrate data densities of .congruent.45
GBits/inch.sup.2 (Betzig et al., Appl. Phys. Lett., 61(2), 142,
1992)--other parameters such as signal-to-noise ratio, reliability and
speed are of major importance for practical interest.
It has even been reported that 1000 bits of information can be stored in a
diffraction limited spot in a photochemical hole-burning medium (W. E.
Moerner, Ed. Persistent spectral hole-burning: science and applications,
Current Topics in Physics, vol.44, Springer Verlag, Berlin 1988) but its
practical use has been hindered by the necessity to cool it at low
temperature, e.g., liquid nitrogen temperature.
The development of either multilayered materials (or even multi-discs) or
holographic techniques is motivated by searches for improving the density
limit by using another spatial dimension; components such as holographic
memory devices using orthogonal phase coding or wavelength multiplexing,
which combines lasers and detectors operating at slightly different
wavelengths in order to increase the signal level, have been developed (D.
A. Parthenopoulos et al., Science, 245, 843, 1989). Another way to improve
3D-optical memory recording and reading concerns the use of, for example,
a photopolymer used as a data storage media. In this method, data may be
recorded as refractive-index variations that arise when
photopolymerisation occurs as a result of two photons excitations. Data
can be read with a differential interference-contrast microscope, and a
memory recording density achieving 1.3 GBits/cm.sup.3 has been reported.
(J. H. Strickler and W. W. Webb, Opt. Lett.,16, 1780, 1991).
SUMMARY OF THE INVENTION
We have now discovered a novel method and apparatus for high-density data
storage in which bit-format data may be read below the diffraction limit,
typically down to 100 angstroms spatial resolution.
We obtain interferometric measurements that allow for the detection of a
small amount of light, and provide a large detection bandwidth, thereby
enabling one to image (see FIG. 4) and reproduce e.g., a 500 Angstrom bit
pattern at frequencies as high as about 30 MHz. An advantage of the
present invention is that it discloses a novel method and apparatus that
enable commercial applications of interferometric near-field storage.
In one aspect, the present invention discloses an apparatus for decoding
high density data encoded in a digital recording media as a series of tags
comprising an information bit pattern including a tracking bit pattern.
The apparatus comprises:
1) a source of electromagnetic radiation for generating an incident wave;
2) means for directing at least a portion of the incident wave to the
media;
3) a scatterer acting as an antenna and capable of re-radiating a signal
wave, said signal wave developing as an interactive coupling between a tag
and said scatterer;
4) means for creating an interference signal based on the signal wave and a
reference wave;
5) a high-speed detector for interrogating at least one of the phase and
amplitude of the interference signal as a read-out signal;
6) means for interpreting and reproducing the read-out signal; and
7) means for detecting the tracking bit pattern and for controlling the
reproduction of the information bit pattern.
In a second aspect, the present invention discloses a method for decoding
high density data encoded in a digital recording media as a series of tags
comprising an information bit pattern including a tracking bit pattern,
the method comprising the steps of:
I) providing an apparatus, said apparatus comprising:
1) a source of electromagnetic radiation for generating an incident wave;
2) means for directing at least a portion of the incident wave to the
media;
3) a scatterer acting as an antenna and capable of re-radiating a signal
wave, said signal wave developing as an interactive coupling between a tag
and said scatterer;
4) means for creating an interference signal based on the signal wave and a
reference wave;
5) a high-speed detector for interrogating at least one of the phase and
amplitude of the interference signal as a read-out signal;
6) means for interpreting and reproducing the read-out signal; and
7) means for detecting the tracking bit pattern and for controlling the
reproduction of the information bit pattern; and
II) accessing the high density data by engaging the media with the
apparatus such that the scatterer can develop the interactive coupling
with each of the series of tags in the presence of the incident wave so
that a binary state can be defined.
