_{1}

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Based on two versions of Maxwell’s Equations we investigate the Poynting vector, the energy transport and the dispersion relation both for right- and left-handed systems. Furthermore, it is shown that the latter systems are necessarily dispersive in contrast to the former ones. In the end we discuss a published example where the mixing of expressions of both versions of Maxwell’s Equations leads to unphysical conclusions. The presentation demonstrates for students how flexible can be the work with different versions of electrodynamics but also how carefully one has to be thereby.

Feynman [^{th} century”. Indeed, these laws were and are fundamental for the development of modern industry. For example, no electronic devise would exist without the knowledge of Maxwell’s famous Equations. However, today, more than 150 years after the finding, we are still hampered by a lot of different versions of Maxwell’s Equations. There are two distinct basic systems namely the SIand the cgs-system where the latter is still used in many textbooks (e.g. [

A survey of the history of negative refractive index going back as far as to 1905 is presented online by [

In the following we investigate two different versions of Maxwell’s Equations in the SI-system. For this purpose we determine the electromagnetic field energy density, the Poynting vector, and the dispersion relations and discuss the distinctions. The two versions are taken from [

where

where e is the dielectric permittivity, m is the magnetic permeability, χ_{E} and χ_{M} are the dielectric and magnetic susceptibility, and σ is the electrical conductivity. For homogeneous and isotropic materials all three quantities are frequency dependent complex functions. In the following unless stated otherwise we neglect the dispersion and treat the quantities as complex numbers. By calculating the divergence of the cross product of

Equation (3) expresses the energy conservation saying that the sum of the divergence of the Poynting vector plus the change of the electromagnetic field energy plus the production of Joule’s heat has to be zero. Note the material properties are completely contained in this first version in the total current.

The Poynting vector and the electromagnetic field energy density u and are given by

Since the fields are quadratic in the expressions in (4) we have to write them in real form. Taking the derivative of u with respect of time we obtain for monochromatic fields

This result indicates that u only consists of the vacuum fields, pointed out by the use of

If we calculate the curl of Equation (3) in (1) and insert Equation (4) we get the dispersion relation of the electric field. By changing the order of the calculation we obtain the relation of the magnetic induction. Both Equations read

These dispersion relations look rather unfamiliar for the first representation of Maxwell’s Equations and they are not very useful. With the aim to get more manageable expressions, we substitute the currents now in (1)

where we used (2) and the relation

In this case the Poynting vector, the field energy density, and the dispersion relations read completely different

For vanishing external charges the dispersion relations will convert into the standard formula.

Assuming that the external current vanishes in the first Equation of (8), we see that the material properties are included now in the fields

where two primes are representing the imaginary part of the dielectric function and of the magnetic permeability, respectively. This formula shows that the energy loss (dissipation) is determined by the imaginary parts of e and m. Note that we used the property Im(e^{*}) = −Im(e). Furthermore, we conclude from (9) that

Let us summarize and compare the most important Equations of both versions.

Both versions are, of course, correct and can be used but one has strictly to note not to mix Equations of different versions. This hint is not redundant because this happens even in peer reviewed papers. Below we will discuss an example where this restriction is not taken into account.

There is still a long-standing discussion on the right notation of Poynting’s vector in the literature. For example, Obukhov and Hehl (2003) derived for it

This expression corresponds to the first Equation of (4). Although this confirms our notation of Poynting’s vector in version 1 this does not imply that the other notation in version 2 is wrong. To approve this we transform the Poynting vector from the first version in the second one in the following

Comparing the 4^{th} with the last row of Equation (12) shows that the two terms,

By this way we can claim that both Poynting vectors are a correct notation in the frame of its respective version of Maxwell’s Equations. Although we can define a lot of different expressions for the Poynting vector we have to assure that the divergence of ^{3} in the SI-system). In fact, there is no unique definition of Poynting’s vector and this seems to be a weak point of electrodynamics. Moreover, one can define an arbitrarily large number of different expressions and no experimental way exists to tell which the correct version is (Feynman, 1989). We will come back to this point by the discussion of the left-handed materials. It should be noted here, however, that the vector potential related to the magnetic field is the more relevant quantity because one can measure effects in regions where

For dispersionless media the group velocity, the phase velocity and the velocity of energy propagation are identical. The latter is defined by the ratio of the time-averaged Poynting vector over the time-averaged energy density

Without restriction of generality let the plane wave propagate along the x direction with E = E_{y} and B = B_{z}

where E_{0} and B_{0} are complex amplitudes. It is in every case possible to rotate the coordinate system in the wanted direction. From (8) we get for the field energy density

Carrying out the time average yields

The magnetic part can be recast by means of the dispersion relation of Equation (8) and with the third Equation of Equation (7). For a system without external current, σ = 0, we find

In analogue manner we calculate the time averaged Poynting vector

Finally, we get for the energy transport velocity from Equation (18) and Equation (19)

with n as the refractive index and

As expected, all three velocities are identical far away of any resonance.

