arXiv:0704.1126v1  [physics.optics]  9 Apr 2007
Negative– and positive–phase–velocity propagation in
an isotropic chiral medium moving at constant velocity
Tom G. Mackaya and Akhlesh Lakhtakiab
a School of Mathematics
James Clerk Maxwell Building
University of Edinburgh
Edinburgh EH9 3JZ, United Kingdom
email: T.Mackay@ed.ac.uk
b CATMAS — Computational & Theoretical Materials Sciences Group
Department of Engineering Science & Mechanics
212 Earth & Engineering Sciences Building
Pennsylvania State University, University Park, PA 16802–6812
email: akhlesh@psu.edu
Abstract
Analysis of electromagnetic planewave propagation in a medium which is a spatiotemporally homoge-
neous, temporally nonlocal, isotropic, chiral medium in a co–moving frame of reference shows that the
medium is both spatially and temporally nonlocal with respect to all non–co–moving inertial frames of ref-
erence. Using the Lorentz transformations of electric and magnetic fields, we show that plane waves which
have positive phase velocity in the co–moving frame of reference can have negative phase velocity in cer-
tain non–co–moving frames of reference. Similarly, plane waves which have negative phase velocity in the
co–moving frame can have positive phase velocity in certain non–co–moving frames.
Keywords: Isotropic chiral medium, Lorentz transformation, negative phase velocity, nonlocality
1 Introduction
Analysis of planewave propagation in a frame of reference that is uniformly moving with respect to the
medium of propagation can lead to the emergence and understanding of new phenomenons. In this respect,
the Minkowski constitutive relations, as widely described in standard books [1, 2], are strictly appropriate
to instantaneously responding mediums only [3]. For realistic material mediums, recourse should be taken
to the Lorentz transformation of electromagnetic field phasors [4, 5].
In an earlier study on plane waves in a medium that is spatiotemporally homogeneous, temporally
nonlocal, isotropic and chiral in a co–moving frame of reference, we reported that planewave propagation
with negative phase velocity (NPV) is possible with respect to a non–co–moving frame of reference, even
though the medium does not support NPV propagation in the co–moving frame [6]. That study applies
strictly only at low translational speeds. In this paper, we demonstrate by means of an analysis based on
the Lorentz–transformed electromagnetic fields in the non–co-moving frame, that our conclusion remains
qualitatively valid for realistic mediums even at high translational speeds.
http://arxiv.org/abs/0704.1126v1
2 Planewave analysis
We consider a spatiotemporally homogeneous, spatially local, temporally nonlocal, isotropic chiral medium,
characterized in the frequency domain by the Tellegen constitutive relations [7]
′ = ǫ0ǫ
′ + i
ǫ0µ0ξ
B′ = −i√ǫ0µ0ξ′E′ + µ0µ′rH′
, (1)
in an inertial frame of reference Σ′. The relative permittivity ǫ′r, relative permeability µ
r and chirality
parameter ξ′ are complex–valued functions of the angular frequency ω′ if the medium is dissipative, and
real–valued if it is nondissipative [1, p. 71]; ǫ0 and µ0 are the permittivity and permeability of free space,
respectively. The electromagnetic field phasors are related by the Maxwell curl postulates as
∇×H′ + iω′D′ = 0
∇×E′ − iω′B′ = 0
. (2)
Our attention is focused on a plane wave, described by the Σ′ phasors
E′ = E′0 exp [i (k
• r′ − ω′t′)]
H′ = H′
exp [i (k′ • r′ − ω′t′)]
, (3)
which propagates in the medium characterized by (1), with wavevector k′ = k′ k̂′ and wavenumber k′ = k0k
There are four possibilities for the relative wavenumber: k′r ∈ {k′r1, k′r2, k′r3, k′r4} where
k′r1 =
r + ξ
k′r2 =
r − ξ′
k′r3 = −k′r1
k′r4 = −k′r4
. (4)
With respect to frame Σ′, the plane wave is assumed to be uniform; i.e., k̂′ ∈ R3.
