Averaging, Scale Statements and Scaling Metrics in Homogeneous and Heterogeneous Electrodynamics

Having as an example of the two scale treatment in electrodynamics consider here the photonic band-gap problem, which is often just the problem of EM propagation in the two-phase porous media. Note, also that in one scale electrodynamics this problem is often solved applying the "innocent" assumption of the infinite domain used.

For the component one of the 2D two-phase (two different optical components) dielectric medium with $\mu =1$ the homogeneous time-harmonic equation we can get for the photonic band-gap study (see more in the "Optics" section - ) is

______ (1)
where $f_{p1}$ is the $\QTR{bf}{H}_{3}$ or $\QTR{bf}{E}_{3}$ polarization determined components of electric or magnetic fields and $\lambda $ is the spectral parameter. Meanwhile, the Upper scale HSP-VAT for the same phase component with the constant coefficients is

______ (2)

Comparison of equation (1) and one of the simplest upper scale HSP-VAT photonic medium equations (2) gives a ground for a number of observations which are critical for understanding and proper application of HSP-VAT scaled mathematical models in electrodynamics and linear optics. When one analyzes (considers), for example, equations (1) and (2) it should be up front a clear understanding of what does the global dependence MATH mean ?:

1) Is this the piece-wise constant in each of the phase function? Is the jump condition present at the interface?

2) Is this the piece-wise constant function with the relatively smooth but very thin transition layer of interface?

3) The most important question is - Are those interface physics phenomena so important qualitatively or quantitatively that their influence should be found in the governing modeling equations ?

The answers to these questions are laying mostly in the physics of another (smaller) scale and they are not ready for pick-up in most of the problems.

Studies (Gratton et al., 1996; Travkin et al., 1995, 1998a, 1999a, 2000a, etc.) concerning the thermal macroscale physics and fluid mechanics in capillary and globular heterogeneous (porous) medium morphologies demonstrated that even when interface transport has no openly declared physical special features as, for example, surficial transport - longitudinal diffusion or adsorption, etc., the input of additional surficial and fluctuation terms in the upper scale HSP-VAT equations solutions can be significant, reaching the same order of magnitude in balance, as traditional diffusion, heat exchange or friction resistance terms input.

That means - the another non-traditional scaled physical effects are exist and we need to take this into consideration when building the two (or more) scale physical model, experiment, optimize the structure.

More importantly to have the lower scale physics of the interface included in the upper scale governing equations when this physics is somehow different from physics of upper scale. In electrodynamics those are, for example, effects of polarization on interface surface, surficial longitudinal waves or current, surface plasmons, etc.

There is the very interesting problem of coupling physics of reflection and transmission of EM incidental wave based on the language of nanoscale surface non-local electrodynamics with the continuum physics of upper scale HSP-VAT description. This is rather natural task within the scaled HSP-VAT description of transport in heterogeneous media.

Meanwhile, there is no other theory or approach which could include within itself the governing equations formulations of the physical phenomena from neighboring lower or upper scale description. We are not talking here about source terms inclusion. The source term inclusion provided usually in the form of some analytical formulae to describe an effect when there is the lack of knowledge or resources, or no vital necessity in analyzing the coupled phenomena of different scales.

The homogenization and fractal methods are inapplicable in most of the situations. Fractal approach is not relevant to most of the morphologies, and the fractal phenomena description is generally too morphological, lacking many physical features presented - as, for example, descriptions in both phases, or description of the phase interchange, etc.

The most sought after characteristics in heterogeneous media transport, which are the effective transport coefficients can be correctly determined using the conventional definition as for the effective conductivity, for example

but only in the fraction of problems, even while employing the DMM-DNM exact solution. The issue is that in majority of problems, as for inhomogeneous, nonlinear coefficients, for example, and in many transient problems having the two-field (EM) DMM-DNM exact solution is not enough to find effective coefficients, because the formulation of correct formulae is beyond the homogeneous methods tools.

Thus, returning to equations (1) and (2) it should be clear understanding that the representative point (RP) of electric field component in (1) is the dot point of approximate size MATH, which includes $\gtrsim 0(10^{6})$ atoms. Then the reasonable representative elementary HSP-VAT volume (REV) at the upper scale for the equation (2) can be about MATH including within itself the thousands of lower scale RP volumes together with their structural elements, etc.

