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Is Huygens Principle just a fundamental way to understand light? It always seemed to me that it was somehow "derived" or that it should be-but is it simply a well-founded theory?

Qmechanic
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  • It's not a quantum theory for one, so it can never be fundamental in the literal sense... – Danu Dec 11 '13 at 08:01
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    Good question. I'd say it's a model that happens to fit rather than derived from anything more fundamental. There's an interesting article discussing some of these issues at http://www.mathpages.com/home/kmath242/kmath242.htm – John Rennie Dec 11 '13 at 08:48
  • @JohnRennie: What do you mean by "rather than derived from anything more fundamental"? If it's more coarse than anything presumably fundamental, then I guess it, as a model, is derivable. At Danu: So quantum theory is necessarily fundamental? At Anonymous: Once one settles on fields and writes them down as propagators, I'd intuitively assume that the principle is equivalent to the requirement of some linearity of amplitude composition. – Nikolaj-K Dec 11 '13 at 08:55
  • Isn't it a statement about being able to decompose any wave on a basis of spherical harmonics? – gatsu Dec 11 '13 at 10:14
  • @gatsu Yes it is. And a rather useful statement as well! – Carl Witthoft Dec 11 '13 at 12:44
  • see https://www.researchgate.net/publication/316994209 for some geometric intuition into HP. – user45664 May 19 '17 at 22:24

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Actually, it can be theoretically derived from D'Alembert equation (that is satisfied by each component of ${\bf E}$ and ${\bf B}$ in absence of sources in view of free Maxwell equations). The idea is to compute the field (any component of ${\bf E}$ or ${\bf B}$) in $p$, when it is generated by a spherical point source localized in $q$ emitting a spherical monochromatic field with fixed (scalar) wavenumber $k$, and between $q$ and $p$ there is a screen with an aperture of known area. The mathematical tool is an integral formula, due to Kirchoff, which produces the solution in $p$ when it is known the value of the field and its normal derivative on a surface surrounding $p$. The surface is chosen to have a part adapted to the screen, including the aperture, and the remaining part is taken far away from $p$. Here, i.e., to fix the value of the field and its normal derivative on the surface, some approximations enter the computation and they usually have physical sense for $k >> d$, where $d$ is the "diameter" of the aperture. This situation is discussed in details in Jackson's textbook. The final formula obtained this way can be shown to be equivalent to directly apply Huygen's principle from scratch.

Valter Moretti
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    Isn't this simply a Green's function method to find the effect of the source? – Abhinav Dec 11 '13 at 13:04
  • I am not an expert on these issues, but I think that certainly the Green's function approach is the starting point. However you next have to perform some appropriate approximation. Birchoff's name here is, in fact, related with the appropriate choice of approximated boundary conditions... – Valter Moretti Dec 11 '13 at 13:46
  • @Abhinav not quite: it's more centred on deriving a field from its values on a boundary. The whole topic is thoroughly treated (if a bit archaically) in Chapter 8 of Born and Wolf "Principles of Optics"; archaic notation and treatment notwithstanding, it's still a great deal more lucid than Jackson IMO. – Selene Routley Dec 11 '13 at 13:49
  • Yes sure, for "boundary conditions" I actually intended "values on a boundary" (the values of the field and its normal derivative on the surface where one integrates and where ranges one argument of the Green function). Thanks for the reference! – Valter Moretti Dec 11 '13 at 13:52
  • Is D'lambert's equation fundamental? Or is it from Maxwell's equations? Those are derived, right? –  Dec 11 '13 at 17:58
  • @Anonymous: D'Alembert equation is the most elementary equation describing non-dispersive waves (the speed of wave propagation does not depend on the frequency of the wave) http://en.wikipedia.org/wiki/Wave_equation. It can be derived, in various contexts, independently from Maxwell equations. It describes, for instance pressure waves in gases and liquid waves, linear waves propagating along mechanical strings and so on. Electromagnetic fields also verify that equation as a byproduct of Maxwell equations in the vacuum. Even weak gravitational waves do! – Valter Moretti Dec 11 '13 at 18:11
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    Actually, at least in classical physics, Maxwell's equations could be considered fundamental equations, they summarize all properties of EM fields, at least in classical (not quantum) physics. They were, more or less, derived by experiments by physicists (not only Maxwell). – Valter Moretti Dec 11 '13 at 18:20
  • I also read that it Huygen could not explain why each of these new point sources only went forward, is there a reason? –  Dec 11 '13 at 23:44
  • Also what do you mean in $p$ and $q$ @VM9? –  Dec 12 '13 at 00:20
  • @Anonymous Actually no. The forward travelling nature s something one must in general put into solutions "by hand". For example, if we know a solution to the wave equation (in a scalar theory as studied by Huygens, Fresnel and so forth) on a plane, the solution is ambiguous: the phases of the constituent plane waves vary like $\exp(i,\vec{k}\cdot\vec{r})$ across the plane and we can't tell from $\vec{k}\cdot\vec{r}$ whether the wavevector has positive or negative $z$ component (I'm here taking the $z$ direction as that in the nominal propagation direction). So you have to assume that .... – Selene Routley Dec 12 '13 at 00:27
  • @Anonymous ... all the $z$-components are positive if you choose to study "forward" travelling waves. If you do this, then you do get a $\cos\theta$ term in the farfield from summing the Huygens propagator over the wavefront by Huygen's principle. For the full vector field $(\vec{E},\vec{H})$, however, you can indeed unambiguously tell the direction of propagation for all the plane wave components and the "tendency to keep going forwards" will be conserved by Maxwell's equations. – Selene Routley Dec 12 '13 at 00:31
  • I think I follow... Also do you now where in Jackson's book he talks about this? I'm a little confused by what $p$ and $q$ are in the answer given. –  Dec 12 '13 at 00:34
  • @Anonymous $q$ is the fixed position of a spherical source emitting spherical waves with wave number $k$, $p$ is any point on the other side of the screen where you want to compute the field. The screen separates $q$ from $p$. There is an aperture on the screen: the wave emitted in $q$ reach $p$ passing through the aperture. – Valter Moretti Dec 12 '13 at 07:28
  • Concerning Jackson's book, unfortunately I have not it now. I just wrote, in my answer, what I was able to remember from some lectures I attended when I was a physics student many many years ago! The book we used was Jackson's one. Perhaps I am taking a mistake. – Valter Moretti Dec 12 '13 at 07:31
  • Look here http://en.wikipedia.org/wiki/Kirchhoff%27s_diffraction_formula there is also the reference pointed out by WetSavannaAnimal aka Rod Vance: Chapter 8 of Born and Wolf "Principles of Optics" – Valter Moretti Dec 12 '13 at 07:33
  • The first figure in that page just illustrates what I intended to say in my answer: my point $q$ is called $P_o$ there, and my $p$ is indicated by $P$. – Valter Moretti Dec 12 '13 at 07:38