Suppose that the inertial frame Σ′ is moving at constant velocity v = vv̂ relative to another inertial
frame Σ. The electromagnetic field phasors in Σ are related to those in Σ′ by the Lorentz transformations
[4, 5]
E = (E′ • v̂) v̂ + γ
I − v̂v̂
′ − v ×B′
B = (B′ • v̂) v̂ + γ
I − v̂v̂
v ×E′
H = (H′ • v̂) v̂ + γ
I − v̂v̂
′ + v ×D′
D = (D′ • v̂) v̂ + γ
I − v̂v̂
′ − v ×H
, (5)
where I = x̂ x̂ + ŷ ŷ + ẑ ẑ is the 3×3 identity dyadic, γ = 1/
1− β2 and the relative translational speed
β = v/c0, with c0 = 1/
ǫ0µ0 being the speed of light in free space. In terms of the Σ phasors, the plane
wave is described by
E = E0 exp [i (k • r− ωt)]
H = H0 exp [i (k • r− ωt)]
. (6)
The phasor amplitude vectors {E0,H0} and {E′0,H′0} are related via the transformations (5), whereas [4]
I + (γ − 1) v̂ v̂
′ + γ vt′, (7)
t = γ
v • r′
, (8)
k = γ
• v̂ +
I − v̂ v̂
′, (9)
ω = γ (ω′ + k′ • v) . (10)
Since k′ ∈ C for a dissipative medium, we have from (9) that k = kRk̂R + ikI k̂I with kR ∈ R, kI ∈ R,
k̂R ∈ R3, and k̂I ∈ R3, but k̂R 6= k̂I in general; i.e., the plane wave is generally nonuniform with respect to
Σ. Similarly, from (10) we have that ω = ωR + iωI with ωR ∈ R and ωI ∈ R. Expressing the Σ phasors as
E = {E0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)]
H = {H0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)]
, (11)
we note that that the periodic propagation of phase is governed by kR and ωR, whereas attenuation or growth
of the wave amplitude is governed by kI and ωI . By writing the phasor amplitudes as E0 = E0R + iE0I and
H0 = H0R + iH0I , the corresponding cycle–averaged Poynting vector may be expressed as
P = exp (−2kI • r)
E0R ×H0R
∫ t0+
cos2 (kR • r− ωRt) exp (2ωIt) dt
− (E0I ×H0R +E0R ×H0I)×
∫ t0+
cos (kR • r− ωRt) sin (kR • r− ωRt) exp (2ωIt) dt
+E0I ×H0I
∫ t0+
sin2 (kR • r− ωRt) exp (2ωIt) dt
for a cycle beginning at time t = t0. The phase velocity is given by
k̂R. (13)
Whether the plane wave has positive phase velocity (PPV) or negative phase velocity (NPV) in the refer-
ence frame Σ is determined by the sign of vp • P: positive for PPV and negative for NPV. Criterions for
determining whether the phase velocity is positive or negative with respect to the reference frame Σ′ are
presented elsewhere [8, 9].
3 Numerical results and discussion
For the sake of illustration, let us consider the cycle–averaged Poynting vector evaluated at the point r = 0
with the temporal averaging starting from t0 = 0; i.e.,
8πωI |ω|2
|ω|2 + ω2I
E0R ×H0R
−ωRωI (E0I ×H0R +E0R ×H0I) + ω2R E0I ×H0I
. (14)
Without loss of generality, let us assume that the plane wave propagates along the z′ Cartesian axis; i.e.,
k̂′ = ẑ′. It follows then from the Maxwell curl postulates (2) that E′
lies in the x′y′ plane with
ŷ′ • E′
= ix̂′ • E′
for k′r = k
r1, k
ŷ′ • E′
= −ix̂′ • E′
for k′r = k
r2, k
. (15)
Further, we take the velocity v to lie in the x′z′ Cartesian plane as per
v̂ = x̂′ sin θ + ẑ′ cos θ . (16)
In Figure 1 the distributions of PPV and NPV in the reference frame Σ for the dissipative scenario
wherein ǫ′r = 6.5 + 1.5i, ξ
′ = 1 + 0.2i and µ′r = 3 + 0.5i are displayed for k
r ∈ {k′r1, k′r2, k′r3, k′r4}. Clearly,
for all four values of k′r, propagation is of the PPV type for β = 0 (i.e., with respect to the Σ
′ frame). As
the relative translational speed increases, the phase velocity of the plane waves corresponding to k′r1 and k
eventually becomes negative provided that π/2 < θ < π. In contrast the plane waves corresponding to k′r3
and k′r4 have NPV at sufficiently large values of β provided that 0 < θ < π/2.
Now, let us look at the scenario where the isotropic chiral medium supports NPV propagation in the
co–moving reference frame. This situation arises when |Re {ξ′} | > Re
, for example [8, 9]. We
take ǫ′r = 6.5 + 1.5i, ξ
′ = 10 + 2i and µ′r = 3 + 0.5i. The corresponding distributions of NPV and PPV are
mapped against β and θ in Figure 2. We see that the plane waves corresponding to relative wavenumbers
k′r1 and k
r3 have NPV when β is small but have PPV when β is sufficiently large. On the other hand, the
plane waves corresponding to relative wavenumbers k′r2 and k
r4 have PPV when β is small but have NPV
when β is sufficiently large.
