# Basic Aspects of Optical Wave Propagation

## Identity of Optical and Radio Waves

Electromagnetics plays a key role in modern radio and optical communication systems, including optical telecommunication systems and LIDAR [1—16]. Therefore, optical communication emphasizes the electromagnetic phenomena described mathematically by Maxwell’s unified theory [1—3,10—14], which we will consider in detail shortly. We should stress here that optical communications started to be investigated just after the invention of the laser in 1960. Atmospheric propagation of optical waves was investigated parallel to improvements in optical lasers and understanding of the problems of optical wave generation in both time and space domains. Thus, solving problems with weather, line-of-site (LOS) clearance, and beam broadening and bending in the real atmosphere contributed toward removal of free-space optical communication as a major aspect in wireless communication.

Since optical waves have the same nature as electromagnetic waves (see Figure

1.1 in Chapter 1), we will start with a physical explanation of electromagnetic waves based on Maxwell’s unified theory [1,2,10—13], which postulates that an electromagnetic field could be represented as a wave. The coupled wave components, the electric and magnetic fields, are depicted in Figure 2.1, from which it

Figure 2.1 Optical wave as an electromagnetic wave with its electrical and magnetic components, wavefront, and direction of propagation.

Basic Aspects of Optical Wave Propagation ? 29

follows that the electromagnetic (EM) wave travels in a direction perpendicular to both EM field components. In Figure 2.1, this direction is denoted as the z-axis in the Cartesian coordinate system by the wave vector k. In their orthogonal space- planes, the magnetic and electric oscillatory components repeat their waveform after a distance of one wavelength along the у-axis and x-axis, respectively (see Figure 2.1).

Both components of the EM wave are in phase in the time domain, but not in the space domain [1,2,10—13]. Moreover, the magnetic component value of the EM field is closely related to the electric component value, from which one can obtain the radiated power of the EM wave propagating along the z-axis (see Figure 2.1).

At the same time, using the Huygen’s principle, well known in electrodynamics [10—13], one can show that an optical wave is the electromagnetic wave propagating only straightforward from the source as a ray with minimum loss of energy and with minimum propagation time (according to Fermat’s principle [2,7,14]) in free space, as a unbounded homogeneous medium without obstacles and discontinuities.

Thus, if we present the Huygen’s concept, as it is shown from Figure 2.2, the ray from each point propagates in all forward directions to form many elementary spherical wavefronts, which Huygens called wavelets.

The envelope of these wavelets forms the new wave fronts. In other words, each point on a wave front acts as a source of secondary elementary spherical waves,

Figure 2.2 Huygens principle for a proof of straight propagation of waves as rays.

described by Green’s function (see References 10—13). These waves combine to produce a new wavefront in the direction of straight propagation. As we will show below, each wave front can be represented by the plane that is normal to the wave vector k (e.g., wave energy transfer). Moreover, propagating forward along straight lines normal to their wave front, waves propagate as light rays in optics, spending minimum energy for passing from the source to any detector, that is, the maximum energy of the ray is observed in a straight direction normal to the wave front (as seen from Figure 2.2). Kirchhoff first mathematically showed this principle based on Maxwell’s general unified theory. Let us now assess all propagation phenomena theoretically using Maxwell’s unified theory.