Up to now, only normal right-handed systems were investigated. For left-handed systems we have to make some changes because the real parts of e, m and n are smaller than zero. As a consequence, the dispersion relation of Equation (8) may be written now as

where Σ is defined by

Furthermore, LHS must be dispersive as can be seen immediately from the Equation for the field energy density. For weak absorption in the so called transparent regime this Equation is given by (Landau and Lifschitz 1980).

Equation (24) has to be regarded as an approximation since the necessary absorbing part is suppressed. The fact that LHS must be absorbing is based on the Kramers-Kronig relation

For

Since the field energy density must be positive we conclude that the terms in (24) containing the dielectric permittivity and the magnetic permeability have to fulfil

For LHS the Poynting vector is changed to

under use of

This can be rewritten as

The denominator of Equation (29) is positive as discussed above and therefore the group velocity is parallel to the Poynting vector and antiparallel to the wave vector. This is in contrast to the behaviour in RHS. In regions of normal dispersion v_{E} is reduced to the group velocity v_{g} if the system is nearly lossless. In the case of dispersion the group velocity is defined by

where we made use of

Calculating the differentiation in the denominator of (30) and few manipulations leads to

It is easy to show that the evaluation of Equation (29) comes to the same result. The denominators of Equation (29) and Equation (32) are positive as discussed above. Therefore the group and energy transport velocity are parallel to the Poynting vector and antiparallel to the wave vector.

For the modelling of LHS the dielectric permittivity and the magnetic permeability are often described by the Drude or Lorentz model ([_{0} we get

where w_{p} is again the plasma frequency and u is the scattering rate. The real part is negative for frequencies well below the plasma frequency and the imaginary one is always positive. For the derivative in Equation (28) follows

Obviously, Equation (34) is only positive as demanded if w > u. This is, however, in agreement with the condition of weak absorption (transparency regime) assumed by the derivation of Equation (24). In the case of Lorentz’s model we obtain

Insertion of the real part in Equation (26) yields

Equation (36) takes positive values between zero and close to the resonance. It has, however, large negative values around w_{0}. The dominant contribution comes from the third term and is proportional to

Because there is no change for the Poynting vector due to

This condition is equivalent to

Let us now calculate Joule’s heat, the energy loss due to the resistance. In an isotropic system Joule’s heat is given by

For the total current we make the ansatz

If one now replace the square of the wave vector by

then follows for

Although this expression has the correct unit (W/m^{3}) it is wrong. The reason therefore is that the total current in Equation (39) belongs to version 1 and contains all the material properties. In this case, however, the fields are simple the vacuum fields and consequently the relative terms in Equation (41) are unity. In reality Equation (41) results from version 2 but there is no current, Equation (39) is zero, if one neglects the free current. Exactly this is implicit assumed in Equation (41). We know, however, that the loss in the dispersion less materials is correct described by the expression (9) in version 2 which is depending only on the imaginary parts. Although it is not quite accurate we can sometimes ignore for RHSs the dispersion but this is not longer possible for LHSs. For LHSs a dispersion is required otherwise we would get a negative energy density (see Equation (24)). Therefore, even when Equation (42) would be right, it is not suited for LHSs. The dielectric permittivity and the magnetic permeability in Equation (42) cannot be simple complex numbers. For materials with dispersion the right side of Equation (13) reads

where the field amplitudes are changing with time much slower than the exponential part. Obviously, a direct conclusion about the sign of the right hand side of Equation (43) is not possible. We know, however, that the derivates in the bracket of Equation (43) are positive for the Drude and Lorentz model in the extent of validity of the theory (weak absorption). Therefore it is physically not justified to switch simple the sign for the real parts in Equation (42), as done by [

In this paper we have derived and compared some electromagnetic properties of leftand right-handed systems. Especially the phase, group and energy transport velocity are considered. Furthermore, we discussed the opportunities presented by the accurate utilization of different versions of Maxwell’s Equations but also the possible pitfalls. An explicit example of the latter case was considered and the published claim that a negative refraction index is impossible due to a negative loss is physically falsified. Additionally it was shown that LHS has to be dispersive.