Software Patent Abstract
The exemplary embodiments of the method, system, software arrangement
and computeraccessible medium according to the present invention
facilitates an analysis of interactions between nonlinear absorbing
materials and an incident (e.g., coherent) electromagnetic wave
based on material properties and characteristics of the incident
beam of the electromagnetic energy. For example, using the exemplary
embodiments of the present invention, it is possible to determine
a laser beam propagation in a variety of multiphoton absorbing materials.
Energy levels associated with such materials, which may be associated
with various electron absorption and/or relaxation phenomena, may
be added to and/or removed from the analysis. Incident laser beams
can vary from continuous wave to attoseconds in duration, and a
numerical solution can be obtained that is radially and/or temporally
dependent. Certain exemplary embodiments of the present invention
can also be used to determine certain contributions of individual
electronic energy levels within the materials to the total absorption.
Software Patent Claims
1. A method for analyzing interactions between a plurality of electromagnetic
radiations and a plurality of absorbing materials, comprising:obtaining
first information related to a plurality of energy states associated
with a particular material of the absorbing materials;obtaining
second information related to a particular electromagnetic radiation
of the radiations; andgenerating a third information based on the
first and second information, where the third information is related
to a characteristic of the particular radiation within the particular
material.
2. The method according to claim 1, wherein the particular electromagnetic
radiation is an electromagnetic wave.
3. The method according to claim 1, wherein the particular electromagnetic
radiation is an electric field.
4. The method according to claim 1, where the particular electromagnetic
radiation comprises a laser pulse.
5. The method according to claim 1, where the particular electromagnetic
radiation comprises a plurality of laser pulses, and where the second
information comprises at least one of a duration, an intensity or
an electric field associated with each of the pulses.
6. The method according to claim 1, where the particular electromagnetic
radiation comprises a continuous electromagnetic wave.
7. The method of claim 1, wherein the particular electromagnetic
radiation is coherent.
8. The method of claim 1, wherein the particular electromagnetic
radiation is a fundamental mode of a cylindrically symmetric waveguide.
9. The method according to claim 1, wherein the first information
comprises at least one of an energy level diagram or an energy level
string.
10. The method according to claim 1, wherein the first information
comprises at least one of an absorption block or a relaxation block.
11. The method according to claim 1, wherein the second information
comprises at least one of a wavelength, a pulse width, an intensity
or an electric field level.
12. The method according to claim 1, further comprising modifying
the first information based on a comparison between at least one
particular parameter associated with the third information and at
least one further parameter obtained from a previously obtained
measurement associated with the particular material.
13. The method according to claim 1, further comprising generating
at least one first relationship based on the first information,
where the first relationship is related to at least one absorption
characteristic of the particular electromagnetic radiation in the
particular material.
14. The method according to claim 13, wherein the at least one
first relationship is associated with at least one population density
of at least one energy level associated with the particular material.
15. The method according to claim 13, wherein the at least one
first relationship generated by providing at least one matrix, and
wherein at least one element of the at least one matrix is based
on the first information.
16. The method according to claim 15, wherein each element of the
matrix is independent of an intensity of the particular electromagnetic
radiation.
17. The method according to claim 15, wherein at least one element
of the matrix is linearly related to an intensity of the particular
electromagnetic radiation.
18. The method according to claim 15, wherein at least one element
of the matrix is based on an .alpha.th power of an intensity of
the particular electromagnetic radiation, where .alpha. is not equal
to zero or 1.
19. The method according to claim 13, further comprising generating
at least one second relationship based on the first information,
where the second relationship is related to at least one propagation
characteristic of the particular electromagnetic radiation in the
particular material.
20. The method of claim 19, wherein the first and second relationships
comprise mathematical equations.
21. The method according to claim 19, wherein the at least one
second relationship is generated by providing at least one vector,
and wherein at least one component of the at least one vector is
based on the at least one matrix.
22. The method according to claim 19, wherein the first and second
relationships are based on a diffraction of the particular electromagnetic
radiation.
23. The method according to claim 19, where the first and second
relationships are based on a stimulated emission within the particular
material.
24. The method according to claim 19, further comprising generating
a fourth information based on the second information using the first
and second relationships, where the fourth information is related
to a population of the energy states.
25. The method according to claim 24, where the fourth information
comprises a population density of entities that are capable of absorbing
at least one photon and being promoted to particular energy states
associated with the particular material.
26. The method according to claim 24, wherein the fourth information
is generated based on a depth within the particular material.
27. The method according to claim 24, wherein the third and fourth
information are generated by determining an intensity of the particular
electromagnetic radiation based on a position relative to a center
of the particular electromagnetic radiation.
28. The method according to claim 27, wherein the position comprises
a radial distance from the center of the particular electromagnetic
radiation.
29. The method according to claim 1, wherein the third information
is generated by determining an intensity of the particular electromagnetic
radiation based on the fourth information, and wherein the third
information is based on a time variable and a depth within the particular
material.
30. The method according to claim 24, further comprising generating
a fifth information relating to a contribution of at least one of
the plurality of energy states to an absorption of the particular
electromagnetic radiation within the particular material.
31. A system for analyzing interactions between a plurality of
electromagnetic radiations and a plurality of absorbing materials,
comprising:a computeraccessible medium which includes thereon a
set of instructions, wherein the set of instructions are configured
to program a processing arrangement to:receive first information
related to a plurality of energy states associated with a particular
material of the absorbing materials;receive second information related
to a particular electromagnetic radiation of the radiations;generate
third information based on the first and second information, where
the third information is related to a characteristic of the particular
radiation within the particular material.
32. A software arrangement for analyzing a plurality of electromagnetic
radiations and a plurality of absorbing materials, comprising:a
first set of instructions which, when executed by a processing arrangement,
is capable of receiving first information related to a plurality
of energy states associated with a particular material of the absorbing
materials;a second set of instructions which, when executed by the
processing arrangement, is capable of receiving second information
related to a particular electromagnetic radiation of the radiations;a
third set of instructions which, when executed by the processing
arrangement, is capable of generating a third information based
on the first and second information, where the third information
is related to a characteristic of the particular radiation within
the particular material.
Software Patent Description
CROSSREFERENCE TO RELATED APPLICATION(S)
[0001]This application claims priority from U.S. Patent Application
Ser. No. 60/813,980, filed Jun. 14, 2006, the entire disclosure
of which is incorporated herein by reference.
FIELD OF THE INVENTION
[0003]The present invention relates to exemplary embodiments of
a method, system and software arrangement which can determine interactions
between an absorbing material and an incident coherent electromagnetic
wave based on both material properties and characteristics of the
incident beam of coherent electromagnetic energy. The absorbing
material can be, e.g., a linear or nonlinear absorber and it may
absorb one or more photons (e.g., N.sub.A.gtoreq.1). In particular,
an exemplary procedure can be provided to determine laser beam propagation
with a wide range of temporal durations in a variety of multiphoton
absorbing materials. For example, incident laser beams can vary
from continuous wave to attoseconds in duration, and a numerical
solution can be obtained that is radially and/or temporary dependent.
In addition, certain contributions of individual electronic energy
levels within the materials to the total absorption can also be
obtained using exemplary methods, systems and/or software arrangements
in accordance with the present invention.
BACKGROUND INFORMATION
[0004]Previously there has been a significant increase in the development
and use of materials that exhibit nonlinear multiphoton behavior.
These materials may be used for such applications as, e.g., a high
precision medical diagnostics tools usage, effective treatments
for various cancers, biological detectors (e.g., markers), threedimensional
("3D") micro and/or nanofabrication, fluorescent imaging
systems, optical limiters, optical storage, semiconductor nanosized
probes, etc.
[0005]Conventional experiments that may be used to characterize
the optical properties of nonlinear materials such as multiphoton
organic/inorganic materials, semiconductors, fluids, gases or nanostructured
materials include, e.g., zscan procedures, optical transmission
techniques, and pumpprobe techniques. Facilities that are equipped
to characterize such materials may require millions of dollars of
equipment including, for example, lasers which can operate at different
wavelengths in the ultraviolet, visible, near infrared ("IR"),
mid IR and far IR regions. A laser beam can have an infinite duration
(e.g., a continuous wave), or a finite duration which can be on
the order of, e.g., nanoseconds ("ns"), picoseconds ("ps"),
or femtoseconds ("fs"). Such facilities can also include
various detectors, measurement electronics and data gathering computers
that may be used to characterize these materials. A laser pulse
duration or width can refer to, e.g., a continuous wave or a wave
having a finite duration.
[0006]Pulse widths provided by the lasers which may be used to
characterize and activate such optical materials can vary by about
12 orders of magnitude. This can make it difficult for a single
numerical code to accurately and robustly characterize all possible
interactions in order to reduce the need for costly experiments.
Additionally, many experiments may need to be performed on a single
material over many orders of magnitude of laser energies, where
different electronic states of the nonlinear material can contribute
to the total absorption behavior at different energy ranges. However,
conventional codes may neglect higher energy levels. This simplification
can yield reasonable results for particular energy ranges and incorrect
results for other ranges.
[0007]Optical transmission measurements can be made using a particular
laser such as, e.g., a Nd:YAG laser, a Ti:sapphire laser, a fiber
laser, a semiconductor laser, a photonic crystal nanolaser, a quantum
cascade laser, etc. The Nd:YAG laser can produce nanosecond pulses,
whereas a Ti:sapphire laser can produce picosecond or femtosecond
pulses. Each individual optical transmission measurement can be
performed using a selected pulse width and a particular wavelength.
However, a further measurement can be required for a different sample
thickness. The number of experiments which may be required for characterizing
these materials over a range of conditions and parameters can be
large, and costs and time associated with such measurements can
also be significant. For example, it may take many months to investigate
a new material. Conventional simulation codes that can be used to
model these measurements may be applicable only to a specific material
interacting with a particular laser system at a certain intensity,
and such codes may use simplifying assumptions that can further
limit their applicability with respect to, e.g., wavelength, pulse
widths, concentration of absorbing particles, and/or sample thickness.
Such codes may not be capable of predicting the effects of variations
in these parameters on the optical transmission behavior of a material
based on one experimental measurement or a limited number of such
measurements.
[0008]Conventional theoretical and/or numerical analyses of a laser
beam transmission through nonlinear absorbing materials can utilize
a number of assumptions that can limit their general applicability.
Such nonlinear absorbing materials are described, e.g., in N. Allard
et al., "The effect of neutral nonresonant collisions on atomic
spectral lines," Rev. Mod. Phys. 54, 11031182 (1982). Shorter
pulsed lasers and multiphoton processes are becoming important
in this field as described, e.g., in U. Siegner et al., "Nonlinear
optical processes for ultrashort pulse generation," in Handbook
of Optics, M. Bass et al., eds., McGrawHill, New York, 2001, vol.
IV, pp. 2531. Thus, there may be a need for a more general approach
which can increase the range of applicability of the equations used
and the assumptions involved.
[0009]Conventional propagation and/or transmission analyses may
neglect several molecular excited states as described, for example,
in Y. R. Shen, The Principle of Nonlinear Optics, Wiley, New York,
1984. These excited states may be used to explain experimental data,
particularly at high incident energy. Approximate theories of propagation
and/or transmission through nonlinear materials have been formulated
by various researchers in conjunction with their particular experimental
data. These approximate theories may require numerical solutions,
and approximate analytic expressions based on simplifying assumptions
may often be used to reduce a required computational time. However,
such approximate numerical solutions may not adequately describe
the laser beam propagation through the material.
[0010]Additionally, because conventional approaches may often be
used in conjugation with specific laser systems (e.g., with a specific
wavelength and pulse duration), the resulting theoretical or numerical
analysis may have a limited applicability. This approach can thus
limit predictive capabilities of the analysis. For example, a theoretical
description for a ns pulsed laser may not be capable of describing
the effects of a ps or fs duration laser pulse interacting with
the same material. Conventional theoretical or numerical analyses
may provide agreement with specific experiments for specific materials
and yield some insight, particularly in absorbers which may be described
using singlephoton processes. However, such conventional analyses
may need to be modified and/or expanded to provide accurate descriptions
and predictions of phenomena involving, e.g., a laser transmission
through absorbers.
[0011]Changing the material or the laser beam characteristics associated
with an absorption interaction may require a different numerical
method and/or computer code to analyze the optical response. For
example, new energy levels in the absorbing material may become
accessible with an increase in laser intensity, and a new set of
coupled equations may be required to describe the laserabsorber
interaction. Because analytical solutions may not be possible, except
in very simple cases, new computer codes may need to be written.
An algorithm and/or code describing two energy levels of an absorber
may not provide accurate results when three or more energy levels
may contribute to a particular laserabsorber interaction. Defining
new algorithms and writing new numerical codes to describe such
absorption interactions can involve, e.g., months or years of effort.
[0012]Multiphotonabsorbing materials may also be used as nonlinear
absorbers, including those described in, e.g., L. W. Tutt et al.,
"A review of optical limiting mechanisms and devices using
organics, fullerenes, semiconductors and other materials,"
Prog. Quantum. Elect. 17, 299305 (1993); J. E. Rogers et al., "Understanding
the onephoton photophysical properties of a twophoton absorbing
chromophore," J. Phys. Chem. A 108, 55145520 (2004); J. W.
Perry, "Organic and metalcontaining reverse saturable absorbers
for optical limiters," in Nonlinear Optics of Organic Molecules
and Polymers, H. S. Nalwa and S. Miyata, eds. (Boca Raton, Fla.:
CRC 1997), pp. 813839; M. J. Potasek et al., "All optical
power limiting," J. Nonlinear Optical Physics and Materials
9, 343365 (2000); M. J. Potasek, "HighBandwidth Optical Networks
and Communication, Photodetectors and Fiber Optics ed. H. S. Nalwa
(Academic Press, 2001) pp. 459543; D. I. Kovsh et al., "Nonlinear
Optical Beam Propagation for Optical Limiting," Appl. Opt.
38, 51685180 (1999); and W. Jia et al., "Optical limiting
of semiconductor nanoparticles for nanosecond laser pulses,"
Appl. Phys. Lett. 85, 63266328 (2004).
[0013]Photon absorbing materials may also be used in applications
such as biological detectors as described in, e.g., S. M. Kirkpatrick
et al., "Nonlinear saturation and determination of the twophoton
absorption cross section of green fluorescent protein," J.
Phys. Chem. B 105, 28672873 (2001), and threedimensional microfabrication
procedures such as those described in, for example, S. Maruo et
al., "Twophotonabsorbed nearinfrared photopolymerization
for threedimensional microfabrication," J. Microelectromechanical
Systems 7, 411415 (1998); B. H. Cumpston et al., "Twophoton
polymerization initiators for threedimensional optical data storage
and microfabrication," Nature 398, 5154 (1999); and G. Witzgall
et al., "Singleshot twophoton exposure of commercial photoresist
for the production of threedimensional structures," Opt. Let.
23, 17451748 (1998).
[0014]Further applications of photon absorbing materials may include
fluorescent imaging systems such as those described in W. Denk et
al., "Twophoton laser scanning fluorescence microscopy,"
Science 248, 7376 (1990), and optical storage systems as described,
for example, in H. E. Pudavar et al., "Highdensity threedimensional
optical data storage in a stacked compact disk format with twophoton
writing and single photon readout," Appl. Phys. Lett. 74, 13381340
(1999); and in P. N. Prasad, "Emerging opportunities at the
interface of photonics, nanotechnology and biotechnology,"
Mol. Cryst. Liq. Cryst. 415, 110 (2004).
[0015]A nonlinear absorbing material in which an excited state
absorption is large, as compared to a ground state absorption, can
be referred to as a reversible saturable absorber ("RSA").
Such materials can exhibit a large absorption at high input laser
energies, but their performance may be limited by an accompanying
linear absorption at low input energy. A transparency (e.g., low
absorption) at low input energy, combined with high absorption at
high input energy, can be achieved with multiphoton absorber ("MPA")
materials in which two or more photons may be absorbed simultaneously.
For examples, the materials that exhibit a large twophoton absorption
("TPA") behavior may be important for a wide range of
applications. Examples of TPA materials are described, for example,
in M. Albota et al., "Design of organic molecules with large
twophoton absorption cross sections," Science 281, 16531656
(1998); and B. A. Reinhardt et al., "Highly active twophoton
dyes: Design, synthesis, and characterization toward application,"
Chem. Mater. 10, 18631874 (1998).
[0016]MPA materials can exhibit complex absorption mechanisms involving
higher level electronic states. For example, TPA may be followed
by excited state absorption ("ESA") which is described,
e.g., in J. Kleinschmidt et al., "Measurement of strong nonlinear
absorption in stilbenechloroform solution, explained by the superposition
of twophoton absorption and onephoton absorption from the excited
state," Chem. Phys. Lett. 24, 133135 (1974). Nonlinear transmission
measurements and Zscan measurements of organic materials can indicate
the presence of ESA. These measurements are described, e.g., in
D. A. Oulianov et al., "Observations on the measurements of
twophoton absorption crosssection," Opt. Comm. 191, 235243
(2001); and S. Guha et al., "Thirdorder optical nonlinearities
of metallotetrabenzoporphyrins and a platinum polyyne," Opt.
Lett. 17, 264266 (1992).
[0017]ESA can be the primary absorption mechanism in a nanosecond
(ns) regime in a TPA material such as, e.g., D.pi.A chromophore
from the AFX group. TPA can be a primary mechanism for populating
the excited states in such materials. However, TPA may dominate
the total absorption behavior in the femtosecond regime. To analyze
and predict the experimentally observable behavior of such materials
under laser irradiation may require a solution to a nonlinear system
of differential equations. Although some material systems can be
described accurately by equations having a simple form which can
be solved analytically, it may be important to have effective and
robust numerical simulation tools to provide useful information
for a wide variety of materials under a broad range of conditions.
[0018]For many RSA and TPA materials such as those described, e.g.,
in G. S. He et al., "Degenerate twophotonabsorption spectral
studies of highly twophoton active organic chromophores,"
J. Chem. Phys. 120, 52755284 (2004); and R. Kannan et al., "Toward
highly active twophoton absorbing liquids. Synthesis and characterization
of 1,3,5triazinebased octupolar molecules," Chem. Mater.
16, 185194 (2004), simulation calculations can be based on a solution
of a coupled system of propagation and rate equations. The rate
equations may be formulated using a phenomenological fivelevel
absorption model which is described, for example, in R. L. Sutherland
et al., "Excited state characterization and effective threephoton
absorption model of twophotoninduced excited state absorption
in organic pushpull chargetransfer chromophores," J. Opt.
Soc. Am. B 22, 19391948 (2005).
[0019]The propagated light in the RSA materials may attenuate as
a result of electron excitations from the ground state and from
singlet and/or triplet excited states. The absorption mechanism
in the TPA materials can be similar to that in RSA materials, except
that two photons can be absorbed during a transition from the ground
state to the first singlet excited state. Depending on the pulse
width and intensity of the incident light, the electron population
densities may change which can alter the transmittance characteristics
of the material. Solving equations describing light propagation
in threephoton absorption ("3PA") materials such as,
e.g., PPAI, which is described, e.g., in D.Y. Wang et al., "Large
optical power limiting induced by threephoton absorption of two
stibazoliumlike dyes," Chem. Phys. Lett. 369, 621626 (2002),
may be less problematic because the absorption model may include
just two levels. In such materials, an incident pulse intensity
may decrease due to simultaneous absorption of three photons from
the ground level to the lowest singlet excited state. However, experimental
investigations of 3PA materials are in an early stage and more complex
nonlinear absorption models should be used for these materials.
[0020]Numerical methods may often be used to solve coupled equations
describing lasermatter interactions, because there are few analytic
solutions for such equations. New numerical code may be written
to describe each energy level diagram representing a particular
material of interest and an associated laser interaction. Such codes
can vary in their degree of sophistication and in any approximations
used, which may limit their applicability to certain lasers, as
well as to particular temporal and/or radial domains. New numerical
codes may be required to describe an increasing variety of possible
interactions between lasers and materials. For example, a large
number of individual computer codes have been written to solve various
approximate subsets of lasermaterial interactions. As more lasers
are developed having new wavelengths and/or pulse widths, many additional
codes or modifications of existing codes may need to be written
to describe them quantitatively.
[0021]Thus, there may be a need overcome the abovedescribed deficiencies
and issues to facilitate the effective and robust numerical simulation
tools to measure, analyze, and predict the behavior of photon absorbing
materials that are exposed to a laser irradiation. Further, there
may be a need for a uniform solver which is capable of modeling
a variety of nonlinear materials having different absorption configurations
under a range of the irradiation conditions such as, e.g., different
wavelengths, pulse widths, sample thicknesses, etc. Such exemplary
simulation tools may provide guidelines for developing new functional
materials, e.g., for designing molecular or semiconductor quantum
dots or wires that may reduce development costs. The numerical method
or computer program for such a simulation tool may not need to be
rewritten when the material or laser conditions are changed.
SUMMARY OF THE INVENTION
[0022]One object of the present invention is to provide a system,
method, software arrangement, and computeraccessible medium for
determining and/or predicting interactions between generic photoactive
materials and electromagnetic waves or electric fields such as,
e.g., a laser pulse or a series of such pulses. The electromagnetic
waves or electric fields may be coherent, and certain properties
of such waves may be provided such as, e.g., pulse duration, intensity,
wavelength, and intervals between successive pulses. The determinations
can be based on one or more energy level diagrams associated with
the material, which can also be provided in a form of an energy
level string. The energy level diagrams and/or energy level strings
can be expressed in terms of absorption blocks and/or relaxation
blocks, and they may be used to formulate relationships such as,
e.g., mathematical equations describing rates of energy level changes
and propagation of the electromagnetic wave through the material.
For example, equations describing propagation and/or absorption
of the electromagnetic wave or electric field in the material can
include, e.g., matrices and/or vectors which can be determined based
on the energy level diagrams or energy level strings. Such energy
level diagrams and/or strings may be modified as appropriate to
provide determined results with an additional accuracy using corresponding
modified rate and propagation equations.
[0023]In exemplary embodiments of the present invention, a numerical
method, system and software arrangement are provided which are capable
of describing interactions between photoactive "generic"
materials (e.g., materials which can be characterized using absorption
and/or relaxation blocks) and an electromagnetic wave or electric
field. Such interactions can include, e.g., propagation phenomena
such as diffraction, stimulated emission and/or cylindrically symmetric
guided waves. For example, a variety of lasergeneric material interactions
can be described quantitatively using a common numerical code when
changes are made to the energy level diagrams or energy level strings
associated with the material, or to the properties of the laser
such as, e.g., wavelength, pulse duration, radial beam diameter,
etc.
[0024]Computational building blocks (e.g. absorption building blocks
or relaxation building blocks) can provide terms to matrices and/or
vectors which can be used to formulate rate and propagation equations.
Mathematical equations describing energy level population dynamics
can include a power series describing the intensity or electric
field and one or more matrices. Such matrices can describe, e.g.,
relaxation rates, the intensity or electric field, and/or an .alpha.th
power of the intensity or electric field. The propagation equation
can include, for example, a series of terms having a form of vectors
multiplied by an intensity of the electromagnetic wave or the electric
field raised to an exponent .beta.. Such vectors may contain absorption
coefficients of the material of interest, where the coefficients
can be related to the energy level diagram or energy level strings
associated with the material.
[0025]In certain exemplary embodiments of the present invention,
the propagation of a shortpulsed laser beam in a multilevel multiphoton
absorbing material can be evaluated, where the propagation is determined
to be in the presence of multiphoton absorption and/or one or more
single photon excited state absorptions. Interactions may also be
determined between the absorbing materials and laser pulses having
a duration or pulse width which can range from nanoseconds to femtoseconds.
[0026]In further exemplary embodiments of the present invention,
contributions of each electronic level to the total absorption within
a material can be determined. This can provide insight into the
roles of and relationships among the various energy levels that
may be present in complex multiphoton absorbing materials. Absorption
profiles and/or intensity distributions may also be determined with
respect to both depth and radius in a material, which can provide
a more accurate prediction of photoninduced effects in absorbing
materials than conventional radially constant techniques used to
predict pulse propagation within such materials.
[0027]In still further exemplary embodiments of the present invention,
the effects of a diffraction on absorption and propagation of the
laser pulse or other coherent wave in the absorbing material can
also be determined. Effects of stimulated emission within the material
can also be determined using certain exemplary embodiments of the
present invention.
[0028]In yet further exemplary embodiments of the present invention,
the propagation of the laser intensity or the electric field through
a cylindrically symmetric guide or core structure, which can contain
or be doped a generic photoactive material, may be described.
