U.S. patent application number 13/865957 was filed with the patent office on 2014-01-16 for reconstruction of nonlinear wave propagation.
This patent application is currently assigned to Opteryx LLC. The applicant listed for this patent is Opteryx LLC. Invention is credited to Christopher Barsi, Jason W. Fleischer, Wenjie Wan.
Application Number | 20140016183 13/865957 |
Document ID | / |
Family ID | 42233594 |
Filed Date | 2014-01-16 |
United States Patent
Application |
20140016183 |
Kind Code |
A1 |
Fleischer; Jason W. ; et
al. |
January 16, 2014 |
RECONSTRUCTION OF NONLINEAR WAVE PROPAGATION
Abstract
Disclosed are systems and methods for characterizing a nonlinear
propagation environment by numerically propagating a measured
output waveform resulting from a known input waveform. The
numerical propagation reconstructs the input waveform, and in the
process, the nonlinear environment is characterized. In certain
embodiments, knowledge of the characterized nonlinear environment
facilitates determination of an unknown input based on a measured
output. Similarly, knowledge of the characterized nonlinear
environment also facilitates formation of a desired output based on
a configurable input. In both situations, the input thus
characterized and the output thus obtained include features that
would normally be lost in linear propagations. Such features can
include evanescent waves and peripheral waves, such that an image
thus obtained are inherently wide-angle, farfield form of
microscopy.
Inventors: |
Fleischer; Jason W.;
(Princeton, NJ) ; Barsi; Christopher; (Yonkers,
NY) ; Wan; Wenjie; (Princeton, NJ) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Opteryx LLC; |
|
|
US |
|
|
Assignee: |
Opteryx LLC
Princeton
NJ
|
Family ID: |
42233594 |
Appl. No.: |
13/865957 |
Filed: |
April 18, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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12629739 |
Dec 2, 2009 |
8427650 |
|
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13865957 |
|
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61200671 |
Dec 2, 2008 |
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Current U.S.
Class: |
359/326 ;
181/175 |
Current CPC
Class: |
G02F 1/353 20130101;
G01N 21/4795 20130101; G01N 21/41 20130101; G01N 2021/1782
20130101; G10K 15/00 20130101; G01N 2021/4173 20130101 |
Class at
Publication: |
359/326 ;
181/175 |
International
Class: |
G02F 1/35 20060101
G02F001/35; G10K 15/00 20060101 G10K015/00 |
Goverment Interests
STATEMENT REGARDING FEDERALLY SPONSORED R&D
[0002] This invention was made with Government support under Grant
No. NSF PHY-0605976 awarded by the National Science Foundation;
Grant No. AFOSR FA9550-07-1-0249 awarded by the Air Force Office of
Scientific Research; Department of Defense grant awarded by the
Army Research Office through a National Defense Science and
Engineering Graduate Fellowship; and Grant No. DE-FG02-08ER55001
awarded by the Department of Energy. The Government has certain
rights in this invention.
Claims
1. (canceled)
2. A physical system, comprising: a medium configured to receive a
first waveform and yield a second waveform; and a processor
configured to obtain information about two of a nonlinear response
of the medium, said first waveform, and said second waveform and
generate a characterization of the remaining one of said nonlinear
response of the medium, first waveform, and second waveform, said
generated characterization determined by numerical propagation of
one of said first and second waveforms.
3. The system of claim 2, wherein one of said waveforms is
optical
4. The system of claim 2, wherein one of said waveforms is
acoustic.
5. The system of claim 2, wherein one of said waveforms is
ultrasonic.
6. The system of claim 2, wherein at least one of the first and
second waveforms is spatially separable.
7. The system of claim 2, wherein at least one of the first and
second waveforms is spatially non-separable.
8. The system of claim 2, wherein the medium is transmissive.
9. The system of claim 2, wherein the medium is reflective.
10. The system of claim 2, wherein the medium is diffractive.
11. The system of claim 2, wherein the medium is a spatial light
modulator.
12. The system of claim 2, wherein the medium comprises a
filter.
13. The system of claim 2, wherein the medium is a
computer-controlled display.
14. The system of claim 2, wherein at least one of the first or
second waveform undergoes a change in the intensity due to the
non-linear response of the medium.
15. The system of claim 2, wherein at least one of the first or
second waveform undergoes an intensity-dependent phase change due
to the non-linear response of the medium.
16. The system of claim 2, wherein the nonlinear response is from a
transducer.
17. The system of claim 2, wherein the nonlinear response results
from noise.
18. The system of claim 2, wherein the numerical propagation
comprises statistical correlation.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of U.S. application Ser.
No. 12/629,739, filed Dec. 2, 2009, entitled "RECONSTRUCTION OF
NONLINEAR WAVE PROPAGATION," which claims priority to and the
benefit of the filing date of U.S. Provisional Application No.
61/200,671, filed on Dec. 2, 2008, entitled "HOLOGRAPHIC
RECONSTRUCTION OF NONLINEAR WAVE PROPAGATION," the benefits of the
filing dates of which are hereby claimed and the specifications of
which are incorporated herein by this reference.
BACKGROUND
[0003] 1. Field
[0004] The present disclosure generally relates to the field of
nonlinear wave propagation, and more particularly, to systems and
methods for modeling and/or calculation of optical wave propagation
and uses thereof.
[0005] 2. Description of the Related Art
[0006] Propagation of waves can occur in various media that can be
classified into either a linear medium or a nonlinear medium. In a
linear medium, dynamics are independent of wave intensity and wave
propagation occurs such that superposition principle holds. In a
nonlinear medium, the superposition principle does not hold. As
generally known, this distinction is important because many
physical systems can be modeled as linear systems. For physical
systems that are generally approximately linear, linear modeling
can provide an approximation of the true physical behavior.
However, all media exhibit nonlinear behavior, if the wave energy
is high enough.
[0007] In a nonlinear medium, a propagating wave may undergo
intensity-dependent phase changes, thereby distorting signals as
they propagate. In certain situations, such distortion of signals
due to the nonlinear propagation is sometimes referred to as "wave
mixing." Among other consequences, such wave mixing due to
nonlinearity results in significant effects such as mode coupling,
generation of new frequencies, and modifications to the signal
phase.
SUMMARY
[0008] In certain embodiments, the present disclosure relates to a
method for characterizing a nonlinear medium. The method includes
providing a known input waveform to said nonlinear medium such that
said input waveform propagates through said nonlinear medium. The
input waveform is representative of two or more dimensional spatial
information. The method further includes characterizing an output
waveform emerging from said nonlinear medium. The method further
includes computationally propagating said output waveform through
said nonlinear medium so as to obtain an estimated waveform that
sufficiently matches said input waveform. The computational
propagation depends on one or more properties of said nonlinear
medium such that obtaining of said estimated waveform results in
characterization of said one or more properties of said nonlinear
medium.
[0009] In certain embodiments, said nonlinear medium comprises a
nonlinear optical medium. In certain embodiments, said
characterizing of said output waveform comprises characterizing a
complex field emerging from said nonlinear medium. In certain
embodiments, an amplitude is measured directly by an imaging
device, and a phase is measured holographically by interference
measurement. In certain embodiments, said input waveform comprises
a coherent waveform.
[0010] In certain embodiments, the method further includes
measuring correlation among various components of said input
waveform. In certain embodiments, said input waveform comprises an
incoherent waveform.
[0011] In certain embodiments, said computationally propagating
comprises numerically evaluating a nonlinear wave equation applied
to a field representative of said output waveform. In certain
embodiments, said field comprises a slowly-varying scalar field. In
certain embodiments, said numerically evaluating comprises
numerically evaluating backward propagation of said scalar field
through said nonlinear medium. In certain embodiments, said
backward propagation of said field is based on said field being
estimated as
.quadrature.(z.sub.i).apprxeq.e.sup.-dzD.quadrature.(z.sub.f=z.sub.i+dz),
where quantity z represents a propagation direction with
z.sub.f>z.sub.i, quantity dz represents an incremental
propagation distance, and quantities D and N represent linear and
nonlinear operators, respectively.
[0012] In certain embodiments, the method further includes storing
information about said one or more properties so as to allow
retrieval and application to an unknown input waveform to
characterize a resulting output waveform, or to a configurable
input waveform that yields a desired output waveform.
[0013] In certain embodiments, the present disclosure relates to an
optical system having a nonlinear optical medium configured to
receive a first waveform and yield a second waveform. The system
further includes a processor configured so as to obtain information
about two of said nonlinear optical medium, first waveform, and
second waveform and generate characterization of the remaining one
of said nonlinear optical medium, first waveform, and second
waveform. The generated characterization includes numerical
propagation of one of said first and second waveforms through said
nonlinear optical medium.
[0014] In certain embodiments, said processor is configured so as
to obtain information about said first waveform and said second
waveform and generate characterization of said nonlinear optical
medium. In certain embodiments, said processor is configured so as
to obtain information about said first waveform and said nonlinear
optical medium and generate characterization of said second
waveform. In certain embodiments, said second waveform comprises an
output waveform emerging from said nonlinear optical medium. In
certain embodiments, said output waveform includes one or more
components originating from within said nonlinear optical medium as
said first waveform propagates therethrough, such that when said
output waveform is incident on a substrate, said one or more
components interacts with said substrate. In certain embodiments,
said one or more components are configured for use in lithography.
