## Abstract

A medium which is an isotropic chiral medium from the perspective of a co-moving observer is a Faraday chiral medium (FCM) from the perspective of a non-co-moving observer. The Tellegen constitutive relations for this FCM are established. By an extension of the Beltrami field concept, these constitutive relations are exploited to show that plane wave propagation is characterized by four generally independent wavenumbers. This FCM can support negative phase velocity at certain translational velocities and with certain wavevectors, even though the corresponding isotropic chiral medium does not. The constitutive relations and Beltrami-like fields are also used to develop a convenient spectral representation of the dyadic Green functions for the FCM.

## 1. Introduction

A fundamental issue in electromagnetics is the variation in the perceived properties of a linear medium according to the observer's inertial frame of reference. Interest in this topic dates from the earliest days of the special theory of relativity, and it remains an active area of research. Electromagnetic fields in moving mediums which are isotropic dielectric–magnetic from the perspective of a co-moving observer have been widely studied (Pappas 1965; Chen 1983; Kong 1986). Recent studies involving an isotropic dielectric–magnetic medium have demonstrated that plane wave propagation with positive phase velocity (PPV), negative phase velocity (NPV) or orthogonal phase velocity can arise depending upon the observer's inertial frame of reference (Mackay & Lakhtakia 2004*a*; Mackay *et al*. in press). The electromagnetics of simply moving plasmas have also been extensively investigated (Chawla & Unz 1971).

In this paper, we consider the electromagnetic fields in linear isotropic chiral mediums moving at constant velocities. A natural formalism for investigating the electromagnetic properties of an isotropic chiral medium, as observed from a co-moving inertial frame of reference, is provided by Beltrami fields. The defining characteristic of a Beltrami field is that the curl of the field is a scalar multiple of the field itself (Lakhtakia 1994*a*,*b*). These fields are useful for the analysis of a wide range of physical phenomena, as in, for example, astrophysics (Chandrasekhar 1956, 1957), hydrodynamics and magnetohydrodynamics (Dritschel 1991), thermoacoustics (Ceperley 1992), chaotic flows (McLaughlin & Pironneau 1991), plasma physics (Yoshida 1991) and magnetostatics (Marcinkowski 1992). In the following sections, an extension of the Beltrami field concept is developed to investigate the electromagnetic properties of an isotropic chiral medium, as observed from a non-co-moving inertial frame of reference.

In the earlier studies involving isotropic chiral mediums moving at constant velocities, the Lorentz-transformed wavevector and the Lorentz-transformed angular frequency have been used to explore Doppler shift and aberration (Engheta *et al*. 1989); the scattering response of an electrically small sphere made of an isotropic chiral medium has been formulated (Lakhtakia *et al*. 1991); and plane wave propagation has been investigated for relatively low translational speeds (Hillion 1993; Ben-Shimol & Censor 1995, 1997). Reflection and transmission coefficients for a uniformly moving isotropic chiral slab have also been calculated using the Lorentz-transformed electromagnetic fields (Hinders *et al*. 1991).

In contrast to these earlier works, the analysis presented in the following sections begins with a derivation of the Tellegen constitutive relations, from the perspective of a non-co-moving observer, for a medium which is an isotropic chiral medium from the perspective of a co-moving observer. By means of the Bohren transform and the consequent introduction of the Beltrami-like fields, these constitutive relations are exploited to consider plane wave propagation—specifically, the propensity for NPV—from the perspective of a non-co-moving observer. The constitutive relations, combined with the Beltrami-like fields, are also used to develop a convenient spectral representation of the dyadic Green functions (DGFs) for the isotropic chiral medium moving at constant velocity.

As regards notational matters, 3-vectors are underlined, whereas 3×3 dyadics are double underlined. The identity 3×3 dyadic is written as . Vectors with the ^{∧} symbol overhead are unit vectors. The operators and deliver the real and imaginary parts, respectively, of complex-valued quantities; the superscript ‘asterisk’ denotes the complex conjugate; and . The permittivity and permeability of free space are denoted by *ϵ*_{0} and *μ*_{0}, respectively; and is the speed of light in free space.

