# Optical Waves Propagation: Deterministic Approach

## Main Equations

The theoretical analysis of optical wave propagation, as a part of the whole electromagnetic spectrum [1—6] (see also Figure 1.1 in Chapter 1), is based on Maxwell’s equations [10—16]. In vector notation, their representations in the uniform macroscopic form are [1—6]:

Here E(r,t) is the electric field strength vector in volts per meter (V/m); H(r,t) is the magnetic field strength vector in amperes per meter (A/m); D(r,t) is the electric flux induced in the medium by the electric field in coulombs/m^{3} (this is why, in the literature sometimes, it is called an “induction” of an electric field); B(r,t) is the magnetic flux induced by the magnetic field in weber/m^{2} (it is also called “induction” of a magnetic field); j(r,*t)* is the vector of electric current density in amperes/ m^{2}; *p(r,t)* is the charge density in coulombs/m^{2}. The curl operator, Vx, is a measure of field rotation, and the divergence operator, V-, is a measure of the total flux radiated from a point.

It should be noted that for a time-varying EM wave field, Equations 2.3 and 2.4 can be derived from Equations 2.1 and 2.2, respectively. In fact, taking the divergence of (2.1) (by use of the divergence operator V-), one can immediately obtain (2.3). Similarly, taking the divergence of Equation 2.2 and using the well-known continuity equation [1—3,10—16]

one can arrive at Equation 2.4. Hence, only Equations 2.1 and 2.2 are independent.

Equation 2.1 is the well-known Faraday law and indicates that a time-varying magnetic flux generates an electric field with rotation; Equation 2.2 without the term *d**D**/dt* (displacement current term [10-13]) limits us to the well-known Ampere’s law and indicates that a current or a time-varying electric flux (displacement current [10-13]) generates a magnetic field with rotation.

Because one now has only two independent Equations 2.1 and 2.2, which describe the four unknown vectors E, D, H, B, three more equations relating these vectors are needed. To do this, we introduce relations between E and D, H and B, j and E, which are known in electrodynamics. In fact, for isotropic media, which are usually considered in problems of land radio propagation, the electric and magnetic fluxes are related to the electric and magnetic fields, and the electric current is related to the electric field, via the constitutive relations [10-13]:

It is very important to note that relations 2.6 are valid only for propagation processes in linear isotropic media, which are characterized by the three scalar functions of any point r in the medium: permittivity e(r), permeability j(r), and conductivity a(r). In relations 2.6 through 2.8, it was assumed that the medium is inhomogeneous. In a homogeneous medium, the functions e(r), j(r), and a(r) transform to simple scalar values e, j, and *a.*

In free space, these functions are simply the constants, that is, *e = e _{0} = *8.854 X 10

^{-12}Farad/meter,

*j = j*4п X 10

_{0}=^{-7}Henry/meter, and

*c =*((e

_{0}j

_{0}) is the velocity of light, which has been measured very accurately.

The system (2.1)-(2.4) can be further simplified if we assume that the fields are time harmonic. If the field time-dependence is not harmonic, then using the fact that Equation 2.1 are linear, we may treat these fields as sums of harmonic

components and consider each component separately. In this case, the time harmonic field is a complex vector and can be expressed via its real part as [10—13]:

where *i* = V— 1, *u* is the angular frequency in radians per second, *u = **2**nf f* is the radiated frequency in Hz = s^{—1}, and A(r,t) is the complex vector (E, D, H, B, or j). The time dependence *<^>e~ ^{iWt}* is commonly used in the literature of electrodynamics and wave propagation. If

*^e*is used, then one must substitute

^{iWt}*—i*for

*i*and

*i*for —i, in all equivalent formulations of Maxwell’s equations. In Equation 2.9,

*e*presents the harmonic time dependence of any complex vector A(r,t), which satisfies the relationship:

^{—,wt}

Using this transformation, one can easily obtain from the system (2.1)

It can be observed that system (2.11)—(2.14) was obtained from system (2.1)— (2.4) by replacing *d/dt* with *—iu.* Alternatively, the same transformation can be obtained by the use of the Fourier transform of system (2.1)—(2.4) with respect to time [1,2,10—15]. In Equations 2.11 through 2.14, all vectors and functions are actually the Fourier transforms with respect to the time domain, and the fields E, D, H, and B are functions of frequency as well. We call them phasors of time domain vector solutions. They are also known as frequency domain solutions of the EM field according to system (2.11)—(2.14). Conversely the solutions of system (2.1) are the time-domain solutions of the EM field. It is more convenient to work with system (2.11)—(2.14) instead of system (2.1)—(2.4) because of the absence of the time dependence and time derivatives in it.