Local-field effects in radiatively broadened magneto-dielectric media: negative
refraction and absorption reduction
Jürgen Kästel and Michael Fleischhauer
Fachbereich Physik, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany
Gediminas Juzeliūnas
Institute of Theoretical Physics and Astronomy, Vilnius University, A Goštauto 12, Vilnius 01108, Lithuania
(Dated: August 11, 2021)
We give a microscopic derivation of the Clausius-Mossotti relations for a homogeneous and
isotropic magneto-dielectric medium consisting of radiatively broadened atomic oscillators. To this
end the diagram series of electromagnetic propagators is calculated exactly for an infinite bi-cubic
lattice of dielectric and magnetic dipoles for a lattice constant small compared to the resonance
wavelength λ. Modifications of transition frequencies and linewidth of the elementary oscillators are
taken into account in a selfconsistent way by a proper incorporation of the singular self-interaction
terms. We show that in radiatively broadened media sufficiently close to the free-space resonance
the real part of the index of refraction approaches the value -2 in the limit of ρλ3 ≫ 1, where ρ is
the number density of scatterers. Since at the same time the imaginary part vanishes as 1/ρ local
field effects can have important consequences for realizing low-loss negative index materials.
PACS numbers:
INTRODUCTION
It is well known that in dense dielectric materials the
induced polarization P alters the field strength Eloc act-
ing on the constituents (i.e. the local field) compared to
the average macroscopic field Em. Macroscopic consider-
ations show that in systems with high symmetry such as
a cubic lattice the two fields are related to each other ac-
cording to Eloc = Em +P/(3ε0) [1, 2]. This leads to the
well-known Clausius-Mossotti relation for the permittiv-
ity ε(ω)
ε(ω) = 1 +
ρα(ω)/ε0
1− ρα(ω)/(3ε0)
where ρ is the density and α(ω) the polarizability of the
oscillators. Similar arguments hold for a purely magnetic
material [3], except that the required densities are usu-
ally much higher due to the smallness of magnetic dipole
moments and polarizabilities. In linear response α(ω) is
well described by a damped-oscillator model [1]
α(ω) = α′ + i α′′ =
ω20 − ω2 − iγω
. (2)
The corresponding (real-valued) parameters such as the
oscillator strength α0, the resonance frequency and
width, ω0 and γ, are determined by the microscopic
model. In general the linewidth γ contains radiative
as well as non-radiative contributions. For purely ra-
diative interaction these parameters are strongly af-
fected by the renormalization of energy levels and spon-
taneous emission processes caused by the interaction
with the vacuum electromagnetic field in the medium
[4, 5, 6, 7, 8, 9, 10, 11]. Since the mode structure of
the electromagnetic field inside a dense medium can be
substantially modified compared to free space, one would
expect that the polarizability entering eq.(1) is different
from that in free space. In a macroscopic approach α(ω)
is however an input function and no conclusion can be
drawn about possible changes due to the different struc-
ture of the vacuum modes inside the medium. To take
into account the modification of transition frequencies
and radiative linewidth in a dense medium in a self-
consistent way requires a microscopic approach.
In the present paper we develop a microscopic ap-
proach to local field effects in dense materials with simul-
taneous dielectric and magnetic response using Greens-
function techniques similar to those used by deVries and
Lagendijk for purely dielectric materials [12]. To this end
we consider an infinitely extended bi-cubic lattice of elec-
tric and magnetic point dipoles with isotropic response
with a lattice constant small compared to the transition
wavelength. We however do not make use of the as-
sumptions made in [12] to renormalize the singular self-
interaction contributions to the lattice T -matrix which
eliminated radiative contributions to linewidth and tran-
sition frequencies altogether. We show that instead the
self-interaction contributions can be summed to yield the
dressed t-matrix of an isolated oscillator interacting with
the vacuum modes of the electromagnetic field in free
space. In this way we derive Clausius-Mossotti relations
for general, radiatively broadened, isotropic magneto-
dielectrica. Apart from non-radiative broadenings, the
electric and magnetic polarizabilities entering these equa-
tions are shown to be exactly those of free space. We
then show that simultaneous local-field corrections to
electric and magnetic fields in purely radiatively broad-
ened magneto-dielectrica have a surprising and poten-
tially important effect: For sufficiently large densities the
http://arxiv.org/abs/0704.0593v2
real part of the refractive index saturates at the level of
−2. At the same time, the imaginary part of the com-
plex index approaches zero inversely proportional to the
density. Thus the medium becomes transparent and left-
handed i.e. displays a negative index of refraction with
low absorption.