A preferred realization of the novel method is disclosed below, and
features utilization of an interferometric near field apparatus providing
super-resolution e.g., 1 nanometer resolution, and fast data transfer rate
thereby enabling decoding of a bit pattern below the diffraction limit.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention is illustrated in the accompanying drawing, (not drawn to
scale), in which:
FIG. 1 shows in overview a principle of an apparatus for high data density
storage comprising multi-pole interactive coupling and interferometric
detection, preferably operating in a reflection mode;
FIG. 2 provides a schematic for describing a preferred embodiment operating
according to the invention;
FIG. 3 illustrates an alternative embodiment of the invention comprising an
integrated optical head including an antenna stripe, for example, made of
a high refractive thin film acting as a probe; and
FIG. 4 illustrates in the practice of the invention, accessing/imaging of a
high density bits pattern having 500 angstroms in spatial dimensions,
impressed in a photoresist onto a silicon substrate by using an e-beam
processing.
DETAILED DESCRIPTION OF THE INVENTION
We develop the detailed description by first disclosing an apparatus in
which access to data may be performed by operating an interferometric
near-field system that is preferably employed as a detector. To this end,
attention is directed to FIG. 1, which shows in overview such a
generalized apparatus 10 operating in a reflection mode.
The FIG. 1 apparatus 10 comprises a source 12 of electromagnetic radiation,
preferably generating an incident electric field E.sub.i, preferably in
the optical spectrum, for example from UV to IR. The electric field
E.sub.i is directed through a conventional interferometer 14 to a focusing
element 16 which preferably comprises an aperture or an objective lens.
The interferometer 14 may comprise e.g., a Michelson, Fabry-Perot or
Twyman-Green apparatus. The driving electric field E.sub.i is now focused
on a bit pattern impressed in a media 18, in turn supported by a substrate
20 for data storage. (Note that in an alternative embodiment comprising a
transmission mode, the driving electric field E.sub.i may be focused
through a transparent substrate illuminating the data bits sensed by a
probe). The substrate 20 for data storage can be of various shapes such as
rectangular with high density data stored transversely in lines or
circular such as an optical disc. The substrate 20 for data storage can be
held on a positioning system 22 operating a rotation or a translation
action.
FIG. 1 also shows a scatterer probe sensor 24 preferably placed with
respect to the bits pattern 18 such that a distance between the probe 24
and at least a portion of the media surface is smaller than the source 12
radiation wavelength, or a multiple of it. For example, a controlled
gap-distance of about 10-100 angstroms may be appropriate in the practice
of the present invention, for example, by way of a piezoelectric electric
actuator 26. This can be accomplished by the positioning system 22,
including a means for positioning the probe 24 and for example a focused
laser beam 12 in close proximity (e.g., 100 angstroms) with the media
surface. The system 22 can also generate a fast translational or
rotational motion between a measurement area of the bits pattern
illuminated with the source 12 and the probe 24.
Note that a suitable scatterer probe may comprise a sharp metallic tip or
an uncoated silicon and/or silicon nitride tip, or a tip coated with a
conductive layer or a molecular system, or a metallic stripe. The probe
preferably comprises a high refractive index material. A capability to
approach the scatterer probe with the media surface in a near field regime
may be realized by e.g., a scanning tunneling microscope (STM), an atomic
force microscope (AFM), an aperture or apertureless near-field optical
microscope, a near-field acoustic microscope, a thermal microscope or a
magnetic force microscope (MFM). A notion of "scanning" references the
fact that probe and media may be in relative motion. Reference may be made
for example to U.S. Pat. Nos. 5,319,977; 4,343,993; 5,003,815; 4,941.753;
4,947,034; 4,747,698; and Applied Physics Letter 65(13), 26 Sep. 1994,
1623. The disclosures of each of these patents and publication are
incorporated herein by reference.
