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A wave travels through a (homogeneous) medium at a given velocity, the velocity depends on the frequency of the energy.... why?
For light waves, dispersion can occur in many ways. Most basically, the refractive index of a material is usually a function of the wavelength of light passing through the medium. This difference in refractive index for different wavelengths leads to a difference in angle of refraction (and velocity), causing dispersion. This is what happens in a prism.
The index of refraction is a property of the material in question. The structure of the medium as well as the what kind of atoms it's made from are very important parameters that decide how the index of refraction will vary with frequency.
Same concept as mechanical waves - the polarizability depends on the frequency, this influences the electric susceptibility and therefore the speed of light. Usually, the magnetic effect is smaller.
It can be, but it does not have to be anisotropic.OK "polarizability" I think that is an anisotropic property.
Electric susceptibilitySo by analogy my first question would be: what is the analogy to the elastic tensor for electromagnetic waves?
Tell me something I don't already know ...
The "anomalous" dispersion is found in and around absorption bands.
Basically, if an oscillator is excited by a wave whose frequency is close to the resonant frequency of the oscillator but slightly lower, the oscillator follows the forcing wave but is behind the forcing wave.
If, however, the oscillator is excited by a wave whose frequency is close to the resonance but slightly higher, the oscillator gets ahead of the forcing wave.
The result is that if you look at light near but slightly redward of an absorption band, you would see unusually high refractive index. If you look at light near but slightly blueward of an absorption band, you would see unusually low refractive index.
Between absorption bands, there is "normal dispersion" - the refractive index increases blueward and does so slowly.
Most transparent colourless substances are between two groups of absorption bands - one in ultraviolet caused by electron excitations, and the other in infrared caused by vibrations of nuclei.
The reason substances have light speed less than in vacuum is that the electronic excitations in ultraviolet get polarized by light even at much lower frequencies, and slow down the light.
In some substances, electrons are notably tightly held - with the result of weak refraction and also weak dispersion. Such as fluorides - the absorption of fluorides is notably far in UV, their electrons have low polarizability, the fluorides have low refractive index and also weak dispersion (big Abbe numbers).
Is that more informative?
It can be, but it does not have to be anisotropic.
Electric susceptibility
OK so polarizability means something different from what I thought you meant. It seems a medium can be polarised by the incoming light. I deal regularly with sound waves (I'm a seismologist) and so that is a foreign concept to me: rocks and minerals polarise sound energy, not the other way around.
Would you agree: you can't polarise a wave in an isotropic medium.
Thanks, I still need to consider the implications.
OK so polarizability means something different from what I thought you meant. It seems a medium can be polarised by the incoming light. I deal regularly with sound waves (I'm a seismologist) and so that is a foreign concept to me: rocks and minerals polarise sound energy, not the other way around.
Would you agree: you can't polarise a wave in an isotropic medium.
Interesting. It seems the propagation of light through a medium involves scattering of light by the atoms in the medium. The time over which this scattering interaction occurs depends on the frequency of the light, thus you have a mechanism for dispersion.
Does that sound OK?
You are implying it is interactions with individual atoms. When EM waves travel through a medium in a coherent way, it interacts with the bulk medium. If individual atoms were involved (as with the classic interaction between a photon and a Hydrogen atom), the phase of the wave would be disrupted because the re-emitted photons would not stay in phase with each other. As it is, the wave maintains its integrity as it travels through the medium.
In an isotropic medium you can have sound waves with different polarizations: longitudinal and transverse. They don't have the same speed usually. These are the S and P waves (so named in geophysics). In anisotropic media you may have more than one kind of transverse waves.
But it's not clear what you mean by "polarization". How do the rocks and minerals "polarise sound energy"? Do you have an example?
"Polarization" as an action (as opposite to a property: linear polarization for example) has to do with a change in the polarization state of the wave. Like when you have "unpolarized" light passing through a piece of polaroid and emerging as linearly polarized.
I suppose it's a matter of semantics and of the same word being used in several ways.
The phase shifts would all be different because there is a statistical distribution associated with absorption and emission of a photon. That's the 'problem'; the wave front would no longer have integrity and you would no longer have a 'ray'. In order to explain what goes on you need to acknowledge that the photons are interacting with the whole thing. (If you really want to consider photons in this process which is essentially a wave phenomenon).Dispersion would manifest as a phase shift, right? So I still don't see the problem with that.
