Maxwell's Equations for nonlinear media

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Discussion Overview

The discussion revolves around the conditions necessary for applying constitutive relations in Maxwell's equations, particularly in nonlinear media. Participants explore the implications of isotropy versus anisotropy in such materials, the nature of dielectric functions, and the effects of non-locality on these relations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the conditions required to use constitutive relations in nonlinear media, questioning whether nonlinear materials must be anisotropic or if isotropic nonlinear materials can exist.
  • One participant suggests that the constitutive relations must be causal and discusses the Taylor series expansion of polarization P in relation to the electric field E, noting the significance of frequency dependence.
  • Another participant raises a question about the implications of assuming a non-local medium, suggesting that the relationship between P and E would change.
  • There is a discussion about the nature of the dielectric function, with some participants asserting that it is a matrix rather than a vector, and questioning the implications of position dependence on the dielectric function.
  • One participant emphasizes that the rules for constitutive equations are phenomenological and cannot be completely derived from first principles, while others discuss the implications of causality and the principle of material frame-indifference.
  • Some participants express confusion regarding the concept of equipresence and its relation to isotropy, debating whether anisotropic media could support different characteristics for observers.
  • References to literature, such as Boyd's book and Truesdell's work, are provided for further exploration of the topic.

Areas of Agreement / Disagreement

Participants express differing views on whether nonlinear media must be anisotropic, with some asserting that isotropic nonlinear materials are possible. The discussion remains unresolved regarding the implications of non-locality and the nature of the dielectric function.

Contextual Notes

Participants note that the constitutive relations are phenomenological and may not be fully derived from first principles. There is also mention of the complexity of dielectric functions in relation to isotropy and anisotropy, as well as the potential for confusion regarding non-local effects.

MathRyan
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What conditions are necessary to use the constitutive relations for Maxwell's equations? I am working in a nonlinear media, but am a little confused about whether I can assume isotropy or not.

If I am assuming the media is nonlinear is it necessarily anisotropic? Or, is it possible to have a nonlinear, isotropic material?

Thanks!
 
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The constitutive relations must be causal. In some cases you can neglect the frequency dependence and simply write down a Taylor series expansion of P as
P~χ(1)Ε+χ(2)Ε^2+χ(3)E^3+... (*)
This relation between P and E is not causal (i.e. instantaneous) but is a good approximation when you are far away from resonance. Beware, this expansion is meaningful only when the terms higher than the linear one become smaller and smaller.

A medium for which at least one of χ(2),χ(3),... are nonzero is nonlinear. If there is no dependence from the position, the medium is isotropic.

Take a look at Boyd's book.
Oh ! and be cautious with the different versions of (*) used by some authors. The units are usually different from one textbook to another.
 
Does anything change if we assume the medium in non-local?
 
Okay, so now I am a little bit confused. Why, if we are assuming a more complicated form of the constitutive relations, is the dielectric function scalar and not a vector function? If there are dependencies on the position shouldn't the dielectric function be a vector function? Is this a mistake on the Wikipedia page?

Does non-local simply mean dependency on position? I thought it was different. I always thought about it as a local disturbance having a global effect (instantaneously). Do the relations on the Wiki still apply?
 
MathRyan said:
What conditions are necessary to use the constitutive relations for Maxwell's equations? I am working in a nonlinear media, but am a little confused about whether I can assume isotropy or not.

If I am assuming the media is nonlinear is it necessarily anisotropic? Or, is it possible to have a nonlinear, isotropic material?

Thanks!
The answer to this question is that non-isotropic and nonlinear are two different things so the medium can be nonlinear and still be istropic.

The other posts are not relevant to this question.
 
The wiki page is correct.

The dielectric constant is not a vector, it is a matrix instead. You can see this from the relation D=εE (D,E are vectors). For an isotropic material this matrix is diagonal and its (nonzero) elements are equal.

In isotropic materials the displacement vector D is parallel to the applied field E and more specifically D_i=εE_i, i =x,y,z; that is, the displacement field D_x is due to the applied field in x-direction E_x and so on. In anisotropic crystals ε is also (or can be chosen to be) diagonal but its elements are not equal.

I don't quite follow you here,

"I always thought about it as a local disturbance having a global effect (instantaneously)."

but the wiki relations are quite general.
 
MathRyan said:
What conditions are necessary to use the constitutive relations for Maxwell's equations? I am working in a nonlinear media, but am a little confused about whether I can assume isotropy or not.

If I am assuming the media is nonlinear is it necessarily anisotropic? Or, is it possible to have a nonlinear, isotropic material?

Thanks!

Let me just add to |squeezed>'s reply:

There are no 'changes' to Maxwell's equations, the constitutive relations are used to relate E, D, B, and H, which must be done to solve the system of equations.

