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We study the controversy about the proper determination of the electromagnetic energy-flux field in anisotropic materials, which has been revived due to the relatively recent experiments on negative refraction in metamaterials. Rather than analyzing energy-balance arguments, we use a pragmatic approach inspired by geometrical optics, and compare the predictions on angles of refraction at a flat interface of two possible choices on the energy flux: and . We carry out this comparison for a monochromatic Gaussian beam propagating in an anisotropic non-dissipative anisotropic metamaterial, in which the spatial localization of the electromagnetic field allows a more natural assignment of directions, in contrast to the usual study of plane waves. We compare our approach with the formalism of geometrical optics, which we generalize and analyze numerically the consequences of either choice.

The location of electromagnetic energy is an elusive subject that has been under discussion since the beginning of electrodynamics [

The problem of the location of energy and the correct expression for the energy flux in the presence of materials acquires additional intricate subtleties related to the description of the energy-exchange mechanism between fields and matter [

Furthermore, in the more general case when the electromagnetic response is linear but not instantaneous, it necessarily depends on frequency and it is dissipative. In this case it is not possible to separate the energy density into material, field and absorption contributions. But even in low-dissipation frequency bands, the correct expression for the Poynting vector (energy flux) depends on the explicit form of the energy-balance equation. Also, in relation to the freedom of choice of Poynting’s vector and the restrictions imposed by other conservation laws: linear and angular momentum, one has to recall that unlike in vacuum, in the presence of material media the relation between Poyting’s vector and the linear-momentum density of the electromagnetic field is still controversial [

Here we will not analyze all different aspects of these longstanding and sometimes subtle questions. We will rather concentrate only in two different proposals for the mathematical expression of the Poynting vector

In this paper, rather than discussing the energetic balance in the material, we propose to look at the con- troversy from the perspective of geometrical optics in an extremely pragmatic approach, based on the fact that the energy flux is not only used to calculate energy balances, but also to quantify light intensity and its direction of propagation. To watch the refraction of a laser beam on a transparent prism is a very common and intuitive experience, in which one could very naturally speak about the “location” of the energy and the direction and “bending” of the energy flux. In contrast, in the idealized case of a plane wave the energy is on the average evenly distributed over all space, and it is therefore unlocalized, making it impossible to use such “intuitive” arguments as above.

For the two fields

Having all this in mind, we tackle the problem by constructing a “ray” of light in order to see how does it refract at an interface between vacuum and an anisotropic metamaterial. One can find different definitions of ray in geometrical optics, for example, one, as a line in the direction of the gradient of the eikonal [

The structure of the paper is as follows: in Section 2 we compare, for each energy-flux proposal, possible interpretations of the energy-balance equations and the terms involved in them; then in Section 3 we present a brief introduction of the electromagnetic properties of anisotropic uniaxial metamaterials with emphasis on the refraction of plane waves at a flat interface; we later state in Section 4 some basic properties of 2D mono- chromatic electromagnetic fields, on which we build our analysis, and make a comparison with the formalism of geometrical optics, which we extend in Section 5. In Section 5.1 we particularize the results and concepts of these two previous sections to a Gaussian beam; we study some its main characteristics, and sketch how to calculate its refraction, to finally display and analyze the corresponding results of the numerical simulations. Section 6 is devoted to our conclusions.

In this section we present briefly the energy-balance equations for the two energy-flux proposals to establish the differences in interpretation of the terms appearing in them. We start with the macroscopic Maxwell’s equations and regard the presence of the material as given by the charge and current densities induced by an external electromagnetic field produced by external sources. Maxwell’s equations, in SI units, can be then written as

where

where 1)

By substituting Equation (5) into Ampère-Maxwell’s law (4) and using the induced charge conservation (6), one can write Equations (1) and (4) as

which together with Equations (2) and (3) form the complete set of the four macroscopic Maxwell’s equations. Here

is called the displacement field, while

is called the magnetic intensity or simply the H field.

If one now calculates

that takes the mathematical form of a conservation law for the energy, and one can interpret

Following the same procedure as above, one can also write the following equation:

In this expression one identifies

We will not discuss further the physical interpretation of the terms that appear in the energy-conservation laws given in Equations (11) and (12); we now rather construct the conceptual and mathematical framework to analyze the energy transport in the refraction of a beam of light at the interface between vacuum and an anisotropic metamaterial. The advantage of dealing with anisotropic metamaterials rather than with crystals, is that in crystals the anisotropy of the electromagnetic response is fixed by the crystalline structure and cannot be changed, while in metamaterials this degree of anisotropy, as well as the signs of the response, can be tailored through the fabrication process.

