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Software Patent Abstract
The exemplary embodiments of the method, system, software arrangement
and computer-accessible 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 computer-accessible 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
CROSS-REFERENCE 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 multi-photon 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), three-dimensional
("3D") micro- and/or nano-fabrication, fluorescent imaging
systems, optical limiters, optical storage, semiconductor nano-sized
probes, etc.
[0005]Conventional experiments that may be used to characterize
the optical properties of nonlinear materials such as multi-photon
organic/inorganic materials, semiconductors, fluids, gases or nanostructured
materials include, e.g., z-scan procedures, optical transmission
techniques, and pump-probe 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, 1103-1182 (1982). Shorter
pulsed lasers and multi-photon 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., McGraw-Hill, New York, 2001, vol.
IV, pp. 25-31. 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 single-photon 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 laser-absorber
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 laser-absorber interaction. Defining
new algorithms and writing new numerical codes to describe such
absorption interactions can involve, e.g., months or years of effort.
[0012]Multi-photon-absorbing 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, 299-305 (1993); J. E. Rogers et al., "Understanding
the one-photon photophysical properties of a two-photon absorbing
chromophore," J. Phys. Chem. A 108, 5514-5520 (2004); J. W.
Perry, "Organic and metal-containing 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. 813-839; M. J. Potasek et al., "All optical
power limiting," J. Nonlinear Optical Physics and Materials
9, 343-365 (2000); M. J. Potasek, "High-Bandwidth Optical Networks
and Communication, Photodetectors and Fiber Optics ed. H. S. Nalwa
(Academic Press, 2001) pp. 459-543; D. I. Kovsh et al., "Nonlinear
Optical Beam Propagation for Optical Limiting," Appl. Opt.
38, 5168-5180 (1999); and W. Jia et al., "Optical limiting
of semiconductor nanoparticles for nanosecond laser pulses,"
Appl. Phys. Lett. 85, 6326-6328 (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 two-photon
absorption cross section of green fluorescent protein," J.
Phys. Chem. B 105, 2867-2873 (2001), and three-dimensional microfabrication
procedures such as those described in, for example, S. Maruo et
al., "Two-photon-absorbed near-infrared photopolymerization
for three-dimensional microfabrication," J. Microelectromechanical
Systems 7, 411-415 (1998); B. H. Cumpston et al., "Two-photon
polymerization initiators for three-dimensional optical data storage
and microfabrication," Nature 398, 51-54 (1999); and G. Witzgall
et al., "Single-shot two-photon exposure of commercial photoresist
for the production of three-dimensional structures," Opt. Let.
23, 1745-1748 (1998).
[0014]Further applications of photon absorbing materials may include
fluorescent imaging systems such as those described in W. Denk et
al., "Two-photon laser scanning fluorescence microscopy,"
Science 248, 73-76 (1990), and optical storage systems as described,
for example, in H. E. Pudavar et al., "High-density three-dimensional
optical data storage in a stacked compact disk format with two-photon
writing and single photon readout," Appl. Phys. Lett. 74, 1338-1340
(1999); and in P. N. Prasad, "Emerging opportunities at the
interface of photonics, nanotechnology and biotechnology,"
Mol. Cryst. Liq. Cryst. 415, 1-10 (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 multi-photon absorber ("MPA")
materials in which two or more photons may be absorbed simultaneously.
For examples, the materials that exhibit a large two-photon 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
two-photon absorption cross sections," Science 281, 1653-1656
(1998); and B. A. Reinhardt et al., "Highly active two-photon
dyes: Design, synthesis, and characterization toward application,"
Chem. Mater. 10, 1863-1874 (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 stilbene-chloroform solution, explained by the superposition
of two-photon absorption and one-photon absorption from the excited
state," Chem. Phys. Lett. 24, 133-135 (1974). Nonlinear transmission
measurements and Z-scan 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
two-photon absorption cross-section," Opt. Comm. 191, 235-243
(2001); and S. Guha et al., "Third-order optical nonlinearities
of metallotetrabenzoporphyrins and a platinum poly-yne," Opt.