[0029]These and other objects, features and advantages of the present
invention will become apparent upon reading the following detailed
description of embodiments of the invention, when taken in conjunction
with the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0030]Further objects, features and advantages of the invention
will become apparent from the following detailed description taken
in conjunction with the accompanying figures showing illustrative
embodiments of the invention, in which:
[0031]FIG. 1 shows five exemplary absorption diagrams, B.sub.0B.sub.4,
which can be used to describe an absorption configuration of generic
materials, together with an energy level diagram that can be utilized
to describe an absorption in a C.sub.60toluene solution;
[0032]FIG. 2a is an exemplary graph of an energy transmittance
T.sub.E as a function of input energy for C.sub.60;
[0033]FIG. 2b is an exemplary graph of the energy transmittance
T.sub.E as a function of input energy for AF455;
[0034]FIG. 2c is an exemplary graph of the energy transmittance
T.sub.E as a function of input energy for PPAI;
[0035]FIG. 3a is an exemplary graph of an evolution of electronic
level population densities and total absorption in C.sub.60 for
an incident fluence value of 0.51 J/cm.sup.2;
[0036]FIG. 3b is an exemplary graph of the evolution of electronic
level population densities and total absorption in C.sub.60 for
an incident fluence value of 2.05 J/cm.sup.2;
[0037]FIG. 3c is an exemplary graph of the evolution of electronic
level population densities and total absorption in C.sub.60 for
an incident fluence value of 14.1 J/cm.sup.2;
[0038]FIG. 3d is an exemplary graph of absolute contributions to
absorption from different active electronic levels corresponding
to the conditions provided in FIG. 3a;
[0039]FIG. 3e is an exemplary graph of the absolute contributions
to absorption from different active electronic levels corresponding
to the conditions provided in FIG. 3b;
[0040]FIG. 3f is an exemplary graph of the absolute contributions
to absorption from different active electronic levels corresponding
to the conditions provided in FIG. 3c;
[0041]FIG. 4a is an exemplary graph of an evolution of electronic
level population densities and total absorption in AF455 for an
incident energy value of 17 .mu.J;
[0042]FIG. 4b is an exemplary graph of the evolution of the electronic
level population densities and the total absorption in AF455 for
the incident energy value of 93 .mu.J;
[0043]FIG. 4c is an exemplary graph of the evolution of the electronic
level population densities and the total absorption in AF455 for
the incident energy value of 0.33 mJ;
[0044]FIG. 4d is an exemplary graph of the absolute contributions
to the absorption from different active electronic levels corresponding
to the conditions provided in FIG. 4a;
[0045]FIG. 4e is an exemplary graph of the absolute contributions
to the absorption from different active electronic levels corresponding
to the conditions provided in FIG. 4b;
[0046]FIG. 4f is an exemplary graph of the absolute contributions
to the absorption from different active electronic levels corresponding
to the conditions provided in FIG. 4c;
[0047]FIG. 5a is an exemplary graph of the electronic level population
densities in PPAI for the incident intensity value of 16.9 GW/cm.sup.2;
[0048]FIG. 5b is an exemplary graph of the electronic level population
densities in PPAI for the incident intensity value of 204.5 GW/cm.sup.2;
[0049]FIG. 6a is an exemplary graph of a numerical solution for
an evolution of a pulse intensity in C.sub.60 as a function of a
radius at .tau.=0 and at depths .eta.={0.00, 0.25, 0.50, 0.75, 1.00}
for an incident fluence value of 2.05 J/cm.sup.2;
[0050]FIG. 6b is an exemplary graph of the numerical solution for
the evolution of the pulse intensity in AF455 as a function of a
radial distance at .tau.=0 and at depths .eta.={0.00, 0.25, 0.50,
0.75, 1.00} for the incident energy value of 131 .mu.J;
[0051]FIG. 6c is an exemplary graph of the numerical solution for
the evolution of the pulse intensity in PPAI as a function of the
radial distance at .tau.=0 and at depths .eta.={0.00, 0.25, 0.50,
0.75, 1.00} for the incident fluence value of 204.5 GW/cm.sup.2;
[0052]FIG. 6d is an exemplary graph of the numerical solution for
evolution of the pulse intensity in AF455 as a function of the radial
distance at .tau.=0 and at depths .eta.={0.00, 0.25, 0.50, 0.75,
1.00} for the incident energy value of 6.6 .mu.J and where R.sub.0=7.07
.mu.m and T.sub.0=204.0 fs;
[0053]FIG. 7 is an exemplary graph of transmittance as a function
of input energy for AF455 in a 0.41 mm slab for femtosecond pulses;
(solid line) using integrated values (as in Eq. (22)), (dashed line)
using certain peak values;
[0054]FIG. 8a is an exemplary graph of an evolution of the electronic
level population densities at the surface of an AF455 0.412 mm slab
for the pulse duration of 144.0 fs;
[0055]FIG. 8b is an exemplary graph of the contributions of active
electron levels to the absorption, superimposed with a total intensity
absorption, in the AF455 slab for the conditions provided in FIG.
8a;
[0056]FIG. 9 is an exemplary diagram of temporal partitions on
a temporal grid which may be used for subsampling calculations
in accordance with certain exemplary embodiments of the present
invention;
[0057]FIG. 10 is an exemplary diagram of an electronic configuration
of an exemplary material which includes a ground state manifold,
a first excited state manifold, and a second excited state manifold;
[0058]FIG. 11a is an exemplary diagram of a single energy state
manifold;
[0059]FIG. 11b is an exemplary diagram of the exemplary single
energy state manifold and vibrational energy levels associated with
the manifold;
[0060]FIG. 11c is an exemplary diagram of the exemplary single
energy state manifold, together with the vibrational energy levels
and rotational energy levels associated with the manifold;
[0061]FIG. 12a is an exemplary energy level diagram associated
with an exemplary nonlinear absorbing material which includes several
manifolds of states, together with substates associated with the
manifolds;
[0062]FIG. 12b is the exemplary energy level diagram shown in FIG.
12a, where each manifold may be represented by a corresponding degenerate
energy level;
[0063]FIG. 13a is an exemplary diagram of a forward absorption
block;
[0064]FIG. 13b is an exemplary diagram of a reverse absorption
block;
[0065]FIG. 14 is an exemplary diagram of a relaxation event and
a corresponding relaxation block;
[0066]FIG. 15 is an exemplary diagram of a nonradiative migration
of an electron and the corresponding relaxation block;
[0067]FIG. 16 is an exemplary diagram of a radiative migration
of the electron and the corresponding relaxation block;
[0068]FIG. 17 is an exemplary energy diagram that is used to represent
various exemplary energy level transitions associated with AF455;
[0069]FIG. 18 is an exemplary energy diagram used to represent
the various exemplary energy level transitions associated with CuTPPS;
[0070]FIG. 19 is an exemplary schematic diagram of photoexcitation
and relaxation of an electron creating an exciton in a semiconductor;
[0071]FIG. 20 is an exemplary energy diagram used to represent
the various exemplary energy level transitions associated with a
semiconductor quantum dot;
[0072]FIG. 21 is a general flow diagram of an exemplary embodiment
of a method according to the present invention;
[0073]FIG. 22a is an illustrative portions of a detailed flow diagram
of an exemplary embodiment of a method according to the present
invention;
[0074]FIG. 22b is a further illustrative portion of a detailed
flow diagram of an exemplary embodiment of a method according to
the present invention; and
[0075]FIG. 23 is a schematic diagram of an exemplary system in
accordance with certain exemplary embodiments of the present invention.
[0076]Throughout the figures, the same reference numerals and characters,
unless otherwise stated, are used to denote like features, elements,
components or portions of the illustrated embodiments. Moreover,
while the present invention will now be described in detail with
reference to the figures, it is done so in connection with the illustrative
embodiments.
DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS
[0077]In certain exemplary embodiments of the method, system and
software arrangement according to the present invention, certain
measured parameters such as, for example, absorption cross sections
and decay rates can be used. For example, certain exemplary procedures
may be used for a numerical calculation of macroscopic rate equations
where these parameters may not be easily derived either from microscopic
quantum mechanical theories or from experimentally measured transmittance
data. To provide a more comprehensive numerical method that can
have a broad range of applicability with respect to both material
characteristics and/or energy characteristics, basic computational
building block diagrams may be used to describe properties of photoactive
materials as described herein below.
[0078]An appropriate wave equation in the presence of an electric
field can be provided by Maxwell's equation in scalar form, which
may be written as:
.gradient. 2 E c ( z , r , t )  1 c 0 2 .differential. 2 .differential.
t 2 E c ( z , r , t ) = 1 0 c 0 2 .differential. 2 .differential.
t 2 P c ( z , r , t ) . ( 1 )
In this exemplary equation, it is assumed that .gradient.E.sub.c=0,
.epsilon..sub.0 can refer to permittivity, and c.sub.0 can represent
the speed of light in vacuum. The electric field E and the induced
nonlinear polarization P can be written as:
E.sub.c(z,r,t)={tilde over (E)}(z,r,t) exp [i(.omega..sub.0tk.sub.0z)]+c.c.;
P.sub.c(z,r,t)={tilde over (P)}(z,r,t) exp [i(.omega..sub.0tk.sub.0z)]+c.c.,
(2)
where .omega..sub.0 (k.sub.0) is a frequency (e.g., wave number)
of the incident electromagnetic wave, and {tilde over (E)}(z,r,t)
and {tilde over (P)}(z,r,t) can represent realvalued slowly varying
envelopes of the electric field and polarization vector, respectively.
These exemplary equations can be simplified using a slowly varying
envelope approximation ("SVEA"), where the pulse envelope
may be assumed to vary slowly in time compared to an optical period.
A paraxial approximation may also be used, where the envelope can
be assumed to vary slowly along the propagation direction. The SVEA
and the paraxial approximation are described, for example, in P.
N. Butcher et al., The Elements of Nonlinear Optics, Cambridge University
Press, Cambridge, UK, 1990.
[0079]Using these approximations, the scalar wave equation in Eq.
(1) may be written as:
( .differential. .differential. z + 1 c 0 .differential. .differential.
t  i 2 k 0 .gradient. .perp. 2 ) E ~ ( z , r , t ) = ik 0 0 P ~
( z , r , t ) , ( 3 )
where .gradient..sub..perp..sup.2, can represent an operator for
the transverse variables. The intensity of the light can be defined
by (z,r,t)=2c.sub.0n.epsilon..sub.0{tilde over (E)}(z,r,t).sup.2,
where n is a linear index of refraction, and photon flux at a carrier
frequency .omega..sub.0 may be defined as {tilde over (.phi.)}(z,r,t)=
(z,r,t)/.omega..sub.0. The incident intensity of the laser pulse
can be written as (z=0,r,t)= .sub.0f(r,t), where f(r,t) may describe
a radial and temporal shape of an incident pulse, or as (z=0,r)=
.sub.0f(r) to describe, for example, a pulse width of very long
or infinite duration, e.g., a temporal continuous wave ("cw").
The term "pulse width" can refer to, for example, either
a finite duration or a very long or infinite duration (e.g., a continuous
wave). The term "laser pulse" can refer to, e.g., a pulse
provided directly by a laser or a cw laser beam which may be pulsed
by an external modulator. .sub.0 can represent a peak value of a
pulse intensity, which may be expressed as .sub.0=2c.sub.0n.epsilon..sub.0{tilde
over (E)}.sub.0.sup.2 with {tilde over (E)}(z=0,r,t)={tilde over
(E)}.sub.0f(r,t) or {tilde over (E)}(z=0,r)={tilde over (E)}.sub.0f(r),
where {tilde over (E)}.sub.0 can represent a peak value of a corresponding
electric field.
[0080]Temperature effects may be ignored in the exemplary procedure
described herein in accordance with certain exemplary embodiments
of the present invention, because they may not be significant in
extremely short time scales (e.g., ns to fs) that can be characteristic
of the absorption processes of interest. Significant thermal effects
may be incorporated using techniques such as those described, e.g.,
in the Kovsh publication. Further, the effects of laser damage in
absorbing materials, which can occur at very high intensities, may
not be directly accounted for. Effects of optical elements such
as, for example, lenses, apertures, beam splitters and/or mirrors
which may be present in an optical path between the laser beam and
the material may be incorporated in the propagating electromagnetic
wave using techniques such as those described, e.g., in P. W. Milonni
and J. H. Eberly, Lasers, New York, N.Y.: John Wiley, 1988, and
in B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, New
York, N.Y.: John Wiley, 1991.
[0081]In certain exemplary embodiments of the present invention,
diffraction effects may be neglected. This approximation can provide
accurate results, for example, if a sample thickness is limited
to at most a few diffraction lengths. Also, diffraction was not
observed in certain absorption experiments described herein. However,
techniques which allow calculation of diffraction effects are also
described herein below.
[0082]An absorbing medium may include two components: a chromophore
and a solvent or polymer that can surround the chromophore. A polarization
vector of the solvent/chromophore material may include a dispersion
term and a Kerr nonlinearity, which can give rise to effects such
as, for example, self(de)focusing, selfsteepening, and a Raman
effect. For the materials that include a solvent and chromophore
that are exposed to input intensities in the ranges described herein,
these effects may not be significant, and can possibly be ignored.
However, such effects can be incorporated into the exemplary techniques
described herein using conventional procedures.
[0083]A polarization vector for the chromophore can be described
by a density matrix. Such vector is described, e.g., in L. Allen
et al., Optical resonance and twolevel atoms, Plenum Press, New
York, 1975; A. I. Maimistov et al., Nonlinear Optical Waves, Kluwer
Academic Publishers, Dordrecht, The Netherlands 1999; and R. L.
Sutherland, Handbook of Nonlinear Optics, Marcel Dekker, New York,
2003.
[0084]A Hamiltonian H of an absorbing system can be described by
an unperturbed Hamiltonian H.sub.0 and an additional Hamiltonian
term H.sub.int (e.g., a perturbation term) such that H=H.sub.0+H.sub.int.
H.sub.int can express an interaction between light and a molecular
system using an electricdipole approximation, e.g., H.sub.int=d.sub.xE.sub.x.ident.dE,
as described, for example, in Moloney et al., "Nonlinear Optics,"
Westview Press, Boulder, Colo., 2004. In such approximation, d can
refer to an electricdipole operator, which can further include
an assumption that the laser is linearly polarized, and d and E
may each be aligned along an xaxis.
[0085]Eigenvalue and Eigenenergy corresponding to an s.sup.th level
may be written as s and .epsilon..sub.s respectively, as described,
e.g., in C. W. Gardiner et al., "Input and output in damped
quantum systems: Quantum stochastic differential equations and the
master equation," Phys. Rev. A, 31, 37613774 (1985). Using
these expressions, the wave function and unperturbed Hamiltonian
can be written as
 .psi. = s a s s , H ^ o s = s s ,
respectively. A density matrix operator may be defined as =.psi..psi.,
and a corresponding equation of motion can be written as
.differential. g ^ .differential. t =  i [ H ^ , g ^ ] , or .differential.
g s 1 s 2 .differential. t =  i s 3 ( H s 1 s 3 g s 3 s 2  g s
2 s 3 H s 3 s 1 ) , ( 4 )
where matrix elements g.sub.s.sub.1.sub.s.sub.2 can represent a
polarization induced by a transition between energy levels s.sub.1
and s.sub.2.
[0086]Photons from incident light can be absorbed by the molecular
system, which may enable the atoms and/or molecules to occupy excited
energy states (e.g., electronic, vibrational, and/or rotational).
Thus, the polarization for n.sub.a atoms or molecules per unit volume,
can be expressed as:
P ~ = n a e ^ .intg. .psi..psi. * R .fwdarw. R .fwdarw. = n a s
1 s 2 g s 1 s 2 d s 2 s 1 = n a Tr ( dg ) , ( 5 )
where can represent a unit electric charge, and R can describe
a distance between separated charges in a dipole moment approximation.
After the excitation to higher energy states, the system may relax
to a ground state through radiative and/or nonradiative transitions.
In a semiclassical approach, the relaxation terms can be added
to the equations of motion of the density matrix using a phenomenological
technique. In general, quantum mechanical determinations of the
relaxation terms may be quite involved as described, e.g., in C.
W. Gardiner et al., "Driving atoms with light of arbitrary
statistics," Phys. Rev. A, 50, 17921806 (1994); M. Lax, "Quantum
noise IV. Quantum theory of noise sources," Phys. Rev. 145,
110129 (1966); and A. Barchielli, "Measurement Theory and
stochastic differential equations in quantum mechanics," Phys.
Rev. A 34, 16421649 (1986).
[0087]For further understanding of the behavior of an ensemble
of radiators (e.g., atoms, molecules, excitons, or impurities in
a crystal) in a field of resonant or nonresonant coherent electromagnetic
waves, it can be beneficial to characterize time scales of the various
processes. The interaction of the radiators with nonresonant atoms
(e.g., those present in a solvent) can give rise to a dephasing
rate .gamma..sub.s.sub.1.sub.s.sub.2, which can be described by
the expression:
.differential. g s 1 s 2 .differential. t =  ( .gamma. s 1 s 2
+ i .omega. s 1 s 2 ) g s 1 s 2 +  i s 3 ( H s 1 s 3 int g s 3
s 2  g s 2 s 3 H s 3 s 1 int ) . ( 6 )
[0088]Rate equations for an exemplary absorbing material such as
C.sub.60 can be determined based on Eq. (4) above. Equations of
motion for densitymatrix elements can be written as:
.differential. g s 1 s 2 .differential. t =  ( .GAMMA. s 1 s 2
+ i .omega. s 1 s 2 ) g s 1 s 2 + i E ~ s 3 ( d s 1 s 3 g s 3 s
2  g s 2 s 3 d s 3 s 1 ) , ( 7 )
where d can be taken along the direction of {tilde over (E)}, .GAMMA..sub.s.sub.s.sub.2
can represent a transverse relaxation time arising from various
nonradiative behaviors such as, e.g., irreversible losses and elastic
scattering, and .omega..sub.s.sub.1.sub.s.sub.2=.omega..sub.s.sub.1.omega..sub.s.sub.2
and .omega.=.omega..sub.0. The decay rate for the offdiagonal terms
can be expressed as .GAMMA..sub.s.sub.1.sub.s.sub.2g.sub.s.sub.1.sub.s.sub.2>>.diffe
rential.g.sub.s.sub.1.sub.s.sub.2/.differential.t+i.omega..sub.s.sub.1.sub
.s.sub.2g.sub.s.sub.1.sub.s.sub.2 for s.sub.1.noteq.s.sub.2. It
may be preferable to denote an absorption cross section from state
s.sub.1 to state s.sub.2 as:
.sigma. s 1 s 2 = .omega..GAMMA. s 1 s 2 d s 1 s 2 2 nc 0 [ .GAMMA.
s 1 s 2 2 + ( .omega. s 1 s 2  .omega. ) 2 ] . ( 8 )
An approximation that .omega..sub.s.sub.1.sub.s.sub.2=.omega..sub.0
can also be used. The equations of motion for the densitymatrix
elements can be written as:
.differential. g 00 .differential. t = .sigma. 01 .phi. ~ ( g 11
 g 00 ) + k 10 g 11 + k 30 g 33 .differential. g 11 .differential.
t = .sigma. 12 .phi. ~ ( g 22  g 11 )  .sigma. 01 .phi. ~ ( g
11  g 00 ) + k 21 g 22  ( k 13 + k 10 g 11 ) .differential. g
22 .differential. t =  .sigma. 12 .phi. ~ ( g 22  g 11 )  k 21
g 22 .differential. g 33 .differential. t = .sigma. 34 .phi. ~ (
g 44  g 33 ) + k 43 g 44  k 30 g 33 .differential. g 44 .differential.
t =  .sigma. 34 .phi. ~ ( g 44  g 33 )  k 43 g 44 ( 9 )
where k.sub.s.sub.1.sub.s.sub.2 can represent longitudinal relaxation
times and {tilde over (.phi.)}(z,r,t)= (z,r,t)/.omega..sub.0. Decay
of the vibrational states may be very fast (e.g., on the order of
femtoseconds), whereby stimulated emission may be negligible and
the term .sigma..sub.s.sub.1.sub.s.sub.2(g.sub.s.sub.1.sub.s.sub.1g.sub.s.sub.2.s
ub.s.sub.2) can be approximated as .sigma..sub.s.sub.1.sub.s.sub.2g.sub.s.sub.2.sub.s.sub.2.
An approximation g.sub.s.sub.2.sub.s.sub.2=N.sub.s.sub.2 may also
be used. The rate equations can thus be written in a form of:
.differential. N ~ 0 .differential. t =  .sigma. 01 .phi. ~ N
~ 0 + k 10 N ~ 1 + k 30 N ~ 3 .differential. N ~ 1 .differential.
t =  .sigma. 01 .phi. ~ N ~ 0  ( .sigma. 12 .phi. ~ + k 13 + k
10 ) N ~ 1 + k 21 N ~ 2 .differential. N ~ 2 .differential. t =
.sigma. 12 .phi. ~ N ~ 1  k 21 N ~ 2 .differential. N ~ 3 .differential.
t =  ( .sigma. 34 .phi. ~ + k 30 ) N ~ 3 + k 43 N ~ 4 .differential.
N ~ 4 .differential. t = .sigma. 34 .phi. ~ N ~ 3  k 43 N ~ 4 (
10 )
The polarization can be described by the equation:
P ~ ( z , r , t ) =  inc 0 0 .omega. 0 ( .sigma. 10 N ~ 1 ( z
, r , t ) + .sigma. 12 N ~ 2 ( z , r , t ) + .sigma. 34 N ~ 3 (
z , r , t ) ) I ~ ( z , r , t ) . ( 11 )
[0089]Combining Eq. (11) with Maxwell's equation, Eq. (1), can
provide a corresponding propagation equation which may be written
as:
( .differential. .differential. z + 1 c .differential. .differential.
t ) I ~ ( z , r , t ) =  ( .sigma. 10 N ~ 0 ( z , r , t ) + .sigma.
12 N ~ 1 ( z , r , t ) + .sigma. 34 N ~ 3 ( z , r , t ) ) I ~ (
z , r , t ) , ( 12 )
where c=c.sub.0/n. This equation may be used to describe propagation
of light in a C.sub.60 solution.
Stimulated Emission
[0090]Equations similar to Eq. (10) above can be derived that include
effects of stimulated emission. In general, stimulated emission
may occur from only one of the electronic levels, and spontaneous
emission can occur from other electronic levels. Thus, the expressions
provided in Eq. (10) can describe one exemplary behavior that may
occur, e.g., stimulated emission, and both stimulated and spontaneous
emission can occur in certain materials. The exemplary equations
that include a description of stimulated emission may be written
as:
.differential. N ~ 0 .differential. t = .sigma. 01 .phi. ~ ( N
~ 1  N ~ 0 ) + k 10 N ~ 1 + k 30 N ~ 3 .differential. N ~ 1 .differential.
t =  .sigma. 01 .phi. ~ ( N ~ 1  N ~ 0 ) + .sigma. 12 .phi. ~
( N ~ 2  N ~ 1 ) + ( k 13 + k 10 ) N ~ 1 + k 21 N ~ 2 .differential.
N ~ 2 .differential. t =  .sigma. 12 .phi. ~ ( N ~ 2  N ~ 1 )
 k 21 N ~ 2 .differential. N ~ 3 .differential. t = .sigma. 34
.phi. ~ ( N ~ 4  N ~ 3 ) + k 43 N ~ 4  k 30 N ~ 3 .differential.
N ~ 4 .differential. t =  .sigma. 34 .phi. ~ ( N ~ 4  N ~ 3 )
 k 43 N ~ 4 ( 13 )
A corresponding propagation equation that includes the effects
of stimulated emission can be written as
[0091] ( .differential. .differential. z + 1 c .differential. .differential.
t ) I ~ ( z , r , t ) = ( .sigma. 01 [ N ~ 1 ( z , r , t )  N ~
0 ( z , r , t ) ] + .sigma. 12 [ N ~ 2 ( z , r , t )  N ~ 1 ( z
, r , t ) ] + .sigma. 34 [ N ~ 4 ( z , r , t )  N ~ 3 ( z , r ,
t ) ] ) I ~ ( z , r , t ) ( 14 )
[0092]As described herein, the relaxation time of the vibrational
states of the electronic levels can be assumed to be very fast (e.g.,
on the order of femtoseconds), so that effects of stimulated emission
may be neglected in such systems. The appropriate dephasing rate
can be determined for a specific material of interest. For organic
molecules that can be provided in solvents, a dephasing time .gamma..sub.s.sub.1.sub.s.sub.2.sup.1
can be between approximately 7 and 70 fs. The dephasing time can
be selected as an upper limit for a laser pulse width T.sub.0, such
that, approximately, T.sub.0>.gamma..sub.s.sub.1.sub.s.sub.2.sup.1.
A lifetime of the lowest excited electronic state .gamma..sub.ss.sup.1
can be approximately 1 ns as described, e.g., in J. Turro, Modern
Molecular Photochemistry, Benjamin, N. Y., 1978. Thus, a laser pulse
width can be selected to be approximately within the range .gamma..sub.s.sub.1.sub.s.sub.1.sup.1>T.sub.0>.gamma..sub.s.sub.1.
sub.s.sub.2.sup.1.
Absorption Energy Diagrams
[0093]An analysis using the density matrix approach, which may
be guided by a phenomenological Jablonski diagram for a single photon
excitation, is described herein below. A similar analysis applicable
to RSA materials including, e.g., copper phthalocyanine, C.sub.60,
and cadmium texaphyrin is described in C. Li et al., "Excitedstate
nonlinear absorption in multienergylevel molecular systems,"
Phys. Rev. A, 51, 569575 (1995). Utilization of a density matrix
approach to investigate a pulse width dependence of the TPA crosssections
of PRL101 measured in the ns and fs regions is described, e.g.,
in A. Baev et al., "General theory for pulse propagation in
twophoton active media," J. Chem. Phys. 117, 62146220 (2002).