The FIG. 1 probe 24 is capable of re-radiating, in the form of a signal
beam -SIG- (E.sub.s +E'.sub.r), an incident radiation beam, the signal
beam comprising carrier beam E.sub.r, combined with bit 18 property
information encoded in the scattered field E.sub.s, for example, as a
probe-bit dipole-dipole coupling. The signal beam -SIG- comprises a
scattered electromagnetic field variation wave E.sub.s, due preferably to
the probe 24 vibrating (or moving relatively) in close proximity to the
media 18 surface.
Note that the FIG. 1 signal beam SIG illustrates an electromagnetic wave
packet representative of media properties and comprising encoded wave
information derivable from a multi-pole interactive coupling between probe
and media.
For example, the incident radiation E.sub.i can drive this action such that
a dipole-dipole coupling interaction may be created between probe dipole
and media dipole.
FIG. 1 further shows that the focusing element 16, for example, a lens,
helps in creating an interference signal -IS- based on the signal beam
(E.sub.s +E'.sub.r) and a reference beam (E.sub.r), and for directing the
interference signal through the interferometer 14. An output signal 28 of
the interferometer 14 can measure either the amplitude of (E.sub.s
+E'.sub.r) or its phase difference with a reference beam E.sub.r. Note
that a self-interference phenomena can be alternatively exploited and
comprises spatially separating the beam 12 in several components having
phase differences that are subsequently made to interfere.
As illustrated in the fundamental FIG. 1, it should be noted that the
incident light can be directed to a scatterer either through the media
(transmission mode) or by reflection at its surface. In this latter case,
particular attention has to be taken to discriminate the scattered signal
against spurious reflected light. For the practical value of the operation
of the invention, only a reflection case is described below.
As alluded to above, the FIG. 1 apparatus may be used in a transmission
mode by setting up a transparent media, and operating mutatis mutandis
visa vis the apparatus of FIG. 1.
Attention is now directed to FIG. 2, which shows details of a preferred
apparatus 30 for the realization of the present method, which details are
consistent with the generalized FIG. 1 apparatus 10.
The FIG. 2 apparatus 30 comprises an interferometer and includes the
following components: an electromagnetic source, preferably a tunable
wavelength. (e.g., 400 nm<.lambda.>2500 nm) laser 32, an optional
acousto-optic modulator 34 in order to prevent spurious back reflection of
light generating laser noise; a special beam expander 36 for relative beam
and measurement area movement; an aperture 38; a means for splitting an
incoming lightwave into first and second lightwaves comprising a pellicule
beam splitter 40; a polarising beam splitter 42; a transmission/collection
optics (preferably a Nomarski Oil/dark-field objective) 44; a Wollaston
prism 46; and, 3 photodetectors PD.sub.n. FIG. 2 shows in association with
the interferometer an optical sensor and a set of electronics 48 (enclosed
by the broken-border box in FIG. 2) that permits both imaging and
scatterer probe-media distance feedback regulation with at least nanometer
accuracy. Preferably using an AFM feedback, one can therefore image while
simultaneously accessing data encoded in a media.
In the FIG. 2 illumination pathway, the laser beam of appropriate
polarisation first passes through the beam steering unit 36 in order to
expand the beam size in accordance with the objective aperture 44. The
laser beam can be adjusted continuously within the beam steering unit 36
by preferably using a piezoelectric positioning system (e.g., x-y-z PZT
tube) allowing small motion with nanometer accuracy. This action, in
conjunction with electronics disclosed below with reference to FIG. 3, can
act as a means for detecting a tracking bit pattern and for controlling a
reproduction of an information bit pattern. The beam steering is also
controlled by the image-collection electronics 48 for relatively
positioning the focused spots, a measurement area of a media 50 and a
scatterer 52, while the scanned beam is traced back and forth.
The expanded laser light passes through the aperture 38 (preferably
matching geometries of the transmission/collection optics 44) in which an
angular discrimination of the incident radiation distribution can select a
total internal reflection illumination. Typically, the pellicle beam
splitter 40 reflects about 10% of the incident radiation to a reference
arm of the interferometer to a detector, preferably a photodiode PD.sub.R
and transmits about 90% of the incident radiation to the polarising
beamsplitter 42.