In certain embodiments, said output waveform includes one or more
components originating from within said nonlinear optical medium as
said first waveform propagates therethrough. The one or more
components have selected information attributable to said first
waveform. In certain embodiments, said output waveform is
configured for transmission to a remote location.
[0015] In certain embodiments, said second waveform comprises a
waveform at least partially within said nonlinear optical medium.
In certain embodiments, said second waveform includes one or more
components originating from within said nonlinear optical medium as
said first waveform propagates through at least a portion of said
nonlinear optical medium, such that said one or more components
interacts with said nonlinear optical medium in a desired manner.
In certain embodiments, said nonlinear optical medium comprises a
data storage medium.
[0016] In certain embodiments, said processor is configured so as
to obtain information about said second waveform and said nonlinear
optical medium and generate characterization of said first
waveform. In certain embodiments, said first waveform comprises a
first component and a second component, with said first and second
components coupling in said nonlinear optical medium to generate a
new nonlinear component that becomes part of said second waveform
even if either or both of said first and second components of said
first waveform to not become part of said second waveform. The new
nonlinear component carries information imparted to it during said
coupling. In certain embodiments, at least one of said first and
second components of said first waveform comprises an evanescent
wave component associated with an object being observed. In certain
embodiments, said evanescent wave is associated with a
subwavelength sized feature on said object. In certain embodiments,
said object is in contact with said nonlinear optical medium so as
to allow coupling of said evanescent wave component with one of
said first and second components so as to yield said new nonlinear
component. In certain embodiments, said nonlinear optical medium is
part of a sample holder for holding said object. In certain
embodiments, said sample holder comprises a microscope slide. In
certain embodiments, said sample holder comprises an enclosure that
encloses at least a portion of said object being imaged.
[0017] In certain embodiments, at least one of said first and
second components of said first waveform comprises a peripheral
component that would be lost and not become part of said second
waveform if propagated through a linear optical medium. In certain
embodiments, said information imparted by said peripheral component
to said new nonlinear component increases effective field of view
said first waveform captured by said second waveform. In certain
embodiments, said nonlinear optical medium comprises a filter
configured to be placed in front of an imaging device so as to
provide said increased effective field of view. In certain
embodiments, devices such as a profilometry device, a tomography
device, or a material testing device can have features of the
foregoing optical system. Such material testing device can be
configured to characterize internal potentials of a material being
tested, or to identify and characterize material defects or induced
defects of a material being tested.
[0018] In certain embodiments, the present disclosure relates to a
computer-readable medium containing machine-executable instructions
that, if executed by a device having one or more processors, causes
the device to perform operations. Such operations include obtaining
a digital representation of a measured output waveform resulting
from propagation of an input waveform through a nonlinear medium.
The operations further include computationally propagating said
digital representation of said measured output waveform through
said nonlinear medium so as to reconstruct a digital representation
of said input waveform, with said computational propagation
depending on one or more properties of said nonlinear medium,
information about said one or more properties stored in said
computer-readable medium or accessible by said machine-executable
instructions.
[0019] In certain embodiments, the present disclosure relates to a
method for characterizing nonlinear wave propagation. The method
includes providing a known input waveform to propagate through a
first nonlinear environment so as to yield a first intensity
distribution. The method further includes providing said known
input waveform to propagate through a second nonlinear environment
so as to yield a second intensity distribution. The method further
includes determining an output waveform based at least in part on a
difference between said first and second intensity distributions.
The method further includes computationally propagating said output
waveform through one of said first and second nonlinear
environments to reconstruct an approximation of said input
waveform.
[0020] In certain embodiments, the method further includes
characterizing said one of said first and second nonlinear
environments based on said computational propagation. In certain
embodiments, the method further includes storing information
representative of said characterization of said one of said first
and second nonlinear environments in a computer-readable medium. In
certain embodiments, said nonlinear wave propagation comprises
nonlinear propagation of electromagnetic radiation.
[0021] In certain embodiments, the present disclosure relates to an
optical system having a nonlinearity component configured to
provide first and second nonlinear propagation environments for an
input waveform to respectively yield first and second output
waveforms. The system further includes an imaging device configured
to detect at least intensity portions of said first and second
output waveforms and generate first and second intensity
distributions, respectively. The system further includes a
processor configured to determine an output waveform based at least
in part on a difference between said first and second intensity
distributions.
[0022] In certain embodiments, said processor is further configured
to computationally propagate said output waveform through one of
said first and second nonlinear propagation environments to
reconstruct an approximation of said input waveform. In certain
embodiments, said nonlinearity component comprises a first medium
and a second medium, said first medium providing said first
nonlinear propagation environment and said second medium providing
said second nonlinear propagation environment. In certain
embodiments, one of said first and second environment comprises a
propagation medium having a substantially zero degree of
nonlinearity.
[0023] In certain embodiments, said nonlinearity component
comprises a tunable nonlinear medium and a controller, where said
controller is configured to provide at least two settings for said
tunable nonlinear medium so as to provide said first and second
nonlinear propagation environments. In certain embodiments, said
output waveform includes phase information retrieved from said
difference between said first and second intensity distributions.
In certain embodiments, said output waveform includes polarization
information retrieved from said difference between said first and
second intensity distributions. In certain embodiments, said input
waveform comprises incoherent wave, and wherein said output
waveform includes correlation information to accommodate said
incoherent wave.
[0024] In certain embodiments, the present disclosure relates to an
apparatus having a nonlinear element configured so as to provide a
nonlinear propagation environment for a wave passing therethrough,
where said nonlinear element is dimensioned to receive an input and
yield an output. The apparatus further includes a computer-readable
medium containing instructions that calculates one of said input
and output if given the other, said calculation achieved
numerically in an iterative manner using one or more parameters
that characterize said nonlinear propagation environment.
[0025] In certain embodiments, said apparatus comprises an imaging
device. In certain embodiments, said instructions calculates said
input based on said output. In certain embodiments, said output
comprises a measured output. In certain embodiments, said apparatus
comprises a microscope. In certain embodiments, said output
comprises a desirable output. In certain embodiments, said
apparatus comprises a lithographic device.
BRIEF DESCRIPTION OF THE DRAWINGS
[0026] FIG. 1 schematically depicts a system having a nonlinear
medium through which an input wavefunction propagates to yield an
output wavefunction;
[0027] FIG. 2 shows that in certain embodiments, the nonlinear
medium of FIG. 1 can be an optical medium having nonlinearity;
[0028] FIG. 3 shows that in certain embodiments, a process can be
implemented to characterize a given nonlinear medium based on a
known input wavefunction and a measured output wavefunction;
[0029] FIG. 4 shows that in certain embodiments, a process can be
implemented to characterize an unknown input wavefunction based on
a characterized nonlinear medium and a measured output
wavefunction;
[0030] FIG. 5 shows that in certain embodiments, a process can be
implemented to obtain a desired output wavefunction based on a
characterized nonlinear medium and a configurable input
wavefunction;
[0031] FIG. 6 shows an example configuration for characterizing a
nonlinear medium based on a known input and a measured output;
[0032] FIGS. 7A-7C show different example object inputs which can
be used in the configuration of FIG. 6 to characterize the
nonlinear medium and to verify the characterization;
[0033] FIGS. 8A and 8B shows intensity and phase distributions of
the input, measured linear propagation outputs, measured nonlinear
propagation outputs, and back-propagation results for a Gaussian
object beam input;
[0034] FIG. 9 shows intensity and phase distributions of the input,
measured linear propagation outputs, measured nonlinear propagation
outputs, and back-propagation results for an object beam passing
through a phase step sometimes referred to as a .pi.-stripe;
[0035] FIG. 10 shows intensity and phase distributions of the
input, measured linear propagation outputs, measured nonlinear
propagation outputs, and back-propagation results for an object
comprising a more complicated USAF 1951 resolution chart;
[0036] FIG. 11 shows that in certain embodiments, an imaging system
can be configured to obtain an image of an object field through a
nonlinear medium;
[0037] FIG. 12 shows a conventional microscope imaging
configuration with various limitations imposed by numerical
aperture;
[0038] FIG. 13A shows an example microscopic imaging configuration
where effective numerical aperture can be increased by use of a
nonlinear medium;
[0039] FIG. 13B shows an example microscope imaging configuration
where information from an evanescent wave associated with a feature
on a sample surface can be coupled with a wave transmitted through
a nonlinear medium that is in contact with the sample surface;
[0040] FIG. 13C shows an example microscope imaging configuration
where information an evanescent wave associated with a feature on a
sample can be coupled with a wave transmitted through a nonlinear
medium where at least the feature on the sample is substantially
embedded in the nonlinear medium;
[0041] FIG. 14 shows that in certain embodiments, a system can be
configured so that a desired output can be obtained by passing a
configurable input through a characterized nonlinear medium;
[0042] FIG. 15 shows that in certain embodiments, the system of
FIG. 14 can be modified so that the desired output and
corresponding effect of the output occurs within a characterized
nonlinear medium;
[0043] FIG. 16 shows that in certain embodiments, the system of
FIG. 14 can be combined with a detection system for detecting the
transmitted output and reconstructing the original input;
[0044] FIG. 17 shows that in certain embodiments, various
functionalities associated with characterization and use of
characterized nonlinear media can be achieved by a nonlinearity
differential component;
[0045] FIG. 18 shows that in certain embodiments, a process can be
implemented to characterize a given nonlinear propagation based on
a known input wavefunction and a difference between two measured
output intensity distributions;
[0046] FIG. 19 shows an example of how two or more different
nonlinear propagation environments can be provided;
[0047] FIG. 20 shows another example of how two or more different
nonlinear propagation environments can be provided;
[0048] FIG. 21 shows an example where a linear propagation
environment and a nonlinear propagation medium can be provided;
[0049] FIG. 22 shows that in certain embodiments, an imaging system
can be configured to obtain an image of an object field via the
nonlinearity difference component of FIG. 17;
[0050] FIG. 23 shows a more specific example of the imaging system
of FIG. 22; and
[0051] FIG. 24 shows an example algorithm that can be implemented
to reconstruct an input wavefunction by an iterative
back-propagation process.