LOCAL-FIELD EFFECTS AND
RENORMALIZATION OF RADIATIVE
SELF-INTERACTION IN DIELECTRIC MEDIA
We start by developing a microscopic scattering ap-
proach to local-field effects in dielectric media taking
into account possible material induced modifications of
radiative linewidth and transition frequencies in a self-
consistent way. To this end we consider a simple cubic
lattice of electric point dipoles with isotropic bare polar-
izability αb
αb(r) = αb
δ(r−R), (3)
whereR denote lattice vectors. The dipoles interact with
the quantized electromagnetic field Ê which obeys the
vector Helmholtz equation
~∇× ~∇× Ê(r, ω)− ω
Ê(r, ω) = µ0ω
P̂ . (4)
In the weak-excitation, i.e. linear response limit, the op-
erator of the microscopic electric polarization P̂ has the
form P̂(r) = αb(r)Ê(r, ω). Solving eq.(4) we can deter-
mine the (isotropic) dispersion relation k = k(ω) from
which the permittivity ε(ω) can be extracted. In the
linear response limit the solution of the quantum me-
chanical interaction problem can most easily be obtained
by means of Greensfunction techniques. In particular it
is sufficient to calculate the scattering T -matrix of the
oscillator lattice. The dispersion relation can then be
obtained via [13, 14, 15]
det T−1 = 0. (5)
The scattering T -matrix obeys a linear Dyson equation
T = V + V G(0)V + · · · = V + V G(0)T, (6)
where G(0)(r, r′, ω) is the free-space retarded propagator
of the electric field which is a solution to the classical
vector Helmholtz equation
~∇× ~∇× G(0)(r, r′, ω)− ω
G(0)(r, r′, ω) =
= 1 δ(r− r′), (7)
V (r, ω) = −ω
2αb(r)
is a linear, isotropic point vertex. Note that integration
over spatial variables was suppressed in eq.(6) for nota-
tional simplicity.
For a cubic lattice of isotropic scatterers, the series can
be summed up to yield [16]
T (k,k′) = −
ei(k−k
R 6=0
RG(0)(R)
where G(0)(R) stands for G(0)(r, r+R, ω0) which due to
the discrete translation invariance is independent on r.
The single-particle scattering t-matrix t(ω) is determined
by the bare polarizability [12]
t(ω)−1 =
+ G(0)(0). (10)
Note that G(0)(0) is diagonal and isotropic. In eq.(9) we
have separated the contribution of the lattice (
R 6=0)
from the multiple scattering events at the same oscilla-
tor (G(0)(0)). This separation is crucial since G(0)(0) is
singular. Rather than eliminating this singularity by a
regularization procedure as done in [12], we note that ex-
pression (10) gives the single-particle scattering t-matrix
t(ω) dressed by the interaction with the vacuum field in
free space. This quantity is experimentally observable
and is related to the single-particle polarizability α(ω) in
free space:
α(ω) = t(ω)
ε0 (11)
αb on the other hand is not observable and thus only a
theoretical notion. At this point other broadening mech-
anisms can be incorporated by adding appropriate non-
radiative decay rates γnon-rad to the polarizability α(ω)
(11) (cf. equation (2) and discussion thereafter).
Obviously, for the radiative part separating the sum∑
RG(0)(R) into G(0)(0)+
R 6=0 e
ik′RG(0)(R) does
the trick of writing the full lattice T -matrix in terms
of the known free space t-matrix. As a drawback we
are left with the sum over the lattice vectors R 6= 0.
Unfortunately this sum can not be evaluated exactly and
has to be treated approximately.
According to Poisson’s summation formula
f(n) =
dxf(x)e−2πikx (12)
the sum over R 6= 0 can be expressed in terms of a real
space integral and a sum over inverse lattice vectors K
of the Fourier transform of the free space Greensfunction
G̃(0)(p)
R 6=0
eikRG(0)(R) =
Ξ(|r|)
(2πa)3
ei(p+k−K)rG̃(0)(p)
Here Ξ(|r|) is some smooth function with Ξ(0) = 0 and
Ξ(|r| > 0) → 1 introduced to prevent the integral from
touching the excluded singular point r = 0.
In the following we restrict the discussion to lattices
with a lattice constant much smaller than the resonant
wavelength, i.e. ka ≪ 1. In this limit the lattice of
oscillators behaves essentially as a homogeneous medium.
Contributions from large K-vectors to the sum, which
reflect the discreteness of the lattice, can be neglected as
long as the singular contribution from the origin has been
excluded. Therefore we only keep the term K = 0 and
assume a Gaussian cutting function Ξ(|r|) = 1− e−r2/δ2 ,
with δ ≪ a. This yields
R 6=0
eikRG(0)(R) ≈ 1
G̃(0)(k)
π3/2δ3
(2π)3
dp p2e−
(k2+p2)e−
k·p̂G̃(0)(p),
where p̂ = p/|p|. Apart from the Gaussian p-integral
which provides a smooth cut-off in reciprocal space, δ
can be treated as a small parameter. That allows to
carry out the integration analytically which in leading
order of δ yields
R 6=0
eikRG(0)(R) ≈ 1
G̃(0)(k)− 1
3ω2/c2
1. (15)
The free-space Greentensor G̃(0)(k) is given by [12]
G̃(0)(k) =
1− |k|2∆k
with ∆k = 1 − k̂ ⊗ k̂ being a projector to directions
orthogonal to k.
With this we are ready to evaluate eq. (5) which reads
in the limit ka ≪ 1
ρα(ω)/ε0
1− |k|2∆k
Solving eq. (17) for the (isotropic) dispersion k = k(ω)
with k(ω) = ε(ω)ω2/c2 finally yields
ε(ω) = 1 +
ρα(ω)/ε0
1− ρα(ω)/3ε0
. (18)
This is the well-known Clausius-Mossotti relation where
for purely radiatively broadened systems α(ω) is the
dressed polarizability of an isolated oscillator interacting
with the free-space electromagnetic vacuum field.
