Invited Paper
What is Negative Refraction?
Martin W. McCall∗, Paul Kinsler and Alberto Favaro
Department of Physics, The Blackett Laboratory
Imperial College of Science, Technology and Medicine
Prince Consort Road, London SW7 2BZ, UK
Dan Censor
Department of Electrical and Computer Engineering
Ben-Gurion University of the Negev, Israel
ABSTRACT
We will review the current status of various intrinsic definitions of negative refraction (i.e.
negative phase velocity, or NPV, propagation which has been variously ascribed to counterposition of (i) the group velocity, (ii) the energy velocity, (iii) the Poynting Vector, with
the wave vector of a plane wave in a medium. A key result is that simultaneously negative
effective and μ can be achieved in a natural medium in motion. However, can this be said
to result in observable phenomena? Recent progress in covariant methods has led to a more
rigorous definition that is tied mathematically to what happens in the medium’s rest frame.
The challenge to produce a definition of NPV propagation that is not restricted to linear
reference frames is also addressed. As well, progress has been made recently in clarifying the
role of causality in deriving conditions for NPV propagation.
1. INTRODUCTION
It is often stated that negative refractive index behavior can only be achieved in artificially
manufactured metamaterials whose electromagnetic response is different from that found in
naturally occurring media.1 The reasoning is that whilst a plasmonic response yields negative
for frequencies below the plasma frequency, magnetic activity in natural materials ceases at
about a GHz,2 so achieving a simultaneously negative value of μ requires an artificially manufactured resonant magnetic response. Much current research is being devoted to achieving
this,3 developing the pioneering work on the split ring geometry.4 However, media in motion
present another paradigm for changing the effective electromagnetic properties of a medium.
Is it possible to induce simultaneously effective negative medium parameters (eff , μeff < 0)
when a medium characterized by positive dispersionless and μ is is set in motion? The affirmative answer that we demonstrate in this paper then raises an important conceptual issue:
if a medium can appear to have negative parameters simply as a result of changing reference
frame, then can the assignment of negative refractive index, n = − |eff |1/2 |μeff |1/2 be said
to identify physical phenomena, such as negative refraction, in any sense that is compatible
with the principle of relativity, which asserts that the laws of physics must look the same
*Tel: +44 (0) 207 594 7738, Fax: +44 (0) 207 594 7714, e-mail: [email protected]
Metamaterials: Fundamentals and Applications II, edited by Mikhail A. Noginov, Nikolay I. Zheludev, Allan D. Boardman,
Nader Engheta, Proc. of SPIE Vol. 7392, 73921M · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.827472
Proc. of SPIE Vol. 7392 73921M-1
in all inertial reference frames? The most utilized criterion for identifying negative index
behavior5 is that of Negative Phase Velocity (NPV) propagation given by (E × H) · k < 0.
However, this criterion not covariant (i.e. it depends on the reference frame), and it must be
modified when considering moving media, as we discuss below. Our initial research9 derived
a criterion that restricted to linear reference frames. Here we present a generalization to
arbitrary reference frames using exterior calculus.
Although (E × H) · k < 0 is the most commonly used criterion, others have also been
discussed, based principally on the opposition of group velocity with phase velocity.6 An
important paper by Stockman7 used causal analysis to draw some conclusions concerning the
nature of global loss of negative index media when losses are minimized at the observation
frequency. Whilst these conclusions were subsequently refined,8 our emphasis here is to point
out that the group velocity based criterion for negative index behavior is distinct from that
based on the power flow. This distinction can be made even more stark and explicit when
again considering media in motion, as we discuss below.
2. DEFINITIONS OF NPV PROPAGATION
The basic concept of a backward wave can be captured through the notion that the flow
of electromagnetic energy opposes the direction of phase advance. Simply stipulating this
opposition for any component of the energy flow and wave vector is insufficient, since this
property is not invariant under spatial coordinate rotations9 and can be satisfied by classical
birefringence.10 More satisfactory is the idea of a negative projection
· k < 0 ,
S
(1)
is a vector representing the direction of flow of electromagnetic energy. We distinwhere S
guish three possibilities for this vector as
=E
×H
S
Poynting vector N P V P
= vg =
S
∂ω
∂k
Group velocity
NPV G
= vE =
S
H
E×
ρem
Energy velocity
NPV E
(2)
where ρem is the electromagnetic energy density. One of our objectives is to demonstrate that
these three criteria, NPVP, NPVG and NPVE are not equivalent. Currently, NPVP is the
most popular, though we shall show that difficulties can arise for this definition if it applied
carelessly.
