Please explain solution (TEM waves in spherical coordinates)

Question

I have a question and a solution, I don’t know how the solution came up with the variables and values highlighted in blue in the solution part.

Question:

Solution:

Answer 1

It's a flat, 2 dimensional plan wave. You have to transform the description to spherical co-ordinates.

By inspection, we see that phi is zero: (it's a plane wave so it's tangential to any point reference). and that the TEM wave angle is a multiple of 45 degrees: (it's got two components of equal magnitude at right angles).

So X and Y are the x:y magnitudes of the vector. z is the magnitude of the combined vector.

It's not a convention I'm familiar with, but our professor had realized that mathematically there was a simpler way of deriving and describing TEM waves that did not require the use of historical concepts like "wave vector", so we didn't spend a lot of time on "wave vectors".