# Propagation of Optical Waves in Free Space

Mathematically, optical wave propagation phenomena can be described by use of both the scalar and vector wave equation presentations. Because most problems of optical wave propagation in wireless communication links are considered in unbounded, homogeneous, source-free isotropic media, we can, with a great accuracy, consider e(r) = e, i(r) *=* ц, a(r) = *a,* and finally obtain from general wave equations:

Because both equations are symmetric, one can use one of them, namely that for E, and by introducing the vector relation V x V x E = V(V ? E) — V^{2}E and taking into account that V-E = 0, we finally obtain

where *k ^{2} = и^{2}ец*.

In special cases of a homogeneous, source-free, isotropic medium, the threedimensional (3D) wave equation reduces to a set of scalar wave equation. This is because in Cartesian coordinates, E(r) = *E _{>}x_{0} + E_{y}y_{0} + E_{z}z_{0},* where x

_{0}, y

_{0}, z

_{0 }are unit vectors in the directions of the

*x, y, z*coordinates, respectively. Hence, Equation 2.16 consists of three scalar equations such as

where Ф(г) can be either E_{x}, *E _{y},* or E

_{z}. This equation fully describes propagation of optical waves in free space.