Wave Propagation Inside a Cylindrical, Left Handed, Chiral World

Isotropic metamaterials, with negative permittivity and permeability, also called left-handed [Veselago, 1958] [Pendry, 2000], [Sihvola 2007] have opened the way to new physical properties, different from those obtained with a conventional material. So, it is natural to inquire how these properties transform when these metamaterials are also chiral because of the importance of chirality in Nature. Some left-handed materials [Pendry 2006] and metamaterials [Grbic & Elefthertades, 2006] have been recently manufactured. We consider here a metachiral circular cylindrical medium with negative permittivity and permeability endowed with the Post constitutive relations [Post, 1962]. Using the cylindrical coordinates r > 0, θ, z and, assuming fields that do not depend on θ, we analyze the propagation of harmonic Bessel beams inside this medium. Two different modes exist charac-terized by negative refractive indices, function of permittivity and permeability but also of chirality which may be positive or negative with consequences on the Poynting vector carefully analyzed. The more difficult problem of wave propagation in a spherical, left handed, chiral world is succinctly discussed in an appendix.


Introduction
Isotropic metamaterials, with negative permittivity and permeability, also called left-handed [Veselago, 1958] [Pendry, 2000], [Sihvola 2007] have opened the way to new physical properties, different from those obtained with a conventional material.So, it is natural to inquire how these properties transform when these metamaterials are also chiral because of the importance of chirality in Nature.Some left-handed materials [Pendry 2006] and metamaterials [Grbic & Elefthertades, 2006] have been recently manufactured.We consider here a metachiral circular cylindrical medium with negative permittivity and permeability endowed with the Post constitutive relations [Post, 1962].Using the cylindrical coordinates r > 0, θ, z and, assuming fields that do not depend on θ, we analyze the propagation of harmonic Bessel beams inside this medium.Two different modes exist charac-terized by negative refractive indices, function of permittivity and permeability but also of chirality which may be positive or negative with consequences on the Poynting vector carefully analyzed.The more difficult problem of wave propagation in a spherical, left handed, chiral world is succinctly discussed in an appendix.

Cylindrical Maxwell's equations and Post's constitutive relations
With the cylindrical coordinates r > 0, θ, z, the Maxwell equations in a circular cylindrical medium are for fields that do not depend on θ We write −|ε|, −|μ| the negative permittivity and permeability [Pendry, 2006] in the metachiral cylindrical medium and the Post constitutive relations are [Post, 1962] in which ξ is the chirality parameter assumed to be real.From (2), (3), we get at once the divergence equation satisfied by the electric field

Cylindrical harmonic Bessel modes
3.1 The Bessel solutions of Maxwell's equations Substituting (3) into (1b) gives Taking into account (1a), Eqs.( 5) become Applying the time derivative operator ∂ t to (5a) gives Using (1a) and the divergence equation ( 4), we have in the first and last terms of (6a) Substituting these two relations into (6a) and introducing the wave operator [Morse & Feshbach, 1953] we get the equation We have similarly in (6b) for the first two terms and for the last one taking into account (1a) so that Eq.( 6b) becomes Finally in (6c) , the first and third terms are according to (1a) and, taking into account these two relations, we get in which ∆ 0 is the wave operator We look for the solutions of Eqs.(8a,b,c) in the form inside the cylindrical medium: in which J 0 , J 1 are the Bessel functions of the first kind of order zero and one respectively while E r , E θ , E z are arbitrary amplitudes.From now on, we use the two parameters Let us now substitute (10) into (8a,b,c).The Bessel functions with k 2 = k r 2 +k z 2 satisfy the following relations with the exponential factor exp(iωt +ik z z) implicit then, using (12a,b) we get the homogeneous system of equations on the amplitudes This system has nontrivial solutions when its determinant is null and a simple calculation gives Leaving aside k 2 −ω 2 n 2 = 0, the equation ( 14) implies or in terms of refractive index m = ck/ω : m 2 ± αcm −cn 2 = 0.These equations have four solutions, two positive and two negative But, it has been proved [Ziolkowski & Heyman, 2001] that in left handed materials, m must be taken negative: m = − |αc ± (α 2 c 2 + 4n 2 c 2 ) 1/2 | so that introducing the γ > I parameter the equation ( 15) has the two negative roots So, there exist two modes with respective wave numbers k 1 , k 2 able to propagate in the meta-chiral cylindrical medium, with two different negative indices of refraction m 1,2 .= ck 1,2 /ω.

Amplitudes of harmonic Bessel beams
The B, D, H components of the electromagnetic field have the form (10), that is.

Energy flow of Bessel waves
Using ( 10), (18) the Poynting vector S = c/8π (E∧H*) where the asterisk denotes the complex conjugation, gives for the first mode Now, according to ( 11) and ( 17): so that since according to (19d), H 1 † = c/ω|μ| − ξ/k 1 , we get and H 1 † > 0 whatever the sign of ξ/|ξ| is.So for k z > 0 (resp.k z < 0) the z-component of the energy flow runs in the direction of the positive ( resp.negative) z axis while according to (10) and ( 18), Bessel waves propagate in the opposite direction with the phase velocity v z = −ω/k z .Consequently S z and v z are antiparallel, but, because S 1,θ is not null, the phase velocity is not strictly antiparallel to the energy flow.

Evanescent waves
It is implicitly assumed in the previous sections that the wave numbers k r , k z are real which implies k r 2 , k z 2 smaller than k 1 2 , k 2 2 with |k 2 | < |k 1 | according to (17).Suppose first k r 2 > k 1 2 , then with the plus (minus) sign in the z > 0 ( z< 0) region to make exp(ik z z) exponentially decreasing, the only solution physically acceptable.Both modes are evanescent but only the second mode if k 1 2 > k r 2 > k 2 2 .Suppose now k z 2 > k 1 2 then k r(1,2) = ±ik s(1,2) , k s(1,2) = (k z 2 − k 1 , 2 2 ) 1/2 (28) and J 0 (±ik s r) = I 0 (k s r) , J 1 (±ik s r) = ±I 1 (k s r) (28a) in which I 0 , I 1 are the Bessel functions of second kind of order zero, one respectively.These functions are expo-nentially growing with r and physically unacceptable in unbounded media..Of course, if k 1 2 > k z 2 > k 2 2 the first mode can exist.

Discussion
Wave propagation in chiral materials is made easy for media equipped with Post's constitutive relations because as electromagnetism, they are covariant under the Lorentz group.In a metachiral material, the refractive index m depends not only on ε, μ but also on the chirality ξ and in cylindrical geometry m may have four different expressions among which only the two negative ones are physically convenient.But, the Poynting vector S depends on the sign of ξ so that S and the phase velocity v may be parallel or antiparallel