## 2. Constitutive relations

We contrast the electromagnetic properties in two different inertial frames of reference, denoted by *Σ*′ and *Σ*, where *Σ*′ moves with constant velocity relative to *Σ*. The space-time coordinates in frame *Σ*′ are related to the space-time coordinates in frame *Σ* by the Lorentz transformation(2.1)wherein(2.2)and the relative speed *β*=*v*/*c*_{0}.

In the absence of sources, the spatio-temporal variations of the (time-domain) electromagnetic fields in frame *Σ*′ are related as(2.3)whereas those in frame *Σ* are related as(2.4)as dictated by the Lorentz covariance of the Maxwell curl postulates. The primed and unprimed fields in equations (2.3) and (2.4) are connected via the Lorentz transformation as (Chen 1983)(2.5)

(2.6)

(2.7)

(2.8)

Let us consider a homogeneous medium, which is an isotropic chiral medium from the perspective of a co-moving observer relative to the frame *Σ*′. In the most general scenario, the medium is spatio-temporally non-local. From the perspective of a co-moving observer relative to *Σ*′, the constitutive relations of the medium may be expressed in the Tellegen form as (Lakhtakia 1994*b*)(2.9)(2.10)with the real-valued (time-domain) constitutive parameters , and . By implementing the spatio-temporal Fourier transforms (Lakhtakia & Weiglhofer 1996)(2.11)and(2.12)along with the convolution theorem (Walker 1988), the frequency-domain constitutive relations emerge as(2.13)

In many applications, the effects of spatial non-locality are negligible in comparison with those of temporal non-locality. The constitutive relations (2.13) may then be approximated as(2.14)for spatially local mediums, wherein(2.15)and(2.16)with , for *Χ*=*ϵ*, *ξ* and *μ*.

Let us now proceed to develop the frequency-domain constitutive relations in frame *Σ*. After using equations (2.5), (2.7) and (2.8) to substitute for , and , respectively, the constitutive relation (2.9) may be expressed in terms of *Σ* fields as(2.17)Similarly, the constitutive relation (2.10) may be expressed in terms of *Σ* fields as(2.18)by using equations (2.5)–(2.7) to substitute for , and , respectively. Implementation of the spatio-temporal Fourier transforms(2.19)and equation (2.12) with equation (2.17) delivers(2.20)and with equation (2.18) yields(2.21)In the derivation of equations (2.20) and (2.21), the principle of phase invariance (Pappas 1965; Kong 1986) has been invoked to obtain the relations(2.22)

The two independent expressions (2.20) and (2.21) which relate and to and can be manipulated to deliver the *Σ* frequency-domain constitutive relations(2.23)Herein, all the 3×3 constitutive dyadics have the form(2.24)The unprimed constitutive parameters emerge as(2.25)(2.26)(2.27)with(2.28)

The constitutive relations (2.23) in *Σ* reduce to those in *Σ*′ specified by equation (2.13) in the limit *v*→0. We note the similarity of these constitutive relations (2.23) to the tensor formulation that is used in plasma physics (Melrose & McPhedran 1991). Interestingly, the constitutive dyadics (2.24) have the same form as that ascribed to a Faraday chiral medium (FCM) (Weiglhofer & Lakhtakia 1998). Hitherto, FCMs have been conceptualized as homogenized composite mediums (Engheta *et al*. 1992; Weiglhofer & Lakhtakia 1998), arising from blending together an isotropic chiral medium with either a magnetically biased ferrite (Weiglhofer *et al*. 1998) or a magnetically biased plasma (Weiglhofer & Mackay 2000). Through the homogenization process, the natural optical activity of isotropic chiral mediums (Lakhtakia 1994*b*) is combined with the Faraday rotation exhibited by gyrotropic mediums (Lax & Button 1962).

