107,887 research outputs found
Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws
We present a new approach to analyze the validation of weakly nonlinear
geometric optics for entropy solutions of nonlinear hyperbolic systems of
conservation laws whose eigenvalues are allowed to have constant multiplicity
and corresponding characteristic fields to be linearly degenerate. The approach
is based on our careful construction of more accurate auxiliary approximation
to weakly nonlinear geometric optics, the properties of wave front-tracking
approximate solutions, the behavior of solutions to the approximate asymptotic
equations, and the standard semigroup estimates. To illustrate this approach
more clearly, we focus first on the Cauchy problem for the hyperbolic systems
with compact support initial data of small bounded variation and establish that
the estimate between the entropy solution and the geometric optics
expansion function is bounded by , {\it independent of} the
time variable. This implies that the simpler geometric optics expansion
functions can be employed to study the behavior of general entropy solutions to
hyperbolic systems of conservation laws. Finally, we extend the results to the
case with non-compact support initial data of bounded variation.Comment: 30 pages, 2 figure
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
Geometric optics of whispering gallery modes
Quasiclassical approach and geometric optics allow to describe rather
accurately whispering gallery modes in convex axisymmetric bodies. Using this
approach we obtain practical formulas for the calculation of eigenfrequencies
and radiative Q-factors in dielectrical spheroid and compare them with the
known solutions for the particular cases and with numerical calculations. We
show how geometrical interpretation allows expansion of the method on arbitrary
shaped axisymmetric bodies.Comment: 12 pages, 6 figures, Photonics West 2006 conferenc
Gravitational helicity interaction
For gravitational deflections of massless particles of given helicity from a
classical rotating body, we describe the general relativity corrections to the
geometric optics approximation. We compute the corresponding scattering cross
sections for neutrinos, photons and gravitons to lowest order in the
gravitational coupling constant. We find that the helicity coupling to
spacetime geometry modifies the ray deflection formula of the geometric optics,
so that rays of different helicity are deflected by different amounts. We also
discuss the validity range of the Born approximation.Comment: 16 pages, 1 figure, to be published in Nuclear Physics
Semilinear geometric optics with boundary amplification
We study weakly stable semilinear hyperbolic boundary value problems with
highly oscillatory data. Here weak stability means that exponentially growing
modes are absent, but the so-called uniform Lopatinskii condition fails at some
boundary frequency in the hyperbolic region. As a consequence of this
degeneracy there is an amplification phenomenon: outgoing waves of amplitude
O(\eps^2) and wavelength \eps give rise to reflected waves of amplitude
O(\eps), so the overall solution has amplitude O(\eps). Moreover, the
reflecting waves emanate from a radiating wave that propagates in the boundary
along a characteristic of the Lopatinskii determinant. An approximate solution
that displays the qualitative behavior just described is constructed by solving
suitable profile equations that exhibit a loss of derivatives, so we solve the
profile equations by a Nash-Moser iteration. The exact solution is constructed
by solving an associated singular problem involving singular derivatives of the
form \partial_{x'}+\beta\frac{\partial_{\theta_0}}{\eps}, being the
tangential variables with respect to the boundary. Tame estimates for the
linearization of that problem are proved using a first-order calculus of
singular pseudodifferential operators constructed in the companion article
\cite{CGW2}. These estimates exhibit a loss of one singular derivative and
force us to construct the exact solution by a separate Nash-Moser iteration.
The same estimates are used in the error analysis, which shows that the exact
and approximate solutions are close in on a fixed time interval
independent of the (small) wavelength \eps. The approach using singular
systems allows us to avoid constructing high order expansions and making small
divisor assumptions
Plane SPDC-Quantum Mirror
In this paper the kinematical correlations from the phase conjugated optics
(equivalently with crossing symmetric spontaneous parametric down conversion
(SPDC) phenomena) in the nonlinear crystals are used for the description of a
new kind of optical device called SPDC-quantum mirrors. Then, some important
laws of the plane SPDC-quantum mirrors combined with usual mirrors or lens are
proved only by using geometric optics concepts. In particular, these results
allow us to obtain a new interpretation of the recent experiments on the
two-photon geometric optics.Comment: 12 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:0810.340
Quasi-isotropic approximation of geometric optics
Modified geometric optics method for solution of Maxwell equation
Supercritical geometric optics for nonlinear Schrodinger equations
We consider the small time semi-classical limit for nonlinear Schrodinger
equations with defocusing, smooth, nonlinearity. For a super-cubic
nonlinearity, the limiting system is not directly hyperbolic, due to the
presence of vacuum. To overcome this issue, we introduce new unknown functions,
which are defined nonlinearly in terms of the wave function itself. This
approach provides a local version of the modulated energy functional introduced
by Y.Brenier. The system we obtain is hyperbolic symmetric, and the
justification of WKB analysis follows.Comment: 29 pages. Some typos fixe
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