As discussed above, we will be dealing with anisotropic uniaxial metamaterials. These are characterized by electric and magnetic response tensors

We will now introduce notation and summarize some of the properties that we will use in this paper; their derivation can be found, for example, in [

The dispersion relations of these modes can be put in terms of

Note that

Finally, it is important to say that, in this medium, the field

so both vectors will only be parallel when there is no anisotropy of the corresponding mode (

Let us consider a plane interface between vacuum and the uniaxial metamaterial, set this interface perpendicular to the optical axis of the metamaterial and fix the z-axis along this direction. Then assume that a plane wave, with its wavevector in the xz plane, impinges from vacuum into the metamaterial. One can immediately see that if the incident wave is p-polarized (

Now we look at the reflection and transmission of plane waves in the presence of uniaxial metamaterials, defined as

where

In terms of these definitions and basic concepts, we now summarize some interesting features of the refraction of plane waves on uniaxial metamaterials. A derivation of all these results can be found in [

1) The angle

2) The angle

and we call this the refraction angle.

3) The refraction of

of

4) The sign of refraction is determined by the sign of

5) The refraction angle, as a function of the incidence angle, is an increasing function if

6) Whenever

7) The critical angle has an inverse behavior in the case

8) There exist critical angles for both polarizations.

9) There is low variation of the refraction angle for

10) In the particular case when

Note especially, on relation with negative refraction, some less restrictive features of these materials due to their anisotropy, for example, the sign of the projection of

With respect to point 3, it is important to note that this refraction problem has a mathematical ambiguity arising from the fact that the dispersion relation (13) is quadratic, and thus two possibilities for

In this work we will be dealing, for simplicity, with the refraction of monochromatic two-dimensional beams, that nevertheless keep most of the physics behind the phenomenon of refraction of actual three-dimensional beams. We consider first an arbitrary two-dimensional monochromatic electric field, defined as a superposition of plane waves in the xz plane,

where re denotes real part. In a given medium, this will be a solution to Maxwell's equations if

We can view this superposition as a series of plane waves traveling along different directions and with different amplitudes, these determined by the function

Recalling now that the magnetic, displacement, and

of a plane wave of wavevector

it is immediate to write the corresponding monochromatic fields associated to the electric field given in Equation (18), as

For s-polarization, the amplitudes

thus in terms of

Note that if we denote

and the same is valid for

For p polarization, one can write an expression for the

where

with the following corresponding expressions for the displacement, electric and magnetic fields,

It is important to note that the linear superposition of plane waves, as the one given in Equation (18) can be

also written as

a factor that is a function only of position. Since in the calculation of the energy densities and energy flux we will be dealing with bilinear products of the form

where we have used

For example, using Equations (22) and (28), the time average of

Also, from Equations (22) and (24) one can easily calculate

Note that this result is general and does not depend on the constitutive relations. On the other hand, for

which clearly differs in direction from

Finally, regarding to the energetic consequences of the choice of energy flux, note that, taking the divergence of

Since

has the value

which, in view of the dispersion relation (13), and following the same reasoning as before with

As we already mentioned in the introduction and in the section concerning the refraction of plane waves, the energy-flux vector (Poynting’s vector) is used, besides the calculation of electromagnetic-energy transport, in determining the “detectable” direction of refraction of plane waves, over the direction given by the angle of refraction of the wavevector. Although in many cases they do coincide, their difference in direction is specially critical in the phenomenon of negative refraction. In our pragmatic approach we will look at the refraction of rays―defined as narrow beams―and then calculate the two expressions for the energy flux:

The first question is how to define the location of the beam in order to visualize it. The first idea could be perhaps to identify it with the transmitted energy flux and visualize it by plotting the transmittance, which is what one usually associates as the measurable quantity in optics experiments. The problem with such definition is that the value of the transmittance depends on the definition of the energy flux, which would lead us to a circular argument. Also, let us recall that the transmittance is proportional to the energy flux perpendicular to the interface, as if the detection of the transmitted power would be accomplished only along the perpendicular direction and not along the direction of the beam. Thus, we choose to look instead at the energy density, which in the absence of dissipation is proportional to