Lett. 17, 264-266 (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 two-photon-absorption spectral
studies of highly two-photon active organic chromophores,"
J. Chem. Phys. 120, 5275-5284 (2004); and R. Kannan et al., "Toward
highly active two-photon absorbing liquids. Synthesis and characterization
of 1,3,5-triazine-based octupolar molecules," Chem. Mater.
16, 185-194 (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 five-level
absorption model which is described, for example, in R. L. Sutherland
et al., "Excited state characterization and effective three-photon
absorption model of two-photon-induced excited state absorption
in organic push-pull charge-transfer chromophores," J. Opt.
Soc. Am. B 22, 1939-1948 (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 three-photon 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 three-photon absorption of two
stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (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 laser-matter 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 sub-sets of laser-material 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 above-described 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 computer-accessible 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 laser-generic 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 short-pulsed laser beam in a multi-level multi-photon
absorbing material can be evaluated, where the propagation is determined
to be in the presence of multi-photon 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 multi-photon 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 photon-induced 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.0-B.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.60-toluene 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 sub-sampling 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 non-radiative 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.0t-k.sub.0z)]+c.c.;
P.sub.c(z,r,t)={tilde over (P)}(z,r,t) exp [-i(.omega..sub.0t-k.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 real-valued 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, self-steepening, 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 two-level 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 electric-dipole 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 electric-dipole operator, which can further include
an assumption that the laser is linearly polarized, and d and E
may each be aligned along an x-axis.
[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, 3761-3774 (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 semi-classical 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, 1792-1806 (1994); M. Lax, "Quantum
noise IV. Quantum theory of noise sources," Phys. Rev. 145,
110-129 (1966); and A. Barchielli, "Measurement Theory and
stochastic differential equations in quantum mechanics," Phys.
Rev. A 34, 1642-1649 (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 non-resonant 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 density-matrix 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 off-diagonal 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 density-matrix
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.1-g.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., "Excited-state
nonlinear absorption in multi-energy-level molecular systems,"
Phys. Rev. A, 51, 569-575 (1995). Utilization of a density matrix
approach to investigate a pulse width dependence of the TPA cross-sections
of PRL-101 measured in the ns and fs regions is described, e.g.,
in A. Baev et al., "General theory for pulse propagation in
two-photon active media," J. Chem. Phys. 117, 6214-6220 (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., N-photon absorbers with both singlet and triplet levels.
[0096]Five exemplary types of absorption mechanisms can be used
to model absorption behavior. These mechanisms 100-140 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.0-N.sub.4, and absorption
cross-sections can be labeled with a .sigma. identifier. As shown
in FIG. 1, upward arrows may represent photo-excitation transitions,
and downward arrows may represent intersystem electron decay events.
In accordance with appropriate exemplary selection rules, single-photon
absorption can occur along singlet-singlet transitions from a ground
state 100 and/or a lowest excited state 110. Single-photon absorption
can also occur along a triplet-triplet 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 singlet-singlet transition 110, or
a singlet-triplet transition 120 from N.sub.1 to N.sub.3. Three-photon
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 100-140 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.60-toluene solution--a nonlinear RSA material--can
be described using a five-level model as described, e.g., in D.
G. McLean et al., "Nonlinear absorption study of a C60-toluene
solution," Opt. Lett. 18, 858-860 (1993). This exemplary model
can be obtained by combining the absorption diagrams 100-120 as
shown in FIG. 1 (e.g., B.sub.0.orgate.B.sub.1.orgate.B.sub.2).
[0099]A five-level absorption model of a chromophore from an AFX
group exhibiting TPA-assisted 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 110-130 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.0-B.sub.4 100-140 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, two-photon .sigma..sub.TPA,
three-photon .sigma..sub.3PA, and, possibly, N.sub.A-photon .sigma..sub.[N.sub.A.sub.]PA
molar cross-sections 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) S-dimensional 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 steady-state estimates of population densities is
described, e.g., in D. G. McLean et al., "Nonlinear absorption
study of a C60-toluene solution," Opt. Lett. 18, 858-860 (1993).
