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Since I first studied the Huygens Principle, the fact that you had to ignore the backward waves arbitrarily has really been bugging me. So I have come across this article (https://www-ee.stanford.edu/~dabm/146.pdf), which acknowledges that issue, and offers a solution by using spatiotemporal dipoles instead of point sources. Is anyone familiar with this article? In this case, can someone explain to me how it works, without using overly complex language, if possible? (English is not my first language)

I tried to understand the article the best I could, but I'm not able to.

Also, what is a spatiotemporal dipole to begin with? Is there a visual representation? And finally, if this article is not right, what is the best explanation for the absence of backward propagating waves?

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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.

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    $\begingroup$ thanks! but I still can't quite picture how it would look like. I'm trying to arrange dipole sources on a Ripple Tank simulator (falstad.com/ripple) so that we get this pattern mentioned by the author, but I just can't (I have selected "Example: Dipole Source" and right clicked on the source do duplicate it multiple times). It still creates backward waves. Thanks anyway. $\endgroup$
    – user137288
    Mar 15, 2017 at 12:55
  • $\begingroup$ @RobertoValente Glad to help. Unfortunately, optics is not my favourite area, so I'm afraid this is all the help I can give you. All the best anyways. $\endgroup$
    – GRrocks
    Mar 15, 2017 at 13:03
  • $\begingroup$ GRocks, are you still around this forum? By the time I asked you this question I could barely understand the article, but now I’m making more sense of it! I’d like to ask one little question however, so are you still here? lol $\endgroup$
    – user137288
    Oct 26, 2017 at 20:37
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So I'm not so sure about whether this is right or not but my book says that more rigorous wave theory (yes, books at my level like to say fancy stuff because I'm just in high school) proves that there is no backward flow of energy during the propagation of a wave. It says that it can be shown mathematically (how it can be done is something that I do not know) that the amplitude of secondary wavelets is proportional to (1+cosθ) where θ is the angle between the ray at the point of consideration and the direction of the secondary wavelets. For a backward wave cosθ will will amount to -1 (as θ is π) and hence the resultant amplitude at any point on the hypothetical backward wavefront would be 0. So a backward wavefront cannot exist.

I'm extremely sorry if the answer sounds naive.

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Miller's expression (6) needs an additional term $1/a$ inside the braces, where $a$ is the radius of the spherical primary wavefront. This term becomes significant if the radius is not very large compared with $1/k$.  In a note called "A tautological theory of diffraction" (in Section 6, which isn't tautological!), I have reworked the derivation to include the additional term. I suspect, and Miller's endnote 12(iii) seems to imply (unbeknown to him), that his spatiotemporal dipoles already yield the $1/a$ correction term; but I am still seeking independent confirmation.*

Whatever may be said about the spatiotemporal dipoles, there is no reasonable doubt about the $1/a$ term itself. I have derived it in two different ways, and it is confirmed by Baker and Copson (cited in my note).

* Update (6 December 2022): No, contrary to my initial and long-held opinion, Miller's spatiotemporal dipoles don't yield the $1/a$ term, unless they are modified by attenuating the inverting monopole. One can also modify the delay of the inverting monopole to allow a surface of integration that is not a primary wavefront. See "Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math", especially the section on "Generalized spatiotemporal-dipole secondary sources".

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