3. CAUSAL ANALYSES
In 2007 Stockman published an important paper7 that was based on the group velocity
definition (2)b. Using the Kramers-Kronig relations11 applied to the causal quantity n2 = μ,
he showed that the phase and group velocity of a plane wave must be related to Im(n2 ) by
the integral relationship
2 ∞ Im(n2 ) 3
c2
=1+
s ds .
(3)
vp vg
π 0 (s2 − ω 2 )2
Proc. of SPIE Vol. 7392 73921M-2
In order for the integral to remain regular at the observation frequency ω, it was posited
that the integrand should vanish. This can be achieved by overcoming loss with gain such
that Im[n2 ] vanishes sufficiently quickly at the observation frequency. The crucial corollary is
that NPVG requires that the right hand side of (3) be negative, and this in turn implies that
Im(n2 ) (i.e. the loss) be significant at other parts of the spectrum. Although this conclusion
is correct, the premise of zero loss at the observation frequency can be avoided. As was shown
in8 the singularity in the integrand can be avoided without necessitating Im(n2 )ω = 0. The
trick is to partially integrate over the regions either side of the singularity (i.e. [0, ω − σ)
and (ω + σ, ∞)) and then let σ → 0. This procedure yields a criterion analogous to Eq. (3)
without the zero-loss condition. The result is
∞
1
ω 1
1
c2
2
2
−1
ω
tanh (z)dz < 0 .
= 1+
Q1 (zω) ln 1 − z dz − ω
Q2 (ωz) + 2 Q2
vp vg
π
z
z
0
0
(4)
where z = s/ω, Q1 (s) = ∂s Im[n2 (s)] and Q2 (s) = ∂s2 Im[n2 (s)]. The key point is that in
(4) it is now the derivatives of the loss spectrum that are significant, not the loss function,
Q0 (ω) = Im[n2 (s)], itself. The criterion of Eq. (4) permits loss, and even significant loss,
at the observation frequency. It is no longer restricted to points of perfect transparency
assumed in deriving (3), and consequently the conclusion that significant loss is necessary at
other parts of the spectrum is also avoided. Of some significance for the current paper is to
1/3
Figure 1. Comparison between NPV propagation criteria. Thick lines show χE where
1/3
χE = P ·k. Thin lines show χG where χG = vg ·k (taking the cube root accommodates vertical
scaling). The various line types correspond to Γ = 0.02 solid, 0.04 dashed, 0.06 dot − dashed.
The double and μ plasmon system has (normalized) plasma frequency of 1.4. Reprinted
figure with permission from8 as follows: Kinsler, P. and McCall, M.W., Phys. Rev. Lett.,
101, 167401, (2008). Copyright (2008) by the American Physical Society.
compare in a simple example, the NPVP and NPVG criteria. We choose a double plasmon
resonance for which
ωp2
μ(ω)
(ω)
.
(5)
=
=1−
0
μ0
ω (ω − iγ)
A simple test is to evaluate the presence of NPVG (and at the same time test the generalized
causality-based criterion in Eq. (4)) is to increase the losses while comparing it against the
Proc. of SPIE Vol. 7392 73921M-3
NPVE condition. The results can be seen in Fig. 1. For sufficiently weak losses the criteria
agree, with both the χE and χG curves remaining below zero. However, as the losses get
stronger χE and χG start to disagree. Nevertheless, we can see that in the preferred region
of ω ≈ 1, where = μ = −1, they disagree only for very large losses. Here the χG criterion
works relatively well because the plasmonic responses vary both smoothly and monotonically;
hence, vg does not change sign and remains in accord with vE .
4. THE CASE OF A MOVING MEDIUM
Take a plane wave traveling with speed (μ)−1/2 along the +z axis in the rest frame of a
scalar isotropic medium. Viewed from a frame moving along the +z axis of the rest frame
with speed v the wave’s phase velocity is, according to the relativistic law of velocity addition
vp =
(μ)−1/2 − v
.