To conclude, by means of the Lorentz–transformed electromagnetic fields, we have demonstrated that a
plane wave with PPV in an isotropic chiral medium can have NPV when observed from a no–co–moving
inertial reference frame. Similarly a NPV plane wave in the co–moving frame can be PPV from a non–co–
moving frame.
Acknowledgements The authors are most grateful to Professor I.M. Besieris (Virginia Polytechnic Institute
and State University) for a discussion on the Minkowski constitutive relations for realistic mediums. TGM
is supported by a Royal Society of Edinburgh/Scottish Executive Support Research Fellowship.
References
[1] H.C. Chen, Theory of electromagnetic waves, McGraw–Hill, New York, NY, USA, 1983.
[2] J.A. Kong, Electromagnetic wave theory, Wiley, New York, NY, USA 1986.
[3] I.M. Besieris and R.T. Compton Jr., Time–dependent Green’s function for electromagnetic waves in
moving conducting media, J. Math. Phys. 8 (1967), 2445–2451.
[4] C.H. Pappas, Theory of electromagnetic wave propagation, McGraw–Hill, New York, NY, USA, 1965.
[5] A. Lakhtakia and W.S. Weiglhofer, Lorentz covariance, Occam’s razor, and a constraint on linear
constitutive relations, Phys. Lett. A 213 (1996), 107–111; correction 222 (1996), 459.
[6] T.G. Mackay and A. Lakhtakia, On electromagnetics of an isotropic chiral medium moving at constant
velocity, Proc. R. Soc. A 463 (2007), 397–418; corrections (submitted).
[7] A. Lakhtakia, Beltrami fields in chiral media, World Scientfic, Singapore, 1994.
[8] T.G. Mackay, Plane waves with negative phase velocity in isotropic chiral mediums, Microwave Opt
Technol Lett 45 (2005), 120–121; corrections: 47 (2005), 406.
[9] T.G. Mackay and A. Lakhtakia, Simultaneous negative– and positive–phase–elocity propagation in an
isotropic chiral medium, Microwave Opt Technol Lett 49 (2007), 1245–1246.
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
Figure 1: The distribution of positive phase velocity (PPV) and negative phase velocity (NPV) in the Σ
reference frame, in relation to β ∈ [0, 1) and θ ∈ [0◦, 180◦), for k′r ∈ {k′r1, k′r2, k′r3, k′r4}. Here, ǫ′r = 6.5+ 1.5i,
ξ′ = 1 + 0.2i and µ′r = 3 + 0.5i.
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
Figure 2: As Figure 1 but with ξ′ = 10 + 2i.

  Analysis of electromagnetic planewave propagation in a medium which is a
spatiotemporally homogeneous, temporally nonlocal, isotropic, chiral medium in
a co-moving frame of reference shows that the medium is both spatially and
temporally nonlocal with respect to all non-co-moving inertial frames of
reference. Using the Lorentz transformations of electric and magnetic fields,
we show that plane waves which have positive phase velocity in the co-moving
frame of reference can have negative phase velocity in certain non-co-moving
frames of reference. Similarly, plane waves which have negative phase velocity
in the co-moving frame can have positive phase velocity in certain
non-co-moving frames.

Introduction
Analysis of planewave propagation in a frame of reference that is uniformly moving with respect to the
medium of propagation can lead to the emergence and understanding of new phenomenons. In this respect,
the Minkowski constitutive relations, as widely described in standard books [1, 2], are strictly appropriate
to instantaneously responding mediums only [3]. For realistic material mediums, recourse should be taken
to the Lorentz transformation of electromagnetic field phasors [4, 5].
In an earlier study on plane waves in a medium that is spatiotemporally homogeneous, temporally
nonlocal, isotropic and chiral in a co–moving frame of reference, we reported that planewave propagation
with negative phase velocity (NPV) is possible with respect to a non–co–moving frame of reference, even
though the medium does not support NPV propagation in the co–moving frame [6]. That study applies
strictly only at low translational speeds. In this paper, we demonstrate by means of an analysis based on
the Lorentz–transformed electromagnetic fields in the non–co-moving frame, that our conclusion remains
qualitatively valid for realistic mediums even at high translational speeds.
