HiWhen the concept of polarization is introduced, it is usually

Re: Polarization

If the electromagnetic wave is transverse, as is the case in free space, then the electric field, magnetic field, and propagation vector of the wave will all be mutually perpendicular. The "polarization" of the wave just describes the direction that the electric field vector points. No matter if a transverse wave is planar, cylindrical, spherical, or whatever, the fields are still perpendicular and polarization still means the same thing. The words "planar", "cylindrical", "spherical", etc. just describe the shape of the wave's wavefront, i.e. the surface over which the field strength is constant. This is a separate concept from polarization. Similarly, "Gaussian" refers to the overall waveshape along its wavefront and is a different concept from polarization.

For example, a line antenna (think radio tower) radiates vertically linearly polarized spherical waves. This means that the waves spread out in all directions - in expanding spheres - but that at any one point in the wave, the magnetic field, electric field, and propagation vector still form an orthogonal triple, with the electric field mostly vertical, and the magnetic field horizontal (depending on the direction of propagation).

 

Re: Polarization

Yes. But you have to remember that in general, the electric field vector has three components, so the total general solution is three sums of plane waves (weighted by coefficients), one in each direction. This is what makes it possible in principle to expand a spherical wave into a sum over plane waves. Plane sinusoidal waves are just used as the particular solutions because they are convenient. We could just as easily express the general solution as a sum over spherical waves, or cylindrical waves, or any complete set of orthogonal functions that obey the wave equation.

 

Re: Polarization

For a spherical wave:

E = E₀ (1/r) e^i(kr-ωt)

The e^i(kr-ωt) part defines it has a sinusoidal traveling wave. Because it contains kr where r is the radial direction of spherical coordinates, instead of the usual kx, that means the peaks of the sine wave are traveling in the r direction, along ever-expanding spheres centered at the origin.

The (1/r) part ensures that energy is conserved. As the wave spreads out in spheres, its strength must diminish because the same energy is smeared over ever-larger areas. For a given radius r, the magnitude of the original expression is constant. This means that the wavefront is spherical

The E₀ part is the polarization vector. It has a magnitude and direction. For free waves, the direction is perpendicular to the direction of propagation and depends on the specific polarization (circular, linear, etc.) prepared. Its magnitude is the peak strength of the field.