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VOLUME 93, NUMBER 2
PHYSICAL REVIEW LETTERS
Homogenization of Magnetodielectric Photonic Crystals
A. A. Krokhin1'2 and E. Reyes2
1Department of Physics, University of North Texas, P.O. Box 311427, Denton, Texas 76203, USA
2Instituto de Fisica, Universidad Aut6noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
(Received 30 July 2003; published 9 July 2004)
We calculate the low-frequency index of refraction of a medium which is homogeneous along axis z
and possesses a periodic dependence of the permittivity e(r) and permeability b(r) in the x-y plane (2D
magnetodielectric photonic crystal). Exact analytical formulas for the effective index of refraction for
two eigenmodes with vector E or H polarized along axis z are obtained. We show that, unlike
nonmagnetic photonic crystals where the E mode is ordinary and the H mode is extraordinary, now both
modes exhibit extraordinary behavior. Because of this distinction, the magnetodielectric photonic
crystals exhibit optical properties that do not exist for natural crystals. We also discuss the limiting case
of perfectly conducting cylinders and clarify the so-called problem of noncommuting limits, w - 0
and E - oo.DOI: 10.1103/PhysRevLett.93.023904
Magnetodielectric photonic crystals (MDPCs)-peri-
odically arranged composites made up of magnetic semi-
conductors (or dielectrics) represent a new class of
artificial materials [1] with tunable by external magnetic
field [2] band gap. In the low-frequency limit (w -- 0),
the wavelength covers many lattice periods thus repeating
the situation that exists in the optics of natural crystals
when the wavelength of visible light is much larger than
atomic spacing. Atoms of any transparent natural crystal
do not exhibit noticeable magnetic susceptibility in the
presence of light; therefore they are considered to be
nonmagnetic. As a result, all equations of crystal optics
have been derived for nonmagnetic (Ukik = ik) materials
[3]. Recently Smith and Schurig [4] have shown that
the electrodynamics of the left-hand medium requires
introduction of the tensor uik in order to describe ade-
quately the effects of macroscopic magnetic inclusions.
Inclusions in a MDPC (with either positive or negative
permeability) are macroscopic magnetically susceptible
"atoms." Metallic inclusions in a dielectric medium may
give rise to the effective magnetic permeability at optical
frequencies [5]. Recently, magnetodielectric photonic
nanocrystals have been fabricated [6]. A review of the
properties of MDPCs is given in Ref. [7].
In this Letter we calculate the effective permittivity
and permeability of MDPCs and demonstrate that optics
of such a periodic medium turn out to be different from
the optics of natural crystals. In a natural crystal, since its
magnetic susceptibility is negligible, optical anisotropy is
due only to the dielectric tensor eik. As a result, the
eigenmodes of Maxwell equations in crystals are classi-
fied as "ordinary" and "extraordinary" waves. The ordi-
nary wave exhibits higher symmetry as compared to the
extraordinary wave. In particular, its index of refraction
is independent on the direction of propagation. In what
follows we show that, in an artificial medium with mac-
roscopic magnetic atoms, the tensor ,Uik 6ik and opticalPACS numbers: 42.70.Qs, 41.20.Jb, 42.25.Lc
anisotropy are determined by both tensors eik and IUik.
Therefore both eigenmodes turn out to be extraordinary
waves; i.e., their indices of refraction depend on the
direction of propagation.
To calculate the effective index of refractionnleff(k) = imck
k--O0(1)
where k = k/k is a unit vector in the direction of propa-
gation, one needs to consider the low-frequency limit in
the wave equation for inhomogeneous medium. Numeri-
cal calculations [1,2] show that the lowest (acoustic)
branch of the dispersion relation exhibits almost linear
behavior from k 0 to the middle of the Brillouin zone.
Therefore the index of refraction (1) is valid not only in
the limit ka < 1 (a is the lattice period), but for much
shorter wavelength ka < 0.5 as well.
For a 2D photonic crystal, the equations for the E- and
H-polarized modes have the following form:-(w/ c)2E(r)E,
(w/c)2p(r)H.(2)
(3)V. [-'(r)VE]
S. [-1(r)VH] =These equations are symmetric with respect to the re-
placement E - H and e ~ . Therefore, in what fol-
lows, we calculate neff for the H mode only and the
effective index of refraction for the E mode is obtained
from that for the H mode by interchanging e and , i.e.,
neff {(r), (r)} e) {(r), E(r)}. Even without fur-
ther calculations, it is clear from Eqs. (2) and (3) that
the E and H modes possess the same symmetry in a
MDPC; i.e., both of them are extraordinary. Unlike this,
if (r) 1, Eq. (2) has higher symmetry. As a result, the
E mode becomes the ordinary mode with neff indepen-
dent on k [8].0031-9007/ 04/93(2)/023904(4)$22.50 2004 The American Physical Society
week ending
9 JULY 2004023904-1
023904-1
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Krokhin, Arkadii A. & Reyes, E. Homogenization of Magnetodielectric Photonic Crystals, article, July 9, 2004; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc107768/m1/1/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.