LOCAL-FIELD EFFECTS FOR
MAGNETO-DIELECTRICS
We now extend the above discussion to the case of a
bi-cubic lattice of electric and magnetic dipole oscillators.
The microscopic, space-dependent bare electric polariz-
ability αbe(r) is then given by
αbe(r) = αbe
δ(r−R) = αbe
eiKr (19)
and, similarly, the bare magnetic polarizability by
αbm(r) = αbm
δ(r−R−∆r) = αbm
eiK(r−∆r)
HereR denotes again the lattice vectors and ∆r the spac-
ing between the electric and magnetic sublattices. The
bare atomic polarizabilities αbe and αbm are assumed
to be scalar for simplicity corresponding to an isotropic
medium. The last expressions in eqn. (19) and (20) give
the bare polarizabilities in reciprocal space, with K being
the reciprocal lattice vectors.
Due to the simultaneous presence of electric and mag-
netic dipole lattices we now have to solve the coupled set
of vector Helmholtz equations for the operators of the
electric and magnetic fields
∇×∇× Ê− ω
Ê = iωµ0∇× M̂+ µ0ω2P̂ (21)
∇×∇× Ĥ− ω
M̂− iω∇× P̂. (22)
In linear response the operator of the polarization P̂
and the magnetization M̂ are proportional to the elec-
tric and magnetic fields respectively, P̂(r) = αbe(r)Ê(r)
and M̂(r) = µ0αbm(r)Ĥ(r).
In the following we will pursue a slightly different ap-
proach to solve the coupled set of equations than used
in the previous section. Taking into account the lattice
symmetry we first write the field variables in the form
Ê(r) =
Ẽ(k−K)ei(k−K)r, (23)
where the dependence on frequency ω was suppressed for
notational simplicity. The subscript denotes integration
over the first Brillouin zone. Substituting this and the
corresponding expression for Ĥ into (21)-(22) gives the
Helmholtz equations in reciprocal space. After some ele-
mentary manipulations the following closed set of equa-
tions is derived:
ραbe/ε0
1− |k−K|2∆k−K
Ẽ(k−K′) = µ0αbm
eiK∆r(k−K)×
1− |k−K|2∆k−K
H̃(k−K′)e−iK
ρµ0αbm
1− |k−K|2∆k−K
H̃(k−K′)e−iK
′∆r = − c
ωµ0αbm
e−iK∆r(k−K)×
1− |k−K|2∆k−K
Ẽ(k−K′)
where ρ = 1/a3 is the particle density. The sum in the
brackets on the left hand sides of eqs. (24,25) can be
rewritten as
1− |k−K|2∆k−K
eikRG(0)(R)
= G(0)(0) +
R 6=0
eikRG(0)(R),
where in the second line we have separated the singular
contribution G(0)(0). One recognizes that this term can
be added to the expressions containing the bare polariz-
abilities in eqs.(24) and (25) yielding the dressed scatter-
ing t-matrices for isolated electric and magnetic dipoles
interacting with the free-space vacuum field:
te(ω)
+ G(0)(0), (26)
tm(ω)
(ω2µ0
+ G(0)(0). (27)
The sum over the Greensfunction excludingR = 0 can be
evaluated in a similar way as in the previous section. If
we again assume a lattice constant a much smaller than
the resonant wavelength, reciprocal K vectors different
from zero can be disregarded. This leads to
ρte(ω)
+ G̃(0)(k)− 1
3ω2/c2
Ê(k) = (28)
µ0αbm
1− k2∆k
Ĥ(k),
ρtm(ω)
+ G̃(0)(k) − 1
3ω2/c2
Ĥ(k) = (29)
c2αbe
ωµ0αbm
1− k2∆k
Ê(k).
Since we are furthermore only interested in propagating,
i.e. transversal modes, we can further simplify the calcu-
lation by projecting onto transversal modes using ∆k
ραe(ω)/ε0
∆kÊ(k) =
µ0αbm
k×∆kĤ(k) (30)
ρµ0αm(ω)
k×∆kĤ(k) =
c2αbe
ωµ0αbm
∆kÊ(k). (31)
Here we have substituted the dressed single parti-
cle t-matrices by the free-space dressed polarizabilities
αe(m)(ω) = te(m)(ω)c
2/ω2ε0
In order to find the dispersion k(ω) = n2ω2/c2 we have
to determine the solution of the secular equation of the
linear set of eqs. (30,31), which results in the condition
ραe(ω)/ε0
× (32)
ρµ0αm(ω)
Solving for the refractive index of the transversal modes
then gives n2 = εµ, where
ε = 1 +
ραe(ω)/ε0
1− ραe(ω)/3ε0
µ = 1 +
ρµ0αm(ω)
1− ρµ0αm(ω)/3
are the relative dielectric permittivity and magnetic
permeability, respectively, both satisfying the Clausius-
Mossotti relations.
Note that for longitudinal modes eqs. (28) and (29) de-
couple. This can be seen by applying the corresponding
projector to longitudinal waves k̂⊗k̂ which leads to a dis-
appearance of the cross-coupling terms. The dispersion
obtained in this way gives either ε = 0 corresponding to
electric excitons [17, 18] or µ = 0 for magnetic excitons.