1 − (μ)−1/2 v/c2
(6)
Fig. 2 shows an example for which (μ)1/2 = 1.5. Now the fields in the frame that is in relative
0.8
0.6
0.4
Phase velocity
0.2
0
0. 2
0. 4
0. 6
0. 8
1
0
0.2
0.4
0.6
Frame speed
0.8
1
Figure 2. Phase velocity vp , vs. frame velocity v, according to Eq. (6), with (μ)1/2 = 1.5.
motion to the medium rest frame are described by the Minkowski constitutive relations12 :
− v × B
− v × H = E
D
c2
(7)
+ v × D
+ v × E = μ H
B
c2
(8)
Proc. of SPIE Vol. 7392 73921M-4
For a plane wave at frequency ω with wave vector k = kv /v the Maxwell curl relations
k × E
= ω B,
k × H
= −ω D
together with Eqs. (7) and (8) determine that
, v × H
= −vvp D
,
= vvp B
v × E
,
=−vE
v × B
vp
=
v × D
v vp H
,
(9)
(10)
where we have used the fact that vp = ω/k. Substituting these relations into Eqs. 7 and 8
determines that
= (vp η)−1 E
, B
= ηvp−1 H
D
(11)
where η = (μ/)1/2 . The effective permittivity and permeability for this case are therefore
(vp η)−1 and ηvp−1 respectively, and the effective refractive index is
−1/2 v/c2
1
−
(μ)
(12)
nef f = (μ)1/2
1 − (μ)1/2 v
If the medium speed (viewed from the moving frame) exceeds the C̆erenkov speed (i.e.
v > (μ)−1/2 ), the effective index is negative. We have thus shown that when a simple
medium is set in motion, a wave directed opposite to the medium flow can experience a
negative refractive index. It is thus possible to achieve a negative refractive index without
metamaterials.
The Poynting vector is given by
2 v
× v × E
= η −1 E
×B
= (ηv)−1 E
×H
= vp η −1 E
,
E
v
(13)
and points in the +z direction whatever the medium speed. By contrast the phase velocity
is negative (i.e. points in the −z direction) above the C̆erenkov speed so that
2 v v −1 ·
k<0,
(14)
E × H · k = η E v
v
since k = ω/vp is negative above the C̆erenkov speed. We have thus shown that the NPVP
criterion, P · k < 0, can also be satisfied by a simple moving medium in motion.
The observation that a moving medium can satisfy the P · k < 0 condition for NPVP is
certainly not new.13–15 The new feature here is that the observation is tied explicitly to the
existence of effectively negative medium parameters that arise as a result of the medium’s
motion.
It is helpful to compare the direction of the Poynting vector in this context with the phase
and group velocities. Since vp = ω/k, the phase velocity points along −z above the C̆erenkov
speed. Since dispersion is absent in this idealized model, we have that vg = ∂ω/∂k =
vp , and the group velocity also points along −z. This example therefore clearly points up
the distinction between NPVP and NPVG. Detailed calculations based on transforming the
electromagnetic energy-momentum tensor between frames shows that the energy velocity also
points along −z. The situation is summarized in Fig. 3. That an NPVP condition can be
Proc. of SPIE Vol. 7392 73921M-5
vp
vg
vp vg
-v
Figure 3. (a) Medium rest frame; v̄p = v̄g = (μ)−1/2 . Both the phase and group velocities
× H,
and point to the right. (b) Medium moving
are parallel with the Poynting vector E
remains fixed by definition). The
with velocity −v, whilst the source strength is fixed (i.e. E
direction of E × H is unchanged, but the directions of vp and vg are now reversed and point
along −z.
Proc. of SPIE Vol. 7392 73921M-6
satisfied by a simple isotropic medium set in motion raises an important and interesting
question: if the condition can be induced by effectively changing reference frame, then can
the condition be said to relate to a physical phenomenon at all? After all, the principle of
relativity forbids a physical phenomenon from occurring in one frame, and not in another.
We attempt to answer this question in the next section.
5. COVARIANT DEFINITION OF NPVP
In16 one of us (MWM) laid out, for the first time, a definition of NPVP that was covariant,
namely was robust to coordinate transformations in both space and spacetime. In addition
to using the covariant generalizations of the Poynting vector and the wave-vector (i.e. the
electromagnetic stress-energy tensor and the 4-wave vector respectively), the new definition
involves the medium 4-velocity U† . Introducing this quantity is crucial, for, in contrast to
vacuum, where all observers are equivalent, the medium 4-velocity singles out a preferred
direction in spacetime, and mathematically ties the definition of NPVP to the rest frame of
the medium in which it occurs.
