Scalar Optics in the Paraxial Approximation (UVic optics, week 2b)

In this lecture, we reduce the multidimensional vector waves studied thus far to a scalar function of one variable. To to this, we apply the approximation that all of the propagation takes place near a fixed direction, given by the optical axis.

[@jamesdennis6120]

Professor Andy Mac is boss.

[@sammyapsel1443]

So in real life, when a EM wave is transmitted say from an atom, it's a spherical wave. And according to the hairy ball theorem, if we inspect the E vector field at some distance r from the atom , there will be some E field somewhere on this shell which is also in the r direction?

[@AndrewMacRae]

Good question! It is actually a dipole pattern. When it is far away from the source the wave-fronts are indeed spherical, BUT the intensity is not constant - it goes to zero along the poles. The point where the polarization is undefined is right where the intensity vanishes. Thus the HB theorem is not violated! A bunch of atoms (that are not phase stable) will produce an incoherent mix of the spherical wavefronts with no steady polarization, but a spherical shape. A good pic of the dipole pattern is, for example, here: www.everythingrf.com/community/what-is-the-difference-between-a-monopole-and-dipole-antenna