This REV volume would contain MATH atoms or MATH atoms, this is even for today's upper end high power supercomputers not a solvable MD problem.

We need to understand that the HSP-VAT allows us to do this calculation by now- at this time, not waiting more for unknown number of years, and what is More Important, using this simulation scaling correctly. Not just summarizing the box atoms movements.

And remember - this is not the MD box modeling. There is no actually a good purpose to calculate such a grand volume of atoms - in any situation this won't be a correct calculation - because this piece of matter already has irregularities included. The REV at the said above scale MATH these irregularities includes. And again, the simple volume averaging of this number of atoms features will not give correct answers if using the OG theorem formulae.

That means also, that when one needs to consider the equation (1) then any single point where it is valid is considered being or in one dielectric phase (for photonic media), or in another phase, or at the interface, and the representative point "size" of it (the volume ) is RP MATH and this is the very generous size for homogeneous matter$.$ The larger size RP might be inhomogeneous by nature of the material. It can include, for example, molecular scale defects, shifts, etc. And strictly speaking the smaller size is needed.

Meanwhile, the probe size and the size of interest in measurement of reflected and transmitted EM fields and energy are much bigger then the size of the REV $\sim $ $1[\mu m^{3}]$ as we determined above. That means, the reflection coefficient for the photonic (and for any microporous) medium volume or the transmission coefficient are the averaged variables by definition

where $P_{r}$ is the reflected power and $P_{i}$ is the incident EM power. That means that the interesting in application and the measured variables as EM wave propagated through the medium volume or surface are averaged over the volume even larger then the REV's size we accepted as resonable. Equation (1) does not match to this scale range. Solving equation (1) in a two-phase medium means that the variable corresponds to the field's point averaged over the volume of RPMATH or less. And just having these fields averaged over the larger volume to look for corresponding size volumes in modeling and in experiment is not sufficient enough for few reasons, some of them we are just discussing above, in this section on HtElectrodynamics, in a few other places in this website, and in

  • - "Fundamentals of Hierarchical Scaled Description in Physics and Technology"

    see also in (Travkin and Catton, 1998a,2001; Travkin et al., 1999, 2000a,b).

    Still, the main reason, why equation (2) together with the lower level micromodel (1) is more correct tool to model the heterogeneous medium EM propagation, is because it has the mechanisms (additional terms) which clear role is to connect physics of lower scale transport, often different from the upper scale, and morphology of the two-phase (or more) medium to its bulk effective properties and upper scale fields.

    It is essential to take into consideration qualitative transformation of homogeneous conventional formulation EM wave propagation problem to heterogeneous scaled (at least on two scales) problem as soon as the numerous structural objects which separate the phases are introduced to the medium for any good reason.

    The processes of EM fields propagation at the lower scale - inside, around and through a single separate element of two-phase medium as layer, globular inclusion, void gap, etc., are not self-sufficient for describing or controlling the characteristics of the medium.

    On the other hand, the formulation of the EM wave propagation problem as the task for one phase field equations, or even as for two-phase homogeneous medium does not characterize or connect the local (lower scale) transport characteristics directly to the performance of the medium or device, e.g. to the upper scale. It does not explain also - how to improve the performance features of heterogeneous device or optimize some of them.

    These ideas can be demonstrated using the example of a fiber Bragg grating, when more than $10^{4}$ gratings along of the optical fiber can be used. This great number of structural elements - as different refractive index gratings in the fiber core, requires the evaluation of reflection coefficient locally in each grating. Meanwhile, more important is the overall reflectance - the reflectance of these $10^{4}$ layers as a medium, which is the pure larger scale physical characteristic.

    Formulation of this problem using the 2 scale two-phase HSP-VAT equations (2) together with the lower scale (1) includes into analysis the interface electric field exchange rate between the phases, effects of irregularities (as Gaussian distribution) in gratings itself, etc. Using the only equation (1) for that purpose ignores interface fields effects (even for constant phase coefficients) and does the reflectance rate evaluation incomplete. It hides also the hard task of coupling the grating performance to the phenomena of temperature dependency, which became the standard problem in heterogeneous thermophysics (Travkin and Catton, 1998a; 2001).

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