Intuitive approach to Huygens' principle

[Cl4r4]

TL;DR Summary

Hi,
Huygens principle has been extended with two independent efforts in order to correct its original feature that gives rise to a back propagating wave. Fresnel proposed the obliquity factor and Miller proposed two kinds of emissions (dephased). Is there an intuitive way to explain the fact that only the forward component must exist?

Hi All,

Huygens principle has been extended with two independent efforts in order to reform its original feature that gives rise to a back propagating wave.

1) Fresnel proposed the obliquity factor $(1/2)(1 + \cos\theta)$.
2) Miller proposed two kinds of emissions (dephased).
D. Miller, "Huygens’s wave propagation principle corrected," Opt. Lett. 16, 1370-1372 (1991).
or: http://www-ee.stanford.edu/~dabm/146.pdf

Is there an intuitive way to explain the fact that only the forward component must exist?

Best regards

 

[tech99]

Every wave front must have another behind it. If we look at the one that is a quarter of a wavelength behind, then the back radiations from the two are 180 degrees out of phase and this cancels rearward propagation.

 

[Cl4r4]

Every wave front must have another behind it. If we look at the one that is a quarter of a wavelength behind, then the back radiations from the two are 180 degrees out of phase and this cancels rearward propagation.

Thank you, tech99.
But this is basically what the work of Miller proposes, isn't it? My question has more to do with the "why". Why every wave front must have another behind it?

 

[tech99]

If a beam of light falls on an absorbing sheet, then the instant a wavefront is absorbed there must be another coming along behind. A second one cannot be supposed to suddenly spring up when the first one disappears - how would it know to do that?

 

[Cl4r4]

But when we speak of absortion we are adding matter, atoms, to this context of a beam of light propagating in space. I think it would be better to try to discuss the application of Huygens principle to light in a scenario with vacuum.

 

[tech99]

If only we could study Huygen's Wavelets in a vacuum, but the detection of an EM wave seems to requires us to use an electric charge, which is matter. Further, the generation, reflection, diffraction, refraction and absorption of EM waves also seems to require electric charge (so far as I am aware). So there seems little chance of ascertaining for sure whether the wavelets exist or whether the wave just behaves as though they existed.

 

[Cl4r4]

Ok, tech99, I think you are correct in this point. So let me suggest that we concentrate on waves that take place on the surface of water. Some times I think that Huygens principle is more an enemy than a friend. I think the two proposals made to adjust it are too much "ad hoc". I still miss the physics in these explanations. Force, causality, inertia, momentum, entropy,..., things like these.

 

[hutchphd]

Ok, tech99, I think you are correct in this point. So let me suggest that we concentrate on waves that take place on the surface of water. Some times I think that Huygens principle is more an enemy than a friend. I think the two proposals made to adjust it are too much "ad hoc". I still miss the physics in these explanations. Force, causality, inertia, momentum, entropy,..., things like these.

Huygens is a very useful conceptual tool for solving the wave equation. It is at heart a Green's Function for free space. But a truly comprehensive solution requires initial conditions and boundary values. The various additions to Huygens are based both on good physics and yet are "ad hoc"...they represent various iterative approximation techniques. You need to know the real physics to use Huygens appropriately.

 

[vanhees71]

I think the answer is simple, if one thinks about the mathematics behind this principle. The most simple and often sufficient model is the Kirchhoff model of wave propagation, where one considers only a scalar quantity (e.g., a component of the electric field assuming a plane wave, neglecting all possible kinds of polarization effects) and simplifies the boundary conditions a bit. The exact theory of diffraction is complicated and analytically solvable only for very simple cases (see Sommerfeld's texbook on optics; Sommerfeld did the exact problem of diffraction as his Habilitationsschrift).

There you solve the Helmholtz equation for a wave mode by assuming that you have some point wave source in front of the obstacle. Then you also assume that this describes the wave everywhere before the obstacle, including the light field within the openings. Then you use the Green's function of the Helmholtz equation to calculate the field in the region after the obstacle. For 3 spatial dimensions this leads to Huygen's principle, i.e., that you can think of every point in the openings as a source of a spherical wave being observed in the region behind the obstacle with the source strength at the openings of the obstacle given by the unperturbed source field before the obstacle. So it's simply that you are only interested on the wave field behind the obstacle.