[0094]Organic molecules may exhibit multiphoton absorption involving
both singlet and triplet states with increasing laser intensities.
This behavior can be difficult to describe based solely on quantum
calculations. Therefore, a phenomenological model based on spectroscopic
and kinetic data can be provided that includes a description of
nonlinear absorbers which further includes state diagrams or Jablonski
diagrams, and is described, e.g., in M. Klessinger et al., Excited
States and Photochemistry of Organic Molecules, VCH, Deerfield Beach,
Fla. 1995. This type of an exemplary model can provide a representation
of population dynamics, which can generate corresponding rate equations.
Experimental data used in this exemplary procedure can includes
an absorption cross section and decay rates of various electronic
levels.
[0095]Exemplary procedures in accordance with exemplary embodiments
of the present invention described herein can provide a description
of the absorption behavior of a variety of nonlinear materials using
e.g., a single generalized numerical method. Several types of absorption
mechanisms may be present within certain nonlinear materials, and
the mechanisms can depend on the number of photons absorbed simultaneously
and/or on the states in which absorption occurs. The exemplary embodiments
of the methods, system, software arrangement and computer accessible
medium according to the present invention can be used to describe,
e.g., Nphoton absorbers with both singlet and triplet levels.
[0096]Five exemplary types of absorption mechanisms can be used
to model absorption behavior. These mechanisms 100140 are shown
in FIG. 1 as transition diagrams and labeled (B.sub.0)(B.sub.4).
Electronic states in FIG. 1 are labeled N.sub.0N.sub.4, and absorption
crosssections can be labeled with a .sigma. identifier. As shown
in FIG. 1, upward arrows may represent photoexcitation transitions,
and downward arrows may represent intersystem electron decay events.
In accordance with appropriate exemplary selection rules, singlephoton
absorption can occur along singletsinglet transitions from a ground
state 100 and/or a lowest excited state 110. Singlephoton absorption
can also occur along a triplettriplet transition 120 from a lowest
triplet excited state, which may be populated by electrons relaxed
along an intersystem crossing link. These exemplary mechanisms do
not explicitly consider ultrafast relaxations that may occur from
vibronic intermediate states.
[0097]TPA can occur from the ground state by simultaneous absorption
of two photons, which may promote electrons to the lowest excited
singlet state. Such transition 130 as shown in FIG. 1 can be followed
by two further transitions: a singletsinglet transition 110, or
a singlettriplet transition 120 from N.sub.1 to N.sub.3. Threephoton
absorption (3PA) may involve a promotion of ground state electrons
to the lowest excited singlet state by simultaneous absorption of
three photons, as shown in the transition diagram 140 of FIG. 1.
[0098]Transition diagrams 100140 of FIG. 1 can represent computational
"building blocks" that may be combined to describing general
absorption behavior of nonlinear absorbing materials. For example,
the absorption in a C.sub.60toluene solutiona nonlinear RSA materialcan
be described using a fivelevel model as described, e.g., in D.
G. McLean et al., "Nonlinear absorption study of a C60toluene
solution," Opt. Lett. 18, 858860 (1993). This exemplary model
can be obtained by combining the absorption diagrams 100120 as
shown in FIG. 1 (e.g., B.sub.0.orgate.B.sub.1.orgate.B.sub.2).
[0099]A fivelevel absorption model of a chromophore from an AFX
group exhibiting TPAassisted excited state absorption is described,
e.g., in the He et al. and Kannon et. al. publications. This exemplary
model, which includes TPA and ESA. can also be obtained, for example,
by combining the absorption diagrams 110130 shown in FIG. 1 (e.g.,
B.sub.1.orgate.B.sub.2.orgate.B.sub.3).
[0100]The term "generic" material can refer to a nonlinear
absorbing material having an absorption energy diagram that may
be described by a combination of one or more basic transition diagrams
such as, e.g., B.sub.0B.sub.4 100140 shown in FIG. 1.
Rate and Propagation Equations
[0101]An absorption energy diagram obtainable as a combination
of transition diagrams can specify the corresponding rate and propagation
equations. For example, in accordance with the exemplary derivation
described herein, a rate equation in a moving time frame (e.g.,
(z,t=t'k.sub.1z)) for a generic nonlinear material can be expressed
in matrix form as:
N ~ ( z , r , t ) t = [ D ^ 0 + .alpha. = 1 N A D ^ .alpha. .alpha.
.omega. 0 I ~ .alpha. ( z , r , t ) ] N ~ ( z , r , t ) , ( 15 )
where N=[N.sub.0,N.sub.1, . . . , N.sub.S].sup.T can represent
a population density vector function N(z,r,t) for a system with
S electronic levels, {circumflex over (D)}.sub.0.ident.{circumflex
over (D)}.sub.0({k.sub.s.sub.1.sub.s.sub.2}), {circumflex over (D)}.sub.1.ident.{circumflex
over (D)}.sub.1({.sigma..sub.s.sub.1.sub.s.sub.2}), {circumflex
over (D)}.sub.2.ident.{circumflex over (D)}.sub.2(.sigma..sub.TPA),
{circumflex over (D)}.sub.3.ident.{circumflex over (D)}.sub.3(.sigma..sub.3PA),
. . . , {circumflex over (D)}.sub.N.sub.A.ident.{circumflex over
(D)}.sub.N.sub.A(.sigma..sub.[N.sub.A.sub.]PA) can be N.sub.A+1
constant S.times.S sparse matrices having decay rates k.sub.s.sub.1.sub.s.sub.2,
single photon .sigma..sub.s.sub.1.sub.s.sub.2, twophoton .sigma..sub.TPA,
threephoton .sigma..sub.3PA, and, possibly, N.sub.Aphoton .sigma..sub.[N.sub.A.sub.]PA
molar crosssections respectively, and (z,r,t) can be a function
of a photon flux density. The propagation equation for such material
may be expressed in a vector form as:
I ~ ( z , r , t ) z =  .beta. = 1 N B ( .sigma. .beta. N ~ ( z
, r , t ) ) I ~ .beta. ( z , r , t )  c ~ I ~ ( z , r , t ) , (
16 )
where .sigma..sub.1.ident..sigma..sub.1({.sigma..sub.s.sub.1.sub.s.sub.2})
, .sigma..sub.2.ident..sigma..sub.2(.sigma..sub.TPA), .sigma..sub.3.ident..sigma..sub.3(.sigma..sub.3PA),
. . . , .sigma..sub.N.sub.B.ident..sigma..sub.N.sub.B(.sigma..sub.[N.sub.B.sub.]P
A) can be N.sub.B constant (mostly sparse) Sdimensional vectors
which may include certain elements of corresponding {circumflex
over (D)}.sub..beta. matrices, and {tilde over (c)} can represent
a linear absorption coefficient. The constant vectors and matrices
in the above equations are described herein below in more detail.
[0102]Certain solutions to the coupled system of Eqs. (15) and
(16) can be formulated using various mathematical and numerical
techniques. For example, a numerical solution of the propagation
equation using steadystate estimates of population densities is
described, e.g., in D. G. McLean et al., "Nonlinear absorption
study of a C60toluene solution," Opt. Lett. 18, 858860 (1993).
An analytic solution of a threelevel approximation for the fivelevel
population density system of RSA C.sub.60 is described, e.g., in
A. Kobyakov et al., "Analytical approach to dynamics of reverse
saturable absorbers," J. Opt. Soc. Am. B. 17, 18841894 (2000).
[0103]Analytic solutions may been formulated in the ns regime for
TPA AF455 as described, e.g., in the Sutherland et al. publication
and in the ps regime for TPA L.sub.34 and for 3PA PPAI dye as described,
e.g., in the Wang et al. publication. A RungeKutta numerical solution
may also be used such as that described, e.g., in I.C. Khoo et
al., "Passive optical limiting of picosecondnanosecond laser
pulses using highly nonlinear organic liquid cored fiber array,"
IEEE J. Sel. Top. Quantum Electron. 7, 760768 (2001). A beampropagation
technique used to model RSA CAP dye in toluene and TPA ZnSe is described,
e.g., in S. Hughes et al., "Modeling of picosecondpulse propagation
for optical limiting applications in the visible spectrum,"
J. Opt. Soc. Am. B. 11, 29252929 (1997). Other exemplary solution
procedures that may be used can include, for example, spectral and
CrankNicholson finite difference methods which can included an
instantaneous Kerr effect, diffraction, thermal effects for RSA
SiNc, and Zscan of a 2PA. An analytic approximation of these exemplary
equations capable of accounting for the effects of long pulses and
a numerical solution of an integrodifferential equation for short
pulses to model general TPA+singletsinglet ESA organic absorbers
can be used as described, e.g., in the Baev et al. publication.
[0104]Exemplary analytical solutions to the coupled system of Eqs.
(15) and (16) can use stringent assumptions about photophysical
properties of the materials and/or the range of temporal pulse durations.
For example, the assumptions that may be used to account for photophysical
properties can include: (a) a "negligible groundstate depopulation
approximation," which assumes that the population density of
the ground state is approximately constant; (b) the excited states
of the singlet and triplet states are proportional to I.sup.2(t),
which can correspond to a quasisteadystate regime where the time
dependence of the population densities approximates that of the
intensity; (c) electronic states N.sub.2 and N.sub.4 may be neglected
or electronic state N.sub.2 may be neglected; (d) repopulation of
the ground state due to the lowest triplet state relaxation may
be ignored; and (e) singlettriplet intersystem crossing and spatial
diffusion may be ignored.
Laser Pulses
[0105]An incident electromagnetic wave (e.g., a laser pulse) interacting
with a nonlinear absorbing material can be characterized using a
variety of parameters which may specify certain properties of the
wave. For example, such parameters can include coherency (or lack
thereof), frequency (e.g., a single frequency or a set of discrete
frequencies), a pulse, or a series of consecutive pulses (e.g.,
a "pulse train"), etc. A single pulse can be further characterized,
e.g., by a temporal pulse width and/or a radial width. Multiple
pulses or a pulse train can be further characterized, e.g., by a
pulse duration, a separation time between pulses, a number of pulses
or overall duration of a pulse train and/or an incident intensity
or energy of each pulse.
[0106]Parameters which may be used to characterize or describe
a laser pulse or other incident electromagnetic wave can be obtained
using various procedures. For example, such parameters can be based
on experimental measurements or manufacturer's specifications. A
frequency may be modified using nonlinear optical techniques. A
temporal pulse width may be modified using further nonlinear optical
techniques such as, e.g., solutions and/or transformlimited nonlinear
pulse compression techniques. The pulses may be created using a
continuous wave laser by applying an external modulator such as,
e.g., an electrooptic modulator. Also, a radial beam waist may
be modified, e.g., by using a lens, an aperture and/or a nonlinear
optical material which may be selffocusing.
[0107]An incident electromagnetic wave that includes multiple pulses
and/or a pulse train can be characterized by a temporal pulse separation
and/or a repetition rate. The pulses in such fields can be created
or modified, e.g., by using intra(laser) cavity procedures such
as, e.g., modelocking, Qswitching, or Qswitched modelocking.
External (laser) cavity procedures may also be used such as, e.g.,
an electrooptic modulator. Exemplary optical procedures such as,
e.g., beam splitting, time delay, and/or recombination may also
be used.
[0108]Certain limitations on a pulse duration (e.g., a ns pulse
duration, a ps pulse duration, a subrange of ns pulse duration,
or a range of up to a few ns) can be assumed or estimated to obtain
certain solutions to these equations. Certain conventional solution
procedures may include a radial variable, although frequently it
is assumed that the spatiallydependent functions are constant in
the radial domain. Further, these conventional procedures may have
been developed to describe the behavior of particular materials
and/or for certain pulse temporal widths.
Solution Techniques
[0109]In accordance with certain exemplary embodiments of the present
invention, a timeresolved radiallydependent finitedifference
numerical scheme can be provided which may be used to describe absorption
and/or relaxation behavior of any generic material interacting with
an incident pulse over a broad range of temporal pulse widths (e.g.,
pulse durations).
[0110]The coupled exemplary system described by Eqs. (15) and (16)
can be converted to a dimensionless form using the following transformations
.eta.=z/L.sub.df, .rho.=r/R.sub.0, .tau.=t/T.sub.0, I(.eta.,.rho.,.tau.)=
(.eta.,.rho.,.tau.)/ .sub.0,N.sub..beta.(.eta.,.rho.,.tau.)=N.sub..beta.(.eta.,.rho.,.tau.)/N,
L.sub.df=.pi.R.sub.0.sup.2n.sub.1/.lamda., where T.sub.0, R.sub.0
are a pulse width and a beam radius, respectively, associated with
the incident pulse shape. N can represent a total population electron
density of the material, which may be independent of time, e.g.,
N = .beta. N ~ .beta. ( .eta. , .rho. , .tau. ) .
The incident pulse can be described by a general formula, I(.eta.=0,.rho.,.tau.)=
.sub.0f(.rho.,.tau.) or I(.eta.=0,.rho.)= .sub.0f(.rho.) for cw.
In additional exemplary embodiments of the present invention, described
in more detail herein below, a standard Gaussian distribution may
be used to describe the form of the incident pulse.
[0111]Using the transformations provided above, Eqs. (15) and (16)
may be rewritten as:
N ( .eta. , .rho. , .tau. ) .tau. = T 0 [ D ^ 0 + .alpha. = 1 N
A D ^ .alpha. I 0 .alpha. .alpha. .omega. 0 I .alpha. ( .eta. ,
.rho. , .tau. ) ] N ( .eta. , .rho. , .tau. ) , ( 17 ) I ( .eta.
, .rho. , .tau. ) .eta. =  L df N .beta. = 1 N B ( .sigma. .beta.
N ( .eta. , .rho. , .tau. ) ) I 0 ( .beta.  1 ) I .beta. ( .eta.
, .rho. , .tau. )  L df c ~ I ( z , r , t ) , ( 18 )
respectively. The mathematical analysis that can be performed to
describe the absorption and relaxation behavior of generic materials
can be based on a solution of Eqs. (17) and (18).
[0112]For example, a family of identical 2D grids .OMEGA., which
may be indexed by a radius .rho., can be defined such that:
={.OMEGA.(.rho..sub.j),.rho..sub.j=j.DELTA..rho.}, (19)
(.rho..sub.j)=(.OMEGA..sub.N(j),.OMEGA..sub.I(j)). (20)
For every .rho..sub.j sample, a member from .OMEGA. can correspond
to a pair of interleaved grids in the .eta..tau. parametric domain.
One such grid can be represented as
N(j)={(.eta..sub.n+1/2,.rho..sub.j,.tau..sub.i+1/2),
n+1/2=(.eta..sub.0+.DELTA..eta./2)+n.DELTA..eta.,.tau..sub.i+1/2=(.tau..su
b.0+.DELTA..tau./2)+i.DELTA..tau.}, (21)
and may be used to sample the population density N(.eta.,.rho.,.tau.).
Another such grid, which can be represented as:
I(j)={(.eta..sub.n,.rho..sub.j,.tau..sub.i),.eta..sub.n=.eta..sub.0+n.DELT
A..eta.,.tau..sub.i=.tau..sub.0+i.DELTA..tau.},
or .OMEGA..sub.E(j)={(.eta..sub.n,.rho..sub.j,.tau..sub.i),.eta..sub.n=.et
a..sub.0+n.DELTA..eta.,.tau..sub.i=.tau..sub.0+i.DELTA..tau.} (22)
may be used to sample the intensity I(.eta.,.rho.,.tau.) or the
electric field E(.eta.,.rho.,.tau.).
[0113]The exemplary dimensionless equations provided in Eqs. (17)
and (18) can be integrated as described herein below, using small
step sizes .DELTA..tau., .DELTA..eta. at .OMEGA..sub.N(j), .OMEGA..sub.I(j)
grid points, respectively. For example, the system of rate equations
can be integrated to yield a spatiallyresolved solution in a current
moving frame (e.g., a reference system of the "pulse rest"),
while the average intensity,
1 2 [ I ( .eta. n , .rho. j , .tau. i ) + I ( .eta. n + 1 , .rho.
j , .tau. i ) ] ,
is held constant. Further, the propagation equation may be solved
over a thin slice [.eta.,.eta.+.DELTA..eta.], by using an available
average population density,
1 2 [ N ( .eta. n + 1 / 2 , .rho. j , .tau. i  1 / 2 ) + N ( .eta.
n + 1 / 2 , .rho. j , .tau. i + 1 / 2 ) ] ,
as an approximation to electronic populations.
[0114]An exemplary procedure to integrate the exemplary coupled
system of rate and propagation equations provided in Eqs. (17) and
(18) can be based on the following mathematical derivation. For
example, the following expression can be derived from Eq. (17) as
follows:
.differential. ln N ( .eta. , .rho. , .tau. ) .differential. .tau.
= T 0 [ D ^ 0 + .alpha. = 1 N A D ^ .alpha. I 0 .alpha. .alpha.
.omega. 0 I .alpha. ( .eta. , .rho. , .tau. ) ] ( 23 )
step can be applied to this equation, which leads to the following
expression:
N ( .eta. , .rho. , .tau. + .DELTA..tau. ) = N ( .eta. , .rho.
, .tau. ) .times. exp ( .intg. .tau. .tau. + .DELTA..tau. T 0 [
D ^ 0 + .alpha. = 1 N A D ^ .alpha. I 0 .alpha. .alpha. .omega.
0 I .alpha. ( .eta. , .rho. , .tau. ' ) ] .tau. ' ) .apprxeq. N
( .eta. , .rho. , .tau. ) exp ( .DELTA..tau. T 0 D ^ 0 + .DELTA..tau.
T 0 .alpha. = 1 N A D ^ .alpha. I 0 .alpha. .alpha. .omega. 0 I
.alpha. ( .eta. , .rho. , .tau. ) .times. 1 2 { I .alpha. ( .eta.
 .DELTA..eta. / 2 , .rho. , .tau. + .DELTA..tau. / 2 ) + I .alpha.
( .eta. + .DELTA..eta. / 2 , .rho. , .tau. + .DELTA..tau. / 2 )
} ) . ( 24 ) ( 25 )
[0115]Calculation of a propagation equation can be performed using
Eq. (18), which can be rearranged to obtain the following expression:
.differential. ln I ( .eta. , .rho. , .tau. ) .differential. .eta.
=  L df N .beta. = 1 N B ( .sigma. .beta. N ~ ( .eta. , .rho. ,
.tau. ) ) I 0 ( .beta.  1 ) I .beta.  1 ( .eta. , .rho. , .tau.
)  L df c ~ ( 26 )
step can be applied to this equation, which leads to the following
expression:
I ( .eta. + .DELTA..eta. , .rho. , .tau. ) = I ( .eta. , .rho.
, .tau. ) .times. exp (  L df N .beta. = 1 N B { .sigma. .beta.
.intg. .eta. .eta. + .DELTA..eta. I 0 ( .beta.  1 ) I .beta. 
1 ( .eta. ' , .rho. , .tau. ) N ( .eta. ' , .rho. , .tau. ) .eta.
' } ) .times. exp (  L df .intg. .eta. .eta. + .DELTA..eta. c ~
.eta. ' ) .apprxeq. I ( .eta. , .rho. , .tau. ) exp (  L df N .DELTA..eta.
.times. .beta. = 1 N B { [ .sigma. .beta. N ( .eta. + .DELTA..eta.
/ 2 , .rho. , .tau.  .DELTA..tau. / 2 ) + N ( .eta. + .DELTA..eta.
/ 2 , .rho. , .tau. + .DELTA..tau. / 2 ) 2 ] .times. .times. I 0
.beta.  1 2 [ I .beta.  1 ( .eta. , .rho. , .tau. ) + I .beta.
 1 ( .eta. + .DELTA..eta. , .rho. , .tau. ) ] }  L df .DELTA..eta.
c ~ ) . ( 27 ) ( 28 )
Equations (25) and (28) may be used when performing a calculation
of the coupled Eqs. (17) and (18).
[0116]The resulting system of coupled equations can be written
as
N n + 1 / 2 , j , i + 1 / 2 ( k ) .apprxeq. exp ( .DELTA..tau.
T 0 D ^ 0 + .DELTA..tau. T 0 D ^ 1 I 0 .omega. 0 1 2 { I n , j ,
i + I n + 1 , j , i ( k ) } + .DELTA..tau. T 0 D ^ 2 I 0 2 2 .omega.
0 1 2 { I n , j , i 2 + I n + 1 , j , i ( k ) 2 } + .DELTA..tau.
T 0 D ^ 3 I 0 3 3 .omega. 0 1 2 { I n , j , i 3 + I n + 1 , j ,
i ( k ) 3 } ) N n + 1 / 2 , j , i  1 / 2 ( k ) ( 29 ) I n + 1 ,
j , i ( k + 1 ) .apprxeq. exp (  L df N .DELTA..eta. { .sigma.
1 N n + 1 / 2 , j , i  1 / 2 ( k ) + N n + 1 / 2 , j , i + 1 /
2 ( k ) 2 }  L df N .DELTA..eta. I 0 { .sigma. 2 N n + 1 / 2 ,
j , i  1 / 2 ( k ) + N n + 1 / 2 , j , i + 1 / 2 ( k ) 2 } 1 2
{ I n , j , i + I n + 1 , j , i ( k ) }  L df N .DELTA..eta. I
0 2 { .sigma. 3 N n + 1 / 2 , j , i  1 / 2 ( k ) + N n + 1 / 2
, j , i + 1 / 2 ( k ) 2 } 1 2 { I n , j , i 2 + I n + 1 , j , i
( k ) 2 }  L df .DELTA..eta. c ~ ) I n , i ( 30 )
These exemplary equations may still contain an interdependence
between the intensity and population densities. Therefore, k=1:K
iterations can be performed to obtain a numerical solution of the
intensity function I(.eta..sub.n,.rho.,.tau.) at a given depth .eta..sub.n.
To simplify notation, only the indices n, j, and i are retained
in the further description below.
[0117]At any depth .eta..sub.n the iterating scheme provided by
Eqs. (29) and (30) may converge very fast using a secondorder Taylor
series expansion of the matrix exponential in Eq. (29). By selecting
a sufficiently small .DELTA..tau., sufficient convergence can be
achieved with a number of iterations k, possibly equal to 2 or 3.
If the pulse is short in the temporal domain, the grid size may
be increased significantly to ensure that eigenvalues of the matrix
in the exponent are less than one, which can allow the use of such
a Taylor expansion.
[0118]An alternate exemplary procedure that may be used is to subsample
the grid on demand, e.g., only at the high intensity areas, which
may be relatively small compared to the entire parametric domain.
For example, at every time step .tau..sub.i1/2.fwdarw..tau..sub.i+1/2
in Eq. (29), the magnitude of the matrix elements in the exponent
can be evaluated. If necessary, M1 additional time subsamples
can be introduced such as, e.g., .tau..sub.i1/2=.tau..sup.0.fwdarw..tau..sup.1.fwdarw.
. . . .tau..sup.M1.fwdarw..tau..sub.i+1/2=.tau..sup.M, where M
can be selected such that the elements of the resulting refined
matrices corresponding to the subsamples are small enough to ensure
the validity of the Taylor series approximation of the matrix exponentials.
To calculate the refined matrices for each .tau..sup.m, the integration
step used to derive Eq. (29) herein can be repeated, using an integration
domain of [.tau..sup.m,.tau..sup.m+1]. To perform this integration,
the average intensity values can also be estimated at {circumflex
over (.tau.)}.sup.m=.tau..sup.m+1/2.DELTA..tau.' samples, where
.DELTA..tau.'=.tau..sup.m+1.tau..sup.m.ident..DELTA..tau./M can
represent the resulting subsampling time procedure. Because any
such sample {circumflex over (.tau.)}.sup.m can be approximately
equal to .tau..sub.i1/2+(m+1/2).DELTA..tau.', I.sub.n,j({circumflex
over (.tau.)}.sup.m).ident.I(.eta..sub.n,.rho..sub.j,{circumflex
over (.tau.)}.sup.m) can be estimated using a linear interpolation
of I.sub.n,j(.tau..sub.i1/2) and I.sub.n,j(.tau..sub.i+1/2) which
can be expressed as:
I n , j ( .tau. ^ m ) = I n , j ( .tau. i  1 / 2 + ( m + 1 / 2
) .DELTA..tau. ' ) = .lamda. m I n , j ( .tau. i  1 / 2 ) + .mu.
m I n , j ( .tau. i + 1 / 2 ) , ( 31 )
where .lamda..sub.m+.mu..sub.m=1, and .lamda..sub.m=(1(m+1/2)/M).
The powers of the intensity values I.sub.n,j.sup..alpha., .alpha.=1
. . . 3, for the midpoint time grid samples .tau..sub.i1/2, .tau..sub.i+1/2
in Eq. (31) can be interpolated using the expression
I n , j .alpha. ( .tau. i  1 / 2 ) .apprxeq. ( I .alpha. ) n ,
j , i ^ .ident. 1 2 ( I n , j , i  1 .alpha. + I n , j , i .alpha.
