Reverse engineering formulas for directing a plane wave

[yefj]

Homework Statement

plane wave

Relevant Equations

plane wave

Hello,There is a model which tunes a plane wave using certain expressions which resembles the spherical to cartesian coordinates.
There are two types of definitions:
propogation normal and electric field vector.
Why they put in propogation normal exactly the spherical to cartesian to represent propogation normal?
What is the link between propogation normal and E-field formulas?
Thanks.

 

[kuruman]

Why they put in propogation normal exactly the spherical to cartesian to represent propogation normal?

It looks like you have a spherical wave. The direction of propagation is radially out and a wavefront is a sphere of radius $R$. This means that the direction of propagation at any point on the sphere is radially out, i.e. normal to the surface. A unit vector normal to the surface of a sphere is $$\mathbf{\hat n}=\sin\!\theta\cos\!\phi~\mathbf{\hat x} +\sin\!\theta\sin\!\phi~\mathbf{\hat y}+\cos\!\theta~\mathbf{\hat z}$$ which is the same as the radial unit vector $\mathbf{\hat r}$ along which the wave propagates.