NEGATIVE REFRACTION AND ABSORPTION
REDUCTION DUE TO LOCAL FIELD EFFECTS
IN MAGNETO-DIELECTRIC MEDIA
It is interesting to consider the implications of the
Clausius Mossotti relations for radiatively broadened me-
dia in the large density limit. Let us first consider a
purely dielectric medium and let us assume that the po-
larizability αe(ω) = α
e(ω) + i α
e (ω) does not depend on
the density, i.e. the medium is radiatively broadened. In
this case one finds
ρ→∞−→ −2 + i
|αe|2
. (35)
In the high-density limit and sufficiently close to reso-
nance the response saturates at a value of −2 with an
imaginary part that vanishes as 1/ρ. At this point the
medium becomes totally opaque since the index of re-
fraction attains an imaginary value n = i
2 indicating
the emergence of a stopping band. This is illustrated
in the left column of Fig 1 for a medium composed of
either electric or magnetic dipole oscillators. For small
densities (ρ|α0|/3 = 1/3) the resonance is centered at ω0
whereas for larger densities (ρ|α0|/3 = 3) the response
shifts to smaller frequencies and is amplified. Eventu-
ally (ρ|α0|/3 = 30) the refractive index becomes almost
purely imaginary in which case light cannot propagate
any longer.
This behavior changes dramatically if we consider me-
dia with overlapping electric and magnetic resonances
described by both an electric polarizability αe(ω) and a
magnetic polarizability αm(ω). Independent application
of Clausius-Mossotti local-field corrections to the permit-
tivity and the permeability leads in the high density limit
n = −2 + i
|αe|2
9α′′m
µ0|αm|2
. (36)
Thus in the spectral overlap region the real part of the
index of refraction approaches the value −2, i.e. attains
a constant negative value. Furthermore the imaginary
part, responsible for absorption losses, approaches zero in
that spectral region as 1/ρ. This rather peculiar behav-
ior is illustrated in the right column of Fig.1. One clearly
recognizes the emergence of a spectral region around the
bare resonance frequency where the real part of the re-
fractive index approaches −2 while the imaginary part is
strongly suppressed.
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
(b)(a)
∆ [γ ] ∆ [γ ]
FIG. 1: (color online) spectrum of the real (solid) and imagi-
nary (dashed) part of the refractive index as well as the real
(dotted) part of the response function(s) ε and/or µ as a func-
tion of the detuning ∆ for a (a) pure dielectric or magnetic
medium for ρ|α0|/3 at ∆ = 0 equal to = 1/3 (top), 3 (middle)
and 30 (bottom) (b) magneto-dielectric medium for ρ|α0|/3
at ∆ = 0 equal to = 1/3 (top), 3 (middle) and 30 (bottom).
Negative refraction of light is currently one of the
most active research areas in photonics [19, 20, 21] due
to fascinating potential applications such as superlens-
ing [22] or electromagnetic cloaking [23, 24, 25]. In
recent years substantial progress has been made in re-
alizing negative refraction in so-called meta-materials
[26, 27, 28, 29]. These are artificial periodic structures
of electric and magnetic dipoles with a resonance wave-
length much larger than the lattice constant which thus
form a quasi-homogeneous magneto-dielectric medium.
In order to achieve a large electromagnetic response, op-
eration close to resonance is needed which is associated
with rather substantial losses. The elimination of these
losses represents one of the main challenges in the field
[30]. We have shown here that in a radiatively broad-
ened medium, i.e. a medium in which density-dependent
broadening mechanism can still be disregarded for suf-
ficiently large densities, local field effects can provide a
negative index of refraction and at the same time effi-
ciently suppress absorption losses.
SUMMARY
In the present paper we have given a rigorous micro-
scopic derivation of Clausius-Mossotti relations for both
the electric and magnetic response in an isotropic, radia-
tively broadened magneto-dielectric medium formed by
a simple bi-cubic lattice of electric and magnetic dipoles.
As opposed to previous microscopic approaches we have
taken into account possible modifications of the single-
particle polarizabilities by the altered electromagnetic
vacuum inside the medium in a self-consistent way. For
a simple bi-cubic lattice it has been shown that the po-
larizabilities entering the Clausius-Mossotti relations are
those of single oscillators interacting with the free-space
vacuum field. We showed that as a consequence of the
local field corrections a radiatively broadened medium
with overlapping electric and magnetic resonances be-
comes lossless with a real part of the refractive index
approaching the value −2 in the high-density limit. The
latter could provide an interesting avenue to construct ar-
tificial materials with negative refraction and low losses.
This work was supported by the Alexander von Hum-
boldt Foundation through the institutional collabora-
tion grant between The Institute of Theoretical Physics
and Astronomy of Vilnius University and the Techni-
cal University of Kaiserslautern. J.K. acknowledges fi-
nancial support by the Deutsche Forschungsgemeinschaft
through the GRK 792 “Nichtlineare Optik und Ultra-
kurzzeitphysik”.
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  We give a microscopic derivation of the Clausius-Mossotti relations for a
homogeneous and isotropic magneto-dielectric medium consisting of radiatively
broadened atomic oscillators. To this end the diagram series of electromagnetic
propagators is calculated exactly for an infinite bi-cubic lattice of
dielectric and magnetic dipoles for a lattice constant small compared to the
resonance wavelength $\lambda$. Modifications of transition frequencies and
linewidth of the elementary oscillators are taken into account in a
selfconsistent way by a proper incorporation of the singular self-interaction
terms. We show that in radiatively broadened media sufficiently close to the
free-space resonance the real part of the index of refraction approaches the
value -2 in the limit of $\rho \lambda^3 \gg 1$, where $\rho$ is the number
density of scatterers. Since at the same time the imaginary part vanishes as
$1/\rho$ local field effects can have important consequences for realizing
low-loss negative index materials.