An analytic solution of a three-level approximation for the five-level
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, 1884-1894 (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 Runge-Kutta numerical solution
may also be used such as that described, e.g., in I.-C. Khoo et
al., "Passive optical limiting of picosecond-nanosecond laser
pulses using highly nonlinear organic liquid cored fiber array,"
IEEE J. Sel. Top. Quantum Electron. 7, 760-768 (2001). A beam-propagation
technique used to model RSA CAP dye in toluene and TPA ZnSe is described,
e.g., in S. Hughes et al., "Modeling of picosecond-pulse propagation
for optical limiting applications in the visible spectrum,"
J. Opt. Soc. Am. B. 11, 2925-2929 (1997). Other exemplary solution
procedures that may be used can include, for example, spectral and
Crank-Nicholson finite difference methods which can included an
instantaneous Kerr effect, diffraction, thermal effects for RSA
SiNc, and Z-scan 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 integro-differential equation for short
pulses to model general TPA+singlet-singlet 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 ground-state 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 quasi-steady-state 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) singlet-triplet 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 transform-limited nonlinear
pulse compression techniques. The pulses may be created using a
continuous wave laser by applying an external modulator such as,
e.g., an electro-optic 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 self-focusing.
[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., mode-locking, Q-switching, or Q-switched mode-locking.
External (laser) cavity procedures may also be used such as, e.g.,
an electro-optic 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 sub-range 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 spatially-dependent 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 time-resolved radially-dependent finite-difference
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 spatially-resolved 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 second-order 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 sub-sample
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.i-1/2.fwdarw..tau..sub.i+1/2
in Eq. (29), the magnitude of the matrix elements in the exponent
can be evaluated. If necessary, M-1 additional time sub-samples
can be introduced such as, e.g., .tau..sub.i-1/2=.tau..sup.0.fwdarw..tau..sup.1.fwdarw.
. . . .tau..sup.M-1.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 sub-samples 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 sub-sampling time procedure. Because any
such sample {circumflex over (.tau.)}.sup.m can be approximately
equal to .tau..sub.i-1/2+(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.i-1/2) 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.i-1/2, .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 sub-sampling 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 sub-ranges
[.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 sub-sampling 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
sub-sampling time on the temporal grid 900.
[0121]An exemplary derivation of the sub-sampling 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 sub-sampling 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 opto-electronic 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,i-1/2}
as a product of individual intensity reductions .LAMBDA..sub.s;.zeta.,
each due to energy levels s with nontrivial absorption cross-sections.
This total intensity reduction may be written as:
.LAMBDA. = s .di-elect 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 ' .di-elect 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.60-toluene 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 two-photon 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 experimentally-observed 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 s-th 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 .di-elect cons. [ .tau.
0 , .tau. 1 ] p ^ s ; { n , j , i } . ( 54 )
EXAMPLE
C.sub.60-Toluene Solution
[0133]The nonlinear material C.sub.60 can be described as a reverse
saturable absorber--a 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 multiphoton-absorbing 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 well-represented 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.
TABLE-US-00001 TABLE 1 Parameters for exemplary multiphoton absorbing
materials. Material/experimental C.sub.60-toluene 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., "Nonlinear-absorbing
fiber array for large-dynamic-range 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 triplet-triplet 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. 3a-c 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. 3a-3c
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.
3d-3f 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. 3d-3f can correspond
to the conditions in FIGS. 3a-3c, 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 triplet-triplet 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 rate-propagation equations, from the
measured data.
TABLE-US-00002 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 sub-columns) 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 AF455--a material which
exhibits two-photon assisted ESA--together 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 110-130 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. 4a-4f show exemplary graphs of absorption in AF455
of a ns scale pulse. FIGS. 4a-4c 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. 4a-4c 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. 4d-4f 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.
4d-4f correspond to the conditions of FIGS. 4a-4c, 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. 4a-4b suggest
a very small depletion of the ground level. This observation supports
the validity of a steady-state 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 steady-state 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 hi |