Is this "correction" to the Huygens Principle legitimate?
From what I understand, by spatio-temporal (meaning having both spatial and time-like properties) 'dipoles', the author is talking about introducing not ONE source of secondary wavelets on the wavefront (As Huygens' theory demands), but actually a twin source along with it, perpendicular to the wavefront, separated by a small distance (it is an analogy to electric dipoles, where you have 2 equal and opposite charges separated by a small distance). Now, both these sources produce near identical waveforms, except that they are OUT OF PHASE in such a manner that the phase difference caused by the time lag of the second source (since they are a distance $d$ apart, the second source requires an additional time $t=d/c$ to reach the first source), is such that the wave resultant in the backward direction is null; i.e. they are completely out of phase.
So, the idea is to replace a single point source by a 'dipole' of light sources; and arrange them so that the wave resultant to the left vanishes, leaving behind a forward wavelet. While this is a good approximation, he argues that taking the limit $d$ goes to $0$ (i.e., bringing together your dipole sources to a point source again), we get the exact situation Huygens' theory requires.
NOTE-(a) Don't get too worked up upon the 'sign' of 'dipole', he just shrugs it off as the sign of the source term in the wave equation. A mathematical idea, not very important physically.
(b) I didn't bother with most of his math, except the wave equation part, ( a bit too dull for my taste :P), so no comments on the math.
Hope this helps.