For the spatially local medium represented by the *Σ*′ constitutive relations (2.14), the corresponding constitutive relations in *Σ* are delivered from equation (2.23) as(2.29)with the 3×3 constitutive dyadics defined as in equation (2.24), but with no dependency on .

As an illustrative example, let us consider the case of the medium specified by the *Σ*′ constitutive parameters *ϵ*′=6.5+i1.5, *ξ*′=1+i0.2 and *μ*′=3.0+i0.5. The corresponding constitutive parameters in *Σ*, as specified in equations (2.25)–(2.27), are plotted in figure 1 against the relative speed *β*∈[0,1). The parameters are constrained such that Re{*Χ*^{t},*Χ*^{z}}→Re{*Χ*′}, Im{*Χ*^{t},*Χ*^{z}}→Im{*Χ*′} and |*Χ*^{g}|→0 as *β*→0, for *Χ*=*ϵ*, *μ* and *ξ*. In figure 1, whereas Re{*Χ*^{t},*Χ*^{g}} and Im{*Χ*^{t},*Χ*^{g}} become vanishingly small as *β* approaches unity for *Χ*=*ϵ* and *μ*, as do Re{*ξ*^{t}} and Im{*ξ*^{t},*ξ*^{g}}, this is not the case for Re{*ξ*^{g}}. The parameters *Χ*^{z} are independent of *β* for *Χ*=*ϵ*, *μ* and *ξ*.

## 3. Plane wave propagation

Let us now consider the propagation of plane waves in spatially local mediums of the chosen kind. A plane wave characterized in frame *Σ* by the wavevector and the angular frequency *ω* is related to a plane wave characterized by the wavevector and the angular frequency *ω*′ in frame *Σ*′ by the relations (2.22). The Doppler shift and aberration arising from the transformation from *Σ*′ to *Σ* have been explored previously (Engheta *et al*. 1989). In the remainder of this section, we exploit the frequency-domain constitutive relations (2.14) and (2.29) for the two frames to consider the plane wave propagation and, in particular, investigate the phenomenon of NPV for the isotropic chiral medium moving at constant velocity. A central element in the analysis is the introduction of the Beltrami and Beltrami-like fields.

The plane wave propagation in frame *Σ*′, with field phasors of the form(3.1)is a well-documented matter (Lakhtakia 1994*b*). It is both mathematically expedient and physically insightful to implement the Bohren transform and introduce the Beltrami field phasors (Bohren 1974)(3.2)with the intrinsic impedance(3.3)Thereby, the frequency-domain Maxwell curl postulates in frame *Σ*′, namely(3.4)may be recast as two uncoupled first-order differential equations, which yield(3.5)for plane waves (3.1). Regardless of the direction of propagation, two wavevectors with and the corresponding wavenumbers are supported, where (Lakhtakia 1994*b*)(3.6)and .

The non-reciprocal bianisotropic nature (Krowne 1984) of the medium specified by equation (2.29) in frame *Σ* leads to more complicated plane wave characteristics than in *Σ*′. Following the strategy used for the frame *Σ*′, it is helpful to use the field phasors(3.7)This enables the frequency-domain Maxwell curl postulates in the frame *Σ*, namely(3.8)to be decoupled as(3.9)for plane waves(3.10)with wavevector and . The 3×3 dyadics in equation (3.9) are given as(3.11)with(3.12)The unprimed constitutive parameters in equation (3.12) are defined as in equations (2.25)–(2.27), but with no dependency on . Whereas are the Beltrami field phasors (Lakhtakia 1994*a*), should be called the Beltrami-like field phasors.

The dispersion relations(3.13)arise immediately from equation (3.9). For an arbitrary direction of propagation specified by the relative orientation angle , the dispersion relations (3.13) may be expressed as the pair of quadratic equations(3.14)wherein the relative wavenumber . The coefficients in these equations are given as(3.15)and(3.16)Thus, four relative wavenumbers emerge as the roots of equation (3.14). While these may be straightforwardly extracted from equation (3.14), explicit algebraic representations of the wavenumbers are generally cumbersome. The following two special cases are noteworthy exceptions. For propagation parallel to the direction of translation (i.e. ), we have(3.17)whereas the relative wavenumbers are delivered as(3.18)for propagation perpendicular to the direction of translation (i.e. ).