Local-field effects in radiatively broadened magneto-dielectric media: negative
refraction and absorption reduction
Jürgen Kästel and Michael Fleischhauer
Fachbereich Physik, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany
Gediminas Juzeliūnas
Institute of Theoretical Physics and Astronomy, Vilnius University, A Goštauto 12, Vilnius 01108, Lithuania
(Dated: August 11, 2021)
We give a microscopic derivation of the Clausius-Mossotti relations for a homogeneous and
isotropic magneto-dielectric medium consisting of radiatively broadened atomic oscillators. To this
end the diagram series of electromagnetic propagators is calculated exactly for an infinite bi-cubic
lattice of dielectric and magnetic dipoles for a lattice constant small compared to the resonance
wavelength λ. Modifications of transition frequencies and linewidth of the elementary oscillators are
taken into account in a selfconsistent way by a proper incorporation of the singular self-interaction
terms. We show that in radiatively broadened media sufficiently close to the free-space resonance
the real part of the index of refraction approaches the value -2 in the limit of ρλ3 ≫ 1, where ρ is
the number density of scatterers. Since at the same time the imaginary part vanishes as 1/ρ local
field effects can have important consequences for realizing low-loss negative index materials.
PACS numbers:
INTRODUCTION
It is well known that in dense dielectric materials the
induced polarization P alters the field strength Eloc act-
ing on the constituents (i.e. the local field) compared to
the average macroscopic field Em. Macroscopic consider-
ations show that in systems with high symmetry such as
a cubic lattice the two fields are related to each other ac-
cording to Eloc = Em +P/(3ε0) [1, 2]. This leads to the
well-known Clausius-Mossotti relation for the permittiv-
ity ε(ω)
ε(ω) = 1 +
ρα(ω)/ε0
1− ρα(ω)/(3ε0)
where ρ is the density and α(ω) the polarizability of the
oscillators. Similar arguments hold for a purely magnetic
material [3], except that the required densities are usu-
ally much higher due to the smallness of magnetic dipole
moments and polarizabilities. In linear response α(ω) is
well described by a damped-oscillator model [1]
α(ω) = α′ + i α′′ =
ω20 − ω2 − iγω
. (2)
The corresponding (real-valued) parameters such as the
oscillator strength α0, the resonance frequency and
width, ω0 and γ, are determined by the microscopic
model. In general the linewidth γ contains radiative
as well as non-radiative contributions. For purely ra-
diative interaction these parameters are strongly af-
fected by the renormalization of energy levels and spon-
taneous emission processes caused by the interaction
with the vacuum electromagnetic field in the medium
[4, 5, 6, 7, 8, 9, 10, 11]. Since the mode structure of
the electromagnetic field inside a dense medium can be
substantially modified compared to free space, one would
expect that the polarizability entering eq.(1) is different
from that in free space. In a macroscopic approach α(ω)
is however an input function and no conclusion can be
drawn about possible changes due to the different struc-
ture of the vacuum modes inside the medium. To take
into account the modification of transition frequencies
and radiative linewidth in a dense medium in a self-
consistent way requires a microscopic approach.
In the present paper we develop a microscopic ap-
proach to local field effects in dense materials with simul-
taneous dielectric and magnetic response using Greens-
function techniques similar to those used by deVries and
Lagendijk for purely dielectric materials [12]. To this end
we consider an infinitely extended bi-cubic lattice of elec-
tric and magnetic point dipoles with isotropic response
with a lattice constant small compared to the transition
wavelength. We however do not make use of the as-
sumptions made in [12] to renormalize the singular self-
interaction contributions to the lattice T -matrix which
eliminated radiative contributions to linewidth and tran-
sition frequencies altogether. We show that instead the
self-interaction contributions can be summed to yield the
dressed t-matrix of an isolated oscillator interacting with
the vacuum modes of the electromagnetic field in free
space. In this way we derive Clausius-Mossotti relations
for general, radiatively broadened, isotropic magneto-
dielectrica. Apart from non-radiative broadenings, the
electric and magnetic polarizabilities entering these equa-
tions are shown to be exactly those of free space. We
then show that simultaneous local-field corrections to
electric and magnetic fields in purely radiatively broad-
ened magneto-dielectrica have a surprising and poten-
tially important effect: For sufficiently large densities the
http://arxiv.org/abs/0704.0593v2
real part of the refractive index saturates at the level of
−2. At the same time, the imaginary part of the com-
plex index approaches zero inversely proportional to the
density. Thus the medium becomes transparent and left-
handed i.e. displays a negative index of refraction with
low absorption.