[0052] These and other aspects, advantages, and novel features of
the present teachings will become apparent upon reading the
following detailed description and upon reference to the
accompanying drawings. In the drawings, similar elements have
similar reference numerals.
DETAILED DESCRIPTION OF CERTAIN EMBODIMENTS
[0053] The present disclosure generally relates to systems and
methods for characterizing nonlinear wave propagation and solving
for various unknowns in propagation of light through a nonlinear
medium. As described herein, such characterization yields a number
of practical applications, some of which can significantly extend
performance capabilities in areas of optics technology.
[0054] Propagation of electromagnetic radiation can occur in
various media that can be classified into either linear medium or
nonlinear medium. In a linear medium, wave propagation occurs such
that superposition principle holds. In a nonlinear medium, the
superposition principle does not hold. As generally known, many
physical systems can be modeled as linear systems. For physical
systems that are generally approximately linear, linear modeling
can provide an approximation of the true physical behavior.
[0055] In a nonlinear medium, a propagating wave may undergo
intensity-dependent phase changes, thereby distorting signals as
they propagate. In certain situations, distortion of signals due to
the nonlinear propagation is sometimes referred to as "wave
mixing." Among other consequences, such wave mixing due to
nonlinearity results in significant effects such as mode coupling,
generation of new frequencies, and modifications to the signal
phase.
[0056] In the context of imaging, the dynamics of wave mixing
within a nonlinear medium cannot be measured directly. In certain
methods, all wave mixing occurring inside a nonlinear medium is
inferred from profiles of the wave before entering the medium
(initial) and exiting from the medium (final). Thus, in such
methods, nonlinear wave dynamics are characterized by varying known
input parameters and comparing measured outputs.
[0057] In the context of material characterization, some parameters
of the nonlinear response of a medium can be inferred from
comparison of known output with known input. For example, a thin
nonlinear medium placed in the path of a focusing optical beam
typically will change the properties of the focus, due to
intensity-dependent changes to the phase. Moving the medium through
the focus then allows successive measurements of the changes due to
such parameters as the change in intensity, absorption, etc., as
described in such papers as Sheik-Bahae, M., Said, A. A., & van
Stryland, E. W., High sensitivity single beam n.sub.2 measurements,
Opt. Lett. 14, 955 (1989). For thick media, it is normally assumed
that significant changes in beam shape do not occur. In various
methods as described herein, physical scanning or beam refocusing
is generally not needed and a greater degree of beam changes, wave
mixing, etc., are allowed. Further, forward- and back-propagation
techniques can be used to computationally recover material
parameters.
[0058] Provided herein are, among others, systems having a
nonlinear medium, characterizing wave propagation within such a
medium, and utilizing such characterization in various practical
applications. In FIG. 1, such a system 100 is depicted as having a
nonlinear medium 102 into which an input wave 104 (represented by a
wavefunction .psi..sub.in) is introduced. The input wave 104
propagates through the medium 102 and emerges from the medium 102
as an output wave 106 (represented by a wavefunction
.psi..sub.out).
[0059] In certain embodiments, as shown in FIG. 2, the nonlinear
medium can be a nonlinear optical medium 112 suitable for receiving
an optical input wave 114 and generating an optical output wave
116. For the purpose of description herein, various nonlinear media
and wave propagations therein are discussed in optics context. It
will be understood, however, that one or more features of the
present disclosure can be applied to nonlinear systems involving
waves outside of what is typically considered to be optical
regime.
[0060] In certain embodiments, the present disclosure relates to
methods for characterizing a nonlinear medium. In certain
embodiments, such characterization of the nonlinear medium includes
using a nonlinear propagation model to back-propagate a measured
output wave through the nonlinear medium to obtain a modeled input
wave that is acceptably close to the known input wave.
[0061] In certain embodiments, such characterization of the
nonlinear medium can be implemented as a process 120 shown in FIG.
3. In a process block 122, an input wave incident on the nonlinear
medium can be characterized. For example, a wavefunction of the
input wave can be characterized. In certain embodiments, such
characterization can include measurement of the input wave prior to
entry into the nonlinear medium. In a process block 124, an output
wave exiting from the nonlinear medium can be measured so as to
yield an output wavefunction. In a process block 126, the output
wavefunction can be numerically back-propagated through the
nonlinear medium so as to substantially reconstruct the input
wavefunction. Because the input wavefunction is known (in process
block 122), accuracy of the numerical back-propagation methodology,
and thus characterization of the nonlinear model of the medium, can
be determined. Thus, in a process block 128, the nonlinear medium
can be characterized based on the numerical back-propagation. In
certain embodiments, such characterization can include an iterative
process to obtain an estimation of one or more parameters
associated with the nonlinear medium. In certain embodiments, such
characterization can include a plurality of calculations of a
wavefunction at various incremental positions within the nonlinear
medium.
[0062] Once characterization of the nonlinear medium is obtained,
an unknown input can be characterized based on a corresponding
output that is either known or measured. Also, an input can be
configured such that its propagation through the characterized
nonlinear medium results in a desired output. FIG. 4 shows a
process 130 that can be implemented to achieve the former; and FIG.
5 shows a process 140 that can be implemented to achieve the
latter.
[0063] In certain embodiments, as shown in the process 130 of FIG.
4, an input that is either unknown or to be further characterized
can be propagated in a process block through a nonlinear medium
that has been characterized. In a process block 134, an output
resulting from the nonlinear propagation can be obtained. In a
process block 136, the input can be characterized based on the
output and information representative of the characterized
nonlinear medium. For example, a measured output can be numerically
back-propagated through the nonlinear medium so as to obtain an
estimation of the input.
[0064] In certain embodiments, as shown in the process 140 of FIG.
5, a desired output can be determined in a process block 142. In a
process block 144, an input can be configured such that propagation
of the input through a characterized nonlinear medium yields an
output that is acceptably close to the desired output. For example,
an input can be calculated using calculation or back-propagation of
the desired output through the medium. In certain embodiments, the
actual output emerging from the nonlinear medium can be
substantially same as the desired output. In a process block 146,
the configured input can be propagated through the nonlinear medium
to obtain an output that is an approximation of or substantially
same as the desired output.
[0065] FIG. 6 shows an example setup 150 that can be utilized to
facilitate characterization of a nonlinear medium 164. The setup
includes a laser 152 that outputs a beam 154. The example laser 152
is a Verdi.TM. type laser operating at a wavelength of
approximately 532 nm. The beam 154 is split by a beamsplitter 156
into first and second beams 158 and 170. The second beam 170 is
used for phase measurement as described below. The first beam 158
is subjected to an optical system 160 that generates an object
signal 162, and the object signal 162 from the optical system 160
acts as an input wave for the nonlinear medium 164. The input wave
162 can be represented by the wavefunction .psi..sub.in described
herein.
[0066] The example nonlinear medium 164 is a photorefractive
strontium barium niobate (SBN: 75) crystal with a self-defocusing
nonlinearity. The crystal 164 has transverse dimensions of
approximately 2 mm.times.5 mm, and a longitudinal dimension of
approximately 10 mm (along the propagation direction). The crystal
164 is oriented so that its crystalline c-axis (depicted as c) is
pointing in the substantially same direction as the polarization of
the laser beam 154. Accordingly, the input wave 162 is
extraordinarily polarized with respect to the crystal 164. To
induce the nonlinearity property, the example crystal 164 is
subjected to a voltage difference of approximately 150 V so as to
yield an electric field of approximately -750 V/cm along the
crystalline axis. To yield a substantially linear response of the
crystal 164, the voltage to the crystal 164 is turned off. Also,
the example crystal 164 can be rotated so that the 5 mm dimension
is along the longitudinal direction. Thus, application or absence
of voltage to the rotated crystal 164 yields nonlinear or linear
propagation distance of 5 mm.
[0067] It will be understood that other types of nonlinear media
can be used.
[0068] The input wave 162 is shown to propagate through the
nonlinear medium 164 and emerge as an output wave 166. The output
wave 166 is combined with a phase-shifted reference beam 180 by a
beamsplitter 182 to yield a combined beam 184. As shown, the
phase-shifted reference beam 180 is based on the second beam 170.