The beam preferably is imaged to a plane wave that overfills the back
aperture of the Nomarski objective 44 which focuses the plane wave to two
diffraction limited spots in the media 50. Because the aperture 38 blanks
the illumination near the center of the beam, the exciting light wave
propagates as an evanescent wave in the area illuminated in the media.
When a scatterer 52 that can operate various motions relative to the media
50 at different frequencies (e.g., resonance frequency) with the help of a
three-coordinate piezoelectric translator 54, is approached typically a
few nanometers close to the media 50, the scatterer 52 is capable of
locally perturbing the wave impinging the smallest spatial asperity (e.g.,
the very end of a pointed STM or AFM tip) of the scatterer 52 resulting in
a coupling mechanism between the scatterer dipole and bit dipole of the
media.
In terms of an electromagnetic field distribution, the scattered electric
field variation due to the vibrating and scanning scatterer 52 in close
proximity to the media 50, may be measured by encoding it as a modulation
in the phase of a second arm of the interferometer. This modulation action
can alternatively be induced by a relative fast motion (e.g., rotation
and/or translation) of the media visa vis the scatterer. Modulation
actions can also be generated as a time variable multi-pole interactive
coupling comprising modulating at least one of the wavelength of the
source 32 or using an external applied electromagnetic field to the
probe-bit area.
The optical signal resulting from the dipole interaction is collected by
the objective 44, and reflected through the polarising beamsplitter 42 to
a Wollaston prism 46 with its axis oriented relative to the Nomarski
prism, in order to optimize the interference of the reflected electric
fields from the two spots, and to measure the phase of the signal beam
(E.sub.s +E'.sub.r) which corresponds to the real part of the scattered
wave E.sub.s.
The light continues through an external lens 56 that focuses the light onto
a photodiode PD.sub.A and PD.sub.B. The imaginary part of the scattered
wave E.sub.s can be detected by orienting the Wollaston prism 46 axis to
be aligned with the Nomarski prism 44 axis, so as to separately detect the
optical powers in the two spots (without mixing) in the differential
photodiode PD.sub.A-B. This detection arrangement preferably operates at
pre-selected frequencies ranging from 100 Hz to 100 MHz, for example, at
least greater than 30 MHz.
The output signal of this differential detector preferably is sent to a
noise suppressor 58 for further noise improvement, by combining the
photocurrent from PD.sub.A-B with that from the reference photodetector
PD.sub.R which is fed a sample of the incident beam. The noise suppressor
output preferably is sent to a lock-in amplifier input 60 in order to
demodulate the resultant near-field AC signal carrying interesting
information about the media. The output of 60 can be sent to a
controller/computer -CC- device for interpreting and reproducing a
read-out signal of a bit information pattern.
The experimental arrangement in FIG. 2, apparatus 30 can incorporate, for
example, directly through objective 44, an optical feedback system for
monitoring the scatter-media surface distance. The scatterer 52 can, as a
probe tip, be resonantly vibrated with the aid of the piezo actuator 54,
and the vibration amplitude can be detected with the help of a second
laser beam 64 at a given wavelength different from that of laser source
32. The optical feedback preferably comprises an assembly of optical
elements, for example, lenses 66 and 68 and dichroic filters 70, 72 in
order to discriminate the two optical paths of different wavelengths for
directing selected light onto the appropriate detectors PD.sub.n. By
adjusting the optics 66, 68 and 70 with respect to the probe tip 52, one
can ensure that the light impinging the rear face of the probe 52 does not
interfere with respect to the light impinging the very end of the probe
tip 52. The optical detection of the back reflected light from the rear
face of the probe 52 is then directed through the electronic set of box
48. The stable operation of the feedback system requires a proper choice
of operating parameters (e.g., scan rate . . . ) and light distribution
(e.g., focus and alignment of said second laser beam relative to said
first beam) in order to minimize any laser noise affecting output signal
quality.
