Fundamentals of Opacity: Understanding EM Wave Absorption

[cesiumfrog]

What is the classical microscopic mechanism for absorption of EM waves?

Consider a free charge isolated in the middle of a well-lit pot of (clear) honey. An external electric field will apply a force to the charge but as the charge develops a velocity it will begin doing work against friction. An incident plane EM wave should cause simple harmonic (transverse) motion of the charge, and the friction should heat the honey, but where does this energy come from?

The transverse motion of the charge should produce its own (symmetric) EM waves. Its portion in one direction may cancel some energy out of the incident beam (this is how mirrors work), but it seems as though the same amount of energy escapes in the (equal and opposite) portion (there is nothing to cancel the scattered/reflected beam). How is this all balanced?

 

[Andy Resnick]

It's radiometry- energy transfer, pure and simple. Energy from the incident EM field is converted into thermal energy. The free charge does not freely move- it's damped due to viscosity, and that damping does not result in re-radiated energy.

The atomic basis of absorption is outside the classical description (obviously)- hence the phenomenological roots of Beer's law.

 

[cesiumfrog]

It's radiometry ... Beer's law

?

Obviously, the energy must come from the incoming EM waves, so the outgoing EM waves must somehow be lessened. I'm asking if anyone can elucidate the mechanism of this lessening?

The atomic basis of absorption is outside the classical description (obviously)

I'm asking about a macroscopic charged object (of finite extent and in an electrically insulating, viscous, continuous medium), not atoms.

 

[Andy Resnick]

?

Obviously, the energy must come from the incoming EM waves, so the outgoing EM waves must somehow be lessened. I'm asking if anyone can elucidate the mechanism of this lessening?


I'm asking about a macroscopic charged object (of finite extent and in an electrically insulating, viscous, continuous medium), not atoms.

Oh- from the first few sentences of your OP, I thought you meant microscopic free charges, not macroscopic dielectrics.

By using the term 'lessened', what you mean is that the EM energy propogating *in a particular direction* is lessened. The mechanism to do this is scattering. Now, what happens to the scattered light? Well, continuous materials have a broad absorption band- that's one point to remember. In conductors, the refractive index is complex becasue of the induced current- you have to start with Maxwell's equations, keeping the charge and current density. So some of the EM energy is converted into current, which is then dissipated by resitive processes- heating.

 

[cesiumfrog]

the EM energy propogating *in a particular direction* is lessened. The mechanism to do this is scattering. [..] you have to start with Maxwell's equations

"Scattering"? Can we be more specific regarding the mechanism of scattering?

Or to rephrase: if I know the precise waveform of the ingoing beam (say it has a Gaussian envelope, or maybe it is an infinite plane wave), how do I calculate the precise waveform of the outgoing EM waves?

 

[Andy Resnick]

In the classical field picture, it's simply matching boundary conditionsfor E and B (or D and H, or the scalar potential, or...) at the scattering object and solving Maxwell's equations. Mie scattering (incoming plane wave, spherical object) has been extensively worked on and extended to a variety of incoming wavefronts (Gaussian, converging, abberated) and objects (concentric spheres, ellipsoids, cylinders). For objects more complicated than spheres and plane waves, people generally use alternate formulations- t-matix scattering codes, etc. There's also Rayleigh-Gans scattering, which considers weakky scattering objects/inhomogeneities.

There's dozens, probably hundreds, of books on the subject. There's a good collection of scattering codes here:

http://www.t-matrix.de/

 

[Andy Resnick]

After re-reading my posts, I think I'm less than clear. Let me try again:

?

Obviously, the energy must come from the incoming EM waves, so the outgoing EM waves must somehow be lessened. I'm asking if anyone can elucidate the mechanism of this lessening?


I'm asking about a macroscopic charged object (of finite extent and in an electrically insulating, viscous, continuous medium), not atoms.

Ok. Again, the major physical mechanisms we are talking about are elastic scattering (for dielectrics) and a complex index of refraction (for conductors).

Elastic scattering: light scatters off of inhomogeneities, any type you care to describe: inclusions, scratches, gradients, discontinuities... Scattering means that some of the light changes direction- sounds obvious enough, but that implies a transfer of momentum, which leads to all kinds of cool technological toys like optical cooling and trapping. The way to calculate the (far field) scattering pattern is to solve Maxwell's equations in all space. The simplest 'real' scattering calculation is Mie scattering, which is the scattering of an incoming plane wave by a single perfect dielectric sphere. This can be simplified to Rayleigh scattering (i.e. 'why' the sky is blue). There's lots of more complicated scattering geometries that have been worked out. It's a lot of math- more than I care to write here. But that website I gave is a good place to start.