By way of a numerical illustration, let us return to the constitutive parameters used in figure 1. The corresponding relative wavenumbers in *Σ*, computed as the roots of equation (3.14), are plotted in figure 2 against the relative speed *β*∈[0,1) and the wavevector orientation angle *θ*∈[0,*π*). For clarity, the wavenumbers in figure 2 are ordered such that . Note that , whereas . At *β*=0, the wavenumbers are independent of *θ*. As *β* increases from zero, the dependencies of the wavenumbers upon *θ* are observed to be highly asymmetric with respect to *θ*=*π*/2. In the limit *β*→1, the *θ*-dependencies of the real parts of the wavenumbers become antisymmetric relative to *θ*=*π*/2, whereas the *θ*-dependencies of the imaginary parts of the wavenumbers become symmetric relative to *θ*=*π*/2.

In relation to plane wave propagation, a topic of considerable present interest is whether the phase velocity is negative or positive (Lakhtakia *et al*. 2003). The NPV is closely related to the phenomenon of negative refraction (Ramakrishna 2005). Plane wave propagation with NPV in *Σ* is signified by (Mackay & Lakhtakia 2004*b*)(3.19)where is the time-averaged Poynting vector; conversely, the PPV in *Σ* is signified by(3.20)In *Σ*′, NPV is signified by and PPV by . Issues concerning NPV propagation for isotropic chiral mediums (Mackay 2005) and FCMs arising as homogenized composite mediums (Mackay & Lakhtakia 2004*b*) have been reported previously.

For the medium of interest here, NPV propagation occurs in *Σ*′, provided that (Mackay 2005)(3.21)For the same medium, by virtue of equation (3.19), NPV propagation occurs in *Σ*, provided that(3.22)the form of the real-valued NPV parameter being provided in appendix A.

Let us return to the numerical example considered in figures 1 and 2. The *βθ*-regimes of NPV and PPV, as determined by evaluatizng for , are mapped in figure 3 with respect to the relative speed *β*∈[0,1} and the wavevector orientation angle *θ*∈[0,*π*). The medium clearly does not support NPV propagation when *β*=*0*, i.e. all plane waves in *Σ*′ must be of the PPV kind. As *β* increases, the *βθ*-regimes supporting NPV propagation in *Σ* emerge in the range *π*/2<*θ*<*π* for the relative wavenumbers and in the range 0<*θ*<*π*/2 for the relative wavenumbers .

Finally, in this section, we note that an alternative derivation of the dispersion relations (3.13) in *Σ* may be developed via the Lorentz transformation of the corresponding dispersion relations in *Σ*′. NPV arises when this Lorentz transformation brings about a change of sign in the angular frequency.

## 4. Dyadic Green functions

The problem of finding the (frequency-domain) field phasors generated by a specified distribution of sources within a linear medium is conveniently tackled by means of the DGFs (Tai 1994). For the isotropic chiral medium in frame *Σ*′, the DGFs are well known (Lakhtakia 1994*b*). However, explicit representations of the DGFs are generally unavailable for anisotropic and bianisotropic mediums (Mackay & Lakhtakia in press). In this section, we exploit the constitutive relations derived in §2, together with the Beltrami-like fields introduced in §3, to establish a convenient spectral representation of the DGFs for the FCM described by equation (2.29).