In another specific embodiment, the FIG. 3 apparatus 74 illustrates an
optical head unit 76 comprising a light source 78, a beam splitting
element 80, a focusing element 82 (e.g., high NA lens or a Nomarski
objective) and a scatterer probe 84, preferably comprising a thin metallic
film.
The optical head 76 is used with a storage media 86 (e.g., rectangular or
circular) preferably held on a positioning mechanism 88, including a means
for positioning the scatterer probe 84 and focused beam 78 in close
proximity (e.g., <100 angstroms) to the media 86 such that a multi-pole
coupling interaction can develop between the scatterer probe 84 and a
bit-feature of the media.
The positioning mechanism 88 can also be used to produce a fast
translational or rotational motion between the sensing optical head unit
76 and the media surface 86. The positioning mechanism 88 is typically
part of a focusing/tracking system capable of small lateral and vertical
motions (e.g., a few micrometers to several centimeters) in order to
displace the sensing head unit 76 at any preselected height and to any
desired track of the media, for example, along a radial direction of the
disc. Typically, this action may be provided by a combination of
mechanical, electromechanical and/or piezo-electrical actuators. A fast
translation is typically obtained by spinning a disc 86 supporting the
encoded data by using a precise spindle mechanism 88, (e.g., air bearing
spindle) a motor 90 and a motor controller 92 coupled with a principal
electronic controller 94.
The light scattered from the probe 84 and the bits of the media 86 is
collected through the objective 82 and supplied to a detector 96 whose
output signal fed the controller 94. The principal controller 94 can
decode tracking bits patterns from, for example, a subcode of data bits
patterns 98 contained on the disc 86. A comparator circuit in association
with the principal controller 94 allows sampling speed and performs a
servo control of the motor controller 92. The principal controller 94
preferably includes a data processing circuit for accessing and converting
the data read from the disc to a suitable code (e.g., audio, image data
signal) for reproducing or displaying data.
In an alternative embodiment, a waveguide or a tapered metallized
single-mode optical fiber element (E. R. Betzig et al., U.S. Pat. No.
5,272,330) can be used as a means for directing the incident light to the
media.
The FIG. 4 shows experimental results obtained in the practice of the
present invention. The first electronic microscopic image 100 is an
example of a bits pattern of a commercially available audio CD having
about 650 MBytes storage capability. Essentially, the bit format
corresponds to an area of approximately 1.6.mu..times.0.8 .mu. and the
closest packing with which individual bits can be resolved by conventional
techniques is about 1 .mu.. The near-field optical images 102, 104 of a
bits pattern impressed in a chromium coated photoresist onto a silicon
substrate by using an e-beam processing were obtained with an apparatus as
disclosed in the present application. The image 102 corresponds to 1
.mu..sup.2 areas resolving small bits having 500 angstroms in spatial
dimensions that are spaced by about 1000 angstroms, yielding a practical
packing density of about 32 GBytes/in.sup.2. This value can easily be
envisioned to about 156 GBytes/in.sup.2 for 200 angstroms bit-format. The
image 104 provides a 0.5 .mu..sup.2 area of data density. Measurements in
a bandwidth of about 30 MHz showing a signal-to-noise ratio of about 50 dB
has been achieved by the inventors demonstrating major advances in speed
and density for optical data storage devices.
As articulated above, we have developed the detailed description of the
novel method and apparatus of the present invention by first disclosing
preferred apparatus (FIGS. 1-3). Utilization of such apparatus can yield
information about high density data encoded in a media. (FIG. 4) We now
turn our attention to how this information can be abstracted in an
intelligible manner, to thereby actualize the steps of the novel method.