So, dielectric materials, given sufficient inhomogeneities, scatter the light and appear opaque. Note that for elastic scattering, there is no heating or absorption of the light by the material. There's inelastic scattering (Raman scattering), but it's usually a very minor process (but exploited for CARS microscopy).

Conductors are different- they have a complex index of refraction. The imaginary part of the refractive index corresponds to absorption (extinction coefficient), and a fundamental explanation for absorption requires a description of quantum mechanics of condensed matter. In the classical picture, it means the electromagnetic field induces a current in the material. The electromagnetic field energy is transduced to an electric current energy, which is then dissipated by the resistive nature of the material. In this way, the material heats up.

There... that's better. Hope this helps.

 

[cesiumfrog]

The way to calculate the (far field) scattering pattern is to solve Maxwell's equations in all space. The simplest 'real' scattering calculation is Mie scattering, which is the scattering of an incoming plane wave by a single perfect dielectric sphere. This can be simplified to Rayleigh scattering [..] There's inelastic scattering (Raman scattering), but [..]

I'll have to research Mie scattering in detail (I'm not looking at elastic nor quantum processes such as Rayleigh/Ramen scattering), I'm interested in how Mie scattering achieves absorption (that is, the asymmetric partial cancellation of original wave without producing an equal reflection wave).

For now I'll persist with this simpler scenario, modelling the "charge in honey" as a classical electrical charge smoothly distributed over the surface of a rigid sphere (held in place by internal non-EM forces), assuming the honey is spatially infinite and electrically totally inert (everywhere isotropic and uniform usual electric permittivity, and also equal throughout the rigid sphere) thus having no effect other than to apply a decelerating force on the sphere (say, proportional to its velocity in a frame where the sphere was initially stationary).

If the sphere is small compared to the wavelength, do you agree this can be modeled by a classical point charge, under the influence of the familiar Lorentz force plus the Abraham radiative back-reaction force plus the friction force (a negative multiple of velocity), in vacuum? And that the result can be obtained by simultaneously solving the Maxwell PDEs (for vacuum except at one non-constant point) with Newton's ODE (those three terms specifying the acceleration of the particle) and with a boundary condition that there be an ingoing plane wave (say, circularly polarised)?

Do you agree the result will involve the charge undergoing an approximately circular motion (with the same period as, and transverse to, the ingoing wave)? That the radiative effect of this (like synchrotron radiation) will be symmetric in both longitudinal directions? That any cancellation of the EM wave in the forward direction will be balanced by EM radiation in the backward direction? Producing zero net decrease of energy in the EM field despite positive work on the non-conservative friction/viscosity-force term?

 

[Andy Resnick]

I'm starting to have trouble following you- scattering and absorption are different phenomena. Also, I don't understand what "asymmetric partial cancellation" means. It seems to be like destructive interference? But that's not what absorption is.

So for the sphere- I don't understand exactly what you mean- is the sphere a dielectric? Is it a conductor? Does it carry a net charge? I'm asking because you seem to write down a simple problem in a very complicated way- radiative back-reaction forces, for example.

If, on the other hand, you are really asking what is the force applied to a free charge by an electromagnetic field and the resultant dynamical evolution of the system, I don't see what that has to do with absorption or opacity.

 

[cesiumfrog]

Are you not aware that Mie scattering theory also deals with absorption? You aren't familiar with the Abraham-Lorentz-Dirac force?

Sorry for describing a simple problem in a complicated way. The simple problem is just a lone electron that is exposed both to an electromagnetic plane wave and to friction. Most of the complexities arose because I do not want to consider the problem quantum mechanically yet (and this decision is valid if the problem is phrased appropriately).

The "paradox" is that conservation of energy appears to be violated in this simple classical model of absorption or inelastic scattering (despite that the same approach appears to describe elastic scattering correctly). (By time reversal symmetry, a related problem might be that I'd like to know what trajectory a charge must trace in order to continuously radiate in one direction more than in the opposite direction.)

 

[Andy Resnick]

Mie scattering can be generalized to include absorptive spheres, that's true. But the simplest description is simply elastic scattering. The Abraham-Lorentz-Dirac force is only relevant for length scales around 1 attometer and 10^-24 seconds, about as far from a sphere in honey (and Mie scattering) as you can get. Mie scattering- most light scattering- is time-independent. I don't know if scattering of a pulse by an object has been worked out in any generality.

In any event, you are starting to discuss concepts that really have no bearing on why a material is opaque, or the mechanism of absorption. Which is fine, but again, I'm losing track of your thought process.