) . ( 32 )
[0119]The numerical integration of the system of rate equations
provided herein may be performed using additional subsampling times.
For a notational simplicity, the following parameter substitutions
can be used in Eq. (24):
D.sub.0=.DELTA..tau.T.sub.0{circumflex over (D)}.sub.0, D.sub..alpha.=.DELTA..tau.T.sub.0{circumflex
over (D)}.sub..alpha.I.sub.0.sup..alpha./.alpha..omega..sub.0, for
.alpha.>0. (33)
Using Eq. (33), Eq. (24) can be expressed as:
N ( .eta. , .rho. , .tau. + .DELTA..tau. ) = N ( .eta. , .rho.
, .tau. ) .times. exp ( 1 .DELTA..tau. .intg. .tau. .tau. + .DELTA..tau.
D 0 + .alpha. = 1 N .DELTA. D .alpha. I .alpha. ( .eta. , .rho.
, .tau. ' ) .tau. ' ) ( 34 )
[0120]An integration domain can be subdivided into M subranges
[.tau..sup.m,.tau..sup.m+1] with .tau..sup.m=.tau.+m.delta..tau.,
and a derivation similar to that described in Eqs. (23)(28) above
may be applied. FIG. 9 illustrates temporal partitions on a temporal
grid 900 that may be used for a subsampling calculation. For example,
dots 910 show temporal values where the intensity values may be
calculated, solid lines 920 show temporal values where the density
values may be calculated, and a dashed line 930 shows an exemplary
subsampling time on the temporal grid 900.
[0121]An exemplary derivation of the subsampling determination
described herein can be provided using only the squared intensity
term in Eq. (34) (e.g., .alpha.=2). The other exponential terms
can be evaluated in the similar manner. The squared intensity term
C2 may be written as Z:
C 2 ( for .alpha. = 2 ) = exp ( 1 .DELTA..tau. .intg. .tau. .tau.
+ .DELTA..tau. D 2 I 2 ( .eta. , .rho. , .tau. ) .tau. ' ) ( 35
) = exp ( 1 .DELTA..tau. m = 0 M  1 .intg. .tau. m .tau. m + 1
D 2 I 2 ( .eta. , .rho. , .tau. ) .tau. ' ) = m exp ( 1 .DELTA..tau.
D 2 .intg. .tau. m .tau. m + 1 I 2 ( .eta. , .rho. , .tau. ) .tau.
' ) ( 36 ) .apprxeq. m exp ( 1 .DELTA..tau. D 2 .DELTA..tau. M 1
2 { I 2 ( .eta.  .DELTA..eta. / 2 , .rho. , ( 37 ) .tau. m + .delta..tau.
/ 2 ) + I 2 ( .eta. + .DELTA..eta. / 2 , .rho. , .tau. m + .delta..tau.
/ 2 ) } ) .apprxeq. m exp [ 1 M D 2 ( .lamda. m 1 2 { I 2 ( .eta.
 .DELTA..eta. / 2 , .rho. , .tau. ) + ( 38 ) I 2 ( .eta. + .DELTA..eta.
/ 2 , .rho. , .tau. ) } + .mu. m 1 2 { I 2 ( .eta.  .DELTA..eta.
/ 2 , .rho. , .tau. + .DELTA..tau. ) + I 2 ( .eta. + .DELTA..eta.
/ 2 , .rho. , .tau. + .DELTA..tau. ) } ) ] .apprxeq. m exp [ 1 M
D 2 ( .lamda. m 1 2 { ( I ) 2 ( .eta.  .DELTA..eta. / 2 , .rho.
, ( 39 ) ( .tau. ) ) + ( I ) 2 ( .eta. + .DELTA..eta. / 2 , .rho.
, ( .tau. ) ) } + .mu. m 1 2 { ( I ) 2 ( .eta.  .DELTA..eta. /
2 , .rho. , ( .tau. + .DELTA..tau. ) ) + ( I ) 2 ( .eta. + .DELTA..eta.
/ 2 , .rho. , ( .tau. + .DELTA..tau. ) ) } ) ] .
[0122]In Eq (37) the intensity terms I.sup.2(*,*,.tau..sup.m+.delta..tau./2)
(where the asterisks `*` represent the variables .eta. and .rho.)
may be replaced by expressions for their corresponding linear interpolation
values .lamda..sub.mI.sup.2(*,*,.tau.)+.mu..sub.mI.sup.2(*,*,.tau.+.DELTA..tau.)
, where .lamda..sub.m=(1(m+1/2)/M) and .lamda..sub.m+.mu..sub.m=1,
to obtain the expression shown in Eq. (38). The expression in Eq.
(39) can be obtained by replacing the terms I.sup.2(*,*,.tau.) and
I.sup.2(*,*,.tau.+.DELTA..tau.) with their average approximations
(I).sup.2(*,*,(.tau.)) and (I).sup.2(*,*,(.tau.+.DELTA..tau.)) evaluated
at .tau.=(i+1/2).DELTA..tau. [e.g., I(*,*,.tau.+(i+1/2).DELTA..tau.)].
These average approximations can be written as:
( I ) 2 (* , * , ( .tau. ) ) .apprxeq. 1 2 { I 2 (* , * , .tau.
 .DELTA..tau. / 2 ) + I 2 (* , * , .tau. + .DELTA..tau. / 2 ) }
, ( 40 ) ( I ) 2 (* , * , ( .tau. + .DELTA..tau. ) ) .apprxeq. 1
2 { I 2 (* , * , .tau.  .DELTA..tau. / 2 ) + I 2 (* , * , .tau.
+ 3 .DELTA..tau. / 2 ) } . ( 41 )
The other terms in Eq. (34) may be evaluated in a similar manner
until the subsampling procedure is complete.
[0123]Using Eqs. (31) and (32), for each affected time sample,
a refined version of Eq. (29) may be written as
N n + 1 / 2 , j , i + 1 / 2 ( k ) .apprxeq. m = 0 M  1 exp { .DELTA..tau.
T 0 M [ D ^ 0 + .alpha. = 1 N .alpha. D ^ .alpha. I 0 .alpha. .alpha.
.omega. 0 .times. ( .lamda. m 1 2 { ( I .alpha. ) n , j , i _ +
( I .alpha. ) n + 1 , j , i _ ( k ) } + .mu. m 1 2 { ( I .alpha.
) n , j , i _ + 1 + ( I .alpha. ) n + 1 , j , i _ + 1 ( k ) } )
] } N n + 1 / 2 , j , i  1 / 2 ( k ) , ( 42 )
where M can be selected to ensure that the elements of the matrix
in the exponent of Eq. (29) are each smaller than a certain threshold
value. Eq. (42) is a general version of Eq. (39) above, which was
derived only for the squared intensity term C2 (e.g., .alpha.=2).
The threshold value .epsilon. may be selected such that 0<.epsilon.<1,
whereby the condition on M can then be written as:
M .ident. M ( n , j , i ) = min s 1 , s 2 { M ' , .DELTA..tau.
T 0 M ' .times. D ^ 0 [ s 1 , s 2 ] + I 0 .omega. 0 1 2 { I n ,
j , i + I n + 1 , j , i ( k ) } D ^ 1 [ s 1 , s 2 ] + I 0 2 2 .omega.
0 1 2 { ( I 2 ) n , j , i + ( I 2 ) n + 1 , j , i ( k ) } D ^ 2
[ s 1 , s 2 ] + I 0 3 3 .omega. 0 1 2 { ( I 3 ) n , j , i + ( I
3 ) n + 1 , j , i ( k ) } D ^ 3 [ s 1 , s 2 ] < } . ( 43 )
[0124]The analysis of nonlinear materials may often be guided by
measurements of their optical transmission. In conventional calculations,
the radial domain may often be assumed to be constant. In exemplary
embodiments of the present invention, both radial and temporal profiles
of the solution may be used to analyze and/or compare transmission
plots of a peak transmittance T.sub..delta. and a conventional integrated
transmittance T.sub.E. The peak transmittance and the integrated
transmittance may be provided by the expressions
T .delta. = .delta. E out ( .rho. * , .tau. * ) .delta. E i n (
.rho. * , .tau. * ) , s . t . ( .rho. * , .tau. * ) = arg max .rho.
, .tau. .delta. E out ( .rho. , .tau. ) , ( 44 ) T E = E out E i
n = T F .intg. 0 + .infin. .rho. ' 2 .pi..rho. ' .intg.  .infin.
+ .infin. .tau. ' I ( .eta. max , .rho. ' , .tau. ' ) 2 .pi. .intg.
0 + .infin. .rho. ' .rho. ' .intg.  .infin. + .infin. .tau. ' I
0  ( .tau. ' ) 2  ( .rho. ' ) 2 , ( 45 )
respectively, where .delta.E.sub.in,out(.rho.,.tau.).apprxeq.T.sub.F.pi.
{square root over (.pi.)}.delta..rho..sup.2.delta..tau.I(.eta..sub.min/max,.rho.,.tau.)
and T.sub.F can represent a cumulative Fresnel transmittance at
the interfaces.
[0125]The integrated value T.sub.E may be conventionally accepted
as a useful parameter for quantifying nonlinear materials because
it can be measured in a laboratory using readily available thermal
detectors, which can average a pulse intensity over both space and
time. The peak intensity can cause damage to optoelectronic detectors
and sensors. Therefore, it may be useful to obtain a numerical verification
of the validity of the laboratory measurements using thermal detectors.
Because a pulse distortion can occur in both the temporal and radial
domains, the computational and exemplary modeling procedures described
herein may also be used to search both radial and temporal domains
of a pulse to find a maximum value of the intensity which may be
used to determine T.sub..delta..
Electronic Level Contributions to Absorption
[0126]Specific contributions from each electronic level to the
total absorption may be estimated, if at all, using conventional
procedures based on the dynamics of the population densities of
electronic levels. However, relative contributions to the absorption
may not be closely correlated with corresponding relative population
densities. An exemplary system in which such a correlation is not
observed, for example, can be AF455 at high input intensities. An
estimation of electronic level contributions to the total absorption
which is based solely on the population densities of the levels
may therefore be inaccurate.
[0127]In exemplary embodiments of the present invention, specific
contributions from each electronic level to the total absorption
may be provided. For example, both population density values and
intensity absorption due to each electronic level may be obtained
for any propagation distance, radius, and time step. This can permit
an accurate calculation of the relative contributions to the total
absorption. To determine these values, a total intensity reduction
.LAMBDA..sub..zeta. may be defined for a specific grid index .zeta.={n+1/2,j,i1/2}
as a product of individual intensity reductions .LAMBDA..sub.s;.zeta.,
each due to energy levels s with nontrivial absorption crosssections.
This total intensity reduction may be written as:
.LAMBDA. = s .dielect cons. S .sigma. .LAMBDA. s ; ; ( 46 ) S
.sigma. = { s .sigma. 1 [ s ] + .sigma. 2 [ s ] + .sigma. 3 [ s
] > 0 } . ( 47 )
[0128]The individual reductions .LAMBDA..sub.s;.zeta. can be obtained
using Eq. (30) as corresponding exponential terms responsible for
intensity decrease at a final iteration, k=K, and may be written
as:
.LAMBDA. s ; .ident. .LAMBDA. s ; n + 1 / 2 , j , i  1 / 2 = exp
(  L df N .DELTA..eta. .beta. = 1 N B I 0 .beta.  1 .times. {
.sigma. .beta. [ s ] N s ; n + 1 / 2 , i  1 / 2 ( K ) + N s ; n
+ 1 / 2 , j , i + 1 / 2 ( K ) 2 } 1 2 { I n , j , i .beta.  1 +
I n + 1 , j , i ( K ) .beta.  1 } ) . ( 48 )
To determine relative absorption contributions, intensity decay
values may be provided which can have a form:
q.sub..zeta.=1.LAMBDA..sub..zeta., (49)
p.sub.s;.zeta.=1.LAMBDA..sub.s;.zeta.. (50)
The total intensity decay values q.sub..zeta. can be used to analyze
which part of a pulse predominantly decreases at a certain depth.
Further, the intensity decay associated with a level s, p.sub.s,.zeta.,
can determine a relative contribution {circumflex over (p)}.sub.s,.zeta.
through application of the following relationship:
p ^ s ; = p s ; / s ' .dielect cons. S .sigma. p s ' ; . ( 51
)
[0129]A nonlinear relationship between the relative contributions
{circumflex over (p)}.sub.s,.zeta. and the population densities
and intensities can indicate that approximating contributions from
specific electronic levels to the total absorption based on available
values of the population densities may not be accurate. The accuracy
of this conventional approximation can be assessed by generating
a plot of total intensity decays q.sub.{.zeta.}, and then superimposing
plots of absolute contributions q.sub.s;{.zeta.} derived from relative
contributions {circumflex over (p)}.sub.s,{.zeta.}.The absolute
contributions of the electronic levels to the total absorption can
be obtained by scaling them to the total intensity decays, e.g.,
using the following relationship:
q.sub.s;{.zeta.}={circumflex over (p)}.sub.s,{.zeta.}q.sub.{.zeta.}.
(52)
[0130]Results of the exemplary computational procedures for determining
absorption behavior described herein may be compared to transmittance
data measured in various nonlinear materials under a variety of
lasing conditions. Certain nonlinear materials can be selected for
comparison to provide a range of such generic materials described
herein. For example, a C.sub.60toluene solution as described, e.g.,
in the McLean et al. publication, may be representative of a typical
single photon absorbers, and it can exhibit reverse saturable absorption.
An AFX chromophore AF455 described, for example, in the Rogers et
al. publication, may represent a typical twophoton absorber. PPAI
dye can represent a 3PA material as described, e.g., in the Wang
et al. publication.
[0131]A comparison of theory with experiments can provide for specifying
a time(in)dependent shape of an incident pulse, (f(.rho.)),f(.rho.,.tau.).
The analytical and numerical techniques described herein can assume
that (f(.rho.)),f(.rho.,.tau.) can be represented by a Gaussian
function, which may be consistent with experimentallyobserved laser
pulse shapes. Using the assumption of a Gaussian pulse shape, the
incident laser intensity or electric field can be expressed as:
I(.eta.=0,.rho.,.tau.)=I.sub.0 exp(.tau..sup.2) exp(.rho..sup.2);
I(.eta.=0,.rho.)=I.sub.0 exp(.rho..sup.2),
or E(.eta.=0,.rho.,.tau.)=E.sub.0 exp(.tau..sup.2/2) exp(.rho..sup.2/2);
E(.eta.=0,.rho.)=E.sub.0 exp(.rho..sup.2/2) (53)
[0132]Individual energy level contributions to the total absorption
may be analyzed by averaging the relative and absolute contributions
expressed in Eqs. (51) and (52) within a portion of the pulse's
time duration. For example, the averaged relative contribution of
an sth level at (.eta..sub.n,.rho..sub.j) within a time range [.tau..sub.0,.tau..sub.1]
can be written as
p ^ s [ .tau. 0 , .tau. 1 ] = i , .tau. i .dielect cons. [ .tau.
0 , .tau. 1 ] p ^ s ; { n , j , i } . ( 54 )
EXAMPLE
C.sub.60Toluene Solution
[0133]The nonlinear material C.sub.60 can be described as a reverse
saturable absorbera material having an ESA cross section that
may be much higher than that of the ground state. The absorption
energy diagram can be expressed as a combination of transition diagrams,
e.g., B.sub.0.orgate.B.sub.2.orgate.B.sub.3. This absorption energy
diagram can be used to uniquely define the vectors and matrices
of the coefficients for the rate and propagation expressions provided
in Eqs. (15) and (16). For example, these vectors and matrices that
describe the absorption behavior of C.sub.60 can be written in the
following form:
D ^ 2 = D ^ 3 = ( 0 ) 5 .times. 5 . ( 55 )
[0134]Exemplary coefficients and experimental parameters that may
be used to describe several multiphotonabsorbing materials are
provided in Table 1. A theoretical absorption behavior of C.sub.60
can be determined using an iteration technique to solve Eqs. (30)
and (42) with K=2. The results of this exemplary procedure are presented
in FIG. 2a, which is an exemplary graph of energy transmittance
T.sub.E, shown in Eq. (45), as a function of input energy in C.sub.60.
Experimentally measured data 200 in FIG. 2a are indicated by .cndot.
(dot) symbols, the thin solid line 210 represents results obtained
using the computational technique described herein, and the thick
solid line 220 represents an original solution provided in the McLean
et al. publication. The measured transmittance data 200 presented
in FIG. 2a is likely wellrepresented by the computational results
210 within an input fluence range from about 3.6.times.10.sup.5
J/cm.sup.2 to about 5 J/cm.sup.2.
TABLEUS00001 TABLE 1 Parameters for exemplary multiphoton absorbing
materials. Material/experimental C.sub.60toluene AF455 parameters
solution.sup.a chromophore.sup.b PPAI dye.sup.c .sigma..sub.3PA
3.2 .times. 10.sup.21 (cm.sup.3/W.sup.2) .sigma..sub.TPA .sup.
0.5 .times. 10.sup.20d (cm.sup.4/GW) .sigma..sub.01 (cm.sup.2)
3.1 .times. 10.sup.18 .sigma..sub.12 (cm.sup.2) 1.6 .times. 10.sup.17
1.68 .times. 10.sup.17 .sigma..sub.34 (cm.sup.2) .sup. 1.4 .times.
10.sup.17e 17.1 .times. 10.sup.17 k.sub.10.sup.1 (ns) 32.5 2.72
k.sub.21.sup.1 (ps) 1.0 1.66.sup.f k.sub.13.sup.1 (ns) 1.35 45.3
k.sub.30.sup.1 (.mu.s) 40.0 0.368 k.sub.43.sup.1 (ps) 1.0 10.0.sup.g
Z.sub.max (mm) 1.0 1.0 10.0 L.sub.df 0.09 2.24 0.81 N.sub.T (nm.sup.3)
1.559 .times. 10.sup.3h.sup. 0.012.sup.i 0.596 .times. 10.sup.3k.sup.
k Energy levels B.sub.0.orgate.B.sub.1.orgate.B.sub.2 B.sub.1.orgate.B.sub.2.orgate.B.sub.3
B.sub.4 diagram R.sub.0 (.mu.m) 33.37.sup.m 13.01 53.03 T.sub.0
(ns) 4.8 1.92 2.1 .times. 10.sup.2 .lamda..sub.0 (nm) 532 800 1064
.sup.aMost parameters are provided in I. C. Khoo et al., "Nonlinearabsorbing
fiber array for largedynamicrange optical limiting application
against intense short laser pulses," J. Opt. Soc. Am. B 21,
1234 1240 (2004); experimental parameters are provided in the McLean
publication. .sup.bMaterial and measurement parameters are provided
in the experimental section of Sutherland et al. .sup.cParameters
are provided in the Wang publication. .sup.dParameter is provided
in the He publication and in the Kannan publication. .sup.eParameter
is provided in the Khoo publication. .sup.fParameter is provided
in the Rogers publication. .sup.gParameter is provided in the Kleinschmidt
publication. .sup.hThis value corresponds to a 2.59 mM solution
of C.sub.60 in toluene .sup.iThis value corresponds to 0.02M .sup.kThis
value corresponds to 0.99 mM of the dye in DMSO .sup.mAll laser
parameters are obtained from corresponding original parameters using
Eq. (53)
[0135]An analytical solution of Eqs. (15) and (16) such as that
described, e.g., in the McLean publication may be less accurate
in describing the evolution of the population densities at high
fluence inputs than the numerical technique described herein. For
example, the results of the numerical procedure 210 in FIG. 2a,
performed in accordance with certain exemplary embodiments of the
present invention, appear to more closely correlate with the experimental
values 200 than does the analytical solution 220 for fluence values
above about 1.4 J/cm.sup.2. Neither solution, however, appears to
match well with the measured data 200 above 5.0 J/cm.sup.2.
[0136]Values of the contributions of individual electronic states
to the total absorption of nanosecond pulses in C.sub.60 can be
provided in Table 2 below. These values were calculated using techniques
described herein in accordance with exemplary embodiments of the
present invention. The nonlinear transmittance of C.sub.60 in the
ns regime that is suggested by the data in Table 2 may be attributed
primarily to a variation in the lowest triplettriplet state absorption
from about 66% to 99.8%, where this value may depending on the input
energy value. The contribution values provided for C.sub.60 in Table
2 (and for AF455 in Table 3 below) can be averaged within the portions
of the pulse duration specified in Tables 2 and 3 using Eq. (54).
[0137]For example, FIGS. 3ac show exemplary graphs of determined
exemplary evolutions of population densities in electronic levels
N.sub.0 300, N.sub.1 310 and N.sub.3 320 in C.sub.60. FIGS. 3a3c
correspond to input fluence values, .PHI..sub.in, of 0.51. 2.05
and 14.1 J/cm.sup.2, respectively. The incident pulse intensity
370 is also shown as a function of time in these figures. FIGS.
3d3f show three exemplary graphs of determined exemplary individual
electronic level contributions q.sub.0 330, q.sub.1 340 and q.sub.3
350 to the absorption in C.sub.60, together with the total absorption
360. The exemplary conditions used to generate FIGS. 3d3f can correspond
to the conditions in FIGS. 3a3c, respectively. The values can be
determined at the entrance of the slab (e.g., .eta.=0) at the pulse
center .rho.=0.
[0138]In addition to the absorption by the triplettriplet state
N.sub.3, the ground level can contribute approximately 23% to the
total absorption at low input pulse intensities. This contribution
may be lost at higher intensities because of a fast bleaching of
the ground level. This may account for the observed agreement between
calculations and experiments for input fluence values less than
about 1.4 J/cm.sup.2, reasonable agreement below about 5.0 J/cm.sup.2,
and a poor agreement of both methods with experimental values above
about 5.0 J/cm.sup.2. Material degradation at high input intensities
may also be present, and can lead to a divergence of numerical solutions,
based on integration of the ratepropagation equations, from the
measured data.
TABLEUS00002 TABLE 2 Individual contributions of electronic states
of C.sub.60 to the total absorption of nanosecond pulses Levels
.PHI..sub.in = 0.51.sup.a .PHI..sub.in = 2.05 .PHI..sub.in = 14.1
<{circumflex over (p)}.sub.0>.sup.b 79.2 23.1 10.4 50.3 4.1
0.4 15.9 0.0 0.0 <{circumflex over (p)}.sub.1> 6.7 10.3 1.7
15.2 8.0 0.3 19.0 0.6 0.0 <{circumflex over (p)}.sub.3> 14.0
66.2 87.4 41.0 87.7 99.2 64.9 99.3 99.9 <q>.sup.c 0.06 0.13
0.17 0.08 0.19 0.21 0.14 0.22 0.22 .sup.aFor each specified input
fluence value .PHI..sub.in (provided in J/cm.sup.2), three values
are provided (in the left, middle, and right subcolumns) which
correspond to the averaged relative contributions within the beginning,
middle, and ending portions of pulse, e.g., <*>.sub.[1,1/3),
<*>.sub.[1/3,1/3), <*>.sub.[1/3,1] as provided in Eq.
(54). .sup.bAveraged relative contribution to the absorption for
an energy level 0, provided in Eq. (54), expressed as a percentage
of the total absorption. .sup.cAveraged intensity decay values obtained
by integrating Eq. (49) using a technique similar to that used to
integrate Eq. (54).
EXAMPLE
AF455
[0139]A theoretical exemplary basis for nonlinear transmittance
in the ns regime of D.pi.A chromophore AF455a material which
exhibits twophoton assisted ESAtogether with experimental transmittance
results is provided in the Sutherland et al. publication. The energy
levels diagram for this material can be represented as a TPA by
combining single photon absorption transition diagrams 110130 shown
in FIG. 1, e.g., B.sub.1.orgate.B.sub.2.orgate.B.sub.3, and using
the following parameters:
.sigma. 1 = [ 0 , .sigma. S , .sigma. T , 0 , 0 ] , .sigma. 2 =
[ .sigma. TPA , 0 , 0 , 0 , 0 ] , .sigma. 3 = 0 D ^ 1 = ( 0 0 0
0 0 0  .sigma. S 0 0 0 0 0  .sigma. T 0 0 0 0 .sigma. T 0 0 0
.sigma. S 0 0 0 ) , D ^ 2 = (  .sigma. TPA 0 0 0 0 .sigma. TPA
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) , D ^ 3 = ( 0 ) 5 .times.
5 , ( 56 )
where the matrix {circumflex over (D)}.sub.0 in Eq. (55) associated
with C.sub.60 may also be used for this material. Calculated transmittance
values 240 in AF455 are shown in FIG. 2b, together with experimental
values 230 provided in the Sutherland publication (summarized in
Table 1). The determined values 240, which were obtained using the
exemplary embodiments of the present invention described herein,
show agreement with the measured data 230 and with an analytical
solution 250 that is also provided in the Sutherland et al. publication.