The second beam 170 is redirected to a phase-shifting mirror 176
via a mirror 172. To shift the reference beam (170) into the
phase-shifted reference beam (180), the phase-shifting mirror 176
is moved in incremental steps of approximately .lamda./4 via a
piezo-actuator 178. In certain embodiments, such shifted images can
allow removal of various artifacts from the interference terms, as
described in, for example, Yamaguchi, I. & Zhang, T.,
Phase-shifting digital holography, Opt. Lett. 22, 1268 (1997).
[0069] As shown in FIG. 6, the combined beam 184 is imaged by an
imaging device 190. As is generally known, a given combined beam
184 yields interference fringes in the corresponding image. The
mirror 176 can be moved (e.g., to four locations) so as to yield
different interference fringe patterns. Such fringe patterns can be
analyzed by known algorithms to produce wavefront shapes.
[0070] In the example shown, the combined beam 184 is magnified
approximately four times by a lens 186, and the magnified beam is
imaged by a charge-coupled-device (CCD) having 520.times.492 array
of 9.9 .mu.m sized pixels. Such magnification allows one to better
resolve various features of the field associated with the combined
beam 184. Although the imaging lens 186 itself gives rise to a
quadratic phase factor, because both reference (180) and object
(166) beams are imaged together, no extraneous fringes in the
interference pattern are produced.
[0071] An image of the combined beam 184 captured in the foregoing
manner is a holographic representation of the output wave 166
emerging from the nonlinear medium 164. This digital holographic
holographic system produced phase information by optical
interference holographic technique. Amplitude information can also
be provided simply by detection of the beam propagated through the
nonlinear medium 164 to the detector 190; and phase information can
be obtained as described herein. Accordingly, the captured image
includes information (amplitude and phase) representative of a full
complex field.
[0072] In certain embodiments, propagation of the complex field
representative of the output wave can be approximated as a scalar
field undergoing paraxial dynamics, where a wavefunction
.psi.(x,y,z) is a slowly varying envelope of the field. In this
approximation, the propagation can be described by a nonlinear form
of Schrodinger equation:
.differential. .psi. .differential. z = [ i 1 2 k .gradient. .perp.
2 + i .DELTA. n ( .psi. 2 ) ] .psi. .ident. [ D + N ( .psi. 2 ) ]
.psi. ( Eq . 1 ) ##EQU00001##
where k=2.pi.n.sub.0/.lamda., .lamda. is the free-space wavelength,
n.sub.0 is the medium's base index, .DELTA.n(|.psi.|.sup.z) is the
nonlinear index change, and D and N are the linear and nonlinear
operators, respectively. A typical choice for
.DELTA.n(|.psi.|.sup.z) is the cubic Kerr nonlinearity, with
.DELTA.n=.gamma.|.psi.|.sup.2 where .gamma. is the nonlinear
coefficient. For any nonlinearity, various issues generally can be
considered. Such issues can include invertibility, integrability,
vector decoupling, noise, nonlinear instabilities, and so on.
[0073] A wavefunction that satisfies Equation 1 and whose evolution
can be calculated numerically using a known Fourier split-step
method (in which the linear and nonlinear operators act
individually for each increment of propagation distance dz), can be
represented as:
.psi.(z+dz).apprxeq.e.sup.dzDe.sup.dzN(.psi.).psi.(z). (Eq. 2)
Equation 2 can be inverted by applying e.sup.-dzNe.sup.-dzD to both
sides to allow calculation of the field at some location z.sub.i
given the field farther along the sample at location z.sub.f that
is greater than z.sub.i. Such inverted equation can be represented
as:
.psi.(z.sub.i).apprxeq.e.sup.-dzN(.psi.)e.sup.-dzD.psi.(z.sub.f=z.sub.i+-
dz). (Eq. 3)
This back-propagation can be treated as an initial-value problem,
in which the output is treated as a starting point. Such
back-propagation works in situations where inverse scattering may
fail, such as in media with non-integrable nonlinearities.
[0074] In certain embodiments, linear and nonlinear propagators
(e.sup.-dzD and e.sup.-dzN(.psi.), respectively) can be transformed
into Fourier representations for the above-referenced numerical
calculation via the known Fourier split-step method. The linear
propagator's Fourier representation can be expressed as in Equation
4. Similarly, the nonlinear propagator can be represented as a
Fourier transform of e.sup.-dz.DELTA.n, where .DELTA.n can be
selected as described herein.
[0075] As is known, the method of back-propagating to overcome
certain issues associated with nonlinear media was demonstrated for
one-dimensional pulses in fiber. Additional details concerning such
one-dimensional technique can be found in, for example, articles by
Goldfarb, G. & Li, G., Demonstration of fiber impairment
compensation using split-step infinite-impulse-response filtering
method, Electron. Lett. 44, 814-815 (2008); and Mateo, E., Zhu, L.
& Li, G., Impact of XPM and FWM on the digital implementation
of impairment compensation for WDM transmission using backward
propagation, Opt. Express 16, 16124-16137 (2008). It will be
emphasized that application of this technique, as described herein,
to two or more dimensional spatial beams containing image
information adds considerable complexity to the inversion, for
example, by introducing degenerate solutions or an anisotropic
response. In certain situations, the increase in dimension can also
add significant new consequences. For example, it is known that one
typically cannot image through an optical fiber, as modes separate
and distort as they propagate, as described in such papers as
Yariv, A., Three-dimensional pictorial transmission in optical
fibers, Appl. Phys. Lett. 28, 88 (1976).
[0076] For the example nonlinear medium 164 (SBN: 75) of FIG. 6,
the photorefractive screening nonlinearity of the crystal is
saturable, with a response
.DELTA. n .varies. r ij I 1 + I , ##EQU00002##
where I is the intensity I=|.psi.|.sup.z, normalized to a
background (dark current) intensity, and .eta..sub.j is the
appropriate electro-optic coefficient. However, recent experiments
have shown that for the defocusing parameters considered here, the
simpler Kerr nonlinearity .DELTA.n=-|.gamma.|I proves sufficient
for modeling. Additional details concerning such approximation can
be found at, for example, an article by Wan, W., Jia, S. &
Fleischer, J. W., Dispersive, superfluid-like shock waves in
nonlinear optics, Nature Phys. 3, 46-51 (2007). Similarly, loss has
been ignored, although the inclusion of linear operators such as
absorption should not affect or substantially affect the
reconstruction process.
[0077] The uniaxial nature of the crystal 164 (SBN: 75) introduces
a slight anisotropy in the nonlinear response. To account for such
anisotropy, a relatively small anisotropic correction (.delta.) of
approximately 7% is introduced to the index of refraction in the
x-direction for the linear propagator. Thus, in a Fourier domain
representation, following correction can be introduced:
- ? ? ( ? ? + ? ? ) -> - ? 4 .pi. ( ? ? + ? ? ) ? indicates text
missing or illegible when filed ( Eq . 4 ) ##EQU00003##
where .delta.=0.07 and k.sub.x and k.sub.y are the wavevectors for
the x- and y-axis. The parameter .delta. can be held constant,
although functional dependence on the intensity did not change the
result. The value of .delta. does, however, can depend on the
applied voltage. In the context of a Gaussian beam, this numerical
approximation affects the Gaussian beam (as describe herein)
generally at its focus only. Furthermore, it does not affect or
substantially affect the other reconstructions as described herein.
Therefore, it is believed that this modification becomes applicable
to high-intensity, focused beams. If based on the foregoing
anisotropic correction, other more complex models of the
photorefractive nonlinearity that include charge transport and
saturation effects may not be as successful at recovering the
input.
[0078] In the example setup 150 of FIG. 6, the optical system 160
includes different configurations for providing different inputs to
the nonlinear medium 164. In certain embodiments, a laser beam
having a substantially Gaussian beam profile can be provided to the
nonlinear medium to calibrate the foregoing back-propagation
algorithm, thereby characterizing the nonlinear medium.
[0079] As shown in FIG. 7A, the back-propagation algorithm can be
calibrated by focusing, via a lens 210 (e.g., a 20 cm plano-convex
lens), a Gaussian beam 158 (approximately 10 .mu.W) onto the input
face of the crystal 164. FIGS. 8A and 8B show various panels (260,
270) at various stages of evolution of the Gaussian beam in linear
propagation, nonlinear propagation, and numerical
back-propagation.
[0080] In FIG. 8A, upper row of panels (a, b, c) show measured beam
intensities for linear propagation (voltage turned off), at z=0
(input face of the crystal), z=5 mm (midpoint of the crystal,
approximated by rotating the crystal), and z=10 mm (exit face of
the crystal). Such locations are depicted in a panel indicated as
262. In FIG. 8B, upper row of panels (h, i, j) show measured phase
measurements (modulo 2.pi.) for linear propagation (voltage turned
off), at z=0 (input face of the crystal), z=5 mm (midpoint of the
crystal, approximated by rotating the crystal), and z=10 mm (exit
face of the crystal). Such locations are depicted in a panel
indicated as 272. In each of the panels a, b, c, h, i, and j, a
scale bar representative of approximately 50 .mu.m is provided at
lower left corner.
[0081] Measured intensities and phases of nonlinear propagation are
shown in middle rows of panels (d and e in FIG. 8A, and k and l in
FIG. 8B). Panels d and k represent intensity and phase measurements
at z=5 mm, respectively, obtained by measurements at the exit face
of a similar crystal having a length of approximately 5 mm (in
place of the 10 mm crystal). Panels e and l represent intensity and
phase measurements at z=10 mm, respectively, obtained by
measurements at the exit face of the 10 mm crystal. In each of the
panels d, e, k, and l, a scale bar representative of approximately
50 .mu.m is provided at lower left corner.