What's obviously completely neglected in this treatment is the part of the wave before the obstacle, which is reflected from the obstacle itself.

 

[Cl4r4]

I think the answer is simple, if one thinks about the mathematics behind this principle. The most simple and often sufficient model is the Kirchhoff model of wave propagation, where one considers only a scalar quantity (e.g., a component of the electric field assuming a plane wave, neglecting all possible kinds of polarization effects) and simplifies the boundary conditions a bit. The exact theory of diffraction is complicated and analytically solvable only for very simple cases (see Sommerfeld's texbook on optics; Sommerfeld did the exact problem of diffraction as his Habilitationsschrift).

There you solve the Helmholtz equation for a wave mode by assuming that you have some point wave source in front of the obstacle. Then you also assume that this describes the wave everywhere before the obstacle, including the light field within the openings. Then you use the Green's function of the Helmholtz equation to calculate the field in the region after the obstacle. For 3 spatial dimensions this leads to Huygen's principle, i.e., that you can think of every point in the openings as a source of a spherical wave being observed in the region behind the obstacle with the source strength at the openings of the obstacle given by the unperturbed source field before the obstacle. So it's simply that you are only interested on the wave field behind the obstacle.

What's obviously completely neglected in this treatment is the part of the wave before the obstacle, which is reflected from the obstacle itself.

Thank you all (vanhees71, hutchphd and tech99),
but regarding this last post by vanhees71 I would like to know if the approach via more envolved integrals like Fresnel-Kirchhoff formula, for instance, produces the due suppression of the backward component.

I am referring to integrals that does not have terms like obliquity factor or delta-type distributions which supresses artificially certain contributions to the final result.

My point is that it seems that some argument based either on symmetry or momentum conservation or entropy (2nd law) or interference or causality or even geometry is still to be found to justify this observational fact. Perhaps the theoretical notion of superposition must be changed (I am not suggesting that the OP has to do with new theories). As I said, I just miss fundamentals in this explanation.

 

[vanhees71]

No, you have to assume that there is no backward component, i.e., that the material of the obstacle is ideally absorptive. The other extreme is some mirror as the surface of the obstacle. Then you get a field before the obstacle that is given as the superposition of the incoming and the reflected waves. The reflected waves can again be calculated using Huygens's principle!

 

[Cl4r4]

Thank you, vanhees71. Ok, I agree with you, but the situations you are describing are in some sense "helping" Huygens principle. I would like to consider a wave in the middle of its media, far from obstacles. Let's think of a sea wave in the middle of the ocean. At a given instant t0 we look at the wave. What will happen just after this instant of time? Will the wave keep its motion in the same sense it was moving just before t0 or will two waves appear propagating in opposite directions? Or something else...
I would like to have some solid physical argument to manouver Huygens principle so that it doesn't collide with the experimental failure. Using either the obliquity factor or the model of two dephased waves seem to be artificial to me. They seem to be taken out from the magician's box.

 

[vanhees71]

Yes, Huygens's principle is heuristic. If you want to really understand wave propagation you have to solve the corresponding equations (hydrodynamics for water waves, (macroscopic) Maxwell equations for light, etc...).

 

[Cl4r4]

But (sorry for this question) in solving these equations for a wave in free space do we naturally scape from having a back propagating wave component appearing out of nothing (or better, out of the body of the ongoing initial direct wave) ?

 

[vanhees71]

You have to consider the physical situation. If you just have some source usually the physically right choice is the retarded solution of the initial-value problem of the wave equation. Usually this is discussed in textbooks about classical electromagnetism in application to electromagnetic waves from a given charge-current distribution (usually with harmonic time dependence, leading to the Helmholtz equation and the corresponding "radiation condition", i.e., the choice of outgoing-wave solutions).

 

[Cl4r4]

It seems then that causallity may be the theoretical agent that keeps the wave in its direction of movement. Suppose we can turn off-on-off the existence of an electron in a certain position of empty space. After this initial stage (non existing eletron - existing electron (for a short time interval) - non existing electron) ) we will see a wave of electromagnetic field propagating in space. After one second the amplitude of this wave must be different from zero basically in a sphere of radius r = 300 000 000 m. So, there is no causal reference to explain the existence of non zero amplitudes elsewhere. Does it seem to be correct?

