# Maxwell's Equations for nonlinear media

1. Jun 3, 2010

### MathRyan

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!

2. Jun 3, 2010

### |squeezed>

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.

3. Jun 3, 2010

### MathRyan

Does anything change if we assume the medium in non-local?

4. Jun 3, 2010

5. Jun 3, 2010

### MathRyan

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?

6. Jun 3, 2010

### Meir Achuz

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.

7. Jun 3, 2010

### |squeezed>

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.

8. Jun 3, 2010

### Andy Resnick

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).

9. Jun 3, 2010

### |squeezed>

Hi Andy

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

Could you explain a bit more this "equipresence" (or give a reference) ?

Thanks

10. Jun 3, 2010

### Andy Resnick

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.

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

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

http://www.math.cmu.edu/math/faculty/noll.html

11. Jun 4, 2010

### PhilDSP

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. Jun 4, 2010

### PhilDSP

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. Jun 4, 2010

### |squeezed>

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 thats 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

14. Jun 4, 2010

### Andy Resnick

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.