[0082] One can see the dramatically different output in the
nonlinear case, particularly after 10 mm of propagation (roughly
three diffraction lengths). In particular, the nonlinear beam has a
depleted central region and high-frequency fringes at the edges, a
profile that does occur in the linear case (in which a Gaussian
beam stays Gaussian). These features can result from wavebreaking
of the central portion of the beam into its own tails and are
similar to the optical shocks demonstrated in articles such as Wan,
W., Jia, S. & Fleischer, J. W., Dispersive, superfluid-like
shock waves in nonlinear optics, Nature Phys. 3, 46-51 (2007).
[0083] The measured output field depicted in panels e and l of
FIGS. 8A and 8B can be back-propagated numerically as described
herein to reconstruct the measured input (panels f and m), as well
as the midpoint (z=5 mm) field (panels g and n). As described
herein in reference to FIG. 6, the imaging device 190 used for
measurements of various fields has a CCD having 520.times.492 array
of pixels. For the digital reconstruction, the frames recorded by
the camera (190) were cropped horizontally, and padded with zeros
vertically to create 512.times.512 pixel frames. In certain
embodiments, the beam propagation code used optimally utilizes the
fast Fourier transform (FFT) algorithm in linear step(s). The
numerical pixel size was matched with the effective (demagnified)
camera pixel size, 2.5 .mu.m, and the propagation step size
(.DELTA.z) was approximately 65 .mu.m for the numerical
back-propagation calculation. A decrease in the step size
(.DELTA.z) to approximately 20 .mu.m showed no appreciable change
in the reconstruction, confirming that numerical convergence had
been achieved.
[0084] To achieve the reconstructed input shown in panels f and m,
the nonlinear coefficient .gamma. is adjusted until the
sum-of-squares error between the two profiles is substantially
reduced or minimized. As shown panels f and m, there is very good
agreement in both phase and intensity with the measured input
(panels a and h). Similarly, and as shown panels g and n, there is
very good agreement in both phase and intensity of the
reconstruction at z=5 mm with the measured input counterparts
(panels d and k). The agreement is particularly pronounced in the
asymmetry of the central portion of the beam, which is slightly
left of center.
[0085] In certain embodiments, a value of the nonlinear coefficient
that yields the foregoing reconstruction can then be fixed and used
for some or all subsequent measurements involving the crystal and
different inputs.
[0086] Although the expanding Gaussian beam experiences significant
changes in both intensity and phase, other nonlinear features or
structures, such as solitons, generally maintain constant-intensity
profiles and acquire only an overall phase change upon propagation.
Normally, experimental observations of these structures follow from
launching a beam close to the initial soliton profile; confirmation
is then reported by comparing the original input to the nonlinear
output after several diffraction lengths. Such an observation
method, however, may say little about evolution towards a
steady-state profile, especially for initial conditions far from
the soliton existence curve. As is known, cut-back methods provide
volumetric information but involve crystal damage or special
geometry. Other methods, such as near-field probes and
scattered-light measurements, are similarly direct but work well
primarily for one-dimensional solitons and cross-sections.
[0087] In contrast, one or more features of the technique disclosed
herein allow reconstruction, and thus effective imaging, of the
beam dynamics all along its propagation, without relying on
coupling mechanisms or material modification.
[0088] As shown in FIGS. 6 and 7B, the back-propagation algorithm
calibrated as described herein can be used to reconstruct an image
of a dark stripe pattern 220. As shown, the laser beam 158 passes
through a spatial light modulator (SLM) or phase step 220 producing
a .pi.-stripe (black, 0 rad; grey, .pi. rad) pattern in the
transmitted beam 222. The beam 222 having phase discontinuity is
imaged and demagnified approximately eight times with a 4f system
(224, 228) so as to yield an object beam 230 (162 in FIG. 6) to be
propagated through the nonlinear medium 162 (FIG. 6).
Reconstruction of the input based on the measured output is
performed similarly as in the Gaussian case, by formatting the
frames recorded by the camera (190) into 512.times.512 pixel
frames.
[0089] FIG. 9 shows various panels 280 corresponding to intensities
and phases (modulo 2.pi.) of input (panels a and e) and outputs of
the object beam 162 in linear propagation (panels b and f) and
nonlinear propagation (panels c and g). Numerically reconstructed
intensity and phase (panels d and h) of the input by
back-propagation calculation are also shown. Where indicated, each
scale bar represents approximately 50 .mu.m. The dotted oval in
panel g indicates appearance of .pi./2 discontinuity that is
successfully eliminated in the reconstructed phase of panel h.
[0090] FIG. 9 further shows, in panel i, reconstruction
cross-sections at various z values (indicated in panel j) using a
backwards propagation algorithm of the internal dynamics of the
dark stripe. Panel j shows simulation of evolution at the various z
values indicated, with an example 5% simulated intensity noise. The
shown scale bar represents approximately 50 .mu.m.
[0091] In FIG. 9, such evolution for a dark stripe (generated by a
.pi.-shift phase discontinuity) is initially too narrow, for its
intensity, to be a dark soliton. In the linear case (panel b), the
stripe expands to roughly three times its original size. In the
nonlinear case (panel c), the output stripe width has narrowed by
.about.30%. Numerical calculations show that this width coincides
with the dark soliton width for the experimental parameters, but
standard experimental techniques can say little about the beam
dynamics. In other words, without showing invariant propagation,
the existence of a soliton could not be proved.
[0092] As described herein, nonlinear digital holography can be
used to reconstruct substantially the entire dynamics along the
propagation path. As with the Gaussian beam case, one can
reconstruct the input to yield reconstructed intensity and phase
(panels d and h). Note that a phase defect at the nonlinear output
(panel g) has been eliminated successfully in the reconstruction
(panel h).
[0093] In panel i of FIG. 9, several cross-sections of the
reconstruction, at intermediate distances within the crystal, are
shown. As the beam propagates, the stripe widens to a dark soliton
profile, radiating energy as it adjusts. The evolution is complex
and can include the following effects: the initial noise smoothens,
owing to the defocusing nonlinearity; the beam profile becomes more
symmetric; diffraction of the central dark stripe is arrested; and
the radiated waves self-steepen and form dispersive shock waves.
Similar profiles have been observed in fiber solitons, but to our
knowledge have not been demonstrated in the spatial case. For
example, two-dimensional (e.g., lateral XY) spatial profiles for
two-dimensional imaging, in the context of some embodiments of the
present disclosure, have not been demonstrated. Also, profiles
associated with nonlinear propagation in unconfined space
(sometimes referred to as free space) have not been
demonstrated.
[0094] As described herein, substantially the entire dynamics is
reconstructed, showing that the central profile settles into its
dark soliton form at about 5 mm, whereas the tails need a longer
propagation distance to relax. This semi-empirical reconstruction
compares favorably with the ideal simulated case, shown in panel
j.
[0095] As shown in FIGS. 6 and 7C, the back-propagation algorithm
calibrated as described herein can be used to reconstruct an image
of a pattern that is even more complex than the stripe pattern of
FIG. 7B. As shown, the laser beam 158 passes through an Air Force
1951 resolution chart 240, producing a transmitted beam 242. The
transmitted beam 242 is imaged with substantial unity magnification
with a 4f system (244, 248) so as to yield an object beam 250 (162
in FIG. 6) to be propagated through the nonlinear medium 162 (FIG.
6). Reconstruction of the input based on the measured output is
performed similarly as in the Gaussian case, by formatting the
frames recorded by the camera (190) into 512.times.512 pixel
frames.
[0096] FIG. 10 shows various panels 290 corresponding to
intensities and phases (modulo 2.pi.) of input (panels a and e) and
outputs of the object beam 162 in linear propagation (panels b and
f) and nonlinear propagation (panels c and g). Numerically
reconstructed intensity and phase (panels d and h) of the input by
back-propagation are also shown. Where indicated, each scale bar
represents approximately 200 .mu.m.
[0097] The Air Force 1951 resolution chart 240 is more complex than
the stripe pattern and the Gaussian. In the linear case, the input
(panels a and e) has diffracted considerably, as shown in panels b
and f, but the characters are still recognizable. After nonlinear
propagation (shown in panels c and g), however, the original
intensity profile has been obliterated almost completely in the
measured output, especially for the numerical symbols, and the
phase profile is severely blurred. Nevertheless, there is very good
agreement between experimental (panels a and e) and reconstructed
phase and intensity (panels d and h) in the reconstructed
image.
[0098] Based on the foregoing example with the complex Air Force
1951 resolution chart, there are a number of applications where
image obliteration and image reconstruction can be implemented. For
example, physical encryption of an image and/or information can be
achieved using similar nonlinear propagation and reconstruction
techniques. As with the dark soliton described in reference to FIG.
9, the reconstruction of the phase is particularly well defined.
This is both surprising and fortunate, because the phase is
typically a much more sensitive quantity and typically carries more
information. Indeed, from the reconstructed phase of panel h, it is
possible to resolve the 10 .mu.m bars of the chart, a resolution
limited mainly by the non-ideal properties (defects and striations)
of the crystal.