 

[Cl4r4]

I am aware that this barrier shaped profile in the existence of the electron tends to introduce Fourier components which are richer than I have initially thought and are in some sense undesirable in this argument...

 

[hutchphd]

The Miller paper is pretty interesting.
https://ee.stanford.edu/~dabm/146.pdf
Worth a careful read!
It is really not a surprising result because one would expect to need to specify more than just the amplitude. To specify the rotation of a motor, for instance, requires more than amplitude as a function of time at a single point. One way to specify is to supply the quadrature signal.
Certainly the exact solution of even the scaler wave equation requires more .

 

[vanhees71]

You need initial conditions (the quantity and its time derivative at one time $t_0$) and maybe, depending on the physical situation, boundary conditions to make the solution of the (inhomogeneous) wave equation unique. If I remember right, there's a good discussion of this in Morse, Feshbach, Methods of theoretical physics.

 

[Cl4r4]

In the context of quantum mechanics, more specifically when solving the problem of the particle in the step potential we introduce "by hand" the cancelation of a term which propagates, say, from the infinity (at right) to the left (suposing that the particle is incident from minus infinity). And we do this in the name of causality. Can it be considered an analogous procedure relatively to these artificial methods to cancel the in-Huygens-expected back progating wave?

 

[vanhees71]

An energy eigenstate is a stationary state. So it's not a causality argument here. Since the Schrödinger equation is of 1st order in time the initial-value problem is anyway uniquely solved by the retarded Green's function.

For the energy eigenfunctions used to solve this initial value problem by Fourier decomposition with respect to these eigenfunctions you need to consider the appropriate boundary conditions that describe the physical situation you want to describe. So from your description I think you want to describe the scattering of particles coming from the right. This means you pose a boundary condition for $x \rightarrow -\infty$ not to have a right-moving wave-component in this region.

Restricting your energy eigenfunctions to the solutions fulfilling this boundary conditions then leads to the description of a wave packet which starts by moving from $-\infty$ to the left. When it hits the potential step, part of it is reflected (moving from the step to the right and interfering with the original left-moving part of the wave) and another part is running further to the left. So far left from the step you have only left-moving parts of the wave by construction, and that's because you wanted to describe this specific situation.

For a complete set of energy eigenfunctions of course you have to consider the other solutions, which have also right-moving parts of the wavefunction for $x \rightarrow \infty$. With them you can describe the corresponding other kinds of scattering processes.

Here you find some animations of such one-dimensional scattering problems with simple steplike potentials:

https://itp.uni-frankfurt.de/~hees/qm1-ss09/wavepack/

The explanations are in German, but it's hopefully also fun to just watch the movies :-).

 

[Cl4r4]

Thank you, vanhees71. Nice animations. But you are then saying that the reason we cut the "right-moving wave-component in this region" is not causality, is it?

 

[vanhees71]

It's just the choice of the solution you need to describe the said type of scattering (incoming "asymptotic free" wave from the right).

 

[hutchphd]

One can call this a boundary value for the problem I think.

 

[Cl4r4]

Sometimes this topic sounds similar to those situations in college where we cut (or limit) the parabolic trajectory because the launched particle could not go bellow the solid ground. It is an external intervention based on reasonable properties or limitations of the system.

 

[bosque]

Thank you all (vanhees71, hutchphd and tech99),
but regarding this last post by vanhees71 I would like to know if the approach via more envolved integrals like Fresnel-Kirchhoff formula, for instance, produces the due suppression of the backward component.

I am referring to integrals that does not have terms like obliquity factor or delta-type distributions which supresses artificially certain contributions to the final result.

My point is that it seems that some argument based either on symmetry or momentum conservation or entropy (2nd law) or interference or causality or even geometry is still to be found to justify this observational fact. Perhaps the theoretical notion of superposition must be changed (I am not suggesting that the OP has to do with new theories). As I said, I just miss fundamentals in this explanation.

See if this helps

Huygens' Principle geometric derivation and elimination of the wake and backward wave​

https://www.nature.com/articles/s41598-021-99049-7

It shows elimination of the backward wave without the obliquity factor.