DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

QUANTIZATION OF LIGHT FIELD IN PERIODIC
DIELECTRICS WITH AND WITHOUT THE
COUPLED MODE THEORY
VLASTA PEŘINOVÁ AND ANTONÍN LUKŠ
The known form (separated variables) of modal functions of a rectangular cavity may be
more easily compared with the modal functions of a rectangular waveguide if a rectangular
waveguide of a finite length is considered (the usual periodic boundary conditions).
Perfect acquaintance with modal functions allows one to understand macroscopic quantization
of the electromagnetic field in a homogeneous or inhomogeneous medium.
Copyright © 2006 V. PeřinováandA.Lukš. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Waveguides are useful optical devices. An optical circuit can be made using them and
various optical couplers and switches. Classical theory of optical waveguides and couplers
has been elaborated in the 1970s [2], performance and the quantum theory have
been gaining importance. Recently, quantum entanglement has been pointed out as another
resource. Quantum descriptions may be very simple, but essentially, they ought
to be based on a perfect knowledge of quantization. By way of paradox, quantization is
based on classical normal modes. Therefore, it is appropriate to concentrate ourselves on
normal modes of rectangular mirror waveguide. It will be assumed that the waveguide is
filled with homogeneous refractive medium.
2. Classical description of the electromagnetic field
Vast literature has been devoted to the solution of the Maxwell equations and their value
for the wave and quantum optics cannot be denied. Depending on the system of physical
units used, the Maxwell equations have several forms. Let us mention only two of
them, appropriate to the SI units and the Gaussian units. The time-dependent vector
fields, which enter these equations, are E(x, y,z,t), the electric strength vector field, and
B(x, y,z,t), the magnetic induction vector field. In fact, other two fields are used, but
they can also be eliminated through the so-called constitutive relations. The so-called
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Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 925–934