[0099] As described herein in reference to FIGS. 6, 7A, 8A, and 8B,
knowledge of an input and measurement of an output allows one to
obtain one or more parameters that characterize a nonlinear medium
through which the input propagates and becomes the output. In
certain embodiments, such characterization of the nonlinear medium
can be achieved by numerically back-propagating the measured output
and reconstructing the input. Such reconstruction process can be an
iterative process, where the input can be considered to be
reconstructed when a given iteration yields an approximation that
is sufficiently close to known input.
[0100] FIG. 24 shows that in certain embodiments, an algorithm 700
can be implemented to perform such reconstruction of the input. As
shown, an input for the algorithm 700 can include a measured output
.psi..sub.out. Such an input is provided to a block 704 where
reverse propagation (or back-propagation) algorithm is performed.
Such an algorithm can be based on Equation 3, where linear and
nonlinear propagators successively evaluate (e.g., at .DELTA.z
intervals) an input along a direction that is reverse of the
initial propagation direction of the input.
[0101] In block 706, a reconstructed input .psi..sub.rec is
obtained from the reverse propagation algorithm 704. The
reconstructed input .psi..sub.rec is then compared to the known
input .psi..sub.in; and in a decision block 708, the algorithm
determines whether the reconstruction (.psi..sub.rec) is
substantially equal to or sufficiently close to the input
(.psi..sub.in). If the answer is "No," then one or more
nonlinearity parameters can be adjusted. For example, in block 710,
the nonlinear index change .DELTA.n (described herein in reference
to Equations 1-3 and FIG. 6) can be adjusted, and the reverse
propagation algorithm 704 can be repeated. If the answer is "Yes,"
then the one or more nonlinearity parameters that yielded the
reconstruction (.psi..sub.rec) can be stored.
[0102] In certain embodiments, such stored one or more parameters
can be retrieved and utilized when the same or substantially same
nonlinear medium is used to characterize an unknown input that
yields a measured output. Similarly, the same nonlinearity
parameter(s) can be retrieved and utilized for configuring an input
that would yield a desired output when propagated through the
nonlinear medium.
[0103] For example, in the situation where the unknown input is to
be characterized based on the measured output, an algorithm similar
to 700 (FIG. 24) can be implemented. As with algorithm 700, the
measured output can be an input, and the reverse propagation
algorithm 704 can numerically propagate the output to yield a
reconstructed input. Because the nonlinear medium has already been
characterized, the retrieved nonlinearity parameter(s) can be
provided for the numerically propagation in the reverse propagation
algorithm 704. The resulting reconstructed input is then taken to
be an approximation of the unknown input.
[0104] In another example where an input is to be configured so
that it would yield a desired output when propagated through the
nonlinear medium, an algorithm similar to the foregoing
reconstruction of the unknown input can be implemented. Here, the
desired output can be treated similar to the measured output case;
and the input can be determined in the same manner as the foregoing
example (reconstructing an unknown input from a measured
output).
[0105] Based on the foregoing measurement and reconstruction
results and known properties of nonlinear propagation, there are
some observations that can be made. The sharpness of the
reconstruction (e.g., reconstruction for the complex Air Force 1951
resolution chart) highlights an irony inherent in self-defocusing
media, where nonlinear mode-coupling of high spatial frequencies
can lead to focusing effects. Although focusing nonlinearities can
couple these modes, noise-induced instabilities can dominate the
signal and may limit the ability to invert Equation 2.
[0106] Some non-limiting examples of beneficial end results that
are or can be provided by various features of the present
disclosure include larger effective numerical apertures and finer
spatial resolution. Such capabilities can be achieved or
implemented in various embodiments of digital imaging.
[0107] FIG. 11 shows a schematic depiction of an imaging
configuration 300 where an input field 302 can be input (arrow 304)
into a nonlinear medium 306 to yield an output field 308. An
imaging device 310 is shown to form one or more images of the
output field 308. Such one or more images can be processed by a
processor 314 in communication with the imaging device 310 so as to
characterize the input field 302.
[0108] As described herein, such characterization can be achieved
by reconstructing the input field 302 via calculated
back-propagation of the output field 308 through the nonlinear
medium 306. In the imaging configuration 300 of FIG. 11, the
nonlinear medium 306 can be one that has already been
characterized. Thus, in certain embodiments, information about the
nonlinear medium 306 can be stored in a computer-readable medium
318 so as to allow the processor 314 to perform the
back-propagation algorithm. As described herein, the information
about the nonlinear medium 306 can include one or more calibration
parameters for the back-propagation algorithm.
[0109] In the imaging configuration of FIG. 11, measurement of an
input is facilitated by its propagation through a nonlinear medium,
followed by numerical back-propagation of a measured output through
a model of the same nonlinear medium. Because the input is passed
through the nonlinear medium, the nonlinear medium can facilitate
recovery of information in the input that otherwise would be lost
during propagation through a linear media.
[0110] As an example, FIG. 12 shows a microscope imaging
configuration 320, where input rays from an object sample 322 pass
through a slide cover 324 to further propagate to an objective lens
330 as rays generally indicated as 342. On the left side of the
example configuration 320, an air gap 326 is provided between the
slide cover 324 and the microscope objective lens 330. Accordingly,
some of the rays passing through the slide cover 324 become lost,
due to, for example, total internal reflection (e.g., ray indicated
as "c"). Of the rays that emerge from the slide cover 324, some are
directed so as to be accepted (e.g., ray "a") by the objective;
while some escape the acceptance range (e.g., ray "b") of the
objective due to, for example, difference in refractive indices
between the slide cover 324 and the air gap 326, and the numerical
aperture of the objective 330.
[0111] To mitigate at least some of such losses of input rays, some
microscopes utilize immersion techniques, where a transmissive
material having refractive index closer to the slide cover 324 is
provided to fill the air gap. In FIG. 12, the transmissive material
such as oil 328 is depicted as filling the gap between the slide
cover 324 and the objective 330; and rays in the oil 328 are
generally indicated as 342. Accordingly, loss due to internal
reflection is reduced. For example, ray "f" that would have been
lost by total internal reflection in the air-gap case is shown to
be transmitted into the oil 328. Also, loss due to refraction
beyond the numerical aperture of the objective is also reduced when
compared to the air-gap case. For example, ray "e" that would have
been lost by refraction beyond the numerical aperture of the
objective in the air-gap case is shown to be accepted by the
objective.
[0112] However, there still are rays that escape the acceptance
range of the objective lens 330. For example, ray "f" is shown to
be directed outside of the objective's acceptance range.
[0113] FIG. 13A shows that in certain embodiments, a microscopic
imaging configuration 350 can include a nonlinear medium 352
disposed between the object sample 322 and the objective lens 330.
As shown, rays 354 indicated as "k," "l," and "m" emerge from the
slide cover 324. Rays "k" and "l" are shown to propagate into the
geometric acceptance range of the objective 330, while ray "m"
escapes the range.
[0114] As is generally known, wave mixing can yield effects such as
mode coupling and generation of daughter waves. Thus, in FIG. 13A,
waves depicted as rays "l" and "m" can mix in the nonlinear medium
352 and generate a daughter wave or wave component indicated as
"n.sup.NL." Thus, while ray "m" is still not captured in the
geometric acceptance range of the objective lens 330, information
about ray "m" imparted to the daughter wave "n.sup.NL" can be
captured by acceptance of the daughter wave "n.sup.NL."
[0115] FIG. 13B shows that in certain embodiments, a microscopic
imaging configuration 600 can include a nonlinear medium 352 in
contact with the object sample 322. As shown, an evanescent wave
602 representative of a feature (which may have a size less than a
wavelength of light) on a surface 604 of the sample 322 can couple
(depicted as 608) with a ray transmitted through the nonlinear
medium 352 into the objective lens 330. In the example
configuration 600, the portion of the surface 604 on which the
feature (e.g., subwavelenth sized feature) is located is in contact
with the nonlinear medium 352.
[0116] FIG. 13C shows that in certain embodiments, a microscopic
imaging configuration 610 can include a nonlinear medium 352 that
at least partially encapsulates the object sample 322. In the
example shown, the sample 322 is embedded in three dimensions
within the nonlinear medium 352. Thus, evanescent waves (e.g., 612)
corresponding to features (which may have a dimension less than a
wavelength of light) on various surfaces of the sample 322 can
couple (depicted as 618) with rays (e.g., 616) transmitted through
at least a portion of the nonlinear medium 352 into the objective
lens 330.
[0117] Thus, in certain embodiments, a super-resolution effect can
result from knowledge of the nonlinear propagation kernel for the
nonlinear medium 352. As described herein, such characterization of
the nonlinear propagation allows reconstruction of the dynamics,
deconvolution of the wave mixing, and therefore recovery of the
underlying (missing) spatial features, thereby yielding better
resolution of such features.
[0118] As described herein, in certain embodiments, this method can
be applied to near-field, subwavelength reconstructions. In this
case, nonlinear wave mixing couples evanescent waves or wave
components with propagating ones. Evolution of such wave mixing and
propagation in nonlinear media can be described by modifying the
nonlinear Schrodinger equation (Equation 1) appropriately, and
back-propagating a reverse solution as described herein. A relevant
equation may be a full wave equation, including a full vector
solution to Maxwell's equations, or any approximation thereof.