Let a source electric current density phasor and a source magnetic current density phasor exist, from the perspective of a non-co-moving observer. Extending the approach adopted in §3 wherein the Beltrami-like fields are introduced to aid the plane wave analysis in *Σ*, we recast as the Beltrami current density phasors(4.1)The Beltrami-like field phasors generated by the source terms (4.1) may then be expressed in terms of the DGFs as(4.2)where *V* is the region containing the source current density phasors. The complementary functions are given by(4.3)wherein and satisfy the relations(4.4)along with(4.5)The DGFs in equation (4.2) are provided as the solutions of the differential equations(4.6)with being the Dirac delta function. By implementing the spatial Fourier transforms,(4.7)with equation (4.6), the components of the spectral DGFs, , with respect to the Cartesian basis vectors, , emerge as(4.8)with(4.9)and(4.10)with(4.11)Having established the spectral DGFs , we obtain the *Σ* field phasors generated by the source phasors as(4.12)and(4.13)

## 5. Discussion

The Tellegen constitutive relations for an isotropic chiral medium moving at constant velocity are presented in equations (2.23) and (2.29). The availability of these constitutive relations facilitates a complete analysis of the plane wave characteristics of the medium, and it also enables the spectral DGFs to be derived in a convenient form. In contrast to earlier studies, the analysis presented herein is not restricted to low relative speeds (Hillion 1993; Ben-Shimol & Censor 1995, 1997). Furthermore, the constitutive relations (2.23) and (2.29) establish that the uniformly moving isotropic chiral medium, in fact, belongs to the category of FCMs.

In §3, the analysis of plane wave propagation in the reference frames *Σ*′ and *Σ* is aided by the introduction of the Beltrami field phasors in equation (3.2) and the Beltrami-like field phasors in equation (3.7), respectively. They facilitate a decoupling of the Maxwell curl postulates. A key property of is that the curl of is a scalar multiple of , as demonstrated in equation (3.5). Such fields are known as the Beltrami fields and their properties are firmly established (Hillion & Lakhtakia 1993; Lakhtakia 1994*a*,*b*). In contrast, the curl of is not generally parallel to , as may be observed from equation (3.9). The Beltrami-like field phasors, , therefore represent an important extension of the usual Beltrami field concept which can be traced back to at least the late 1880s (Beltrami 1889; Silberstein 1907; Trkal 1919).

Owing to their relatively large parameter space, linear bianisotropic mediums support a richer palette of plane wave properties than do anisotropic and isotropic mediums, as has been highlighted lately by the investigations of NPV propagation (Mackay & Lakhtakia 2004*b*) and optical singularities (Berry 2005). The plane wave study presented in §3 reveals that the bianisotropic FCM described by the constitutive relations (2.29) generally supports four independent wavenumbers for each direction of propagation, from the perspective of a non-co-moving observer (the exception being propagation perpendicular to the direction of translation for which only two independent wavenumbers are supported). This contrasts with the two independent wavenumbers supported by the isotropic chiral medium from the perspective of a co-moving observer. We see in figure 3 that an isotropic chiral medium, which does not support NPV propagation from the perspective of a co-moving observer, does support NPV propagation from the perspectives of a certain class of non-co-moving observers, provided that the relative speed is sufficiently high. This finding is consistent with the results presented for an isotropic dielectric–magnetic medium moving at constant velocity (Mackay & Lakhtakia 2004*a*). This is also consistent with a study which showed that a FCM arising as a homogenized composite medium can support NPV propagation, provided that the gyrotropy parameter of the gyrotropic constituent medium is sufficiently large (Mackay & Lakhtakia 2004*b*).

While explicit representations of the DGFs are available for isotropic mediums, these are generally not available for anisotropic and bianisotropic mediums (Mackay & Lakhtakia in press). However, as in §4, the field phasors for the FCM described by the constitutive relations (2.29) may be formulated in terms of the spectral DGFs. By exploiting the constitutive relations (2.29) and the Beltrami-like field phasors , a convenient representation of the spectral DGFs is established in equations (4.7)–(4.11).

## Acknowledgments

T.G.M. acknowledges EPSRC for support under grant GR/S60631/01.

## Footnotes

- Received June 17, 2006.
- Accepted August 17, 2006.

- © 2006 The Royal Society