[0140]FIGS. 4a4f show exemplary graphs of absorption in AF455
of a ns scale pulse. FIGS. 4a4c show exemplary determined evolutions
of population densities in electronic levels N.sub.0 400, N.sub.1
410, N.sub.2 420 and N.sub.3 430 in AF455. FIGS. 4a4c illustrate
input energy values, E.sub.in, of 17 .mu.J, 93 .mu.J, and 0.33 mJ,
respectively. The incident pulse intensity 480 is also shown as
a function of time in these figures. FIGS. 4d4f show exemplary
graphs of determined individual electronic level contributions q.sub.0
440, q.sub.1 450 and q.sub.3 460 to the absorption in AF455, together
with the total absorption 470. The conditions used to generate FIGS.
4d4f correspond to the conditions of FIGS. 4a4c, respectively.
The exemplary values are determined at the entrance of the slab
(e.g., .eta.=0) at the pulse center .pi.=0.
[0141]The population density graphs shown in FIGS. 4a4b suggest
a very small depletion of the ground level. This observation supports
the validity of a steadystate approximation used by Sutherland
et al. for the population densities in a ns pulse regime, and can
account for the agreement between the numerical solution obtained
using the techniques described herein and their analytical solution.
A steadystate assumption may not be appropriate for higher input
energies, because there may be a considerable increase of the population
densities N.sub.1 410 and N.sub.3 430 as shown in FIG. 4c for a
pulse duration in the ns regime. For a pulse duration in the fs
regime, the ground state 80 may be depleted rapidly, as shown in
FIG. 8a. Therefore, the exemplary embodiments of the present invention
described herein may be more accurate for higher input intensities
as they can encompass the dynamics of population densities, and
do not require the use of a steady state approximation.
[0142]An analysis of absolute contributions from different electronic
levels as shown, e.g., in the exemplary graphs in FIGS. 4d4f, can
provide certain information relating to absorption behavior within
an AF455 sample for ns pulses. For example, most absorption can
be seen to occur during the latter half of the propagating pulse,
and can result in advancing the leading edge of a pulse as it propagations
through a sample. For relatively small input energies, the ground
level can be a main contributor to the total absorption, where approximately
61% of the average contribution to absorption can occur during the
latter portion of the pulse. Details of this exemplary effect are
provided, e.g., in Table 2 which provides the individual contributions
averaged over the leading, central, and trailing parts of the propagating
pulse at .eta.=0 for the three indicated input energy values.
[0143]Table 3 herein provides exemplary data for contributions
of individual states to the absorption of nanosecond pulses in AF455.
Contribution of the ground state in TPA materials such as AF455
for highenergy pulses may falls abruptly as shown, for example,
in columns 2 and 3 of Table 3. The main contributors to absorption
can be ESA of the lowest singletsinglet and the lowest triplettriplet
states with corresponding individual contributions of approximately
53.1% and approximately 74.7% for the central and latter portions
of a pulse, respectively. (A contribution of 77.2% in the {circumflex
over (p)}.sub.0 level due to TPA in the early portion of a pulse
may be less important because the associated absolute intensity
decay value q can be 10 times less than corresponding values in
other portions of the pulse).
TABLEUS00003 TABLE 3 Individual contributions to the total absorption
of nanosecond pulses by the electronic states of AF455. Levels E.sub.in
= 17.0 .mu.J E.sub.in = 93.0 .mu.J E.sub.in = 330.0 .mu.J <{circumflex
over (p)}.sub.0> 94.7 61.2 13.5 77.2 23.7 2.9 49.2 8.8 0.9 <{circumflex
over (p)}.sub.1> 4.5 25.6 19.5 20.0 53.1 22.3 45.6 68.5 23.0
<{circumflex over (p)}.sub.3> 6.9 13.2 66.9 2.8 23.2 74.7
5.2 22.3 76.0 <q> 8.8e5 8.8e4 6.4e4 5.8e4 0.01 0.01 3.0e3
0.08 0.12
EXAMPLE
PPAI
[0144]With the advent of lasers capable of providing shorter pulses
and higher intensities, materials that exhibit three or more photon
absorption processes can be studied. These materials may be of interest
in areas such as laser micro or nanomachining because of their
ability to reduce the feature size of structures in substrate materials.
In contrast to the nonlinear materials described herein, there may
have been relatively little prior investigation of these materials
including, e.g., a 3PA dye PPAI described in the Wang publication.
Experimental investigations of this material indicate a high threephoton
absorption crosssection in the near infrared region, so that all
the parametric vectors and matrices may be zero or negligibly small,
except for the following two:
.sigma. 3 = [ .sigma. 3 PA , 0 ] , D ^ 3 = (  .sigma. 3 PA 0 .sigma.
3 PA 0 ) . ( 57 )
[0145]The energy level absorption diagram for PPAI may include
only a single building block B.sub.4 140 shown in FIG. 1, where
the appropriate parameters are provided in Table 1. FIG. 2c shows
a numerical solution 270 of the absorption behavior of PPAI, obtained
using techniques in accordance with exemplary embodiments of the
present invention. The solution 270 shows agreement with experimental
transmittance measurements 260 provided in the Wang publication.
Moreover, the present solution 270 appears to provide a better representation
of the experimental data 260 in the high input intensity region
than does the analytic solution 280 provided in the Wang publication.
[0146]FIGS. 5a and 5b show two exemplary graphs of absorption in
PPAI of a ps scale pulse having an incident intensity value, I.sub.in,
of 16.9 GW/cm.sup.2 and 204.5 GW/cm.sup.2, respectively. The evolution
of the electronic level population densities levels N.sub.0 500
and N.sub.1 510, are shown at the entrance of the slab, .eta.=0,
and at .rho.=0. The incident pulse intensity 520 is also shown in
these graphs.
[0147]The analytical model described in the Wang publication for
absorption ion PPAI includes the assumption that there is a constant
population density of the ground level. This assumption may be reasonable
for a low intensity pulse, which may be suggested in the graph of
FIG. 5a. However, this assumption may not be reasonable for a high
intensity pulse, as suggested by the timeresolved population density
solution 500, 510 depicted in FIG. 5b. This solution shows that
the population of the N.sub.0 state 500 may decrease to less than
half the initial value in the latter portion of the pulse. Dynamics
of the population density of the N.sub.1 state 510 shown in FIG.
5b also suggests that excited state absorption may significantly
affect the transmittance at high input intensities.
RadiallyDependent Solutions
[0148]The exemplary numerical procedures described herein can include
a radial dependence of various parameters, which can provide significant
insight into absorption behavior of materials. Many conventional
determinations can neglect a radial dependence, and may only approximate
absorption behavior, e.g., by a single averaged value of the various
parameters that may be a function, e.g., only of time.
[0149]FIGS. 6ad show exemplary graphs of numerical solutions of
evolution of a pulse intensity as a function of radius at .tau.=0
and at different depths .eta.={0.00, 0.25, 0.50, 0.75, 1.00} for
different MPA samples. For example, FIG. 6a shows an exemplary radial
dependence of absorption behavior of a ns scale pulse in C.sub.60
with an input fluence .PHI..sub.in=2.05 J/cm.sup.2. FIG. 6b shows
an exemplary radial dependence of absorption behavior of a ns scale
pulse in AF455 with an input energy E.sub.in=131 .mu.J. FIG. 6c
shows an exemplary radial dependence of absorption behavior of a
ps scale pulse in PPAI for an incident energy intensity value I.sub.in=204.5
GW/cm.sup.2. FIG. 6d shows an exemplary radial dependence of absorption
behavior of a fs scale pulse in AF455 with an input energy E.sub.in=6.6
.mu.J, R.sub.0=7.07 .mu.m, and T.sub.0=204.0 fs;
[0150]For MPA materials, e.g., AF455 and PPAI, not accounting for
radial dependence of parameters in mathematical models when calculating
absorption behavior may be a reasonable simplification. For example,
the exemplary graph of pulse intensity in PPAI, shown in FIG. 6c,
may be approximately flat soon after a pulse enters a sample (e.g.,
.eta.>0.25). An initially Gaussianshaped pulse in AF455 may
gradually relax towards a plateau shape after propagation through
the second half of the sample (e.g., .eta.>0.5), as shown in
FIGS. 6b and 6d. However, the Gaussian shape of an incident pulse
propagating through the C.sub.60 sample, shown in FIG. 6a, may likely
not relax significantly in a radial direction until it reaches the
end of the material. Therefore, assuming radial invariance of absorption
behavior may not be an accurate assumption in certain materials
such as C.sub.60. Because the determinations that include a radial
dependence may be time consuming, it may be desirable to determine
if radial variations may play a significant role in absorption under
certain conditions. If radial variations are not important, then
simpler and less time consuming determinations can be performed
using a procedure that assumes radial invariance which may still
provide sufficiently accurate results.
[0151]There may be additional reasons to include radial dependence
of parameters in absorption calculation techniques. The exemplary
techniques described herein include determinations of near field
properties of the laser beam pulse. However, certain detectors may
conventionally be placed at a far field where radial distortions
may cause problems. Such distortions may not be adequately described
by conventional theories which neglect a radial dependence of the
laser beam pulse. Based on the results shown, e.g., in FIGS. 2a2c,
a radial dependence of the beam pulse may not be significant for
the experiments described herein. However, radial distortions may
become more prominent in newer materials, and therefore it may be
necessary to retain the radial dependence. Therefore, the radial
dependence of the laser beam can be included in the exemplary determinations
described herein. This can allow for the determination of radial
distortions at a far field using a HuygensFresnel principle as
described, e.g., in P. W. Milonni et al., Lasers, John Wiley, New
York, 1988.
[0152]The radial distortion may also affect any selffocusing/defocusing
that can arise if a material has a large nonlinear index of a refraction
or if the input intensity is sufficiently large to induce such an
effect. The results shown, e.g., in FIGS. 2a2c, indicate that self(de)focusing
may have little significance in the experiments described here.
However, if more powerful lasers are used, it may be important to
include the radial dependence of the beam characteristics to obtain
accurate predictions of absorption behavior.
[0153]Further exemplary embodiments of the present invention may
be used to describe absorption of a plurality of coincident pulses
(e.g., a "pulse train"). For example, an intensity or
electric field of an incident pulse train can be described by the
following equation:
I ( 0 , r , t ) = n = 0 N I n f ( r , t  nt r ) , Or E ( 0 , r
, t ) = n = 0 N E n f ( r , t  nt r ) , ( 58 )
where I.sub.n or E.sub.n is an initial peak intensity or peak electric
field of an nth pulse, t.sub.r is a pulse separation, N is a number
of pulses in the pulse train, and f(r,tnt.sub.r) describes a shape
of the incident pulse.
[0154]A pulse having a Gaussian shaped may be expressed by the
function
I(0,r,t)=I.sub.0 exp(t.sup.2/T.sub.0.sup.2) exp(r.sup.2/R.sub.0.sup.2),
or E(0,r,t)=E.sub.0 exp(t.sup.2/2T.sub.0.sup.2)exp(r.sup.2/2R.sub.0.sup.2)
(59)
where I.sub.0 or E.sub.0 is a peak intensity or peak electric field,
R.sub.0 is a 1/e pulse radius, and T.sub.0 is a 1/e temporal pulse
halfwidth. The input intensity corresponding to N pulses may be
expressed as
I ( 0 , r , t ) = n = 0 N I n exp [  ( t  nt r ) 2 / T 0 2 )
exp [  r 2 / R 0 2 ] , or E ( 0 , r , t ) = n = 0 N E n exp [ 
( t  nt r ) 2 / 2 T 0 2 ) exp [  r 2 / 2 R 0 2 ] . ( 60 )
This equation may be used instead of the expression provided, e.g.,
in Eq. (53) herein to determine the absorption effects resulting
from a series of incident pulses.
Ultrashort Pulses
[0155]Unlike conventional numerical or analytical procedures, the
exemplary procedures according to the present invention described
herein can permit investigation of a variety of nonlinear absorption
phenomena, including situations that may not have been measured
experimentally such as, e.g., absorption of shortpulsed lasers.
For example, certain exemplary embodiments of the present invention
may be used to analyze interactions of ultrashort pulses with nonlinear
materials such as, e.g., interactions of fs range pulses with AF455,
a system that may not yet have been studied experimentally.
[0156]An ultrashort pulse may be characterized, e.g., by a Gaussian
intensity distribution and parameters R.sub.0=7.07 .mu.m, T.sub.0=204.1
fs, and .lamda..sub.0=800 nm. \ Using these parameters and the data
for AF455 provided in Table 1 herein, a transmittance of approximately
0.002 was obtained at the exit of an AF455 slab (z.sub.max=1.5L.sub.df.apprxeq.0.412
mm) for an input energy of approximately 100 .mu.J. FIG. 7 shows
a graph of determined integrated transmittance 700 and peak transmittance
710 over a range of input energy values. The transmittance value
of 0.002 for a fs scale pulse can be almost two orders of magnitude
smaller than that obtained by using ns pulses. By increasing the
slab thickness, an even lower transmittance may be achieved hypothetically,
although possibly not with a sensitivity as strong as that of diffraction
with respect to a longer sample thickness. Thus, AF455 may represent
a desirable candidate for a high intensity nonlinear absorbing material.
Energy Level Population Dynamics
[0157]Still further exemplary embodiments of the present invention
may be used to examine population dynamics and the contribution
of individual energy levels to the total absorption. For example,
FIG. 8a illustrates the evolution of electron population densities
in levels N.sub.0 800, N.sub.1 810, and N.sub.2 820 at a sample
entrance for an incident energy of 6.6 .mu.J. The ground level of
a material exposed to such a high energy pulse can be quickly depleted
and may not repopulate because of slow decay rates k.sub.1,0 and
k.sub.3,0. The averaged values of ground level excitation contributions
to the total absorption are calculated as approximately 95% within
the first portion of pulse, approximately 48% in the second portion
of the pulse, less than 0.01% in the final portion. The depopulation
of the ground level can increase the electron density of the singletexcited
state N.sub.1 followed by fast excitation to N.sub.3, so that in
the second half of the pulse the contribution of this state to the
total absorption can be approximately 51%.
[0158]FIG. 8b shows a graph of the contributions of q.sub.0 830
and q.sub.1 840 to the total absorption 850 in AF455. The relative
magnitudes of q.sub.0 830 and q.sub.1 840 in FIG. 8b may suggest
that TPA can be more important than ESA for this material. For example,
the averaged total absorption q 850 which may be expressed, as e.g.,
in Eq. (49), in a first portion of a pulse may be approximately
0.22, and in a second portion of the pulse it may be approximately
0.08. Thus, AF455 may be primarily a TPA material for pulses in
the fs regime, whereas it may exhibit primarily TPAassisted ESA
behavior for pulses in the ns regime as shown, for example, in Table
3. Because an intersystem crossing time 1/k.sub.1,3 may be on the
order of nanoseconds, the lowest triplettriplet state N.sub.3 may
not contribute significantly to the total absorption. For example,
the individual contribution the lowest triplettriplet state N.sub.3
can be less than 10.sup.4% and is thus not shown in FIG. 8b. The
numerical determinations of the absorption behavior of AF455 indicate,
e.g., that the maximum values of N.sub.3 and N.sub.4 can be approximately
10.sup.6.
[0159]The peak transmittance, shown in Eq. (44), can be below the
integrated transmittance for all input energy values, as shown in
FIG. 7. Therefore, the peak intensity of the transmitted pulse may
not cause any damage to a detector placed behind an AF455based
limiter.
[0160]For example, direct measurements of the nonlinear Kerr coefficient
have not been obtained for chromophores such as AF455. At high incident
intensities, the Kerr nonlinearity may give rise to selfphase modulation,
which in the presence of diffraction may lead to self(de)focusing.
In the presence of the dispersion, this nonlinearity may lead to
a temporal pulse reshaping. If these phenomena are significant for
a particular nonlinear material under certain conditions, then their
effects can be accounted for, e.g., using techniques described in
the Potasek publications and the Kovsh publication.
Stimulated Emission
[0161]Further exemplary embodiments of the present invention may
be used, e.g., to describe stimulated emission effects from a twophoton
absorption (TPA) state with nonlinear excitedstate absorption (ESA).
The emission may be to a conduction band or it may be a free electron
excitation. An exemplary material that can exhibit stimulated emission
effects and which may be described using exemplary embodiments of
the present invention is a Green Fluorescent Protein (GFP) as described,
e.g., in S. Kirkpatrick et. al., "Nonlinear Saturation and
Determination of the TwoPhoton Absorption Cross Section of Green
Fluorescent Protein", J. Phys. Chem. B 2001, 2867 (2001).
[0162]For example, an exemplary electronic configuration of a material
is shown in FIG. 10 which includes certain exemplary states: e.g.,
a ground state manifold 1000, a first excited state manifold 1010,
which may be reached through a TPA event and can be referred to
as a TPAstate, and a second excited state 1020, which can be reached
via a one photon absorption event from the first excited state manifold
1010. The second excited state 1020 can be referred to as an ESAstate,
and it may be a conduction band or a free electron excitation. A
stimulated emission can occur between the first excited state 1010
and the ground state 1000. A spontaneous emission can occurs between
the second excited state 1020 and the first excited state 1010,
or between the second excited state 1020 and the ground state 1000.
[0163]Parameters k.sub.10 1030 and k.sub.21 1040 shown in FIG.
10 can represent decay rates of the TPAstate 1010 and the ESAstate
1020, respectively, and the parameter k.sub.20 1050 can represent
a decay rate from second excited state 1020 to the ground state
1000.
[0164]The corresponding matrices that may be used in the rate equation
provided in Eq. (15) can be written as:
D ^ 0 = [ 0 k 10 k 20 0  k 10 k 21 0 0  ( k 20 + k 21 ) ] ; D
^ 1 = [ 0 0 0 0  .sigma. ESA 0 0 .sigma. ESA 0 ] ; D ^ 2 = ( 
g 10 .sigma. TPA .sigma. TPA 0 g 10 .sigma. TPA  .sigma. TPA 0
0 0 0 ) . ( 61 )
[0165]The parameter g.sup.10=g.sup.1/g.sup.0 may also be used,
where g.sup.1 and g.sup.0 can represent an electron degeneracy of
electronic levels 1 and 0, respectively. In the present example,
all other matrices: {circumflex over (D)}.sub.N.sub.A(.sigma..sub.[N.sub.A.sub.]PA)=0;
.sigma..sub.1=(0,.sigma..sub.ESA,0), .sigma..sub.2=(g.sup.10.sigma..sub.TPA,.sigma..sub.TPA,0);
and all other vectors .sigma..sub.N.sub.B(.sigma..sub.[N.sub.B.sub.]PA)=0.
[0166]A phenomenological propagation equation for the intensity
I, similar to Eq. (16), can be expressed as:
I ~ z =  .sigma. TPA I ~ 2 ( N ~ 1  g 10 N ~ 0 )  .sigma. ESA
I ~ N ~ 1 . ( 62 )
Using a change of variables as in Eq. (18), leads to a following
form of the propagation equation:
I .eta. =  .sigma. TPA I 2 ( N 1  g 10 N 0 )  .sigma. ESA IN
1 . ( 63 )
Diffraction Effects
[0167]The exemplary embodiments of the present invention described
herein may not necessarily account for effects of diffraction during
multiphoton absorption in nonlinear materials. Diffraction effects
can be accounted for by modifying the rate and propagation expressions
provided in Eqs. (15) and (16). These propagation expressions are
expressed in terms of intensity. However, it may be preferable for
certain interactions to express the propagation equations in terms
of a corresponding electric field. Both real and imaginary portions
of the electric field can be calculated as described, e.g., in the
Potasek publications.
[0168]It can be convenient to express the electric field in terms
of a normalized function:
{tilde over (E)}(z,r,t)={tilde over (Q)}(z,r,t)Q.sub.0', Q.sub.0.sup.2.ident..epsilon..sub.0Q.sub.0'.sup.2.
(64)
Using the transformations provided in Eq. (17), the electric field
can be written in dimensionless parameters:
E(.eta.,.rho.,.tau.)=Q(.eta.,.rho.,.tau.)Q.sub.o'. (65)
[0169]To include effects of diffraction in the model, the intensity
can be expressed in terms of a classical complex electric field,
E.sub.c, provided in Eq. (2), using the expression:
=.epsilon..sub.0nc.sub.0E.sub.c.sup.2. (66)
The exemplary system of equations can be made dimensionless using
the transformations provided herein to obtain Eq. (17), together
with the following relationship:
E c , E c = 2 Q 0 '2 Q , Q = 2 0 Q o 2 Q , Q , ( 67 )
which can lead to the following expression for the field intensity:
(.eta.,.rho.,.tau.)=2nc.sub.0Q.sub.0.sup.2Q(.eta.,.rho.,.tau.),Q(.eta.,.rh
o.,.tau.).ident.2nc.sub.0Q.sub.0.sup.2 Q(.eta.,.rho.,.tau.), (68)
with
Q.ident.Q,Q. (69)
[0170]Using the transformations described above, the rate equation
corresponding to Eq. (15) can be expressed as:
N ( .eta. , .rho. , .tau. ) .tau. = T 0 [ D ^ 0 + .alpha. = 1 N
A D ^ .alpha. .alpha. .omega. 0 ( 2 nc 0 Q 0 2 ) .alpha. Q _ .alpha.
( .eta. , .rho. , .tau. ) ] N ( .eta. , .rho. , .tau. ) . ( 70 )
Eq. (70) can be expressed in a more compact form using the following
substitutions:
D 0 .ident. T 0 D ^ 0 , D .alpha. .ident. T 0 D ^ .alpha. .alpha.
.omega. 0 ( 2 nc 0 Q 0 2 ) .alpha. . ( 71 )
which can lead to the following equation:
N ( .eta. , .rho. , .tau. ) .tau. = [ D 0 + .alpha. = 1 N A D .alpha.
Q _ .alpha. ( .eta. , .rho. , .tau. ) ] N ( .eta. , .rho. , .tau.
) , ( 72 )
[0171]A rate operator may be introduced as
( .eta. , .tau. ) .ident. 0 + .alpha. = 1 N A .alpha. Q _ ( .eta.
, .tau. ) , ( 73 )
with .sub.0.ident.D.sub.0 and .sub..alpha..sup. Q(.eta.,.tau.).ident.D.sub..alpha.
Q.sup..alpha.(.eta.,.rho.,.tau.), which can lead to the following
form of the rate equation:
N ( .eta. , .rho. , .tau. ) .tau. = ( .eta. , .tau. ) N ( .eta.
, .rho. , .tau. ) , ( 74 )
Using the transformation provided in Eq. (68), a dimensionless
propagation equation corresponding to Eq. (16) which does not contain
a term to account for diffraction effects can be written as:
Q ( .eta. , .rho. , .tau. ) .tau. = {  L df N [ .beta. = 1 N B
( .sigma. ~ .beta. N ( .eta. , .rho. , .tau. ) ) ( 2 nc 0 Q 0 2
) .beta.  1 Q _ .beta.  1 ( .eta. , .rho. , .tau. ) ]  c ~ L
df } Q ( .eta. , .rho. , .tau. ) , ( 75 )
or, equivalently, as:
Q ( .eta. , .rho. , .tau. ) .tau. = {  .beta. = 1 N B ( .sigma.
.beta. N ( .eta. , .rho. , .tau. ) ) Q _ .beta.  1 ( .eta. , .rho.
, .tau. )  c } Q ( .eta. , .rho. , .tau. ) , ( 76 )
by using the substitutions .sigma..sub..beta..ident.L.sub.dfN(2nc.sub.0Q.sub.0.sup.2).sup..beta.1{t
ilde over (.sigma.)}.sub..beta. and c.ident.{tilde over (c)}L.sub.df.
[0172]Eq. (76) may be rewritten in an operator form by introducing
absorption operators .PHI..sub..beta..sup.Q, .PHI..sub..beta..sup.N
and .PHI..sub.L to account for absorption due to the intensity,
electronic density, and the linear absorptions, respectively, where
.PHI..sub..beta..sup.N(.eta.,.tau.).ident..sigma..sub..beta.N(.eta.,.rho.
,.tau.), .PHI..sub..beta..sup.Q(.eta.,.tau.).ident. Q.sup..beta.1(.eta.,.rho.,.tau.),
and .PHI..sub.L.ident.c. The resulting equations may be expressed
as:
.PHI. ( .eta. , .tau. ) .ident.  .beta. = 1 N B ( .sigma. .beta.
N ( .eta. , .rho. , .tau. ) ) Q _ .beta.  1 ( .eta. , .rho. , .tau.
)  c =  .beta. = 1 N B .PHI. .beta. N ( .eta. , .tau. ) .PHI.
.beta. Q ( .eta. , .tau. )  .PHI. L , and ( 77 ) Q ( .eta. , .rho.
, .tau. ) .eta. = .PHI. ( .eta. , .tau. ) Q ( .eta. , .rho. , .tau.
) . ( 78 )
[0173]An absorption operator that accounts for diffraction effects
(e.g., a diffraction operator) can be written as:
.PSI. df ( .rho. , .tau. ) .ident. i 4 ( 1  ib .differential.
.differential. .tau. ) .gradient. .rho. 2 , with ( 79 ) b .ident.
1 .omega. 0 T 0 . ( 80 )
A Fourier transform of this operator can have the following form:
.PSI. df ( .rho. , .omega. ) .ident. i 4 ( 1  b .omega. ) .gradient.