In the search of a criterion to determine how a monochromatic field refracts, one may require to define the direction of propagation of the field. At this respect, we derived the following result which we find interesting, and, to our knowledge, unnoticed yet. Let us start considering the simplest case of an isotropic, homogeneous, non-magnetic medium in which

We recognize in

Since the electric field in Equation (22) can be also written as

homogeneous, isotropic, non-magnetic medium, the time average of the field

Going a little bit further, note that the dependence on the material in the expressions for the electric field

This same result does not hold for all materials while regarding the energy flux as given by

The real part of

can write

One can see that the first term in the right hand side points along the direction of the gradient of phase of the electric field as in the case of a homogeneous nonmagnetic material, but now, due to absorption, the field

Nevertheless, the very general result that for any monochromatic electromagnetic field and for any material the direction of

The analogous result for p polarized light might not be as obvious, but is also quite interesting. Using the expressions for the fields given in Equations (25) and (27) one can write,

Without magnetic absorption, both fields are parallel, even in anisotropic media. Moreover, none of them has the property of pointing in the direction of maximum change of the phase of

where we have written

be mathematically clarified by the fact that Maxwell's equations in regions free of external sources together with the constitutive relations are invariant under the interchange of

We now use the results for 2D monochromatic fields to construct a localized beam. We start by regarding an s-polarized beam localized along the z-axis, and impose a boundary condition over the magnitude E of the electric field at

From Equation (22) we get that

tified as the spatial Fourier transform of

Thus, the electric field in any point at any time is given by

This is a 2D Gaussian beam, confined in the x direction and extended along the z direction. Regarding its composition as a superposition of plane waves, note that the plane wave corresponding to wavevector

We will be plotting

We are interested in the refraction of an incident beam from vacuum to an anisotropic metamaterial, but with an arbitrary angle of incidence

and the relationship between these two coordinate systems is given by

Replacing these rotated variables in Equation (43) we get the following expression for the incident beam on the

where the axis of the beam lies along the line

are related through the dispersion relation―and amplitudes given by

Note that the center of the beam remains in the same position.

Given the incident field in Equation (45) and setting the location of the uniaxial metamaterial in

To this purpose, we follow the next steps to refract and reflect a given mode of the incident beam:

1) For a given mode-characterized in the integral by

2) From the resultant wave vector

rotating it as required in Equation (44).

3) Calculate the z-component of this mode by using the dispersion relation in the corresponding medium (vacuum or metamaterial), and assigning

a) a negative sign for the reflected mode.

b) the sign of

4) Multiply the amplitude of this mode by the transmission or reflection coefficient in Equation (15), as a function of the parallel (x-component) of the wavevector.

To summarize this, we have, in terms of

the expressions for the reflected and transmitted fields:

It is worth to note that the reflected and transmitted beams are―due to the presence of the transmission and reflection amplitudes inside these integrals―not Gaussian beams any more. This makes them no longer have the symmetries of the incident beam. Thus, we need a criterion to define the direction of propagation of the transmitted and reflected beams. It seems plausible to define this direction tracing a circle of radius r from the center of the beam, and, for each r, look for the local maximum of

It is convenient for both, calculations and analysis, to express the above relations regarding the composition of the beam in terms of dimensionless quantities. For this, we define

In terms of these quantities, Equation (43) can be expressed equivalently as,

Naturally, there are analogous dimensionless quantities for the reflected and transmitted beams (47). In terms of

which are dimensionless measures of the averages of

We will now take a look at the results of numerical simulations of the refraction of the Gaussian beam. These computations were obtained through a custom c program and plotted in gnuplot with a little help of bash. The source code can be freely downloaded from our page^{1}. For the plotting, we present here some numerical results with effective-medium anisotropic parameters from actual metamaterial experimental reports [

The first material is a laminate metamaterial (LM) made up of a succession of sheets of silver and silica. We took the effective properties at 400 nm of the seven-layered version. This material does not respond mag- netically but has an electrical anisotropic permittivity. Its parallel component for this wavelength is

The second metamaterial is a split ring resonator (SRR). SRR’s were the first constructed metamaterials in which negative refraction was observed. In order to obtain an isotropic response they were built by placing equal resonators on the cells of a cubic lattice. This SSR omitted the isotropization process, placing the resonators in parallel sheets, thus obtaining an uniaxal anisotropic metamaterial. At a microwave frequency of 1.8 GHz the effective properties (again, ignoring the imaginary part) are