According to the covariant definition, NPVP is said to occur in a medium whose 4-velocity
is U whenever
β
<0,
(15)
−Tαβ U α K⊥
where the summation convention over space time
1
ν
Tμ = Re F μα (Gνα )∗ −
2
is the electromagnetic stress-energy
Cartesian coordinates as
⎛
0 −Ex −Ey −Ez
⎜ Ex
0
Bz −By
(Fαβ ) = ⎜
⎝ Ey −Bz
0
Bx
Ez By −Bx
0
indices α = 0, 1, 2, 3 has been used. Here,
∗ 1 ν
αβ
δ Fαβ G
,
(16)
4 μ
tensor, and Fαβ and Gμν are the field tensors given in
⎞
⎞
0
Dx
Dy
Dz
⎜ −D
0
Hz −Hy ⎟
x
⎟ . (17)
Gαβ = ⎜
⎝ −Dy −Hz
0
Hx ⎠
−Dz Hy −Hx
0
⎛
⎟
⎟ ,
⎠
The other two quantities in (15) are U α , the four velocity of the medium, and K⊥ , a perpendicular projection of the plane wave’s four wave vector K α = (ω, k), constructed as
Kμ U μ
β
β
(18)
Uβ .
K⊥ = K −
Uα U α
In the medium’s rest frame, where U α = (1, 0, 0, 0), it is straightforward to show that (15)
reduces to NPVP (i.e. Eqs. (1) and (2a)). By construction (15) is frame independent.
Since the covariant criterion of Eq. (15) is mathematically tied to the rest frame of the
medium, it is fully equivalent to the criterion
Prf · krf < 0 ,
†
(19)
vectors in Minkowski
Whilst vectors in Euclidean space are denoted with an over-arrow (A),
spacetime will be set in boldface (A).
Proc. of SPIE Vol. 7392 73921M-7
where the rf subscripts indicate vectors calculated in the rest frame of the medium. Succinctly,
NPVP occurs if and only if it occurs in the medium rest frame. This definition is consistent
with the principle of relativity, but cuts across much literature where the NPVP criterion
P · k < 0 has been calculated in frames in relative motion to the medium.17,18
A further comment should be made in relation to the criterion of Eq. (15). The form of
the stress-energy tensor given in Eq. (16) is only valid in linear reference frames, i.e. those
obtained from Cartesian-Lorentz coordinates in locally flat Minkowski spacetime. In the next
section we attempt to remove this restriction by positing a further refinement of the definition
of NPVP that is pre-metric.
6. PRE-METRIC DEFINITION OF NEGATIVE PHASE VELOCITY
PROPAGATION
Recently we have enquired as to the whether a formulation of NPVP is possible without the
use of the spacetime metric. If U is the four-velocity vector of a medium, the electromagnetic
four-momentum density supported by the medium can be stated as
Σ[U] :=
1
[F ∧ (UH) − H ∧ (UF )] ,
2
(20)
where F is the Faraday 2-form, H is the field excitation twisted 2-form 19 (p.165), and indicates left contraction19 (p.33). The other key quantity, also a 1-form, is the projected
(dual) 4-wave vector, given by Eq. (18). In the absence of a metric this definition must be
modified to
K = K − (U K)dτ ,
(21)
where the only restriction on the 1-form dτ is that it satisfies the normalization Udτ = 1.
Without a metric, combining the 3-form Σ[U] with the 1-form K to produce a metric free
NPV criterion must be facilitated by converting one of these quantities
to a vector and then
19
contracting. This conversion is mediated by the operator (p.39), wherein
I U := − (Σ[U ])
(22)
converts Σ[U] to a twisted vector density of weight w = +1. Finally, our pre-metric NPV
criterion can be stated as
(23)
I u(K ) = − (Σ[U ])(K ) < 0 .
The additional flexibility offered by Eq. (23) is that it is not restricted to linear frames as
was Eq. (15).