In case of a medium, "polarization" refers to the medium acquiring electric dipole momentum under influence of electric field - whether due to long-distance separation of charges (in conductors) or short distance induced electric moments.
The mechanical analogue of "polarization" is "strain". And the analogue of "polarizability" is "compressibility".
Note that elastic phenomena are more complex than electromagnetic. Elasticity tensor has what, 21 components? Even in completely isotropic environment, light has 1 speed, but sound has 2 (compression wave and shear wave speed).
The phase shifts would all be different because there is a statistical distribution associated with absorption and emission of a photon. That's the 'problem'; the wave front would no longer have integrity and you would no longer have a 'ray'. In order to explain what goes on you need to acknowledge that the photons are interacting with the whole thing. (If you really want to consider photons in this process which is essentially a wave phenomenon).
Polarisation has a rather specific meaning with waves and only applies to transverse waves (not P waves or sound). The 'polarisation' of molecules by an EM field is not the same thing and could better be termed displacement in this case.
You are implying it is interactions with individual atoms. When EM waves travel through a medium in a coherent way, it interacts with the bulk medium. If individual atoms were involved (as with the classic interaction between a photon and a Hydrogen atom), the phase of the wave would be disrupted because the re-emitted photons would not stay in phase with each other. As it is, the wave maintains its integrity as it travels through the medium.
But the wave DOES interact with individual atoms. And yes, it causes scattering.
Look at gases. The molecules in atmospheric or lower pressure gases are too far from another to allow light to interact with more than one molecule at a time, except on the rare occasions when molecules are undergoing collision.
The phases of light scattered from different air molecules are indeed out of phase.
Now, a wave of light encounters many air molecules over one wave. With the result that while the retarding/refracting effects of air molecules add up over many molecules, the scattered waves being out of phase undergo destructive interference. This is not complete because the positions of air molecules are random. There is Rayleigh scattering in air. But it is still relatively weak - much of the light can pass through long column of air and is left over from scattering, yet appreciably retarded.
I was actually thinking of solids and liquids (which have high refractive indices but isn't Rayleigh scattering due to polarisation of the molecules in a gas and not the atoms? It's an elastic scattering phenomenon and I don't know how atomic energy levels would work apart from at frequencies corresponding to line spectra.
Well I want to get down to the nub of what it is that causes the dispersion. So far the only explanation given involves absorption and emission of photons, if there is a better way of looking at it then I'm all ears.
The mechanical analogue of "polarization" is "strain". And the analogue of "polarizability" is "compressibility".
The differences in light speed come because medium is usually "softer" than vacuum. In a medium, electric field causes movement of charges, and the field from the moved charges usually opposes and weakens the inducing field. Just like in a mechanical medium, a softer medium with less resistance to compression or shear will transmit sound at a lower speed, so in case of electromagnetic field a medium with higher permittivity (weaker electric field) transmits the wave at a lower speed.
I find this interesting but conflicting with the understanding I had so far.
The speed of a mechanical wave in a medium is:
(a) Directly proportional to restoring (elastic) force of the material, i.e. how much the medium resists strain or deformation and hence with which force it recover the equilibrium position.
(b) Inversely proportional to inertia of the material.
For example:
- In a string, (a) is tension and (b) is the longitudinal density.
- In sound, (a) is compressibility and (b) is volume density.
In the case of EM waves, I had thought that permittivity (which is related to how easily the material polarizes) plays the role (b) of inertia, not (a). In fact, permittivity is placed in the denominator of the formula, like density, thus clearly denoting that the speed is inversely proportional to this factor. Of course, one could say that speed is proportional to how low permittivity is, but that is a convoluted approach.
While it is true that dipoles align against the E field and hence weaken it, it does not seem that such is the reason for an EM wave’s retardation, since the latter is an oscillation, so the dipoles align in one direction and then in the opposite direction, thus acting like an antenna that, instead of attenuating the E field, simply re-radiates the wave, thus leaving the E field unaffected.
Is this understanding correct or should I change it?
Compare an oscillator circuit including a capacitor.
A dielectric with dielectric permittivity will decrease the voltage of a capacitor, while leaving the charges unchanged. This also means that the increased capacity of the capacitor decreases the frequency of the oscillator circuit. That is the restoring force side - not the inertia side of the equation.