.rho. 2 , F ~ ( .differential. n .differential. .omega. n ) .ident.
(  .omega. ) n . ( 81 )
[0174]By adding together the nonlinear and linear absorption terms,
the propagation equation can be expressed in terms of the electric
field as:
Q ( .eta. , .rho. , .tau. ) .eta. = { .PHI. ( .eta. , .tau. ) +
.PSI. ( .rho. , .tau. ) } Q ( .eta. , .rho. , .tau. ) , ( 82 )
This propagation equation includes complex values of Q(.eta.,.rho.,.tau.),
whereas the rate equation provided in Eq. (74) can be expressed
in terms of real vectors N(.eta.,.rho.,.tau.) and real values of
Q(.eta.,.rho.,.tau.).
[0175]A numerical scheme to solve the rate equation in Eq. (74)
is described herein. An iteration formula that may be used to solve
this equation numerically can be written as:
N n + 1 / 2 , j , i + 1 / 2 .apprxeq. exp ( .delta. D 0 + .alpha.
= 1 N A .delta. D .alpha. 1 2 { Q _ n , j , i .alpha. + Q _ n +
1 , j , i .alpha. } ) N n + 1 / 2 , j , i  1 / 2 , ( 83 )
where .delta.D.sub..alpha..ident..DELTA..tau.D.sub..alpha., .alpha..epsilon.[0
. . . N.sub.A]. This equation can be expressed in operator form
as:
N _ _ i + 1 / 2 n + 1 / 2 .apprxeq. .DELTA..tau. Q _ [ n ] N _
_ i  1 / 2 n + 1 / 2 , or N _ _ i + 1 / 2 n + 1 / 2 .apprxeq. exp
( .DELTA..tau. { 0 + .alpha. = 1 N A .alpha. Q _ [ n ] } ) N _ _
i  1 / 2 n + 1 / 2 , ( 84 )
with N.sub.i+1/2.sup.n+1/2.ident.[N.sub.n+1/2,0,i1/2, . . . ,
N.sub.n+1/2,j,i1/2, . . . , N.sub.n+1/2,N,i1/2] and N.sub.n+1/2,j,i1/2.ident.N(.eta..sub.n+1/2,.rho..sub.j,.tau..sub.i1/2),
where expressions for .eta..sub.n+1/2,.rho..sub.j, and .tau..sub.i1/2
are provided in Eq. (21). The parameter
.alpha. Q _ [ n ] .ident. D .alpha. 1 2 { Q _ n , j , i .alpha.
+ Q _ n + 1 , j , i .alpha. }
may be considered as a discretization of .sub..alpha..sup. Q(.eta.,.tau.)
and can be used in the expression for the rate operator provided
in Eq. (73), with Q.sub.n,j,i.ident. Q(.eta..sub.n,.rho..sub.j,.tau..sub.i),
where expressions for .eta..sub.n and .tau..sub.i are provided in
Eq. (22).
[0176]A general solution for the propagation equation provided
in Eq. (82) can be expressed as:
Q ( .eta. + .DELTA. .eta. ) = exp { .intg. .eta. .eta. + .DELTA..eta.
( .PHI. ( .eta. ' , .tau. ) + .PSI. ( .rho. , .tau. ) ) .eta. '
} Q ( .eta. ) = exp { .intg. .eta. .eta. + .DELTA..eta. .PHI. (
.eta. ' , .tau. ) .eta. ' + .PSI. ( .rho. , .tau. ) .DELTA..eta.
} Q ( .eta. ) . ( 85 )
[0177]A symmetric splitstep (Fourier) technique may be used to
solve this propagation equation. For example, a value at a subsequent
depth sample can be determined from the value at a current depth
sample by first applying a diffraction operator along .DELTA..eta.
while neglecting a rate operator, and then applying the rate operator
to the resulting Q value while neglecting the diffraction operator.
Mathematically, this can be equivalent to a commuting of the rate
and the diffraction operators, which can be expressed as:
Q ( .eta. + .DELTA. .eta. ) .apprxeq. exp ( .intg. .eta. .eta.
+ .DELTA..eta. .PHI. ( .eta. ' , .tau. ) .eta. ' ) .DELTA..eta..PSI.
( .rho. , .tau. ) Q ( .eta. ) . ( 86 )
[0178]Such operators may not generally commute, and an error introduced
by this approximation can be on the order of .DELTA..eta.. To decrease
the computational error, a symmetric version of the splitstep procedure
can be used, which may be described by the following equation:
Q ( .eta. + .DELTA. .eta. ) .apprxeq. .DELTA..eta. 2 .PSI. ( .rho.
, .tau. ) exp ( .intg. .eta. .eta. + .DELTA..eta. .PHI. ( .eta.
' , .tau. ) n ' ) .DELTA..eta. 2 .PSI. ( .rho. , .tau. ) Q ( .eta.
) . ( 87 )
[0179]An expression for the raterelated operator can be derived,
e.g., using the following approximation:
exp ( .intg. .eta. .eta. + .DELTA..eta. .PHI. ( .eta. ' , .tau.
) .eta. ' ) = exp ( .intg. .eta. .eta. + .DELTA..eta. .eta. ' {
 .beta. = 1 N B .PHI. .beta. N ( .eta. ' , .tau. ) .PHI. .beta.
Q ( .eta. ' , .tau. )  .PHI. L } ) = exp (  .PHI. L .DELTA..eta.
) .beta. = 1 N B exp ( .intg. .eta. .eta. + .DELTA..eta. .PHI. .beta.
N ( .eta. ' , .tau. ) .PHI. .beta. Q ( .eta. ' , .tau. ) .eta. '
) .apprxeq. exp (  .PHI. L .DELTA..eta. ) .beta. = 1 N B exp (
.DELTA..eta. { .PHI. .beta. N ( .eta. + .DELTA..eta. / 2 , .tau.
 .DELTA..tau. / 2 ) + .PHI. .beta. N ( .eta. + .DELTA..eta. / 2
, .tau.  .DELTA..tau. / 2 ) 2 } .times. { .PHI. .beta. Q ( .eta.
+ .DELTA..eta. , .tau. ) + .PHI. .beta. Q ( .eta. , .tau. ) 2 }
) . ( 88 )
[0180]Substituting the approximation for the rate operator provided
in Eq. (88) into Eq. (87) can provide a form of the propagation
equation which may be expressed as:
Q ( .eta. + .DELTA..eta. ) .apprxeq. .DELTA..eta. 2 .PSI. ( .rho.
, .tau. ) exp (  .DELTA..eta..PHI. L ) .times. .beta. = 1 N B exp
( .DELTA..eta. { .PHI. .beta. Q ( .eta. + .DELTA..eta. / 2 , .tau.
 .DELTA..tau. / 2 ) + .PHI. .beta. Q ( .eta. + .DELTA..eta. / 2
, .tau.  .DELTA..tau. / 2 ) 2 } .times. { .PHI. .beta. Q ( .eta.
+ .DELTA..eta. , .tau. ) + .PHI. .beta. Q ( .eta. , .tau. ) 2 }
) .DELTA..eta. 2 .PSI. ( .rho. , .tau. ) Q ( .eta. ) . ( 89 )
[0181]A solution for the diffraction equation can be obtained by
using a CrankNicholson scheme. For example, an approximation to
the diffraction operator provided in Eq. (79) can be expressed as:
.PSI. df ( .rho. , .tau. ) .apprxeq. .PSI. ( .rho. ) = 4 .gradient.
.rho. 2 . ( 90 )
Applying a differential operator exp(.PSI.(.rho.).DELTA..eta./2).smallcircle.Q(.eta.)
can be equivalent to solving the following differential equation:
.differential. Q ( .eta. , .rho. ) .differential. .eta. = 1 2 .PSI.
( .rho. ) Q ( .eta. , .rho. ) , ( 91 )
Substituting the expression for the diffraction operator provided
in Eq. (90) can yield the following equation to be solved:
.differential. Q ( .eta. , .rho. ) .differential. .eta. = a m .gradient.
.rho. 2 Q ( .eta. , .rho. ) , ( 92 )
where a.sub.m.ident.1/2.times.i/4 and a time variable can be omitted
when applying the diffraction operator. A forward difference approximation
for the .eta. derivative can be written as
.differential. Q .differential. .eta. = Q j n + 1  Q j n .DELTA..eta.
. ( 93 )
[0182]A CrankNicholson scheme can be used to approximate the Laplacian
operator, e.g.:
.gradient. .rho. 2 Q ( .eta. , .rho. ) .apprxeq. 1 2 { .gradient.
.rho. 2 Q ( .eta. + .DELTA..eta. , .rho. ) + .gradient. .rho. 2
Q ( .eta. , .rho. ) } ; ( 94 ) .gradient. .rho. 2 Q ( .eta. n ,
.rho. ) = ( 1 .rho. .differential. .differential. .rho. + .differential.
2 .differential. .rho. 2 ) Q ( .eta. n , .rho. ) .apprxeq. [ 1 .rho.
j Q j + 1 n  Q j  1 n 2 .DELTA..rho. + Q j + 1 n  2 Q j n + Q
j  1 n .DELTA..rho. 2 ] . ( 95 )
Applying these operators can lead to the following iteration scheme:
Q j n + 1  Q j n .DELTA..eta. .apprxeq. a m 4 .rho. j .DELTA..rho.
[ Q j + 1 n + 1  Q j  1 n + 1 + Q j + 1 n  Q j  1 n ] + a m
2 .DELTA..rho. 2 [ Q j + 1 n + 1  2 Q j n + 1 + Q j  1 n + 1 +
Q j + 1 n  2 Q j n + Q j  1 n ] . ( 96 )
[0183]After grouping the terms containing identical (or substantially
similar) Q samples, the diffraction equation can be expressed in
a discrete form as:
u.sub.j1.sup.jQ.sub.j1.sup.n+1+u.sub.j.sup.jQ.sub.j.sup.n+1+u.sub.j+1.su
p.jQ.sub.j+1.sup.n+1=v.sub.j1.sup.jQ.sub.j1.sup.n+v.sub.j.sup.jQ.sub.j.s
up.n+v.sub.j+1.sup.jQ.sub.j+1.sup.n, (97)
with:
u j  1 j = a m .DELTA..eta. 2 .DELTA..rho. ( 1 2 .rho. j  1 .DELTA..rho.
) ; u j j = 1 + a m .DELTA..eta. .DELTA..rho. 2 ; u j + 1 j = 
a m .DELTA..eta. 2 .DELTA..rho. ( 1 2 .rho. j + 1 .delta..rho. )
; v j  1 j =  a m .DELTA..eta. 2 .DELTA..rho. ( 1 2 .rho. j 
1 .DELTA..rho. ) ; v j j = 1  a m .DELTA..eta. .DELTA..rho. 2 ;
and v j + 1 j = a m .DELTA..eta. 2 .DELTA..rho. ( 1 2 .rho. j 
1 .DELTA..rho. ) . ( 98 )
Equations (97) and (98) may be applicable only for "internal"
indices 0<.rho..sub.j<.rho..sub.N.sub.r, e.g., 0<.rho..sub.1.ltoreq..rho..sub.j.ltoreq..rho..sub.N.sub.r.sub.1<.r
ho..sub.N.sub.r, and boundary cases may need to be evaluated separately.
[0184]A solution to the differential equation that includes diffraction
effects can be obtained at a boundary where .rho.=0. For example,
a derivative .differential.Q/.differential..rho. can be zero at
.rho.=0 in a system having cylindrical symmetry such as, e.g., a
circular pulse footprint. A limit of the expression Q'(.rho.)/.rho.
can be obtained when .rho..fwdarw.0 by using a Maclaurin expansion,
e.g.:
Q ' ( .rho. ) = Q ' ( 0 ) + Q '' ( 0 ) .rho. + o ( .rho. 2 ) ;
( 99 ) Q ' ( .rho. ) .rho. = Q '' ( 0 ) + o ( .rho. 1 ) > Q
'' ( 0 ) , ( 100 )
Using Eqs. (99) and (100), the diffraction equation at .rho.=0
can be written as
[0185] .differential. Q .differential. .eta. = 2 a m .differential.
2 Q .differential. .rho. 2 . ( 101 )
[0186]A CrankNicholson technique can be applied to estimate the
second derivative term for .rho..fwdarw.0, e.g.:
.differential. 2 Q .differential. .rho. 2 ( .eta. n , 0 , .tau.
) .apprxeq. 1 2 { .differential. 2 Q .differential. .rho. 2 ( .eta.
+ .DELTA..eta. , 0 , .tau. ) + .differential. 2 Q .differential.
.rho. 2 Q ( .eta. , 0 , .tau. ) } . ( 102 )
A secondorder approximation to the second derivative can be written
as:
.differential. 2 Q .differential. .rho. 2 ( .eta. n + 1 , 0 , .tau.
) .apprxeq. 1 .DELTA..rho. 2 { Q j + 1 n + 1  2 Q j n + 1 + Q j
 1 n + 1 } . ( 103 )
[0187]Using Eqs. (102) and (103), the diffraction equation at the
center of a pulse can be written in an approximate form as:
Q j n + 1  Q j n .DELTA..eta. .apprxeq. 2 a m 2 .DELTA..rho. 2
[ Q j + 1 n + 1  2 Q j n + 1 + Q j  1 n + 1 + Q j + 1 n  2 Q
j n + Q j  1 n ] . ( 104 )
The derivative of Q can be zero with respect to radial distance
at .rho.=0, because the center of the pulse can be an extreme value.
For example, parameter values can be approximated at j=1 as Q.sub.1.sup.n+1=Q.sub.1.sup.n+1
and Q.sub.1.sup.n=Q.sub.1.sup.n. Therefore the diffraction equation
may be rewritten as follows:
Q 0 n + 1  Q 0 n .DELTA. .eta. .apprxeq. a m .DELTA. .rho. 2 [
2 Q 1 n + 1  2 Q 0 n + 1 + 2 Q 1 n  2 Q 0 n ] . ( 105 )
[0188]Eq. (105) can be used to obtain the following general equation
which is applicable at a center of a pulse (e.g., at .rho.=0):
u 0 0 Q 0 n + 1 + u 1 0 Q 1 n + 1 = v 0 0 Q 0 n + v 1 0 Q 1 n ,
where ( 106 ) u 0 0 = 1 + 2 a m .DELTA. .eta. .DELTA. .rho. 2 ;
u 1 0 =  2 a m .DELTA. .eta. .DELTA. .rho. 2 ; v 0 0 = 1  2 a
m .DELTA. .eta. .DELTA. .rho. 2 ; and v 1 0 =  2 a m .DELTA. .eta.
.DELTA. .rho. 2 . ( 107 )
[0189]A solution to the diffraction equation at an outer boundary
of a pulse, e.g., at .rho.=.rho..sub.N.sub.r, can be obtained using
a technique similar to that used to find a solution at the center
of the pulse described herein. For example, a value at a nonexisting
sample point .rho.=.rho..sub.N.sub.r.sub.+1 can be reconstructed
by evaluating a linearly interpolation at values of .rho. near the
boundary, e.g.:
Q N r + 1 n  Q N r n .DELTA. .rho. = Q N r n  Q N r  1 n .DELTA.
.rho. . ( 108 )
This linear interpolation can lead to a relationship, such as:
Q.sub.N.sub.r.sub.+1.sup.n=2Q.sub.N.sub.r.sup.nQ.sub.N.sub.r.sub.1.sup.n
. (109)
Eq. (109) can be substituted into the general diffraction equation,
Eq. (97) to provide a diffraction equation having a form
[0190]( .sub.N.sub.r.sub.1.sup.N.sup.r .sub.N.sub.r.sub.+1.sup.N.sup.r)Q.sub.N.sub.r.sub.1.sup.n+1+(
.sub.N.sub.r.sup.N.sup.r+2 .sub.N.sub.r.sub.+1.sup.N.sup.r)Q.sub.N.sub.r.sup.n+1=({tilde
over (v)}.sub.N.sub.r1.sup.N.sup.r{tilde over (v)}.sub.N.sub.r+1.sup.N.sup.r)Q.sub.N.sub.r.sub.1.sup.n+({tilde
over (v)}.sub.N.sub.r.sup.N.sup.r+2{tilde over (v)}.sub.N.sub.r.sub.+1.sup.N.sup.r)Q.sub.N.sub.r.sup.n.
(110)
Eq. (110) can be expressed in a simpler form as:
u N r  1 N r Q N r  1 n + 1 + u N r N r Q N r n + 1 = v N r 
1 N r Q N r  1 n + v N r N r Q N r n , where ( 111 ) u N r  1
N r = a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r ; u N r N r = 1
 a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r ; v N r  1 N r = 
a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r ; and v N r N r = 1 +
a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r . ( 112 )
[0191]Eqs. (97), (106) and (111) can be combined to provide a propagation
procedure that includes diffraction effects, which can be calculated
as a solution of the following linear system of equations with a
tridiagonal matrix:
UQ n + 1 = VQ n , where ( 113 ) Q * = [ Q o * , , Q j  1 * , Q
j * , , Q N r * ] ' and ( 114 ) U = [ u 0 0 u 1 0 u 0 1 u 1 1 u
2 1 u 1 2 u 2 2 u 3 2 u j  1 j u j j u j + 1 j u N r  1 N r u
N r N r ] . ( 115 )
[0192]The matrix V can be expressed in a form analogous to that
of the matrix U in Eq. (115). A conventional technique applicable
to tridiagonal matrix systems can be used to solve Eq. (113) such
as, e.g., an LU decomposition using forward and backsubstitution
for a tridiagonal system.
[0193]A solution to a linear equation such as that in Eq. (113)
can be stable, and a zero pivoting may not be required for a matrix
which has a diagonal dominance at its rows. For example, a diagonal
dominance condition can be expressed for a tridiagonal matrix by
the relationship as follows:
u.sub.j.sup.j>u.sub.j1.sup.j+u.sub.j+1.sup.j. (116)
In the exemplary embodiments of the present invention described
herein, using the approximation .rho..sub.j=j.DELTA..rho., the condition
provided in Eq. (116) may generally hold for the central and internal
sample points, e.g.:
[0194] u j  1 j + u j + 1 j = a m .DELTA. .eta. 2 .DELTA. .rho.
( 1 2 j .DELTA..rho.  1 .DELTA..rho. + 1 2 j .DELTA..rho. + 1 .DELTA..rho.
) = a m .DELTA. .eta. .DELTA. .rho. 2 < 1 + a m .DELTA. .eta.
.DELTA. .rho. 2 = u j j . ( 117 )
However, Eq. (117) can likely use a constraint on incremental delta
values used because of a condition at a boundary sample point, e.g.:
[0195] 1  a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r > a m
.DELTA. .eta. 2 .DELTA. .rho..rho. N r , ( 118 )
which may be true if, e.g.:
a m .DELTA. .eta. 2 .DELTA. .rho..rho. N r .rho. N r = 1 = a m
.DELTA. .eta. 2 .DELTA. .rho. < 1 , ( 119 )
which can provide a constraint, e.g.:
.DELTA..eta. .DELTA..rho. < 2 a m . ( 120 )
[0196]An exemplary solution to the propagation equation provided
in Eq. (89) can likely use a specification of an operator .PHI..sub..beta..sup.Q(.eta.+.DELTA..eta.,.tau.),
which may depend on an unknown .eta.+.DELTA..eta. sample point.
For example, an iteration procedure can be applied, wherein this
sample point at a current iteration is drawn from results of previous
iterations. A determination of a subsequent sample point value for
Q can be written as:
Q n + 1 .rarw. ( N = i + 1 / 2 n + 1 / 2 ( k ) N = i + 1 / 2 n
+ 1 / 2 ( k ) Q n + 1 ( k + 1 ) .ident. Q rd n + 1 ( k + 1 ) .PSI.
/ 2 Q r n + 1 ( k + 1 ) .PHI. Q mid n ) k = 1 K .PSI. / 2 Q n (
.ident. Q n + 1 ( 1 ) ) . ( 121 )
According to Eqs. (83)(84), an iteration step for the rate equation
can have a form:
N = n + 1 / 2 , i + 1 / 2 ( k ) .apprxeq. exp ( .DELTA..tau. Q
_ ( k ) [ n ] ) N = n + 1 / 2 , i  1 / 2 ( k ) , where ( 122 )
Q _ ( k ) [ n ] = 0 + .alpha. = 1 N A .alpha. Q _ ( k ) [ n ] ,
and ( 123 ) .alpha. Q _ ( k ) [ n ] = D .alpha. 1 2 { Q _ n , j
, i .alpha. + Q _ n + 1 , j , i ( k ) .alpha. } . ( 124 )
[0197]The propagation equation can be evaluated by performing three
determinations for each iteration: a determination of the diffraction
at k=0 using Eq.
[0198](125), and two further determinations of a rate using Eq.
(126) and a diffraction effect using Eq. (127). Eqs. (125)(127)
can be written as
Q mid n = U  1 VQ n , ( 125 ) Q r n + 1 ( k + 1 ) = exp (  .DELTA..eta..PHI.
L ) .beta. = 1 N B exp ( .DELTA..eta..PHI. .beta. N ( k ) [ n +
1 / 2 , i  1 / 2 ] .PHI. .beta. Q ( k ) [ n ] ) Q mid n , and (
126 ) Q n + 1 ( k + 1 ) .ident. Q rd n + 1 ( k + 1 ) = U  1 VQ
r n + 1 ( k + 1 ) , ( 127 )
where the following discrete forms of propagation operators .PHI..sub..beta..sup.N(.eta.,.tau.)
and .PHI..sub..beta..sup.Q(.eta.,.tau.) may be used in Eq. (126):
.PHI. .beta. N ( k ) [ n + 1 / 2 , i  1 / 2 ] .ident. .sigma.
.beta. { N n + 1 / 2 , j , i  1 / 2 ( k ) + N n + 1 / 2 , j , i
+ 1 / 2 ( k ) 2 } , ( 128 ) .PHI. .beta. Q ( k ) [ n ] .ident. Q
_ n , j , i .beta.  1 + Q _ n + 1 , j , i ( k ) .beta.  1 2 ,
and ( 129 ) .PHI. L .ident. c . ( 130 )
[0199]A computational procedure for determining the propagation
of an electromagnetic pulse in a nonlinear absorbing material which
accounts for diffraction effects can be described using the following
equations. Based on Eqs. (71), (83), (84), or (122)(124), a rate
equation can be used which can have the following form:
N = n + 1 / 2 , i + 1 / 2 ( k ) .apprxeq. exp ( .DELTA..tau. {
0 + .alpha. = 1 N A .alpha. Q _ ( k ) [ n ] } ) N = n + 1 / 2 ,
i  1 / 2 ( k ) = exp ( 0 + .alpha. = 2 N A .alpha. { Q _ n , j
, i .alpha. + Q _ n + 1 , j , i ( k ) .alpha. } 2 ) N = n + 1 /
2 , i  1 / 2 ( k ) , with ( 131 ) 0 .ident. .DELTA..tau. T 0 D
~ 0 , and .alpha. .ident. .DELTA..tau. T 0 D ~ .alpha. .alpha. .omega.
0 ( 2 nc 0 Q 0 2 ) .alpha. . ( 132 )
[0200]Using the results provided in Eqs. (125)(130) above together
with comments provided herein following Eq. (76), a mathematical
expression to describe propagation of a pulse can be written in
the following form:
Q n + 1 ( 1 ) .ident. Q n , ; Q mid n = U  1 VQ n ; ( 133 ) Q
r n + 1 ( k + 1 ) = exp (  .beta. = 1 N B .PHI. .beta. { N n +
1 / 2 , j , i  1 / 2 ( k ) + N n + 1 / 2 , j , i + 1 / 2 ( k )
2 } { Q _ n , j , i .beta.  1 + Q _ n + 1 , j , i ( k ) .beta.
 1 2 }  .phi. L ) Q mid n ; ( 134 ) Q n + 1 ( k + 1 ) .ident.
Q rd n + 1 ( k + 1 ) = U  1 VQ r n + 1 ( k + 1 ) ; with ( 135 )
.PHI. .beta. .ident. .DELTA..eta. L df N ( 2 nc 0 Q 0 2 ) .beta.