http://arxiv.org/abs/0704.1126v1
2 Planewave analysis
We consider a spatiotemporally homogeneous, spatially local, temporally nonlocal, isotropic chiral medium,
characterized in the frequency domain by the Tellegen constitutive relations [7]
′ = ǫ0ǫ
′ + i
ǫ0µ0ξ
B′ = −i√ǫ0µ0ξ′E′ + µ0µ′rH′
, (1)
in an inertial frame of reference Σ′. The relative permittivity ǫ′r, relative permeability µ
r and chirality
parameter ξ′ are complex–valued functions of the angular frequency ω′ if the medium is dissipative, and
real–valued if it is nondissipative [1, p. 71]; ǫ0 and µ0 are the permittivity and permeability of free space,
respectively. The electromagnetic field phasors are related by the Maxwell curl postulates as
∇×H′ + iω′D′ = 0
∇×E′ − iω′B′ = 0
. (2)
Our attention is focused on a plane wave, described by the Σ′ phasors
E′ = E′0 exp [i (k
• r′ − ω′t′)]
H′ = H′
exp [i (k′ • r′ − ω′t′)]
, (3)
which propagates in the medium characterized by (1), with wavevector k′ = k′ k̂′ and wavenumber k′ = k0k
There are four possibilities for the relative wavenumber: k′r ∈ {k′r1, k′r2, k′r3, k′r4} where
k′r1 =
r + ξ
k′r2 =
r − ξ′
k′r3 = −k′r1
k′r4 = −k′r4
. (4)
With respect to frame Σ′, the plane wave is assumed to be uniform; i.e., k̂′ ∈ R3.
Suppose that the inertial frame Σ′ is moving at constant velocity v = vv̂ relative to another inertial
frame Σ. The electromagnetic field phasors in Σ are related to those in Σ′ by the Lorentz transformations
[4, 5]
E = (E′ • v̂) v̂ + γ
I − v̂v̂
′ − v ×B′
B = (B′ • v̂) v̂ + γ
I − v̂v̂
v ×E′
H = (H′ • v̂) v̂ + γ
I − v̂v̂
′ + v ×D′
D = (D′ • v̂) v̂ + γ
I − v̂v̂
′ − v ×H
, (5)
where I = x̂ x̂ + ŷ ŷ + ẑ ẑ is the 3×3 identity dyadic, γ = 1/
1− β2 and the relative translational speed
β = v/c0, with c0 = 1/
ǫ0µ0 being the speed of light in free space. In terms of the Σ phasors, the plane
wave is described by
E = E0 exp [i (k • r− ωt)]
H = H0 exp [i (k • r− ωt)]
. (6)
The phasor amplitude vectors {E0,H0} and {E′0,H′0} are related via the transformations (5), whereas [4]
I + (γ − 1) v̂ v̂
′ + γ vt′, (7)
t = γ
v • r′
, (8)
k = γ
• v̂ +
I − v̂ v̂
′, (9)
ω = γ (ω′ + k′ • v) . (10)
Since k′ ∈ C for a dissipative medium, we have from (9) that k = kRk̂R + ikI k̂I with kR ∈ R, kI ∈ R,
k̂R ∈ R3, and k̂I ∈ R3, but k̂R 6= k̂I in general; i.e., the plane wave is generally nonuniform with respect to
Σ. Similarly, from (10) we have that ω = ωR + iωI with ωR ∈ R and ωI ∈ R. Expressing the Σ phasors as
E = {E0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)]
H = {H0 exp [− (kI • r− ωIt)]} exp [i (kR • r− ωRt)]
, (11)
we note that that the periodic propagation of phase is governed by kR and ωR, whereas attenuation or growth
of the wave amplitude is governed by kI and ωI . By writing the phasor amplitudes as E0 = E0R + iE0I and
H0 = H0R + iH0I , the corresponding cycle–averaged Poynting vector may be expressed as
P = exp (−2kI • r)
E0R ×H0R
∫ t0+
cos2 (kR • r− ωRt) exp (2ωIt) dt
− (E0I ×H0R +E0R ×H0I)×
∫ t0+
cos (kR • r− ωRt) sin (kR • r− ωRt) exp (2ωIt) dt
+E0I ×H0I
∫ t0+
sin2 (kR • r− ωRt) exp (2ωIt) dt
for a cycle beginning at time t = t0. The phase velocity is given by
k̂R. (13)
Whether the plane wave has positive phase velocity (PPV) or negative phase velocity (NPV) in the refer-
ence frame Σ is determined by the sign of vp • P: positive for PPV and negative for NPV. Criterions for
determining whether the phase velocity is positive or negative with respect to the reference frame Σ′ are
presented elsewhere [8, 9].