7. REFRACTION THROUGH A VELOCITY DISTRIBUTION
All the criteria examined thus far have concerned infinite plane waves in an infinite homogeneous medium. The Principle of Relativity, stating that the laws of physics must look the
same in all inertial reference frames, then acts as a powerful sieve in distinguishing criteria
for NPV propagation that are consistent with this principle and those that are not. However,
when the medium is finite, as with the existence of an interface to a neighboring medium, or
where the medium is inhomogeneous, the translational symmetry is broken, and arguments
based on changing reference frame can no longer be applied in a simple way. Whilst the
Proc. of SPIE Vol. 7392 73921M-8
appearance of an effective negative index as a result of medium motion can be classified
as an illusory observation with no physical consequences when considering an infinite uniform medium, physical consequences can be expected to arise when the medium is moving
inhomogeneously, i.e. there is a velocity distribution. Here we report our initial progress.
Light traveling in a moving medium can be treated using a ray model, essentially by
writing down the proper time for a massless particle traveling in a moving medium, and
then minimizing path lengths using a variational principle. This approach20,21 gives rise to
ray equations, where for laminar medium flow oriented entirely parallel to the x axis, and
ignoring the z dimension, we find that for a refractive index given by χ + 1 = 1/c2 μ = 1/n2 ,
and with τ = ct , we have
(24)
∂λ x = − χγ 2 βp + 1 + χγ 2
2 2
2
∂λ τ = − χγ β − 1 p + χγ β
(25)
(∂λ y)2 = W (y)
W (y) = − (1 + χ) 1 − p2 + χγ 2 (1 + pβ)2 ,
(26)
(27)
where λ parameterizes the ray paths. There is also an equation for ∂λ τ , but it plays no role
in the ray path. Here an important parameter is p, which is a constant of the ray path, and
can be defined by the initial “shooting” conditions φ0 , χ0 , so that from dx/dy = tan φ0 gives
√
1
1 + χ0
=±
.
(28)
p=±
sin φ0
n0 sin φ0
Examination of these equations enables us to characterize the set of all possible ray paths.
For simple medium flows, these fall into four categories determined by (a) whether n ≥ 1 or
n < 1, and (b) whether the velocity field is a function of x or y. Analyses of the ray equations
shows for what regions of parameter space propagation is allowed (i.e. where W > 0), and
where the ray might reverse direction (either in x or in y)22 ; the ray equations can be used
to generate parameter boundaries and conditions between the different types of trajectory.
The n ≥ 1 and v ≡ v(y) case was partly treated in,20 but in general there are three
types of ray paths. Reversed rays start off at an angle to, but along with the material flow,
and these will bend back on themselves as the flow becomes sufficiently fast. Deflected rays
start off at a steep angle to, but against the material flow, these are deflected towards being
true anti-parallel. Guided rays start off at a shallow angle too, but against the material flow.
These can be bent back on themselves in y, so that they would eventually exit the flow.
The n ≥ 1 and v ≡ v(x) case exhibits two types of ray trajectory. For v(x) directed away
from the interface to the flow, and increasing in speed, light rays will be deflected until they
are perpendicular to the flow. For a v(x) directed towards the interface, and increasing in
speed, light rays will be deflected until they are parallel to the flow.
8. CONCLUSION
This paper has surveyed the various definitions of negative phase velocity propagation, finding
that they are not equivalent. The distinctions are sharply illustrated by the example of a
dispersionless moving medium whereby, for propagation contra to the medium flow, the
Proc. of SPIE Vol. 7392 73921M-9
v=c
1.0
0.8
0.6
B(y)
0.4
0.2
0.0
−4
v=0
−3
−2
x
−1
0
1
√
Figure 4. Typical ray paths for light travelling in an ordinary medium with n = 2,
whose laminar velocity flow increases linearly with height. All the light rays enter the flow
at the same point, but have different angles of incidence. Dot-dashed lines are guided rays,
continuous lines are rays that penetrate the shear flow without reversing direction, and dashed
lines are those that reverse direction.
phase and group velocity reverse above the C̆erenkov speed (μ)−1/2 , but the Poynting vector
maintains its direction. This scenario yields a negative effective refractive index. A definition
of NPV that is mathematically tied to the medium rest frame is more satisfactory (Eq. (15)),
but is restricted to linear frames of reference. This restriction is removed in the criterion
expressed in Eq. (23) which represents the most mathematically rigorous definition devised
to date. The presence of a velocity distribution breaks the translational symmetry inherent in
any intrinsic definition of NPV that is restricted to infinite plane waves in an infinite uniform
medium.
9. ACKNOWLEDGEMENTS
This work was supported by EPSRC Grant nos. EP/E031463/1.
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