 1 .sigma. ~ .beta. ; .phi. L .ident. .DELTA..eta. c ~ L df ; and
( 136 )
[0201]By introducing a parameter
.xi. = a m .DELTA. .eta. .DELTA..rho. 2
and by applying Eq. (98), the nonzero elements of internal rows
of matrices U and V, i.e. u.sub.j.sup.i and v.sub.j.sup.i such that
i.epsilon.{1, . . . , N.sub.r} and j.epsilon.{i1, . . . , i+1},
may be written as:
u j  1 j = .xi. 2 ( 1 2 j  1 ) ; ( 137 ) u j i = 1 + .xi. ; (
138 ) u j + 1 j =  .xi. 2 ( 1 2 j + 1 ) ; ( 139 ) v j  1 j = 
u j  1 j ; ( 140 ) v j j = 1  .xi. = 2  u j j ; and ( 141 ) v
j + 1 j = .xi. 2 ( 1 2 j + 1 ) =  u j + 1 j . ( 142 )
[0202]By applying Eq. (107), the nonzero elements of the first
rows of matrices U and V, i.e. u.sub.j.sup.i and v.sub.j.sup.i such
that i=0 and j.epsilon.{0,1}, may be written as:
u.sub.0.sup.0=1+2.xi.; (143)
u.sub.1.sup.0=2.xi.; (144)
v.sub.0.sup.0=12.xi.=2u.sub.0.sup.0; (145)
and
v.sub.1.sup.0=2.xi.=u.sub.1.sup.0. (146)
[0203]By applying Eq. (112), the nonzero elements of the last rows
of matrices U and V, i.e. u.sub.j.sup.i and v.sub.j.sup.i such that
i=N.sub.r and j.epsilon.{N.sub.r1,N.sub.r}, may be written as:
u N r  1 N r = .xi. 2 N r ; ( 147 ) u N r N r = 1  .xi. 2 N r
; ( 148 ) v N r  1 N r =  .xi. 2 N r =  u N r  1 N r ; and (
149 ) v N r N r = 1 + .xi. 2 N r = 2  u N r N r . ( 150 )
Guided Waves
[0204]Conventional optical measurements may use unguided transmission
of light. An optical beam may broaden in diameter over propagation
distances greater than a diffraction length because of diffraction.
Light may be directed over transmission distances greater than the
diffraction length without incurring diffractive effects by using
guided wave optics. Guided wave structures can operate by confining
light within the structure using total internal reflection. This
can be achieved by providing a first dielectric medium having a
particular refractive index embedded within a second dielectric
medium having a lower refractive index. Certain exemplary embodiments
of the present invention may be used to describe optical effects
of cylindrically symmetric shaped wave guides. The first dielectric
medium, which may be provided in a shape of a core or inner cylinder,
may be a gas, liquid, fluid, gel or solid, and it may include a
multiphoton absorber. The second dielectric medium, which may be
provided as a cladding or outer cylinder surrounding the core, may
have a lower index of refraction than the core material.
[0205]Certain exemplary embodiments of the present invention may
be used to describe a fundamental mode of such wave guides. For
example, if a wave guide has a length such that a polarization of
the light may not be maintained, then a polarizationpreserving
wave guide can be used. An effective core area can be obtained from
a modal distribution F(x,y) for the fundamental mode of the wave
guide as described, e.g., in the Agrawal publication, e.g., .sub.eff=(.intg..sub..infin..sup..infin..intg..sub..infin..sup..infin.
F(x,y).sup.2dxdy).sup.2/.intg..sub..infin..sup..infin..intg..sub..infin
..sup..infin.F(x,y).sup.4dxdy. Here, an intensity can be described
by a function varying along .eta. and .tau. only, and may be expressed
as (.eta.,.tau.)=2nc.sub.0Q.sub.0.sup.2 Q(.eta.,.tau.)/ .sub.eff.
Energy Levels and Transitions
[0206]In certain exemplary embodiments of the present invention,
a more general formulation of the exemplary computational "building
blocks" shown in FIG. 1 can be used to provide information
and/or data relating to interactions between electromagnetic waves
and generic photoactive materials. For example, an "energy
level" can refer to one or a group of energy states which can
contribute, e.g., to macroscopic physical phenomena such as those
described herein using mathematical models. Energy levels may apply
to atoms, molecules, and/or solids. Such energy levels can include,
but are not limited to, electronic, vibrational, rotational, and
continuum levels of an atom, molecule, and/or solid, a conduction
band, and a valence band of semiconductors and metals.
[0207]The computational "building blocks" can refer to
a transition (e.g., a promotion or a relaxation) of an entity including,
but not limited to, an electron or an exciton, between different
energy levels. As described herein, a "generic material"
can refer to a material whose energy level diagrams can be represented
as union of a set of computational building blocks. A "block
diagram expression" can refer to a notation for computational
building blocks which may include an expression for an absorption
block, e.g., .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha., and for
a relaxation block, e.g.,
R type s 1 s 2 m 1 m 2 .
An "energy diagram string" .chi..sub.material can refer
to a string of one or more diagram expressions of computational
building blocks which may represent the energy level diagram of
a given material. An exemplary energy diagram string can be represented
mathematically as
.chi. material .dielect cons. ( B s 1 s 2 .alpha. m R type s 1
s 2 m 1 m 2 ) + . ( 151 )
where (ab).sup.+ can represent a regular expression which may specify
a set of textual strings having an arbitrary number of a's and b's
in interchanged fashion, e.g., a, b, ab, ba, aba, abb, bbb etc.
A "qset" .sup.mM can refer to a set of energy levels
characterized by a common value m which may be defined by some function
of a certain quantum number (e.g., a principal quantum number, a
spin quantum number, an angular momentum number, etc).
[0208]The five exemplary absorption diagrams B.sub.0B.sub.4 100140
shown in FIG. 1 can represent a reduced subset of the set of all
possible computational building blocks. A detailed description for
a more general set of building blocks can be provided, for example,
by describing associated schematic energy diagrams, descriptive
notations, and a technique for building a coupled system of ratepropagation
equations based on an energy diagram string, .chi..sub.material,
which can be expressed in terms of individual building blocks as
shown in Eq. (151) above.
[0209]Two types of computational building blocks can be used to
describe generic materials, e.g., absorption blocks and relaxation
blocks. The absorption blocks can be associated with transitions
from a lower energy level to a higher one, whereas the relaxation
blocks can be associated with transitions from a higher energy level
to a lower one. Such building blocks can be used to describe energy
level diagrams associated with a variety of photoactivated materials.
They can also be used to uniquely define terms in a corresponding
coupled system of ratepropagation equations which can describe
interactions between, e.g., electromagnetic pulses and absorbing
materials.
[0210]An energy level diagram can be used to develop a set of computational
building blocks. Energy levels can be associated with, e.g., electronic
states, vibrational and/or rotational electronic substates in atoms,
molecules and solids, and/or bands and discrete levels in semiconductors
and metals. An energy level diagram can also define possible electron
and/or exciton transitions between such energy levels by describing
types of transitions (e.g., radiative and/or nonradiative) and
corresponding absorption crosssections or decay time values.
[0211]An energy level diagram may be constructed using quantum
calculations, e.g., calculations based on Schrodinger equations,
experimental measurements, and/or hypothetical formulations, e.g.,
constructing a proposed diagram or adding elements to an existing
diagram.
[0212]A Schrodinger equation can include all possible energy levels
and/or states that a material system can be in at a particular moment
in time and a particular place in space, and can describe the evolution
of such material system. For example, a particle of mass M (e.g.,
an electron) having a potential energy V.sub.P({right arrow over
(r)},t) can be described by a wavefunction .PSI..sub.SE({right arrow
over (r)},t) that satisfies the following Schrodinger equation:
 2 2 M .gradient. 2 .PSI. SE ( r .fwdarw. , t ) + V P ( r .fwdarw.
, t ) .PSI. SE ( r .fwdarw. , t ) = i .differential. .PSI. SE (
r .fwdarw. , t ) .differential. t . ( 152 )
[0213]Systems containing many particles, e.g., atoms, molecules,
liquids or solids, can be described using more complex but similar
equations as described, e.g., in B. E. A. Saleh et al., Fundamentals
of Photonics, John Wiley & Sons, Cambridge University Press,
1991. pp. 384591; and in C. Kittel, Introduction to Solid State
Physics, John Wiley, New York, 1967.
[0214]The probability of locating a particle within a volume d.upsilon.
about the position {right arrow over (r)} in a time interval between
t and t+dt can be written as:
P({right arrow over (r)},t)d.upsilon.dt=.PSI..sub.SE({right arrow
over (r)},t).sup.2d.upsilon.dt. (153)
Allowable energy levels E.sup.n of the particle can be obtained
by neglecting timevarying interactions, using the following form
of a wavefunction, expressed as a product of spatial and temporal
terms:
SE({right arrow over (r)},t)=.psi./({right arrow over (r)})exp[i(E.sup.n/)t],
(154)
The spatial term .psi.({right arrow over (r)}) can be described
by the following timeindependent Schrodinger equation:
 2 2 M .gradient. 2 .psi. ( r .fwdarw. ) + V P ( r .fwdarw. )
.psi. ( r .fwdarw. ) = E n .psi. ( r .fwdarw. ) , ( 155 )
and the energy levels E.sup.n can be determined as eigenvalues
of the operator equation
({right arrow over (r)})=E.sup.n.psi.({right arrow over (r)}),
(156)
where the operator H can be obtained from Eq. (155) as
H ^ =  2 2 M .gradient. 2 + V P ( r .fwdarw. ) . ( 157 )
[0215]The size of the energy level diagram containing most or all
possible energy levels may be infinite, so that a reasonable simplification
can be used to limit the number of levels contained therein. For
example, the size of an energy level diagram can be reduced by including
only such energy levels which may be important in describing the
evolution of a particular system (or which may contribute the most
to a particular physical phenomenon of interest), and/or those which
may be revealed or explained by experimental measurements. Conventional
energy level diagrams may be constructed in this way, and can provide
an appropriate agreement between mathematical models based on such
diagrams and experimental observations.
[0216]Certain materials can exhibit measured characteristics and/or
behavior (e.g., optical transmittance) which may not fully agree
with the existing model that can be limited to a particular set
of energy levels. For example, additional energy levels may need
to be added to an existing model as new experimental techniques
and/or conditions are provided such as, e.g., a new laser frequency,
a higher laser intensity obtained using a reduced temporal pulse
width, etc. Such further hypothetical energy level diagrams can
be provided based on, e.g., experience, judgment and/or comparisons
with energy diagrams of similar materials. An exemplary mathematical
model based on a new energy level diagram can be tested against
the experimental measurements. An example of this exemplary approach
is, e.g., C.sub.60, which was originally described using an energy
diagram having only lowlying singlet states to explain experimental
absorption data. However, later experiments did not agree with this
exemplary simple model, and an additional set of triplet states
was added to the energy level diagram. Determinations based on this
additional energy level diagram exhibited a better agreement with
the experimental measurements.
[0217]The energy levels can be obtained as eigenvalues of the Schrodinger
timeindependent operator equation provided in Eq. (156). There
can be an infinite number of discrete energy levels (e.g., electronic
levels of an isolated atom) associated with a material based on
a solution to this equation. The materials that include many atoms
(e.g., molecules or crystals) can have additional energy levels
associated with a certain electronic energy level. To simplify construction
of an energy level diagram, a set of energy levels provided by the
Schrodinger equation can be restricted to those levels required
to generate reasonable agreement between calculations and experimental
measurements.
[0218]For example, FIGS. 11a11c show exemplary energy levels which
can be used in energy level diagrams. A certain vibrational and/or
rotational energy levels associated with it which may be degenerate
with respect to an energy level 1100 as shown in FIG. 11a. The electronic
energy level 1100 can have vibrational levels 1110, 1120 associated
therewith, and rotational levels may degenerate with respect to
these vibrational levels, as shown in FIG. 11b. Exemplary details
of the vibrational energy levels v.sup.i 1130 and rotational energy
levels u.sub.j.sup.i 1140 are shown in FIG. 11c. There may be a
small difference between the vibrational and rotational energy levels
1130, 1140 as compared to an energy difference between various electronic
levels 1100. The vibrational and rotational energy levels can be
referred to as "sublevels" or as a "manifold of
states" associated with a particular electronic energy level
or "state."
[0219]In certain exemplary embodiments of the present invention,
a small, finite set of energy levels may be used to describe certain
energy level diagrams. The term "energy level" can represent
either an energy level E.sup.n obtained as an eigenvalue of the
operator equation provided in Eq. (156), or a degenerate energy
level such as the exemplary level 1100 shown in FIG. 11a. An energy
level can be included in an energy level diagram, in accordance
with certain exemplary embodiments of the present invention, if
such energy level provides a significant contribution to a macroscopically
observable physical phenomenon such as, e.g., an experimental measurement
or a known absorption mechanism.
[0220]In many systems that include an absorbing material interacting
with a laser operating in the UV to near IR spectral regions, the
vibrational and/or rotational energy sublevels within a manifold
of states may exhibit very short life times compared to a temporal
pulse width of the laser. In such exemplary systems, electrons from
higher energy substates can relax to a lowest possible substate
within a manifold in a time that is much shorter than a duration
of a pulse, so that the contributions of these higher energy substates
to a macroscopic physical phenomena may be ignored. FIG. 12a shows
an exemplary illustration of a detailed energy level diagram 1200
which includes several manifolds of states 1210. Because the substates
within each manifold may have a relatively short life time, each
manifold 1210 can be represented by a degenerate energy level 1230
in the corresponding energy diagram 1220 shown in FIG. 12b.
[0221]The following exemplary criteria may be used to select and/or
build energy levels associated with particular materials to form
an energy level diagram, which may be used in certain exemplary
embodiments of the present invention: [0222]1. levels that are reachable
and/or accessible by entities energetically promoted by one or more
incident photons; [0223]2. levels that can provide a significant
contribution to a macroscopic physical phenomenon, such as one indicated
or suggested by experimental measurements; [0224]3. levels which
can be degenerate energy levels or combinations of degenerate levels;
and [0225]4. levels provided based on theoretical predictions.
[0226]According to the first exemplary criterion above, energy
levels may be selected, which can be reached and/or accessed by
entities (e.g., electrons or excitons) that may absorb incident
photons. The second exemplary criterion can allow a selection, from
the energy levels which may satisfy the first exemplary criterion,
of a smaller number of energy levels which, when using a mathematical
model, may be used to describe and/or account for macroscopic phenomena
such as, e.g., measured physical absorption processes. The third
exemplary criterion can allow combination of certain energy levels
to form degenerate energy levels, e.g., by a combination of vibrational
and/or rotational sublevels within a particular manifold of electronic
states. The third exemplary criterion can also allow a combination
of such degenerate levels into a further "joint" degenerate
energy level. The fourth exemplary criterion can allow the building
blocks to be used to expand conventional energy level diagrams or
to build new energy level diagrams by introducing additional levels.
This exemplary procedure can provide a more accurate description
of absorption phenomena in a particular material without requiring
a new algorithm or numerical method to solve a set of equations
describing the expanded energy level diagram for the material. For
example, when using the lasers having greater intensity and/or spectral
ranges, additional terms may be added to an existing energy level
diagram for a particular material to achieve consistency between
the predictions and the experimental measurements.
[0227]The energy level can be classified by associating a quantum
parameter with it which may stay the same during certain transitions,
and can change in value during other transitions. For example, energy
levels having a common value of a certain quantum parameter q can
be grouped into a qset which may be denoted by the symbol .sup.qM.
For example, Q.sub.P(N.sub.i) can represent a value of a quantum
parameter associated with energy level N.sub.i. The qsets corresponding
to this value can be represented as:
qM={N.sub.iQ.sub.P(N.sub.i)=q}. (158)
[0228]An electron spin multiplicity, which may be represented as
M.sub.s, is a quantum parameter which may give rise to different
sets of energy levels. This exemplary parameter may be used to classify
energy levels in conventional optics formulations. Such energy level
classification may often be used in the optics literature. Exemplary
procedures described herein for creating the energy diagrams can
avoid the imposition of restrictions which may be based on energy
level classification. For example, a spin multiplicity can depend
on spin quantum numbers, and may be expressed as
M s ( N i ) = 2 e .dielect cons. N i spin ( e ) + 1
for electrons e located in a common energy level N.sub.i. In this
exemplary classification, e.g., singlet states having a spin multiplicity
M.sub.s=1 can be denoted by .sup.1M, triplet states can be denoted
by .sup.3M, and in general:
mM={N.sub.iM.sub.s(N.sub.i)=m}. (159)
[0229]An absorption block, which may be denoted by a symbol .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.,
can represent a mechanism that includes simultaneous absorption
of .alpha. photons by an entity (e.g., an electron or an exciton)
and dislocation of an entity (e.g., an electron or an exciton) from
a particular energy level N.sub.s.sub.1 to another energy level
N.sub.s.sub.2 within the same qset .sup.mM. Two types of photon
absorption mechanisms can be distinguished. For example, a forward
absorption, for which .alpha.>0, can refer to an absorption mechanism
in which a photoactivated entity (e.g., an electron or an exciton)
may be promoted from a lower energy level to a higher energy level.
A reverse absorption can refer to an absorption mechanism in which
an electron may relax from a higher energy level to a lower energy
level by reemitting one or more photons that may be coherent with
the incident light. This exemplary mechanism, for which .alpha.<0,
can include the phenomenon of stimulated emission.
[0230]An exemplary schematic diagram of a forward absorption block
1300 that represents absorption of .alpha. photons is shown in FIG.
13a. This diagram shows energy levels 1310, 1320 involved in the
absorption, and a molar cross section parameter .sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2,
1330. A block diagram expression .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.
1340 which may represent the forward absorption block 1300 is also
shown in FIG. 13a, and may include a qset m associated with the
states.
[0231]An exemplary schematic diagram of a reverse absorption block
1300 that represents emission of .alpha. photons is shown in FIG.
13b. The emission can occur between two energy levels 1360, 1370,
and a molar cross section parameter .sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2
1380 can be associated with the reverse absorption. A block diagram
expression 1390 is shown in FIG. 13b which may be used to represent
the reverse absorption block. This expression 1390 is similar to
the expression 1340 for a forward emission block shown in FIG. 13a,
and a minus sign may be associated with the .alpha. term to indicate
that the expression 1390 represents a reverse emission event.
[0232]For example, a general block diagram expression .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.
1340 shown in FIG. 13a can include indices s.sub.1,s.sub.2 which
may represent certain energy levels that an electron is promoted
"from" and "to" respectively (or relaxed "to"
and "from" respectively if .alpha.<0). The parameter
.alpha. can represent the number of photons that may be absorbed
to promote, e.g., an electron from level s.sub.1 to level s.sub.2,
where .alpha. can be negative if the expression is used to represent
a reverse absorption block that includes a relaxation from level
s.sub.2 to level s.sub.1. The parameter m can represent an index
of a qset which may include both the source level s.sub.1 and the
destination level s.sub.2. The parameters m,.alpha.,s.sub.1, and
s.sub.2 can be referred to as indices of the absorption block .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.,
whereas .sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2 can represent
a parameter of the absorption block. Optionally, the parameter .alpha.
which may appear in a block diagram expression may be omitted if
it is equal to one, e.g., .sup.mB.sub.s.sub.1.sub.s.sub.2.ident..sup.mB.sub.s.sub.1.sub.s.sub.2.sup
.1. Further, a parameter of an absorption block can be included
in the corresponding block diagram expression if more clarity is
desired, e.g., .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.[.sigma..sub.[.alpha.]PA.sup.s
.sup.1.sup.s.sup.2].
[0233]The use of the computational building blocks can allow a
grouping of all absorption blocks into a single group B. The elements
in a group B can describe the various possible types of absorption
events in generic materials. An exemplary group B may be written
as:
B={.sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha..alpha..epsilon.{
. . . ,1,1,2, . . . };s.sub.1,s.sub.2,m.epsilon.{0,1,2, . . . }}.
(160)
[0234]For example, the absorption blocks can be provided for a
TPA chromophore AF455 using the techniques described herein. A conventional
chromophore from an AFX group of TPA materials such as, e.g., AF455,
may have two qsets of electronic levels associated with it: singlet
states and triplet states. An energy diagram 150 for AF455 is shown
in FIG. 1 and labeled as (TPA+ESA). Electronic states 160 on the
left side of the energy diagram 150 can be associated with a singlet
qset .sup.1M={N.sub.0,N.sub.1,N.sub.2}, and states 170 on the right
side of the energy diagram 150 can be associated with a triplet
qset .sup.3M={N.sub.3,N.sub.4}. An absorption portion of the energy
diagram for AF455 can be represented as an energy level diagram
string, using the computational building block notation described
herein, as:
AF455.sub.B=.sup.1B.sub.01.sup.2.orgate..sup.1B.sub.12.orgate..sup.3B.sub
.34, (161)
where, for example, .sup.1B.sub.01.sup.2 can represent a simultaneous
absorption of two photons and promotion of one electron from a ground
level to a lowest excited level in the singlet qset .sup.1M 160;
.sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2 in the singlet qset
.sup.1M 160 can correspond to .sigma..sub.TPA; .sup.3B.sub.34 can
represent an absorption of one photon and promotion of one electron
from the lowest excited state to the higher excited state in the
triplet qset .sup.3M 170; .sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2
in the triplet qset .sup.3M 170 can correspond to .sigma..sub.34;
and .chi..sub.AF455.sub.B can represent the energy level diagram
string for AF455 using the computational building blocks that may
be restricted to the set of absorption building blocks B.
[0235]A second type of the computational building block can be
referred to as a relaxation block, and can be denoted by
R type s 1 s 2 m 1 m 2 .
A relaxation block can represent various types of electron, exciton
and/or phonon relaxations which may occur between energy states
associated with a single qset or with different qsets. In contrast
to the reverse absorption described herein, whereas one or more
photons that may be coherent with the incident light can be emitted
such as in stimulated emission, the relaxation can lead to the emission
of incoherent photons. For example, a relaxation block can represent
an event in which an entity (e.g., an electron, exciton or phonon)
may migrate from a higher energy level N.sub.s.sub.1.epsilon..sup.m.sup.1M
to a lower energy level N.sub.s.sub.2.epsilon..sup.m.sup.2M. This
exemplary migration may be accompanied by the emission of the radiation,
which can include a radiative transfer that can be represented by
a wave sign `.about.` in the diagram expression. Alternatively,
the transfer may be a nonradiative (e.g., heat) relaxation that
can be represented by a bar `` in the diagram expression. The relaxation
block can include a parameter k.sub.s.sub.1.sub.s.sub.2 that may
represent a relaxation decay rate.
[0236]An exemplary relaxation event 1400 and a corresponding relaxation
block 1410 are shown in FIG. 14. The indices in the relaxation block,
R type s 1 s 2 m 1 m 2 ,
include s.sub.1 and s.sub.2, which can represent the index of the
energy level that an electron relaxes from and to, respectively.
The "type" can identify a particular category of relaxation,
which can include a radiative transfer (`.about.`) or a nonradiative
transfer (``). The type may be blank if the relaxation category
is not known. Indices m.sub.1 and m.sub.2 can represent indices
of the qsets that include the source and destination electronic
levels, respectively. Alternatively, e.g., a single index of a qset
may be provided if an electron relaxationoccurs between electronic
levels of a particular qset, e.g.,
R type s 1 s 2 m .ident. R type s 1 s 2 mm .
[0237]FIG. 15 shows an exemplary diagram 1500 of a nonradiative
migration of the electron which can be associated with an intersystem
crossing decay rate k.sub.s.sub.1.sub.s.sub.2 1510. This exemplary
relaxation diagram can be represented by the relaxation block .sup.m.sup.1.sup.m.sup.2
R.sub.s.sub.1.sub.s.sub.2 1520.
[0238]FIG. 16 shows an exemplary diagram 1600 of a radiative migration
of the electron which can be associated with the intersystem crossing
decay rate k.sub.s.sub.1.sub.s.sub.2 1610. This exemplary relaxation
diagram can be represented by the relaxation block .sup.m{tilde
over (R)}.sub.s.sub.1.sub.s.sub.2 1620 which may correspond, e.g.,
to a fluorescence emission during migration of an electron along
states having a common spin multiplicity m.
[0239]A group of most or all possible relaxation building blocks
associated with a particular material can be grouped into a set
R of relaxation blocks, which may be expressed as:
R = { R type s 1 s 2 m 1 m 2  type .dielect cons. { ' ~ ' , '
 ' , ' } ; s 1 , s 2 , m 1 , m 2 .dielect cons. { 0 , 1 , 2 ,
} } . ( 162 )
[0240]An exemplary energy diagram 1700 that can represent various
energy level transitions associated with AF455 is shown in FIG.
17. This diagram 1700 includes relaxation links 17101730. A relaxation
portion of the energy diagram string that can be used to describe
AF455 can be written as:
AF455.sub.R=.sup.1{tilde over (R)}.sub.10.orgate..sup.1R.sub.21.orgate..sup.3R.sub.43.orgate..sup.13
R.sub.13.orgate..sup.31{tilde over (R)}.sub.30. (163)
[0241]Eq. (163) can be combined with Eq. (161) to provide a complete
diagram string .chi..sub.AF455, which can be written as:
AF455=.sup.1B.sub.01.sup.2.orgate..sup.1B.sub.12.orgate..sup.3B.sub.34.org
ate..sup.1{tilde over (R)}.sub.10.orgate..sup.1R.sub.21.orgate..sup.3R.sub.43.orgate..sup.13
R.sub.13.orgate..sup.31{tilde over (R)}.sub.30. (164)
[0242]Another exemplary material that can behave as a nonlinear
absorber is, e.g., a CuTPPS chromophore contained within a gel host.
For example, CuTPPS entrapped within an aluminosilicate host is
described, e.g., in X. D. Sun et al., "Nonlinear Effects in
Chromophore Doped SolGel Photonic Materials", J. Solgel Sci.
Technol., 9, 169181, (1997). This exemplary nonlinear material
can have six energy levels and three or more qsets. Such material
can absorb light at 584 nm and may exhibit upconverted photon emission
at 434 nm provided by a relaxation links.
