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We study temporal reflection of an optical pulse from the refractive-index barrier created by a short pump soliton inside a nonlinear dispersive medium such as an optical fiber. One feature is that the soliton’s speed changes continuously as its spectrum redshifts because of intrapulse Raman scattering. We use the generalized nonlinear Schrödinger equation to find the shape and spectrum of the reflected pulse. Both are affected considerably by the soliton’s trajectory. The reflected pulse can become considerably narrower compared to the incident pulse under conditions that involve a type of temporal focusing. This phenomenon is explained through space–time duality by showing that the temporal situation is analogous to an optical beam incident obliquely on a parabolic mirror. We obtain an approximate analytic expression for the reflected pulse’s spectrum and use it to derive the temporal version of the transformation law for the parameter associated with a Gaussian beam.
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On the Maxwell‐Bloch system in the sharp‐line limit without solitons
Abstract We study the (characteristic) Cauchy problem for the Maxwell‐Bloch equations of light‐matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically‐motivated initial‐boundary conditions are satisfied. In particular, we present a proper Riemann‐Hilbert problem that generates the uniquecausalsolution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading‐order asymptotics to self‐similar solutions that satisfy a system of ordinary differential equations related to the Painlevé‐III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self‐similar solutions. We make this observation precise and for the first time we relate the PIII self‐similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially‐unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.
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- PAR ID:
- 10509348
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 1
- ISSN:
- 0010-3640
- Page Range / eLocation ID:
- 457 to 542
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/cpa.22136