Since the optical dipole interaction varies as r.sup.-3, a measured signal
primarily derives from the scatterer. One can therefore assume that the
scatterer can be modeled as a sphere of radius a, and polarizability
.alpha..sub.t and that the bit feature that is being decoded has a
polarizability .alpha..sub.f and radius a (although the theory could
easily be generalized for any arbitrary feature size). If the scatterer
and media are immersed in a driving electric field E.sub.i (caused by the
incident radiation), and .epsilon. is the dielectric permittivity of the
surrounding medium, the following coupled equations for the induced dipole
moments P.sub.t and P.sub.f in the scatterer and bit feature respectively
(FIG. 1) can be written as:
P.sub.t =.alpha..sub.t .epsilon.(E.sub.i +E.sub.f) (1)
P.sub.f =.alpha..sub.f .epsilon.(E.sub.i +E.sub.t) (2)
Here, E.sub.t and E.sub.f are the corresponding near-fields generated by
the dipole moments of scatterer and bit-feature respectively. For the case
where the spacing r between scatterer and bit-feature is greater than the
diameter 2a, the dipole approximation can be used and the following
expressions for E.sub.t and E.sub.f can be written:
##EQU1##
Substituting for E.sub.t and E.sub.f in equations (1) and (2) and solving
for P.sub.t and P.sub.f it is found, after eliminating terms of order less
than r.sup.-3, that:
##EQU2##
Equations (5) and (6) show very clearly how the scatterer polarization
couples with the bit-feature polarization to generate a polarization
modulation term
##EQU3##
It is this polarization modulation that produces a modulation in the
scattered electric field E.sub.s as the scatterer bit-feature spacing r is
modulated by vibrating the scatterer or moving the bit-feature relative to
the scatterer. As mentioned earlier, equations (5) and (6) are derived for
the case where r>2a the linear dimension of the dipole (for a sphere, this
linear dimension is comparable to its diameter). The corresponding
equations for arbitrary r can be obtained using quasi-static theory simply
by changing r to (r.sup.2 +a.sup.2)3/2 in equations (5) and (6). More
general expressions for the polarization modulation .DELTA.P and the
polarizability modulation .DELTA..alpha. are thus:
##EQU4##
From equations (7) and (8), .DELTA.P and .DELTA..alpha. decreases rapidly
from their maximum values as the scatterer bit-feature spacing is
increased (.about.1/r.sup.3). As it will be described later, for
situations where ka<<1,
##EQU5##
being the optical propagation constant in a medium of refractive index
n)the scattered electric field modulation .DELTA.E.sub.s is directly
proportional to .DELTA..alpha.; one can therefore expect to see a strong
decrease in .DELTA.E.sub.s as the scatterer bit-feature dipole-dipole
coupling decreases with increasing r. Furthermore, equations (7) and (8)
show that .DELTA.P and .DELTA..alpha. are proportional to the product of
the complex polarizability of the scatterer .alpha..sub.t and that of the
bit-feature .alpha..sub.f. Consequently, the phase of the scattered field
component .DELTA.E.sub.s can change drastically depending on the complex
polarizability of the scatterer as previously observed.
Now, the modulation .DELTA.E.sub.s of the scattered field E.sub.s caused by
the polarizability modulation .DELTA..alpha. can be calculated by applying
the scattering matrix treatment used by van de Hulst (Light Scattering by
Small Particles, Wiley, New York 1957) to study light scattering from
small particles. For an incident field E.sub.i, the spherically scattered
wave has electric field E.sub.s at a distance d in the far field given by
##EQU6##
where the relevant scattering matrix component S (which has both real and
imaginary components) can be written in terms of the polarizability
.alpha.:
##EQU7##
and for a simple sphere of radius a, and complex refractive index m
(relative to the surrounding medium)
##EQU8##
Note that imaginary terms of order k.sup.5 and higher order terms in the
expansion for S have been omitted as we are dealing with scattering from
very small particles (i.e., ka<<1).
As just shown, the reflected wave from the back surface of the media is a
concentric spherical wave of amplitude
##EQU9##
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