 

[cesiumfrog]

I'm trying to understand the fundamental process of absorption of plane waves. (I concede that material opacity is not always due to absorption... not sure where you're pulling "1 attometre" from.) Please let me describe a simple example in which absorption occurs by a process I do not yet understand:

A Van de Graaff generator prepares opposite charge on two highly (say "super-") conductive spheres. Let us separate the spheres by a great distance and there attach each sphere to the handle of its own crankshaft.

Now, if work is done to turn one crankshaft (causing one charged sphere to move in circles) then an EM wave will propagate out (in all directions including toward the second sphere) causing the second sphere to also move in circles (albeit with less force and requiring that the sphere isn't too heavy) and can thereby do work through the second crankshaft.

Since energy has been transferred, some of the EM wave must have been absorbed, but how?

In particular, I note that since the second sphere is undergoing circular motion it must emit it's own radiation but that it should do so in all directions (so if one component does destructively interfere with the first sphere's wave, then an equivalent part will go in the opposite direction) hence I don't see how the necessary net "lessening" occurs of the total energy propagating in the EM fields.

 

[granpa]

Are you not aware that Mie scattering theory also deals with absorption? You aren't familiar with the Abraham-Lorentz-Dirac force?

Sorry for describing a simple problem in a complicated way. The simple problem is just a lone electron that is exposed both to an electromagnetic plane wave and to friction. Most of the complexities arose because I do not want to consider the problem quantum mechanically yet (and this decision is valid if the problem is phrased appropriately).

The "paradox" is that conservation of energy appears to be violated in this simple classical model of absorption or inelastic scattering (despite that the same approach appears to describe elastic scattering correctly). (By time reversal symmetry, a related problem might be that I'd like to know what trajectory a charge must trace in order to continuously radiate in one direction more than in the opposite direction.)

I understand you perfectly. I have no idea why Andy cant. I think its a very good question.

 

[cesiumfrog]

I understand you perfectly. [..] I think its a very good question.

Ah, its inspiring to hear I'm not crazy.

Consider a plane wave passing across a thin sheet (conductor or charged insulator). Motion of charge produces secondary plane waves (symmetrically in both directions). By addition of phasors, the condition for absorption of energy is $0 < r < - cos \Delta\theta < 1$, that is, not only must the secondary wave be roughly 180 degrees out of phase with the incident beam but also its amplitude (having ratio r to the incident beam) must be less (this is what distinguishes reflection from absorption, and may sound obvious, but I hadn't looked at it correctly until now). It arises because intensity isn't linear with amplitude.

The induced velocity of charge lags the acceleration (which in turn is approximately in phase with the governing incident electric field) by 90 degrees (simple harmonic motion) and, assuming steady state is maintained, any work done by the charge (e.g. friction) must be balanced by the electric field doing work. Note that such work depends on the force component in phase with velocity (Newtonian mechanics).

So! If there is friction, then the velocity must lag differently such that the incident electric field acquires a component 90 degrees out of phase from the acceleration, and hence the component in phase with acceleration is lessened (the case of circular polarisation seems easiest to visualise). Hence, the amplitude of S. H. motion is lessened, hence the amplitude of radiation is lessened (though I won't bother to quantify this here). And this causes more wave-intensity to be canceled than was reflected.

 

[Andy Resnick]

I'm trying to understand the fundamental process of absorption of plane waves. (I concede that material opacity is not always due to absorption... not sure where you're pulling "1 attometre" from.) Please let me describe a simple example in which absorption occurs by a process I do not yet understand:

<snip>

From the Larmor formula for the power radiated by an accelerated charge:

$$E_{rad} = \frac{2e^{2}a^{2}T}{3c^{2}}$$

Compared with the change in kinetic energy of that particle due to the acceleration:

$$E_{0} = m(aT)^{2}$$

In order for the effect of radiated power to be significant process we can replace T with a characteristic time $\tau$, and

$$\tau = \frac{2}{3}\frac{e^{2}}{mc^{3}}$$ ~ 10^-24 seconds. The corresponding distance light moves in that time is 10^-15 m, or a fm (and not an am, sorry.)

 

[cesiumfrog]

Thankyou Andy, your responses kept me motivated to not stop thinking about this topic until I was satisfied I had resolved it (post 14).

Now you seem to be calculating the amount of time that it would take any moving electron to decelerate at a constant rate if it did so by its own emission of radiation (which strikes me as rather unphysical in requiring a specially increasing external magnetic field for it to occur). I don't actually see why you thought this (by the way, shouldn't it be 10^-23?) is supposed to be "characteristic" let alone pertinent to a situation that isn't even specific to the single electron charge (as emphasised in the post you just replied to, 12). Anyway, hopefully the fundamental mechanism of opacity (reflection/scattering and absorption) has been elucidated for you too. I wonder whether one can test the model's prediction that 100% absorption cannot be achieved by a single-layer (thin) media?