The 'rules' for constitutive equations are not (AFAIK) complete; they are phenomenological in nature and cannot (yet) be completely derived from first (microscopic) principles.

One rule is causality- that gives the Kramers-Kronig relations between the real and imaginary components of the permittivity and permeability (which are phenomenological parameters used to relate E and D, or B and H).

The permittivity/permeability can be complex, space- and time-varying, scalar or tensor. The material can have 'fading memory' (or not), be linear or nonlinear.

Another rule for constitutive relations is 'equipresence', or the principle of material frame-indifference. That is, all observers will agree on the same constitutive law.

'Causality' is actually two rules: one is 'local action' (in determining the local field values, values beyond a certain neighborhood may be disregarded) and 'determinism' (the present value of the field is dependent only on the past values of the field).
 
Hi Andy

Andy Resnick said:
The 'rules' for constitutive equations are not (AFAIK) complete; they are phenomenological in nature and cannot (yet) be completely derived from first (microscopic) principles.

But since you can get them using the Lorentz model (or even QMally), is this still considered to be phenomenological ?
Andy Resnick said:
Another rule for constitutive relations is 'equipresence', or the principle of material frame-indifference. That is, all observers will agree on the same constitutive law.

'Causality' is actually two rules: one is 'local action' (in determining the local field values, values beyond a certain neighborhood may be disregarded) and 'determinism' (the present value of the field is dependent only on the past values of the field).
Could you explain a bit more this "equipresence" (or give a reference) ?

Thanks
 
  • #10
|squeezed> said:
Hi Andy

But since you can get them using the Lorentz model (or even QMally), is this still considered to be phenomenological ?

That is not true in general, but only for certain idealized materials (at low temperatures). When someone published a QM derivation of the permittivity of (say) concrete, I'll agree with you.


|squeezed> said:
Could you explain a bit more this "equipresence" (or give a reference) ?

Thanks

Truesdell's volume "The nonlinear theory of fields" p 56-86 is as good a place to start as any.

http://books.google.com/books?id=dp...v=onepage&q=truesdell field nonlinear&f=false

His student, Walter Noll, also did a lot of work:

http://www.math.cmu.edu/math/faculty/noll.html
 
  • #11
Andy Resnick said:
Another rule for constitutive relations is 'equipresence', or the principle of material frame-indifference. That is, all observers will agree on the same constitutive law.

It would seem to me that that rule would require the medium to be isotropic. An anisotropic medium could support different characteristics for observers at different positions. Is that not so? An example would be the fields near a water molecule. They are not all at symmetric with respect to the angle of an observer.
 
  • #12
MathRyan said:
Does anything change if we assume the medium in non-local?

In a chapter of Jackson's EM textbook where he analyzes the Kramers-Kronig relations he also gives a general solution to a permittivity value/function that involves non-local factors in a classical setting.
 
  • #13
PhilDSP said:
It would seem to me that that rule would require the medium to be isotropic. An anisotropic medium could support different characteristics for observers at different positions. Is that not so? An example would be the fields near a water molecule. They are not all at symmetric with respect to the angle of an observer.

Andy says that the characterization of a medium (isotropic/anisotropic) is universal. Its all about ε,μ and not about the fields that come out from Maxwell's eqs (at least that's my understanding). We know the relation of the fields between two different coordinate systems, the material properties must be ε=ε', μ=μ'.

If in an observer "sees" three different refractive indices (extreme case) then i don't see* how another (moving) guy could see an isotropic medium. If you have a BBO (anisotropic) crystal and you produce the 2nd harmonic, then i don't see why a moving observer (say in a train) can't do the same.


* i didn't go through this one :biggrin:
 
  • #14
PhilDSP said:
It would seem to me that that rule would require the medium to be isotropic. An anisotropic medium could support different characteristics for observers at different positions. Is that not so? An example would be the fields near a water molecule. They are not all at symmetric with respect to the angle of an observer.

Equipresence is not 'frame indifference'. Equipresence means: A variable present as an independent variable in one constitutive relation should be present in all constitutive relations. For example, if the permittivity is dependent on density in one constitutive relation, it must depend on density in all constitutive relations.

The specific relationship may depend on observers- for example, Newton's law is not frame-independent, and we must sometimes add in 'ficticious' forces to reconcile two frames of reference.

"material indifference" is something else- that means the response of the material is independent of the observer. This is not coordinate invariance (which means the equations must be of tensor form), and is usually trivially satisfied.

|squeezed> is correct- a birefringent material will be birefringent in all reference frames, a nonlinear material will be nonlinear in all reference frames, etc.
 

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