What causes dispersion?

[billiards]

A wave travels through a (homogeneous) medium at a given velocity, the velocity depends on the frequency of the energy.... why?

 

[mfb]

The velocity depends on the material - how elastic it is (for a mechanical wave) and similar properties. And those values can depend on the frequency (and on the amplitude, but that is a different effect). As an example, a material could be very elastic with a low frequency (it has enough time to change its shape in some way), but stiff at a high frequency.

 

[billiards]

Thanks, that makes sense. How about for light waves then?

 

[dmriser]

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.

 

[billiards]

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.

OK so why does the refractive index change for different frequencies (or wavelengths)?

(I already know about waveguides, I'm interested in the case of homogeneous media.)

 

[dmriser]

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.

 

[mfb]

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.

 

[billiards]

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.

Tell me something I don't already know ...

 

[billiards]

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.

OK "polarizability" I think that is an anisotropic property. I can totally understand how a material would be anisotropic, and even, how the length scale of the anisotropic fabric would affect how anisotropic the material seemed to be to different wavelengths.

However, in the case of sound waves, we did not need to invoke anisotropy. The material responded differently at different elastic frequencies and that was enough to explain dispersion. So I do not understand why you are bringing this polarizability into the equation and introducing it by saying "same concept as mechanical waves".

Indeed, I can even see how anisotropy can cause dispersion. However, this would be the case of the inhomogeneous medium, which is basically dispersion caused by waveguides, and I am specifically interested in the more "intrinsic" dispersion of a homogeneous medium.

Now I do think the concept probably is analagous for elastic and electromagnetic waves. The elastic tensor is a function of frequency, due I guess to inelastic effects dominating at certain frequencies by interesting mechanism that we have not dealt with ... (I would be interested to know more about these mechanisms if anybody can comment) ...

So by analogy my first question would be: what is the analogy to the elastic tensor for electromagnetic waves?

 

[snorkack]

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?

 

[mfb]

OK "polarizability" I think that is an anisotropic property.

It can be, but it does not have to be anisotropic.

So by analogy my first question would be: what is the analogy to the elastic tensor for electromagnetic waves?

Electric susceptibility

 

[sophiecentaur]

Tell me something I don't already know ...

The fact is that we don't actually know what you don't know. That is not a very polite response. Try to be a bit nicer, if you can. It will elicit better responses, young man. I am correct about your gender, I think.

 

[billiards]

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?

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?

 

[billiards]

It can be, but it does not have to be anisotropic.

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.

Electric susceptibility

Thanks, I still need to consider the implications.

 

[nasu]

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.

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.

 

[snorkack]

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.

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).

 

[sophiecentaur]

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.

 

[billiards]

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.

Dispersion would manifest as a phase shift, right? So I still don't see the problem with that.

 

[billiards]

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.

Yeah I agree with everything you said ... I guess I was taliking about polarisation as an action -- the waves of course have an initial polarisation from source. An anisotropic rock can either misalign the particle motion from the ray direction for P-waves (thereby changing the polarisation in the ray frame) or propagate "fast" and "slow" polarised energy at different speeds for S-waves thereby changing linearly polarised energy into elliptically polarised energy.

 

[sophiecentaur]

Dispersion would manifest as a phase shift, right? So I still don't see the problem with that.

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).

 

[sophiecentaur]

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.

 

[billiards]

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).

Oh right. I guess that makes sense. In effect the wave passes through the medium and distorts the medium, thereby imprinting the signal of the wave onto the medium for an instant (and any measuring device that happens to be attached). I guess I'm so used to thinking of rocks as being "fixed" that I unconsciously attributed the polarisation to the wave (which is a valid way of looking at it in my experience). I don't have a problem with thinking of the medium as being polarised for the instant in which the wave travels through ... it seems like just another way of looking at the same thing ... Which raises the issue of what exactly is a wave anyway?

But anyway, I'm keen to talk about dispersion.

 

[billiards]

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).

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.

 

[billiards]

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.

I would imagine the polarisation of light would refer not to physical displacements, but rather some vector perturbation in the electromagnetic field.


Also I believe it is correct to consider that P waves are polarised longitudinally (at least in isotropic media!).

 

[snorkack]

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.

 

[sophiecentaur]

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.

 

[snorkack]

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.

Monoatomic gases like noble gases or quicksilver vapour undergo Rayleigh scattering, refraction/retardation and far UV absorption qualitatively in the same manner as multiatomic gases like nitrogen or oxygen. The orbits of electrons are perturbed/distorted by external electric field, and if the frequency is sufficient then the electronic excited states of the molecules span the whole molecule.

 

[sophiecentaur]

Thanks for clearing that up.

 

[snorkack]

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.