Other methods can also be used.
[0119] In certain embodiments, this method has potential to work
based on recent observations of the Goos-Hanchen effect (described
in, for example, Goos, F. & Hanchen, H., Ein neuer and
fundamentaler versuch zur totalreflexion, Ann. Phys. 1, 333-346
(1947)), in which a totally internally reflected beam is spatially
translated from the incident beam. This translation, owing to phase
shifts of evanescent waves at the interface, is normally on the
order of a wavelength but can be enhanced significantly by
nonlinearity. Additional details concerning such effects and
enhancements can be found at, for example, Artmann, K., Berechnung
der seitenversetzung des totalreflektierten strahles, Ann. Phys. 2,
87-102 (1947); Tomlinson, W. J., Gordon, J. P., Smith, P. W. &
Kaplan, A. E., Reflection of a Gaussian beam at a nonlinear
interface, Appl. Opt. 21, 2041-2051 (1982); Emile, O., Galstyan,
T., Le Floch, A. & Bretenaker, F, Measurement of the nonlinear
Goos-Hanchen effect for Gaussian optical beams, Phys. Rev. Lett.
75, 1511-1513 (1995) and Jost, B. M., Al-Rashed, A.-A. R. &
Saleh, B. E. A., Observation of the Goos-Hanchen effect in a
phase-conjugate mirror, Phys. Rev. Lett. 81, 2233-2235 (1998).
Combined with the reconstruction algorithm, this coupling of
near-field behavior with far-field propagation can be extended to
imaging in certain embodiments. Moreover, unlike traditional
point-by-point scanning techniques, certain embodiments of the
nonlinear digital holography disclosed herein include an inherently
wide-angle, farfield form of microscopy.
[0120] From the opposite perspective, such nonlinear wave mixing
could allow subwavelength lithography with super-wavelength-scale
initial patterns. In certain embodiments, such an application of
nonlinear wave mixing can be treated as a system 370 shown in FIG.
14. The system 370 includes a source 372 that provides a known
input wave 374 that passes through a nonlinear medium 376 so as to
yield a desired output wave 378. In certain embodiments,
propagation properties in the medium 376, including nonlinear wave
mixing, can be controlled based on knowledge of the nonlinear
propagation. Thus, the input wave 374 can be configured accordingly
so as to yield the desired output 378.
[0121] Similar to the example configuration described in reference
to FIG. 11, information concerning the nonlinear propagation can be
stored in a computer readable medium 390 in communication with a
processor 380. The processor 380 can utilize such information and
control the source to generate the configured input wave 374.
[0122] In certain embodiments, a lithographic device can include an
arrangement of an optical element and a nonlinear medium similar to
those in the microscope examples described in reference to FIGS.
13A-13C. Such a device can be operated in reverse so that an input
configured based on a characterized nonlinear medium and passed
through that nonlinear medium yields a desired output. Similar to
the example advantages obtainable in the microscope, an effective
numerical aperture of the device can be realized, thereby providing
an increased resolution. Furthermore, evanescent wave interactions
on a substrate (e.g., photoresist an/or semiconductor substrate)
can be induced by the reversed process so as to allow, for example,
formation of or interaction with subwavelength size features.
[0123] In FIG. 14, the output wave 378 is depicted as emerging from
the nonlinear medium 376. In certain situations, however, it may be
desirable to generate and utilize results of nonlinear propagation
in the nonlinear medium itself. Thus, in certain embodiments as
shown in FIG. 15, a system 400 can include a source 402 that
provides a known input wave 404 that enters a nonlinear medium 406.
The entering wave begins to interact with the nonlinear medium 406,
and such interaction can be characterized and controlled based on
propagation properties stored in a computer readable medium 420 in
communication with a processor 410. Thus, the input wave 404 can be
configured to yield desired interaction in the nonlinear medium
406.
[0124] In certain embodiments, systems such as the example 400
shown in FIG. 15 can be utilized to achieve holographic recording
and readout operations. By way of a non-limiting example, recording
of information at different depths of media can be achieved. In
certain embodiments, such recording medium can be the nonlinear
medium 406, the source 402 can provide signals 404 representative
of data records. Thus, recording to and reading from double or
multiple layers of medium such as DVD (or similar optical medium)
can be achieved. During such recording at a deeper layer, the
intervening layer(s) may be affected by the recording beam.
However, such interaction in the intervening layer(s) occurs in a
known manner (which can be calculated by, for example,
back-propagation method) as described herein, and can be
compensated algorithmically during the write and/or readout
process.
[0125] Within the foregoing recording and reading context, quality
of signals associated with such operations can be improved by
knowledge of the interactions occurring in the recording media. As
described herein, such recording can be configured so as to yield a
desired recording beam configuration. Such a configuration can
include wave mixing to yield subwavelength recording capability
with super-wavelength-scale beam, so as to yield a higher density
recording.
[0126] In certain embodiments, a recording system described in
reference to FIG. 15 can have components similar to those found in
the lithographic system described herein. For example, a
combination of a lens and nonlinear medium facilitates an increase
in numerical aperture of the system and/or provides capability to
induce evanescent wave based interactions for formation of
subwavelength recording features.
[0127] In certain embodiments, features associated with
configurations where a desired output is obtained based on
nonlinear propagation can be combined with features associated with
configurations where an input is characterized, based again on
nonlinear propagation. FIG. 16 shows an example system 430 where a
desired output 440 is generated by propagation of an input 434
through a nonlinear medium 436 in a manner similar to that
described in reference to FIG. 14. In certain embodiments, such
output 440 can be propagated linearly to a remote location and be
detected by a detector 460 under the control of a processor 470.
The processor 470 can be provided with information concerning the
nonlinear propagation (e.g., by having access to a computer
readable medium 450), so as to allow reconstruction of the input
434.
[0128] In certain embodiments, the transmitted output 440 can be in
a form suitable for reconstruction of an image. In certain
embodiments, the transmitted output can be in a form that carries
information that cannot be reconstructed without knowledge of the
nonlinear propagation. Encryption and steganography are some
non-limiting examples of possible applications.
[0129] Based on the present disclosure, it should be appreciated
that various features can be applied to any situations where
improved characterization of input or output waves is desirable. By
way of non-limiting examples, some applications where nonlinear
reconstruction and enhanced imaging can be applied include:
enhanced field of view; improved depth determination;
three-dimensional imaging via holography and nonlinear medium
immersion; profilometry; tomography; edge detection; improved
resolution; subwavelength resolution imaging; and improved axial
resolution.
[0130] Additionally, various features of the present disclosure can
be applied to material characterization wherever nonlinear
propagation is involved. By way of non-limiting examples, such
applications can include: nonlinear measurements to match model
parameters; characterization of internal potentials, including
those that are optically induced; identification and
characterization of material defects; and identification and
characterization of induced defects.
[0131] Additionally, various features of the present disclosure can
be applied to provide improved sensitivity in certain optical
applications. By way of non-limiting examples, such applications
can include: amplification of features and correction of
aberrations.
[0132] In the present disclosure, some of the examples are
described in the context of coherent waves. For example, various
observations associated with the setup 150 of FIG. 6 are in the
context of coherent light such as a laser. It will be understood,
however, that some or all of the features as described herein can
also be applied to situations involving incoherent waves. For
incoherent cases, correlation information can be obtained and
incorporated appropriately in a known manner so as to allow, for
example, numerical back-propagation an output to digitally
reconstruct the incoherent input field. For such cases, known
generalizations of the nonlinear Schrodinger equation (Equation 1)
may be used, such as propagation of a mutual coherence function or
radiation transport equations.
[0133] In various examples of the present disclosure, including the
experimental setup described in reference to FIGS. 6-10,
characterization of unknown inputs and configuring of desired
outputs are described in the context of all-or-none presence or
absence of a nonlinear medium. It will be appreciated, however,
that similar techniques can also be implemented in situations where
a propagation medium is characterized based on some difference in
nonlinearity of the propagation medium. Such a difference can arise
from, for example, differences in applied voltage, temperature,
polarization, etc., or by use of a different nonlinear medium. In
such a context, the all-or-none situation can be considered to be
an example case of the nonlinearity-difference approach, where the
"none" portion corresponds to substantially zero nonlinearity.
[0134] FIG. 17 shows a system 500 having a nonlinearity
differential component 502 into which an input wave 504
(represented by a wavefunction .psi..sub.in) is introduced. The
input wave 504 propagates through the component 502 and emerges as
an output wave 506 (represented by a wavefunction
.psi..sub.out).
[0135] In certain embodiments, such nonlinearity differential
component can be characterized by a process 510 shown in FIG. 18.
In a process block 512, a first output intensity distribution can
be obtained with the nonlinearity differential component at first
linearity. In a process block 514, a second output intensity
distribution can be obtained with the nonlinearity differential
component at second linearity. In a process block 516, complex
output field can be obtained based on difference between the first
and second output intensity distributions. In a process block 518,
the complex output field thus obtained can be back-propagated
through the medium having one of the first and second
nonlinearities to reconstruct the input wavefunction. In a process
block 520, the medium can be characterized based on the calculated
back-propagation.