3 Numerical results and discussion
For the sake of illustration, let us consider the cycle–averaged Poynting vector evaluated at the point r = 0
with the temporal averaging starting from t0 = 0; i.e.,
8πωI |ω|2
|ω|2 + ω2I
E0R ×H0R
−ωRωI (E0I ×H0R +E0R ×H0I) + ω2R E0I ×H0I
. (14)
Without loss of generality, let us assume that the plane wave propagates along the z′ Cartesian axis; i.e.,
k̂′ = ẑ′. It follows then from the Maxwell curl postulates (2) that E′
lies in the x′y′ plane with
ŷ′ • E′
= ix̂′ • E′
for k′r = k
r1, k
ŷ′ • E′
= −ix̂′ • E′
for k′r = k
r2, k
. (15)
Further, we take the velocity v to lie in the x′z′ Cartesian plane as per
v̂ = x̂′ sin θ + ẑ′ cos θ . (16)
In Figure 1 the distributions of PPV and NPV in the reference frame Σ for the dissipative scenario
wherein ǫ′r = 6.5 + 1.5i, ξ
′ = 1 + 0.2i and µ′r = 3 + 0.5i are displayed for k
r ∈ {k′r1, k′r2, k′r3, k′r4}. Clearly,
for all four values of k′r, propagation is of the PPV type for β = 0 (i.e., with respect to the Σ
′ frame). As
the relative translational speed increases, the phase velocity of the plane waves corresponding to k′r1 and k
eventually becomes negative provided that π/2 < θ < π. In contrast the plane waves corresponding to k′r3
and k′r4 have NPV at sufficiently large values of β provided that 0 < θ < π/2.
Now, let us look at the scenario where the isotropic chiral medium supports NPV propagation in the
co–moving reference frame. This situation arises when |Re {ξ′} | > Re
, for example [8, 9]. We
take ǫ′r = 6.5 + 1.5i, ξ
′ = 10 + 2i and µ′r = 3 + 0.5i. The corresponding distributions of NPV and PPV are
mapped against β and θ in Figure 2. We see that the plane waves corresponding to relative wavenumbers
k′r1 and k
r3 have NPV when β is small but have PPV when β is sufficiently large. On the other hand, the
plane waves corresponding to relative wavenumbers k′r2 and k
r4 have PPV when β is small but have NPV
when β is sufficiently large.
To conclude, by means of the Lorentz–transformed electromagnetic fields, we have demonstrated that a
plane wave with PPV in an isotropic chiral medium can have NPV when observed from a no–co–moving
inertial reference frame. Similarly a NPV plane wave in the co–moving frame can be PPV from a non–co–
moving frame.
Acknowledgements The authors are most grateful to Professor I.M. Besieris (Virginia Polytechnic Institute
and State University) for a discussion on the Minkowski constitutive relations for realistic mediums. TGM
is supported by a Royal Society of Edinburgh/Scottish Executive Support Research Fellowship.
References
[1] H.C. Chen, Theory of electromagnetic waves, McGraw–Hill, New York, NY, USA, 1983.
[2] J.A. Kong, Electromagnetic wave theory, Wiley, New York, NY, USA 1986.
[3] I.M. Besieris and R.T. Compton Jr., Time–dependent Green’s function for electromagnetic waves in
moving conducting media, J. Math. Phys. 8 (1967), 2445–2451.
[4] C.H. Pappas, Theory of electromagnetic wave propagation, McGraw–Hill, New York, NY, USA, 1965.
[5] A. Lakhtakia and W.S. Weiglhofer, Lorentz covariance, Occam’s razor, and a constraint on linear
constitutive relations, Phys. Lett. A 213 (1996), 107–111; correction 222 (1996), 459.
[6] T.G. Mackay and A. Lakhtakia, On electromagnetics of an isotropic chiral medium moving at constant
velocity, Proc. R. Soc. A 463 (2007), 397–418; corrections (submitted).
[7] A. Lakhtakia, Beltrami fields in chiral media, World Scientfic, Singapore, 1994.
[8] T.G. Mackay, Plane waves with negative phase velocity in isotropic chiral mediums, Microwave Opt
Technol Lett 45 (2005), 120–121; corrections: 47 (2005), 406.
[9] T.G. Mackay and A. Lakhtakia, Simultaneous negative– and positive–phase–elocity propagation in an
isotropic chiral medium, Microwave Opt Technol Lett 49 (2007), 1245–1246.
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
Figure 1: The distribution of positive phase velocity (PPV) and negative phase velocity (NPV) in the Σ
reference frame, in relation to β ∈ [0, 1) and θ ∈ [0◦, 180◦), for k′r ∈ {k′r1, k′r2, k′r3, k′r4}. Here, ǫ′r = 6.5+ 1.5i,
ξ′ = 1 + 0.2i and µ′r = 3 + 0.5i.
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
0 90 ° 180 °
Figure 2: As Figure 1 but with ξ′ = 10 + 2i.
