| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 26, 2016 |
| Latest Amendment Date: | July 26, 2016 |
| Award Number: | 1615859 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Pedro Embid
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2016 |
| End Date: | July 31, 2021 (Estimated) |
| Total Intended Award Amount: | $234,997.00 |
| Total Awarded Amount to Date: | $234,997.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
110 8TH ST TROY NY US 12180-3590 (518)276-6000 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
NY US 12180-3522 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This collaborative project expands the research programs of the Principal Investigators on mathematical models of optical phenomena. It comes in response to the NSF initiative on "Optics and Photonics". The interaction between light and optical media is one of the most fruitful areas of study in applied physics and provides the basic mechanism underlying devices such as lasers and optical amplifiers. For decades, it has been providing a rich source of new physical phenomena, among the latest being "slow light", the recently-observed slowing-down of light pulses to the speed of a bicycle. Slow light can potentially be used in devices such as optical memory. This project is aimed at understanding the physical mechanisms underlying the slow light phenomenon by using a remarkable, highly accurate mathematical model that can be solved with explicit formulas. The validity of this model and its explicit solutions will be verified using numerical simulations of more realistic models and careful comparisons with experiments. Interdisciplinary training in applied mathematics and nonlinear optics will be provided to graduate and undergraduate students, and a lively, challenging research and training environment for both student groups will be established.
The slowdown of light pulses is modeled as the interaction between an optical pulse and an active medium with two or three working levels, the latter being a prototypical case known as the Lambda configuration. This interaction is described by completely integrable Maxwell-Bloch equations with non-vanishing boundary conditions, a new twist. This project is a mathematical study of novel dynamics generated by the interaction of light with two-level media and the Lambda-configuration medium, and includes: (i) developing a systematic, completely integrable theory of the dynamics for the two-level and Lambda-configuration Maxwell-Bloch equations with non-zero boundary conditions, (ii) using the analytical results of step (i) to describe phenomena related to slow light, (iii) numerical studies of dynamical phenomena in more general cases in which the two-level and Lambda-configuration Maxwell-Bloch equations are not integrable. The completely-integrable description of slow light involves two new aspects: (1) non-zero boundary conditions, (2) non-trivial evolution of the spectral data. The understanding of the first aspect will be extended from the Nonlinear Schroedinger equation to the Maxwell-Bloch equations by studying scattering and inverse-scattering problems with the spectral parameter on a Riemann surface. The second aspect is complicated by the presence of the former and involves a careful derivation of how spectral data evolves from the initial state of the medium and finding correct cancellations of highly oscillatory terms. In addition to generating new models and descriptions of the dynamics exhibited by light interacting with active optical media, the project will advance the theory of completely integrable systems.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
In the recent 20 or so years, experiments have measured bullets of light that would propagate at the speed of a bicycle. This incredibly slow speed was observed using laser beams propagating in media such as rarefied gasses and crystals. Slow and stopped light has potential applications in optical memory. Mathematically, light propagation in pertinent media is described by special sets of equations called Maxwell-Bloch equations. In certain cases, such as when losses are small, these equations can, in some sense, be solved exactly. When losses are present or even large, they can be solved numerically. In this work, we have undertaken the task of trying to understand, quantify, and predict slow light propagation mathematically. In one experiment, the researchers injected a light pulse into a ruby crystal, and it traveled slowly through it. The question we answered was: did it begin traveling at the speed of light (such as it does in the air) and then slowed down to a crawl as it traveled through the crystal, or did it already start traveling slowly. Our numerical simulations clearly show that the answer is the latter. For other cases, we developed new exact solutions of the Maxwell-Bloch equations, which look like one or more light pulses. Most of them were not computed or observed before, and some of them traveled slowly. In addition, we developed novel methods for simulating Maxwell-Bloch equations, and also described dispersion properties waves emerging in optics.
Last Modified: 10/01/2021
Modified by: Gregor Kovacic
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