[0136] In certain embodiments, different output
intensities--obtained from, for example, two or more different
nonlinearities and/or differences between nonlinear and linear
propagation--can be recorded; and full complex waveforms at the
output can be reconstructed. As an example, output intensities can
be measured after linear and nonlinear propagation, giving
I.sub.out.sup.lin=|.PSI..sub.out.sup.lin|.sup.2 and
I.sub.out.sup.NL=|.PSI..sub.out.sup.NL|.sup.2, respectively. A
phase can be chosen for .PSI..sub.out.sup.lin, after which the full
wavefunction can be back-propagated to the input using a linear
propagation algorithm so as to yield a reconstructed wavefunction
.PSI..sub.in. The reconstructed wavefunction .PSI..sub.in is then
forward-propagated using a nonlinear propagation algorithm,
creating a trial output wavefunction .PSI..sub.out.sup.NL. The
reconstructed amplitude of .PSI..sub.out.sup.NK is then replaced by
the measured amplitude .PSI..sub.out.sup.NL. The process is then
repeated: back-propagation with the nonlinear algorithm to
reconstruct a wavefunction .PSI..sub.in, followed by
forward-propagation with the linear algorithm to create
.PSI..sub.out.sup.lin. This is then replaced by the measured
amplitude of .PSI..sub.out.sup.lin. This two-way reconstruction is
then iterated until the unknown phase is recovered. Additional
details about such phase retrieval algorithms can be found at, for
example, Gerchberg, R. W. & Saxton, W. O., A practical
algorithm for the determination of the phase from image and
diffraction plane pictures, Optik 35, 237 (1972); and Fienup, J.
R., Phase retrieval algorithms: a comparison, Appl. Opt. 21, 2758
(1982).
[0137] In situations involving incoherent light, there is an
overall envelope phase, and other parameters such as correlation
data can be utilized to characterize a statistical distribution
representative of the incoherent light. These parameters are well
known and described in such papers as Sun, C., Dylov, D. V., and
Fleischer, J. W., Nonlinear focusing and defocusing of partially
coherent spatial beams, Opt. Lett. 34, 3003 (2009). The foregoing
phase retrieval and reconstruction method can be applied to the
reconstruction of the incoherent light distribution as well.
[0138] In certain embodiments, the nonlinearity differential
component 502 of FIG. 17 can be implemented in a number of ways.
FIG. 19-21 show non-limiting example configurations where two or
more nonlinearities can be provided.
[0139] In FIG. 19, an example configuration 530 includes a tunable
nonlinear medium 532 whose nonlinear property can be adjusted by a
controller 534. For example, application of different electrical
potentials can yield different nonlinearities. As shown, an input
wavefunction 536 can be provided at each of at least two
nonlinearity settings to obtain at least two outputs 538.
[0140] In FIG. 20, an example configuration 540 includes two
separate nonlinear media 542, 544. For the purpose of description
herein, it will be assumed that the nonlinear media 542 and 544
have been characterized or are characterizable (for example, using
methods described herein). An input 546 propagated through the
first medium 542 yields a first output 548. The same input 546
propagated through the second medium 544 yields a second output
550.
[0141] In FIG. 21, an example configuration 640 includes a linear
medium 642 and a nonlinear medium 644. An input 646 propagated
through the linear medium 642 yields a first output 648. The same
input 646 propagated through the nonlinear medium 644 yields a
second output 650.
[0142] In certain embodiments, one or more features described in
reference to FIGS. 17-21 can be implemented to obtain improved
images, and/or to obtain desired outputs, in manners similar to
those described in reference to FIGS. 11-16. By way of an example,
FIG. 22 shows an imaging system 560 having an imaging device 568
that receives an output 566 of a nonlinearity difference component
564. The component 564 can be configured as described herein in
reference to FIGS. 17-21 so as to receive an input wave 562 for two
or more propagations therein. The imaging device 568 and/or the
nonlinearity difference component 564 can be controlled by a
processor 570; and such control can be facilitated by information
about nonlinearities stored in a computer readable medium 572.
[0143] FIG. 23 shows a more specific example of the imaging system
560 of FIG. 22. A system 580 can include a camera whose field of
view or numerical aperture is depicted as 584. In a first imaging
operation, the camera 582 can obtain an image of without a
nonlinear medium. Then, in a second imaging operation, the camera
can obtain another image with an optical element 590 formed from a
nonlinear medium. Intensity images obtained from the two operations
can be combined to yield a complex field, and such a field can be
back-propagated through the nonlinear medium so as to reconstruct
the input associated with the second imaging operation.
[0144] As described herein, such reconstruction can yield
information that would otherwise be absent. For example,
information about waves outside of the field of view 584 can be
retained by mixing of such waves and generation of nonlinear waves
that survive and become part of the complex image. Thus, the
nonlinear element 590 can provide an expanded field of view 592 for
the camera 582.
[0145] In another example, information associated with fine or
microscopic features in the input region that would otherwise be
absent can be retained and transmitted by similar wave mixing.
Thus, combined with the expanded field of view and/or numerical
aperture feature(s), imaging via nonlinear element 590 can provide
nonlinear digital images that are inherently wide-angle (e.g.,
expanded field of view and/or increased numerical aperture) and far
field form of microscopy rich with fine details.
[0146] As described herein, some of the numerical propagation
calculations are in the context of backward or reverse propagation.
Such algorithms are used to describe, for example, reconstruction
of an unknown input based on a measured output, or configuring an
input based on a desired output. It will be understood, however,
various features of the present disclosure can be implemented
and/or achieved by numerical propagation calculations in a forward
direction, where an output is constructed or reconstructed based on
an input.
[0147] As described herein, some of the examples are in the context
of electromagnetic radiation, and more particularly, in the context
of light. It will be understood that various features of thepresent
disclosure can be implemented and/or achieved in wave-related
situations, including, for example, electromagnetic radiation
outside of the light range and sound-related situations such as
acoustics and ultrasonic applications.
[0148] In one or more example embodiments, the functions, methods,
algorithms, techniques, and components described herein may be
implemented in hardware, software, firmware (e.g., including code
segments), or any combination thereof. If implemented in software,
the functions may be stored on or transmitted over as one or more
instructions or code on a computer-readable medium. Tables, data
structures, formulas, and so forth may be stored on a
computer-readable medium. Computer-readable media include both
computer storage media and communication media including any medium
that facilitates transfer of a computer program from one place to
another. A storage medium may be any available medium that can be
accessed by a general purpose or special purpose computer. By way
of example, and not limitation, such computer-readable media can
comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage,
magnetic disk storage or other magnetic storage devices, or any
other medium that can be used to carry or store desired program
code means in the form of instructions or data structures and that
can be accessed by a general-purpose or special-purpose computer,
or a general-purpose or special-purpose processor. Also, any
connection is properly termed a computer-readable medium. For
example, if the software is transmitted from a website, server, or
other remote source using a coaxial cable, fiber optic cable,
twisted pair, digital subscriber line (DSL), or wireless
technologies such as infrared, radio, and microwave, then the
coaxial cable, fiber optic cable, twisted pair, DSL, or wireless
technologies such as infrared, radio, and microwave are included in
the definition of medium. Disk and disc, as used herein, includes
compact disc (CD), laser disc, optical disc, digital versatile disc
(DVD), floppy disk and blu-ray disc where disks usually reproduce
data magnetically, while discs reproduce data optically with
lasers. Combinations of the above should also be included within
the scope of computer-readable media.
[0149] For a hardware implementation, one or more processing units
at a transmitter and/or a receiver may be implemented within one or
more computing devices including, but not limited to, application
specific integrated circuits (ASICs), digital signal processors
(DSPs), digital signal processing devices (DSPDs), programmable
logic devices (PLDs), field programmable gate arrays (FPGAs),
processors, controllers, micro-controllers, microprocessors,
electronic devices, other electronic units designed to perform the
functions described herein, or a combination thereof.
[0150] For a software implementation, the techniques described
herein may be implemented with code segments (e.g., modules) that
perform the functions described herein. The software codes may be
stored in memory units and executed by processors. The memory unit
may be implemented within the processor or external to the
processor, in which case it can be communicatively coupled to the
processor via various means as is known in the art. A code segment
may represent a procedure, a function, a subprogram, a program, a
routine, a subroutine, a module, a software package, a class, or
any combination of instructions, data structures, or program
statements. A code segment may be coupled to another code segment
or a hardware circuit by passing and/or receiving information,
data, arguments, parameters, or memory contents. Information,
arguments, parameters, data, etc. may be passed, forwarded, or
transmitted via any suitable means including memory sharing,
message passing, token passing, network transmission, etc.
[0151] Although the above-disclosed embodiments have shown,
described, and pointed out the fundamental novel features of the
invention as applied to the above-disclosed embodiments, it should
be understood that various omissions, substitutions, and changes in
the form of the detail of the devices, systems, and/or methods
shown may be made by those skilled in the art without departing
from the scope of the invention. Components may be added, removed,
or rearranged; and method steps may be added, removed, or
reordered. Consequently, the scope of the invention should not be
limited to the foregoing description, but should be defined by the
appended claims.
[0152] All publications and patent applications mentioned in this
specification are indicative of the level of skill of those skilled
in the art to which this invention pertains. All publications and
patent applications are herein incorporated by reference to the
same extent as if each individual publication or patent application
was specifically and individually indicated to be incorporated by
reference.
* * * * *