Ignoring photons, look at field as continuous phenomenon.

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.

Now, a mechanical analogue of dispersion would be elasticity dependent on time/frequency. For example, a substance which on rapid deformation exerts high elastic stress - but once the deformation ends, will plastically relax to a lower stress (but not to zero) over some timescale.

If such substance were subjected to waves with high frequency, it would transmit them at a high speed due to high stiffness.

If the waves are very low frequency, the substance would be relaxed all the time, so it would exert only the final stiffness and accordingly transmit sound at the lower speed.

If the substance were subjected to waves at the frequency in the timescale of relaxation then large amounts of energy would be turned to heat, since large force is exerted to compress the substance, but only small force would be restoring it.

 

[Saw]

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. We also talk about optical density of the material. I also found somewhere the formula below, which I don’t fully understand and which someone might be able to explain, where the slowing down of light is related to mass density (M) together with other factors like how far the electron orbits from the nucleus (R) …

$$% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % aH3oaAdaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaaabaGaeq4T % dG2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaaaacqGH9aqpda % qadaqaamaalaaabaGaeqyWdiNaamOtamaaBaaaleaacaWGbbaabeaa % aOqaaiaaiodacqaH1oqzdaWgaaWcbaGaaGimaaqabaGccaWGnbaaaa % GaayjkaiaawMcaamaabmaabaWaaSaaaeaacaaIYaGaamyzamaaCaaa % leqabaGaaGOmaaaakiaadkfadaahaaWcbeqaaiaaikdaaaaakeaaca % aIZaGaeuiLdqKaamyraaaaaiaawIcacaGLPaaaaaa!51C4! \frac{{{\eta ^2} - 1}}{{{\eta ^2} + 1}} = \left( {\frac{{\rho {N_A}}}{{3{\varepsilon _0}M}}} \right)\left( {\frac{{2{e^2}{R^2}}}{{3\Delta E}}} \right)$$

Ultimately, what all this would mean is that light slows down because, every time it interacts either with an atom or, as the answer to the FAQ says, with the bulk of a material, it makes a brief stop: more stops and longer ones mean more delay.

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?

 

[Studiot]

The simple wave equation is

∂²η/∂x² = (1/c²)(∂²η/∂t²

The solution of this has velocity c independent of frequency. This is called the phase velocity.

However more complicated wave equations exist where the solution does not include a constant c but have velocity dependent upon frequency.

eg

∂⁴η/∂x⁴ = -(ρS/YI²)(∂²η/∂t²

These commonly arise from non linearities in the restoring force.

Since different frequency waves then travel at different velocities, a wave shape distorting effect called dispersion occurs.
The power in a wave depends upon its shape so if the shape changes the power changes, hence the term dispersion.

Sorry I don't have acces to my latex editor at the moment.

Please note SophieCentaur's comment that polarisation only occurs in trnasverse waves, not longitudinal ones and also that power loss only occurs at the instant of polarisation and not thereafter.

With waves with dispersion as every frequency has a different phase velocity we sometimes group velocity which refers to a (small) renge of frequencies that have a small range of velocities and therfore a signal containing only this range of frequencies will pass a substantial distance undistorted. They will, of course disperse eventually.

 

[snorkack]

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.

Likewise consider that the Hertz antenna is a limiting case of oscillator circuit - the plates are converted to ends of the antenna in surrounding medium. The frequency would respond to the permittivity of the surrounding medium in the same way.

 

[Studiot]

@snorkack

I'm sorry, I don't follow where any of your examples are dealing with dispersion?

 

[Saw]

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.

You bring in a good comparison, the role of the same dielectric when faced not with an electromagnetic wave but with a similar phenomenon, an oscillating current.

I fully agree that in this second example capacitance (which is a function of the permittivity of the dielectric material and of the geometry of the capacitor) plays against capacitative reactance, which is usually defined as a kind of inertia that opposes change in tension. To sum up, permittivity plays against inertia. Somehow we could say that here higher permittivity means better “conductivity” of the oscillation that the AC is. And the same applies to frequency: the higher the frequency, the lower the reactance and the higher the “conduction” of the oscillation.

However, the fact is that in the situation that is the object of our discussion (the dielectric is faced with an EM wave) permittivity and frequency play the opposite roles. The higher the permittivity and the higher the frequency, the lower the velocity, which looks more like inertia (since it plays against velocity) than like restoring force (which would play in favor of velocity). Someone could argue: “but you are comparing apples with pears: in one case you talk about the preservation of changes in tension and frequency; in the other, of the preservation of velocity". And I would agree. That is why I tended to believe that the two situations admit different solutions: in one case (AC - tension), permittivity acts against inertia, in the other (EM wave – velocity) it would be on inertia side.