About the Bidimensional Beer-Lambert Law

Source: arXiv

ABSTRACT In acoustics, ultrasonics and in electromagnetic wave propagation, the
crossed medium can be often modelled by a linear invariant filter (LIF) which
acts on a wide-sense stationary process. Its complex gain follows the
Beer-Lambert law i.e is in the form exp [-\alphaz] where z is the thickness of
the medium and \alpha depends on the frequency and on the medium properties.
This paper addresses a generalization for electromagnetic waves when the beam
polarization has to be taken into account. In this case, we have to study the
evolution of both components of the electric field (assumed orthogonal to the
trajectory). We assume that each component at z is a linear function of both
components at 0. New results are obtained modelling each piece of medium by
four LIF. They lead to a great choice of possibilities in the medium modelling.
Particular cases can be deduced from works of R. C. Jones on deterministic
monochromatic light. keywords: linear filtering, polarization, Beer-Lambert
law, random processes.

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    ABSTRACT: In acoustic, ultrasonic or electromagnetic propagation, crossed media are often modelled by linear filters with complex gains in accordance with the Beer-Lambert law. This paper addresses the problem of propagation in media where polarization has to be taken into account. Because waves are now bi-dimensional, an unique filter is not sufficient to represent the effects of the medium. We propose a model which uses four linear invariant filters, which allows to take into account exchanges between components of the field. We call it bi-filter because it has two inputs and two outputs. Such a circuit can be fitted to light devices like polarizers, rotators and compensators and to propagation in free space. We give a generalization of the Beer-Lambert law which can be reduced to the usual one in some cases and which justifies the proposed model for propagation of electromagnic beams in continuous media. Comment: 26 pages, 2 figures
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