LOCAL-FIELD EFFECTS AND
RENORMALIZATION OF RADIATIVE
SELF-INTERACTION IN DIELECTRIC MEDIA
We start by developing a microscopic scattering ap-
proach to local-field effects in dielectric media taking
into account possible material induced modifications of
radiative linewidth and transition frequencies in a self-
consistent way. To this end we consider a simple cubic
lattice of electric point dipoles with isotropic bare polar-
izability αb
αb(r) = αb
δ(r−R), (3)
whereR denote lattice vectors. The dipoles interact with
the quantized electromagnetic field Ê which obeys the
vector Helmholtz equation
~∇× ~∇× Ê(r, ω)− ω
Ê(r, ω) = µ0ω
P̂ . (4)
In the weak-excitation, i.e. linear response limit, the op-
erator of the microscopic electric polarization P̂ has the
form P̂(r) = αb(r)Ê(r, ω). Solving eq.(4) we can deter-
mine the (isotropic) dispersion relation k = k(ω) from
which the permittivity ε(ω) can be extracted. In the
linear response limit the solution of the quantum me-
chanical interaction problem can most easily be obtained
by means of Greensfunction techniques. In particular it
is sufficient to calculate the scattering T -matrix of the
oscillator lattice. The dispersion relation can then be
obtained via [13, 14, 15]
det T−1 = 0. (5)
The scattering T -matrix obeys a linear Dyson equation
T = V + V G(0)V + · · · = V + V G(0)T, (6)
where G(0)(r, r′, ω) is the free-space retarded propagator
of the electric field which is a solution to the classical
vector Helmholtz equation
~∇× ~∇× G(0)(r, r′, ω)− ω
G(0)(r, r′, ω) =
= 1 δ(r− r′), (7)
V (r, ω) = −ω
2αb(r)
is a linear, isotropic point vertex. Note that integration
over spatial variables was suppressed in eq.(6) for nota-
tional simplicity.
For a cubic lattice of isotropic scatterers, the series can
be summed up to yield [16]
T (k,k′) = −
ei(k−k
R 6=0
RG(0)(R)
where G(0)(R) stands for G(0)(r, r+R, ω0) which due to
the discrete translation invariance is independent on r.
The single-particle scattering t-matrix t(ω) is determined
by the bare polarizability [12]
t(ω)−1 =
+ G(0)(0). (10)
Note that G(0)(0) is diagonal and isotropic. In eq.(9) we
have separated the contribution of the lattice (
R 6=0)
from the multiple scattering events at the same oscilla-
tor (G(0)(0)). This separation is crucial since G(0)(0) is
singular. Rather than eliminating this singularity by a
regularization procedure as done in [12], we note that ex-
pression (10) gives the single-particle scattering t-matrix
t(ω) dressed by the interaction with the vacuum field in
free space. This quantity is experimentally observable
and is related to the single-particle polarizability α(ω) in
free space:
α(ω) = t(ω)
ε0 (11)
αb on the other hand is not observable and thus only a
theoretical notion. At this point other broadening mech-
anisms can be incorporated by adding appropriate non-
radiative decay rates γnon-rad to the polarizability α(ω)
(11) (cf. equation (2) and discussion thereafter).
Obviously, for the radiative part separating the sum∑
RG(0)(R) into G(0)(0)+
R 6=0 e
ik′RG(0)(R) does
the trick of writing the full lattice T -matrix in terms
of the known free space t-matrix. As a drawback we
are left with the sum over the lattice vectors R 6= 0.
Unfortunately this sum can not be evaluated exactly and
has to be treated approximately.
According to Poisson’s summation formula
f(n) =
dxf(x)e−2πikx (12)
the sum over R 6= 0 can be expressed in terms of a real
space integral and a sum over inverse lattice vectors K
of the Fourier transform of the free space Greensfunction
G̃(0)(p)
R 6=0
eikRG(0)(R) =
Ξ(|r|)
(2πa)3
ei(p+k−K)rG̃(0)(p)
Here Ξ(|r|) is some smooth function with Ξ(0) = 0 and
Ξ(|r| > 0) → 1 introduced to prevent the integral from
touching the excluded singular point r = 0.
In the following we restrict the discussion to lattices
with a lattice constant much smaller than the resonant
wavelength, i.e. ka ≪ 1. In this limit the lattice of
oscillators behaves essentially as a homogeneous medium.
Contributions from large K-vectors to the sum, which
reflect the discreteness of the lattice, can be neglected as
long as the singular contribution from the origin has been
excluded. Therefore we only keep the term K = 0 and
assume a Gaussian cutting function Ξ(|r|) = 1− e−r2/δ2 ,
with δ ≪ a. This yields
R 6=0
eikRG(0)(R) ≈ 1
G̃(0)(k)
π3/2δ3
(2π)3
dp p2e−
(k2+p2)e−
k·p̂G̃(0)(p),
where p̂ = p/|p|. Apart from the Gaussian p-integral
which provides a smooth cut-off in reciprocal space, δ
can be treated as a small parameter. That allows to
carry out the integration analytically which in leading
order of δ yields
R 6=0
eikRG(0)(R) ≈ 1
G̃(0)(k)− 1
3ω2/c2
1. (15)
The free-space Greentensor G̃(0)(k) is given by [12]
G̃(0)(k) =
1− |k|2∆k
with ∆k = 1 − k̂ ⊗ k̂ being a projector to directions
orthogonal to k.