Some points to take into account when looking at the results of the simulations are:

1) Due to the dimensionless representation we are using, the units of length in the plots are the width of the beam. Therefore, a same plot with larger larger units of length is equivalent to a thinner beam and vice-versa. In all the figures presented here, we use a parameter

2) The fields

First of all and in order to clarify the idea we have been discussing about the refraction of a light beam, we show in

The symmetry of the beam described in the preceding section makes us expect that in some approximation the propagation of the beam is represented by the propagation of the main mode. Thus, we also indicate the direction of

We present the results for the refraction of the beam at a vacuum-LM interface in

There are some features of these results that we would like to remark:

1) Unlike

A stationary field is established by this interference, just as it happens in the interference between incident and reflected plane waves on an interface, case in which the interference term is a function exclusively of z. This characteristic is somewhat preserved in the beam although it is highly localized (these plots are just windows of

2) Away from the interference zone, the direction of both

viewed from far away, we would only notice an abrupt change in direction from the incidence to the refraction angle.

3) As expected, both

4) The “rays” of

5) In all the simulations that we displayed, the line traced by the local maxima of

6) The magnitude of both

7) In

And last, perhaps the most important observations:

8) For all cases,

9) Some of the basic refraction properties of the propagation of plane waves in uniaxial metamaterials re- ferred in Section 3 are preserved in the case of the beam: a) Negative refraction is obtained when

10) In the metamaterial, the field

This results reveal that for this beam the main wave represents an astonishingly good approximation to the beam in geometrical terms. In general, it is important to remark that such agreement is by no means obvious, since the energy and energy flux are not linear quantities; in fact, it does not happen in other less symmetrical beams, which we do not treat here for the sake of brevity.

The point labeled 6 about

direction of

Let us define

Written in this way, we can recognize the term

On the other hand, note, from Equation (35) that the essential difference between

The numerical analysis of these two quantities

The discussion of the intensity predictions of the two choices of the Poynting vector also leads to an intere- sting question: Since we define intensity as proportional to the energy density, one could ask if there exists a device capable of responding to this quantity. Consider an idealized “intensity detector” consisting of a small plane screen, whose detection result is the integration of the intensity over such surface. Center this detector in a point along the axis of the Gaussian beam. First, put the screen aligned with the axis and take a measure with this device. Afterwards, put the screen in the orthogonal position (remember this is a 2D beam) and take a second measure. Since in the first case the axis coincides with the line of maxima of intensity, the measure is necessarily greater than in the second. But our experience with detectors tells us this is not the case; in fact, it is exactly opposite. This is important because, since

It is also important to stress that the results we show here make evident that in general the ray directions in the formalism of geometrical optics and the notion of a ray as an idealized narrow beam (characterized by its intensity) are not equivalent.

We discussed the choice between two possible expressions for the Poynting vector: 1)

1) For any monochromatic 2D field and in any medium (even absorbing ones) there is a “ray” formalism which extends the eikonal formalism. The directions of those “rays” are given, in s polarization, by

2) The directions of the rays, defined in this work as idealized narrow beams, coincide within the simulations presented here with the Poynting vector if we define it as

a) The “ray” formalism described in conclusion 1 (and therefore the eikonal formalism) is not equivalent to the “intuitive” notion of light ray given by idealized narrow beams.

b) Following the geometrical criterion proposed here, the field

c) The definition of light ray as an idealized narrow beam and the results obtained here allow us to associate the light rays with the field lines of the

We would like to thank Vadim A. Markel for stimulating discussion at the early stages of this project; Augusto García-Valenzuela and Roberto Alexander-Katz for their comments and the full review of the paper; and to Víctor Romero-Rochín for very interesting discussions related to topics about energy conservation. One of us (CP-L) must acknowledge that the work presented here was supported by a graduate scholarship granted by Consejo Nacional de Ciencia y Tecnología (México).

Carlos Prieto-López,Rubén G. Barrera, (2016) Electromagnetic-Energy Flow in Anisotropic Metamaterials: The Proper Choice of Poynting’s Vector. Journal of Modern Physics,07,1519-1539. doi: 10.4236/jmp.2016.712139