[0243]An exemplary energy diagram 1800 that can represent various
energy level transitions associated with CuTPPS is shown in FIG.
18. The energy levels of CuTPPS can be N.sub.0,N.sub.1,N.sub.2,N.sub.3,N.sub.4,N.sub.5.
These energy levels may be grouped into three qsets, as shown in
FIG. 18: .sup.1M={N.sub.0,N.sub.1,N.sub.3} 1810, .sup.2M={N.sub.2}
1820, and .sup.4M={N.sub.4,N.sub.5} 1830. Based on these qsets,
the energy diagram string for CuTPPS can be expressed as:
CuTPPS=.sup.1B.sub.01.orgate..sup.1B.sub.13.orgate..sup.4B.sub.45.orgate..
sup.1{tilde over (R)}.sub.10.orgate..sup.12 R.sub.32.orgate..sup.14
R.sub.14.orgate..sup.21{tilde over (R)}.sub.21.orgate..sup.2{tilde
over (R)}.sub.20.orgate..sup.4{tilde over (R)}.sub.54.orgate..sup.41{tilde
over (R)}.sub.40. (165)
[0244]A further example of generation of an energy level string
can be provided for a semiconductor quantum dot using the exemplary
procedures described herein. For example, an exemplary diagram 1900
of photoexcitation of an exciton in a semiconductor is shown in
FIG. 19 and described, e.g., in B. E. A. Saleh et al., Fundamentals
of Photonics, John Wiley & Sons, Cambridge University Press,
1991, pp. 384591. In this exemplary material system, an exciton
recombination can provide both a stimulated and a spontaneous emission.
A ratio of electron degeneracy between level 1 1910 and level 0
1920 can be a unity in this exemplary analysis. As shown in FIG.
19, an open circle 1930 can represent an electron "hole",
and a solid black circle 1940 can represent an electron. The exciton
can include the electronhole pair 1930, 1940.
[0245]A coupled set of ratepropagation equations can be used to
describe propagation of an electric field through a semiconductor
quantum dot. This set of equations may be written as:
{ N 0 .tau. =  .sigma. 01 I .omega. 0 ( N 0  N 1 ) + k 10 N 1
N 1 .tau. = .sigma. 01 I .omega. 0 ( N 0  N 1 )  k 10 N 1 , and
( 166 ) I .eta. =  .sigma. 01 I ( N 1  N 0 ) , ( 167 )
[0246]An energy level diagram 2000 which can be used to describe
the quantum dot illustrated in FIG. 19, which may be based on Eqs.
(166) and (167) is shown in FIG. 20. This energy level diagram that
may be used to describe absorption and emission by a quantum dot
can include three basic blocks and two energy levels 2010, 2020
associated with a single same qset. The energy levels are .sup.1M={N.sub.0,N.sub.1},
and the energy level string .chi..sub.InGaAs can be written as:
semiconductor=.sup.1B.sub.01.orgate..sup.1B.sub.01.sup.1.orgate..sup.1R.s
ub.10, (168)
where .sup.1B.sub.01.sup.1 can represent an exemplary absorption
block that can be associated with inverse absorption.
[0247]Each computational building block in the set B.orgate.R (which
includes both absorption and relaxation blocks) can define a unique
corresponding term in the coupled system of ratepropagation equations
provided in Eqs. (15) and (16), which may be used to describe propagation
of an electric field through a sample of a photoactive material.
The computational building blocks described herein may be analogous
or similar to computational molecules that can be used to formulate
multidimensional finite difference numerical schemes.
[0248]A particular energy diagram string associated with a photoactivated
material mat can include a component e in it, such that .chi..sub.mat
can be written in a form .chi..sub.mat= . . . .orgate.e.orgate.
. . . . The component e can represent one of the types of computational
building blocks described herein, e.g., the absorption building
block or the relaxation building block. Each type of the computational
building block can provide certain terms in a matrix {circumflex
over (D)}.sub..alpha. and a vector .sigma..sub..beta. which appear
in the rate and propagation equations provided in Eqs. (15) and
(16).
[0249]For example, e can represent an absorption building block,
e=.sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha., which may be associated
with an absorption parameter .sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2.
The value of this parameter can be included at corresponding places
in the matrix {circumflex over (D)}.sub..alpha. and vector .sigma..sub..beta.,
which can lead to the following expressions:
{circumflex over (D)}.sub..alpha.[s.sub.1,s.sub.2]={circumflex
over (D)}.sub..alpha.[s.sub.1,s.sub.2]sgn(.sigma..sub.[.alpha.]PA.sup.s.sup.1
.sup.s.sup.2).sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2, (169)
{circumflex over (D)}.sub..alpha.[s.sub.2,s.sub.1]={circumflex
over (D)}.sub..alpha.[s.sub.2,s.sub.1]+sgn(.sigma..sub.[.alpha.]PA.sup.s.sup.1
.sup.s.sup.2).sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2, (170)
and
[s.sub.1]=.sigma..sub..alpha.[s.sub.1]+sgn(.sigma..sub.[.alpha.]PA.sup.s.s
up.1.sup.s.sup.2).sigma..sub.[.alpha.]PA.sup.s.sup.1.sup.s.sup.2.
(171)
[0250]Alternatively, e can represent a relaxation building block,
e = R t s 1 s 2 m 1 m 2 ,
which may be associated with a decay rate k.sub.s.sub.1.sub.s.sub.2.
The value of this decay rate can be included in the matrix {circumflex
over (D)}.sub.0 to yield the following expressions:
{circumflex over (D)}.sub.0[s.sub.1,s.sub.2]={circumflex over (D)}.sub.0[s.sub.1,s.sub.2]k.sub.s.sub.1.sub.s.sub.2,
(172)
and
{circumflex over (D)}.sub.0[s.sub.2,s.sub.1]={circumflex over (D)}.sub.0[s.sub.2,s.sub.1]+k.sub.s.sub.1.sub.s.sub.2.
(173)
[0251]The matrices {circumflex over (D)}.sub.0, {circumflex over
(D)}.sub..alpha. and the vector .sigma..sub..beta. which appear
in the rate and propagation equations provided in Eqs. (15) and
(16) may be constructed for a particular material using the following
procedure. For example, each matrix {circumflex over (D)}.sub.0,
{circumflex over (D)}.sub..alpha. and the vector .sigma..sub..beta.
may be defined initially to have all elements equal to zero. The
energy diagram string .chi..sub.mat, for a given photoactive material
can be obtained, e.g., using the exemplary procedures described
herein. Each computational building block appearing in .chi..sub.mat
can then be used to modify certain elements in the matrices {circumflex
over (D)}.sub.0, {circumflex over (D)}.sub..alpha. and the vector
.sigma..sub..beta.. For example, Eqs. (169)(171) can be applied
for each absorption block that appears in the energy diagram string,
and Eqs. (172) and (173) can be applied for each relaxation block.
This exemplary procedure can be repeated until all component e in
the energy diagram string (e.g., all absorption and relaxation blocks)
have been used to modify appropriate elements in the matrices {circumflex
over (D)}.sub.0, {circumflex over (D)}.sub..alpha. and the vector
.sigma..sub..beta. using Eqs. (169)(173). The resulting matrices
and vector may be provided in Eqs. (15) and (16), and these exemplary
equations may be solved using exemplary procedures described herein
to determine the absorption behavior of the particular material.
[0252]Certain restrictions may be applied when identifying and/or
providing computational building blocks associated with a particular
photoactive absorbing material. For example, each energy level,
excluding three special states N.sub.0, N.sub.V, and N.sub.e, can
be associated with a unique group of states, e.g., a qset. For
a particular material, all such qsets .sup.mM may be identified
by an expression which can be written as M={.sup.mMm=m.sub.1, .
. . , m.sub.M.sub.M}.
[0253]For example, N.sub.0 can represent a ground state and it
may belong to more then one qset. This condition can be written
as:
N.sub.0.epsilon..sup.mM, .Ainverted.m.epsilon.{j.sub.1, . . .
,j.sub.M.sub.0}.OR right.{m.sub.1, . . . ,m.sub.M.sub.M}. (174)
N.sub.V can represent a valence band, which may be present in all
qsets. This condition may be expressed mathematically as:
N.sub.V.epsilon..sup.mM, .Ainverted.m.epsilon.{m.sub.1, . . .
,m.sub.M.sub.M}. (175)
N.sub.e can represent a conduction band, which may also be present
in all qsets, e.g.:
[0254]N.sub.e.epsilon..sup.mM, .Ainverted.m.epsilon.{m.sub.1,
. . . ,m.sub.M.sub.M}. (176)
[0255]Each absorption building block .sup.mB.sub.s.sub.1.sub.s.sub.2.sup..alpha.
identified or proposed for a particular material can be subjected
to certain exemplary constraints. These constraints can include,
e.g.: [s.sub.1,s.sub.2,m].epsilon..sub.0.sup.3; .alpha..epsilon.\{0}
with m.epsilon.{m.sub.1, . . . , m.sub.M.sub.M}; s.sub.2.noteq.0;
N.sub.s.sub.1.noteq.N.sub.e; and N.sub.s.sub.2.noteq.N.sub.V. These
constraints can assist to provide consistent building block formulations
for describing the absorption behavior of the nonlinear materials
in accordance with certain exemplary embodiments of the present
invention.
[0256]In a similar manner, each relaxation building block
R t s 1 s 2 m 1 m 2
identified or proposed for a particular material can also be subjected
to certain exemplary constraints. For example, such constraints
on relaxation building blocks can include: [s.sub.1,s.sub.2,m.sub.1,m.sub.2].epsilon..sub.0.sup.4,
m.epsilon.{m.sub.1, . . . , m.sub.M.sub.M}; (m.sub.1,m.sub.2).epsilon.{m.sub.1,
. . . , m.sub.M.sub.M}.sup.2; s.sub.1.noteq.0; N.sub.s.sub.2.noteq.N.sub.e;
and N.sub.s.sub.1.noteq.N.sub.V.
[0257]In certain exemplary embodiments of the present invention,
steadystate solutions of the rate and propagation equations described
herein may provide an approximation of population density dynamics.
Such exemplary solution can correspond to a solution of a nonlinear
system of rate equations for population densities in which the righthand
sight of the equations are set equal to zero, e.g., .differential.N/.differential.t=0.
Update procedures provided in Eqs. (169), (170), (172) and (173)
may be omitted when obtaining such steadystate approximate solutions.
Exemplary Computational Procedures
[0258]An exemplary flow diagram of a method 2100 according to certain
exemplary embodiments of the present invention is shown in FIG.
21. For example, initial data may be provided for the determination
of an interaction between, e.g., an electromagnetic wave and a nonlinear
absorbing material (step 2110). Such data can include, for example,
an identification of the absorbing material together with material
properties, which may include an energy level diagram or an energy
diagram string. The initial data may also include properties of
the coherent electromagnetic wave which may be, e.g., one or more
laser pulses, etc. Properties of the coherent electromagnetic wave
can include a wavelength or a plurality of wavelengths, pulse duration
and/or number, interval between a plurality of pulses, intensity,
fluence, radius or diameter of a collimated or focused beam at an
incident surface of the absorbing material, etc. Numerical parameters
may also be provided and can include, e.g., temporal and/or spatial
resolution and/or extent of numerical determinations to be performed.
[0259]After initial parameters can be established (step 2110),
an initial depth z within the material can be initialized to a value
of 0 (step 2120). This can correspond to the incident coherent electromagnetic
wave first contacting an outer surface of the absorbing material.
Propagation and rate equations such as, e.g., Eqs. (15) and (16)
can be formulated based on the parameters such as the energy level
diagram or an energy diagram string associated with the absorbing
material using procedures such as, e.g., those described herein
(step 2130).
[0260]An intensity of the coherent electromagnetic wave or laser
pulse(s) can then be determined as a function of time t and, optionally,
radial distance r from the center of the incident wave at a depth
between z and z+dz (step 2140). The determinations intensity values
can be based at least in part on intensity values determined at
a depth z, which can be initially set equal to 0 (step 2120), e.g.,
at the material surface. The parameter dz can represent an incremental
depth interval used in the exemplary computational techniques described
herein. The determination of intensity as a function of r and t
may be performed using several iterations of the rate and propagation
equations to obtain convergent and/or consistent values at a depth
interval between z and z+dz. After the intensity is obtained at
the depth z, contributions of each energy level in the material
to absorption within that depth interval can optionally be determined
(step 2150).
[0261]If the depth z is greater than or equal to a maximum depth
z.sub.max of interest (step 2160), the absorption results can be
provided to, e.g., a database, data file and/or a display arrangement
(step 2180). The maximum depth z.sub.max can be, e.g., a sample
thickness of the absorbing material. The absorption results can
include, e.g., electric field, intensity, energy level populations
and/or contributions of individual levels to absorption. These exemplary
results can be provided as a function of depth within the absorbing
material, time and optionally radial distance from a center of the
incident wave.
[0262]If the depth z is less than the maximum depth z.sub.max of
interest (step 2160), the current depth can be incremented by an
amount dz (step 2170). The electric field intensity can then be
determined at a depth interval between the new depth z and z+dz
(step 2140) and the absorption contributions by individual electronic
levels within this new depth interval can also be determined (step
2150). This exemplary procedure can be repeated at increasing depths
until the maximum depth z.sub.max is reached (step 2160).
[0263]A detailed exemplary flow diagram of a method 2200 according
to particular exemplary embodiments of the present invention is
shown in FIGS. 22a22b. For example, the material parameters associated
with a nonlinear absorbing material of interest may be provided
(step 2205). Such material parameters can include, for example,
a sample thickness (which may be set equal to the maximum depth
of interest, z.sub.max) and/or an index of refraction. The material
parameters can also include a concentration of lightactivated atoms,
molecules and/or radiators in gas, liquid or solid. Further material
parameters that may be provided include, e.g., cross sections for
single and/or multiphoton absorptions, a linear absorption coefficient,
and/or decay rates for energy levels associated with the material.
Certain material parameters may be optionally provided in a form
of one or more absorption blocks and/or relaxation blocks, and/or
an energy diagram string.
[0264]Parameters associated with an incident laser pulse or pulses,
or with another form of incident electromagnetic wave, may also
be provided (step 2210). Such parameters can include, e.g., one
or more central wavelengths or a carrier frequency, a temporal pulse
width (e.g., a pulse duration), a beam radius or diameter or other
physical dimension associated with the incident pulse or wave, and
optionally an incident pulse function if the pulse intensity is
not uniform over its crosssectional area and/or over its temporal
functional shape. If the laser or coherent electromagnetic wave
is provided, e.g., as a plurality of incident pulses, such parameters
can be provided for each pulse. Further parameters can include,
e.g., a number of pulses and a temporal interval between successive
pulses.
[0265]Numerical parameters which can be used to obtain a solution
to the appropriate absorption equations may also be provided (step
2215). These numerical parameters may include, e.g., a time step
or temporal resolution dt, a depth interval dz, and/or a radial
distance increment dr. Other numerical parameters that may be provided
include a number of iterations K and/or a subsampling threshold
value .epsilon., which are described herein.
[0266]An intensity of the incident electromagnetic wave (e.g.,
a laser pulse) can be determined (step 2220) using, for example,
provided electromagnetic wave parameters (step 2210).
[0267]Matrices, vectors and constants associated with propagation
and rate equations may then be formulated (step 2225), and can be
based at least in part on material parameters such as, e.g., an
energy level diagram or an energy diagram string associated with
the absorbing material. The rate equation may have a form as provided,
e.g., in Eqs. (15), (72) or (74). The propagation equation may have
a form as provided, e.g., in Eqs. (16), (76) or (82). The matrices,
vectors and/or constants may be formulated, for example, using exemplary
procedures such as those provided in Eqs. (169)(173).
[0268]Computational grids may then be provided for solving the
rate and propagation equations numerically (step 2230). These grids
may have a form, e.g, such as provided in Eq. (21) which may be
used to solve the population density equation, and Eq. (22) may
be used to solve the intensity or electric field propagation equation.
The size scale of these can be based at least in part on the provided
numerical parameters (step 2215), and they can be related to, e.g.,
a spatial and/or a temporal resolution at which the intensity and/or
population densities may be determined.
[0269]Discrete values of the intensity or electric field of the
incident coherent electromagnetic wave (e.g., a laser pulse) can
be determined at a first surface of the material (e.g., z=0) (step
2235) using, for example, provided numerical parameters (step 2215)
and a provided incident intensity of the electromagnetic wave (step
2220). The incident intensity or electric field may be determined
as a function of time and/or location at certain points on the provided
grids (step 2230). An initial spatial distribution of intensity
or electric field at entrance of the material sample may be expressed
in terms of, e.g., a radial distance from a center of a pulse, polar
coordinates, rectangular coordinates, etc., and such spatial distribution
may further be provided at various times. An initial spatial and/or
temporal distribution of intensity or electric field at an initial
surface of the material sample may be obtained from experimental
measurements, or by applying analytic or numerical calculations
of various optical configurations which may be present outside of
the surface of the material. Such measurements or calculations can
be converted into discrete values and used as initial beam or field
parameters in certain exemplary embodiments of the present invention.
[0270]An incremental depth dz may be added to a current depth z
within the material (step 2240). The total depth z+dz can then be
compared to a maximum depth, e.g., z.sub.max (step 2245), and if
z+dz is greater than or equal to z.sub.max then various determined
absorption values associated with interactions between the material
and the coherent electromagnetic wave can be provided, e.g., to
an output device or storage medium (step 2299). The distance z.sub.max
can correspond to a sample thickness or to some other depth at which
it may be provided that interactions in the material are not calculated
further.
[0271]Absorption interactions, which may include changes in intensity
of the coherent electromagnetic wave and/or population densities
of energy levels within the material, can be determined within a
portion of the material that lies between the depths z and z+dz.
The parameter k, which can represent a number of iterations to be
performed within the portion of material between z and z+dz, may
be set equal to zero initially and vectors N.sub.1/2,j.sup.n+1/2[k]
can be initialized to unit vectors (step 2250). Such vectors can
be associated, for example, with population densities of electronic
levels within the portion of the material.
[0272]The population densities can then be updated to new values
which may be present after a subsequent time procedure has elapsed
(step 2255) using, e.g., Eq. (29), (83), (84), (131) or (132). A
parameter M which may be related to the time step can then be determined
(step 2260). If M is greater than 1 (step 2265), then population
density changes over the time step may be redetermined using a refined
set of substeps (step 2270). This determination can be performed,
e.g., using Eq. (42).
[0273]When the population densities are determined over the time
step, the intensity distribution of the coherent electromagnetic
wave can be modified based on the new population densities (step
2275) using, e.g., Eq. (30), (133)(136). Optionally, a contribution
of each electronic level in the material to the total absorption
within the portion of the material between z and z+dz may also be
determined (step 2280). These contributions may be calculated, e.g.,
using Eqs. (48)(52), or (54).
[0274]The parameter k can then be increased by 1 (step 2285), and
compared to a further parameter K (step 2290). K can represent,
for example, a preselected number of computational iterations to
be performed at a particular depth interval within the material,
e.g., between z and z+dz. If k is less than K (step 2290), then
another iteration can be performed to update values of the population
densities (step 2255) and intensity distribution (step 2275) over
the depth interval. If k is greater than or equal to K (step 2290),
then the preselected number of computational iterations may have
been performed over the depth interval between z and z+dz. The current
determined values of population densities (step 2255), intensity
distribution (step 2275), and energy level contributions to the
total absorption (step 2280) associated with the depth interval
between z and z+dz in the material may then be stored (step 2293).
These values may be stored, e.g., by writing them to a computerreadable
medium, printing them, providing them to a data analysis program
and/or displaying them on a screen.
[0275]The depth z may then be increased by an amount dz (step 2296),
and the new depth z may then be compared to a maximum depth, e.g.,
z.sub.max (step 2245). If z+dz is greater than or equal to z.sub.max
then various absorption values and population densities associated
with interactions between the material and the coherent electromagnetic
wave may have been calculated throughout the region of interest
in the material, and these values can be can be provided, e.g.,
to an output device or storage medium (step 2299).
[0276]The interaction between a nonlinear absorbing material and
an coherent electromagnetic wave such as, e.g., a laser pulse, which
may be determined using exemplary embodiments of the present invention,
can optionally be compared to experimental measurements of such
interactions. Such comparisons are shown, e.g., in FIGS. 2a2c.
If agreement between certain determined and experimental values
associated with the interaction are not sufficiently close for a
particular purpose, material parameters used in certain exemplary
embodiments of the present invention may be modified to provide
better agreement. Such parameters can be provided, e.g., in step
2110 of FIG. 21 and step 2205 of FIG. 22a. The material parameters
may be modified by changing certain characteristics of an energy
level diagram and/or one or more relaxation and/or absorption blocks
associated with the material or, alternatively, adding additional
energy levels and/or relaxation and/or absorption blocks to the
existing material parameters. Such modifications may be made, e.g.,
based on physical models, expected improvements in agreement between
the determinations and experiments, etc.
[0277]The results of the exemplary computational procedures shown
in FIGS. 21, 22a and 22b, which may include a radial or other spatial
dependence of the calculated values, can be compared to results
obtained using conventional techniques which may assume a radially
constant solutions. Such a comparison can provide information relating
to whether a local peak intensity may exceed a critical value for
a material and cause damage. Such localized values may not be provided
by conventional techniques. Thus exemplary embodiments of the present
invention may be used to provide more reliable prediction of damage
in absorbing materials which may be caused by interactions with,
e.g., one or more incident laser pulses.
[0278]In certain exemplary embodiments of the present invention,
an absorbing material may include a plurality of layers, where two
or more layers may each be associated with a different set of material
properties. Each layer w may also have a particular thickness z.sub.w
associated with it. To determine the absorption interactions in
such multilayered absorbing materials, an exemplary method such
as that shown in FIGS. 21, 22a and/or 22b can be used with respect
to an upper layer that is first contacted by an incident coherent
electromagnetic wave (e.g., w=1). This exemplary procedure may be
applied throughout the first layer, e.g., until a depth of z.sub.1
is reached, at which point a second layer (e.g., w=2) which may
be formed of a different material may be present. An intensity or
electric field distribution can thus be determined at a depth z.sub.1,
which can be based on interactions within the first layer. This
distribution can be used as an initial condition to determine interactions
within the second layer where the depth used in the determinations
can range, e.g., from z.sub.1 to z.sub.2. Material properties associated
with the second layer may be used in this second set of determinations.
This exemplary procedure may be continued for additional layers,
if present, to provide information relating to interactions between
an coherent electromagnetic wave and a multilayered absorbing material.
Optionally, reflection phenomena and/or enhanced absorption phenomena
which can occur at an interface between two materials may also be
accounted for in the calculations using conventional techniques.
Exemplary System
[0279]An exemplary embodiment of a system according to the present
invention is shown in FIG. 23. For example, an input arrangement
2310 may be used to provide information to a computer 2330. Such
information can include, e.g., parameters associated with a particular
laser pulse or series of pulses, identification of a particular
absorbing material, thickness of a sample of such material, etc.
The input arrangement 2310 can include, but is not limited to, a
keyboard, a mouse, or an arrangement capable of reading information
from a computeraccessible medium such as, e.g., a hard drive, a
CDROM, a DVDROM, a flash memory, a network connection, etc.
[0280]The exemplary system may further include a database 2320,
which can also be configured to communicate with the computer 2330.
The database 2320 can include, for example, numerical parameters
that may be used to calculate an interaction between a nonlinear
absorbing material and an coherent electromagnetic wave as described
herein. The numerical parameters can include, e.g., incremental
depth, radius and/or time intervals that may be used when performing
calculations such as those described herein. The database 2320 can
also include material parameters including, but not limited to,
energy level diagram information, relaxation block parameters, absorption
block parameters, etc. Some or all of such material parameters may
optionally be provided to the computer 2330 using the input arrangement
2310.
[0281]The computer 2330 can include a processing arrangement, memory,
etc. It may be configured, e.g., to determine an interaction between
a nonlinear absorbing material and an coherent electromagnetic wave
using exemplary techniques described herein and shown, e.g., in
FIGS. 21 and 22a22b. Information associated with such interaction
may be communicated to an output device 2340. Such information can
include, e.g., intensity or electric field distributions as a function
of spatial position within the absorbing material and/or time, energy
level populations, and/or contributions of individual levels to
the total absorption. The output device 2340 may include, but is
not limited to, a video monitor, a printer, a data storage medium,
and the like. The computer 2330 can include a hard drive, CD ROM,
RAM, and/or other storage devices or media which can include thereon
software, which can be configured to execute the exemplary embodiments
of the method of the present invention. Such storage devices or
media may optionally contain the information associated with the
database 2320.
[0282]The foregoing merely illustrates the principles of the invention.
Various modifications and alterations to the described embodiments
will be apparent to those skilled in the art in view of the teachings
herein. It will thus be appreciated that those skilled in the art
will be able to devise numerous systems, arrangements and methods
which, although not explicitly shown or described herein, embody
the principles of the invention and are thus within the spirit and
scope of the present invention. In addition, all publications, patents
and patent applications referenced herein, to the extent applicable,
are incorporated herein by reference in their entireties.