With this we are ready to evaluate eq. (5) which reads
in the limit ka ≪ 1
ρα(ω)/ε0
1− |k|2∆k
Solving eq. (17) for the (isotropic) dispersion k = k(ω)
with k(ω) = ε(ω)ω2/c2 finally yields
ε(ω) = 1 +
ρα(ω)/ε0
1− ρα(ω)/3ε0
. (18)
This is the well-known Clausius-Mossotti relation where
for purely radiatively broadened systems α(ω) is the
dressed polarizability of an isolated oscillator interacting
with the free-space electromagnetic vacuum field.
LOCAL-FIELD EFFECTS FOR
MAGNETO-DIELECTRICS
We now extend the above discussion to the case of a
bi-cubic lattice of electric and magnetic dipole oscillators.
The microscopic, space-dependent bare electric polariz-
ability αbe(r) is then given by
αbe(r) = αbe
δ(r−R) = αbe
eiKr (19)
and, similarly, the bare magnetic polarizability by
αbm(r) = αbm
δ(r−R−∆r) = αbm
eiK(r−∆r)
HereR denotes again the lattice vectors and ∆r the spac-
ing between the electric and magnetic sublattices. The
bare atomic polarizabilities αbe and αbm are assumed
to be scalar for simplicity corresponding to an isotropic
medium. The last expressions in eqn. (19) and (20) give
the bare polarizabilities in reciprocal space, with K being
the reciprocal lattice vectors.
Due to the simultaneous presence of electric and mag-
netic dipole lattices we now have to solve the coupled set
of vector Helmholtz equations for the operators of the
electric and magnetic fields
∇×∇× Ê− ω
Ê = iωµ0∇× M̂+ µ0ω2P̂ (21)
∇×∇× Ĥ− ω
M̂− iω∇× P̂. (22)
In linear response the operator of the polarization P̂
and the magnetization M̂ are proportional to the elec-
tric and magnetic fields respectively, P̂(r) = αbe(r)Ê(r)
and M̂(r) = µ0αbm(r)Ĥ(r).
In the following we will pursue a slightly different ap-
proach to solve the coupled set of equations than used
in the previous section. Taking into account the lattice
symmetry we first write the field variables in the form
Ê(r) =
Ẽ(k−K)ei(k−K)r, (23)
where the dependence on frequency ω was suppressed for
notational simplicity. The subscript denotes integration
over the first Brillouin zone. Substituting this and the
corresponding expression for Ĥ into (21)-(22) gives the
Helmholtz equations in reciprocal space. After some ele-
mentary manipulations the following closed set of equa-
tions is derived:
ραbe/ε0
1− |k−K|2∆k−K
Ẽ(k−K′) = µ0αbm
eiK∆r(k−K)×
1− |k−K|2∆k−K
H̃(k−K′)e−iK
ρµ0αbm
1− |k−K|2∆k−K
H̃(k−K′)e−iK
′∆r = − c
ωµ0αbm
e−iK∆r(k−K)×
1− |k−K|2∆k−K
Ẽ(k−K′)
where ρ = 1/a3 is the particle density. The sum in the
brackets on the left hand sides of eqs. (24,25) can be
rewritten as
1− |k−K|2∆k−K
eikRG(0)(R)
= G(0)(0) +
R 6=0
eikRG(0)(R),
where in the second line we have separated the singular
contribution G(0)(0). One recognizes that this term can
be added to the expressions containing the bare polariz-
abilities in eqs.(24) and (25) yielding the dressed scatter-
ing t-matrices for isolated electric and magnetic dipoles
interacting with the free-space vacuum field:
te(ω)
+ G(0)(0), (26)
tm(ω)
(ω2µ0
+ G(0)(0). (27)
The sum over the Greensfunction excludingR = 0 can be
evaluated in a similar way as in the previous section. If
we again assume a lattice constant a much smaller than
the resonant wavelength, reciprocal K vectors different
from zero can be disregarded. This leads to
ρte(ω)
+ G̃(0)(k)− 1
3ω2/c2
Ê(k) = (28)
µ0αbm
1− k2∆k
Ĥ(k),
ρtm(ω)
+ G̃(0)(k) − 1
3ω2/c2
Ĥ(k) = (29)
c2αbe
ωµ0αbm
1− k2∆k
Ê(k).
Since we are furthermore only interested in propagating,
i.e. transversal modes, we can further simplify the calcu-
lation by projecting onto transversal modes using ∆k
ραe(ω)/ε0
∆kÊ(k) =
µ0αbm
k×∆kĤ(k) (30)
ρµ0αm(ω)
k×∆kĤ(k) =
c2αbe
ωµ0αbm
∆kÊ(k). (31)
Here we have substituted the dressed single parti-
cle t-matrices by the free-space dressed polarizabilities
αe(m)(ω) = te(m)(ω)c
2/ω2ε0
In order to find the dispersion k(ω) = n2ω2/c2 we have
to determine the solution of the secular equation of the
linear set of eqs. (30,31), which results in the condition
ραe(ω)/ε0
× (32)
ρµ0αm(ω)
Solving for the refractive index of the transversal modes
then gives n2 = εµ, where
ε = 1 +
ραe(ω)/ε0
1− ραe(ω)/3ε0
µ = 1 +
ρµ0αm(ω)
1− ρµ0αm(ω)/3
are the relative dielectric permittivity and magnetic
permeability, respectively, both satisfying the Clausius-
Mossotti relations.
