ELECTROMAGNETIC THEORY OF LIGHT

ELECTROMAGNETIC THEORY
OF LIGHT

As we saw in the introductory chapter, integrated photonics devices are based on

optical waveguides with depth of the order of microns, comparable to the wavelength

of the optical radiation used in these devices (visible and near infrared). This fact

implies that the performance of the optical chips cannot be analysed in terms of ray

optics, but instead the light must be treated as electromagnetic waves. Therefore, the

electromagnetic theory of light is necessary to properly describe the behaviour of the

different optical elements that are present in any integrated photonics device. In some

cases, the vectorial nature of the electromagnetic waves can be simplified, and a scalar

treatment of the optical waves is often enough for a reasonable description of the

phenomena involved.

In this chapter we present the basics of the electromagnetic theory of light. We

derive the wave equation starting from Maxwell’s equations for light propagation in

free space, and then the wave equation in dielectric media is obtained, by introducing

the definition of refractive index. The solution for the temporal part of the wave

equation admits solution in the form of harmonic functions, which is then used to

derive a wave equation for monochromatic waves, where only the spatial dependence

of the electromagnetic field needs to be considered. The so-called Helmholtz equation

is indeed the starting equation for the analysis of optical waveguides. We then study

the properties of plane waves, as a particular solution of the Helmholtz equation, and

describe the behaviour of electromagnetic waves from the point of view of the vectorial

nature of the electric and magnetic fields, in terms of their polarisation character. Losses

in passive waveguides, as well as gain in active waveguides (lasers and amplifiers)

are also important topics when dealing with light propagation in optical waveguide

structures. To describe the behaviour of electromagnetic radiation in absorbing/gain

media in a general manner, we derive a more general wave equation by defining a

complex refractive index.

Optical waveguides are inherently inhomogeneous structures, in the sense that different

media with different optical constants are necessary to achieve light confinement.

The present chapter deals with the behaviour of light at dielectric interfaces,

and the reflected and transmitted waves are described by defining the reflection and

transmission coefficients, where the two types of incident waves (TE and TM polarised

waves) are examined separately. Also, the energy relations between incident, reflected

and transmitted waves are derived. Finally, the important phenomenon of total internal

reflection, being a key topic in the understanding and description of optical waveguides,

is discussed.