Note that for longitudinal modes eqs. (28) and (29) de-
couple. This can be seen by applying the corresponding
projector to longitudinal waves k̂⊗k̂ which leads to a dis-
appearance of the cross-coupling terms. The dispersion
obtained in this way gives either ε = 0 corresponding to
electric excitons [17, 18] or µ = 0 for magnetic excitons.
NEGATIVE REFRACTION AND ABSORPTION
REDUCTION DUE TO LOCAL FIELD EFFECTS
IN MAGNETO-DIELECTRIC MEDIA
It is interesting to consider the implications of the
Clausius Mossotti relations for radiatively broadened me-
dia in the large density limit. Let us first consider a
purely dielectric medium and let us assume that the po-
larizability αe(ω) = α
e(ω) + i α
e (ω) does not depend on
the density, i.e. the medium is radiatively broadened. In
this case one finds
ρ→∞−→ −2 + i
|αe|2
. (35)
In the high-density limit and sufficiently close to reso-
nance the response saturates at a value of −2 with an
imaginary part that vanishes as 1/ρ. At this point the
medium becomes totally opaque since the index of re-
fraction attains an imaginary value n = i
2 indicating
the emergence of a stopping band. This is illustrated
in the left column of Fig 1 for a medium composed of
either electric or magnetic dipole oscillators. For small
densities (ρ|α0|/3 = 1/3) the resonance is centered at ω0
whereas for larger densities (ρ|α0|/3 = 3) the response
shifts to smaller frequencies and is amplified. Eventu-
ally (ρ|α0|/3 = 30) the refractive index becomes almost
purely imaginary in which case light cannot propagate
any longer.
This behavior changes dramatically if we consider me-
dia with overlapping electric and magnetic resonances
described by both an electric polarizability αe(ω) and a
magnetic polarizability αm(ω). Independent application
of Clausius-Mossotti local-field corrections to the permit-
tivity and the permeability leads in the high density limit
n = −2 + i
|αe|2
9α′′m
µ0|αm|2
. (36)
Thus in the spectral overlap region the real part of the
index of refraction approaches the value −2, i.e. attains
a constant negative value. Furthermore the imaginary
part, responsible for absorption losses, approaches zero in
that spectral region as 1/ρ. This rather peculiar behav-
ior is illustrated in the right column of Fig.1. One clearly
recognizes the emergence of a spectral region around the
bare resonance frequency where the real part of the re-
fractive index approaches −2 while the imaginary part is
strongly suppressed.
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
(b)(a)
∆ [γ ] ∆ [γ ]
FIG. 1: (color online) spectrum of the real (solid) and imagi-
nary (dashed) part of the refractive index as well as the real
(dotted) part of the response function(s) ε and/or µ as a func-
tion of the detuning ∆ for a (a) pure dielectric or magnetic
medium for ρ|α0|/3 at ∆ = 0 equal to = 1/3 (top), 3 (middle)
and 30 (bottom) (b) magneto-dielectric medium for ρ|α0|/3
at ∆ = 0 equal to = 1/3 (top), 3 (middle) and 30 (bottom).
Negative refraction of light is currently one of the
most active research areas in photonics [19, 20, 21] due
to fascinating potential applications such as superlens-
ing [22] or electromagnetic cloaking [23, 24, 25]. In
recent years substantial progress has been made in re-
alizing negative refraction in so-called meta-materials
[26, 27, 28, 29]. These are artificial periodic structures
of electric and magnetic dipoles with a resonance wave-
length much larger than the lattice constant which thus
form a quasi-homogeneous magneto-dielectric medium.
In order to achieve a large electromagnetic response, op-
eration close to resonance is needed which is associated
with rather substantial losses. The elimination of these
losses represents one of the main challenges in the field
[30]. We have shown here that in a radiatively broad-
ened medium, i.e. a medium in which density-dependent
broadening mechanism can still be disregarded for suf-
ficiently large densities, local field effects can provide a
negative index of refraction and at the same time effi-
ciently suppress absorption losses.
SUMMARY
In the present paper we have given a rigorous micro-
scopic derivation of Clausius-Mossotti relations for both
the electric and magnetic response in an isotropic, radia-
tively broadened magneto-dielectric medium formed by
a simple bi-cubic lattice of electric and magnetic dipoles.
As opposed to previous microscopic approaches we have
taken into account possible modifications of the single-
particle polarizabilities by the altered electromagnetic
vacuum inside the medium in a self-consistent way. For
a simple bi-cubic lattice it has been shown that the po-
larizabilities entering the Clausius-Mossotti relations are
those of single oscillators interacting with the free-space
vacuum field. We showed that as a consequence of the
local field corrections a radiatively broadened medium
with overlapping electric and magnetic resonances be-
comes lossless with a real part of the refractive index
approaching the value −2 in the high-density limit. The
latter could provide an interesting avenue to construct ar-
tificial materials with negative refraction and low losses.
This work was supported by the Alexander von Hum-
boldt Foundation through the institutional collabora-
tion grant between The Institute of Theoretical Physics
and Astronomy of Vilnius University and the Techni-
cal University of Kaiserslautern. J.K. acknowledges fi-
nancial support by the Deutsche Forschungsgemeinschaft
through the GRK 792 “Nichtlineare Optik und Ultra-
kurzzeitphysik”.
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