PHYSICAL REVIEW LETTERS
PRL 99, 057202 (2007)
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3 AUGUST 2007
Simulations of Ferrite-Dielectric-Wire Composite Negative Index Materials
Frederic J. Rachford
Code 6365, Naval Research Laboratory, Washington, D.C. 20375, USA
Douglas N. Armstead
Physics Department, Westminster College, New Wilmington, Pennsylvania 16172, USA
Vincent G. Harris and Carmine Vittoria
Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA
(Received 16 January 2007; published 30 July 2007)
We perform extensive finite difference time domain simulations of ferrite based negative index of
refraction composites. A wire grid is employed to provide negative permittivity. The ferrite and wire grid
interact to provide both negative and positive index of refraction transmission peaks in the vicinity of the
ferrite resonance. Notwithstanding the extreme anisotropy in the index of refraction of the composite,
negative refraction is seen at the composite air interface allowing the construction of a focusing concave
lens with a magnetically tunable focal length.
DOI: 10.1103/PhysRevLett.99.057202
PACS numbers: 85.70.Sq, 42.15.Eq, 42.79.Bh, 85.70.Ge
In the last several years much interest has been focused
on negative index material (NIM) composites starting with
the pioneering experiments of the UCSD group [1– 4]. In
1968 Veselago [5] predicted that a material possessing
simultaneous negative permittivity " and permeability would possess a negative index of refraction n, and consequently display unique properties such as an oppositely
directed phase velocity and pointing vector and an inverse
Doppler response. Subsequently others [6 –8] have demonstrated that composites of split ring resonators (SRRs)
and conducting wire arrays can be constructed to possess
an effective composite negative " and . Potentially NIMs
can be used to produce superior lenses, phase compensators, etc. [9–12]. The index of refraction must be frequency
dependent since strong frequency dispersion is required to
produce negative constituent parameters. Most NIM constructs to date rely on metallic resonant structures such as
SRRs and geometric arrays of wires. They typically display fixed narrow operational frequency bandwidths. In
attempting to overcome bandwidth and tunablity constraints of the prior NIM constructs we have extensively
investigated ferrite based designs that can be frequency
tuned with the application of a magnetic field. Within the
limitations of this Letter we report several interesting
properties of barium-M (BaM) ferrite and Mylar composites threaded by a monodirectional square wire grid. A
more comprehensive report of our simulations will be
published elsewhere.
Our composites consist of BaM ferrite laminae with inplane anisotropy to provide a low loss < 0 and a monodirectional square wire grid to provide an " < 0.
Combining the ferrite and wire grid we can produce a
composite with simultaneous average < 0 and " < 0 to
yield n < 0 similar to that previously demonstrated in SRR
or wire structures [1]. We choose wire grid spacings and
0031-9007=07=99(5)=057202(4)
wire diameters to tune [13] a high pass cut off slightly
above the ferrite antiresonance. We assume that the ferrite
magnetization, the millimeter wave polarization, and the
wire grid axis are coaligned. The ferrite based permeability
employed in these simulations was derived from measured
ferrimagnetic resonance (FMR) data for in-plane anisotropy BaM ferrite films [14] as grown in our laboratory. The
frequency dependent ferrite permeability for linearly polarized radiation passing through a BaM ferrite is given by
[15,16]
4MS H Ha 4MS 2
;
H Ha 4MS 2 ! iH=22
(1)
where MS is the saturation magnetization, Ha is the magnetocrystalline anisotropy field, is the gyromagnetic
ratio, H is the ferromagnetic resonance line width, and
! 2f is the excitation frequency. This frequency dependent permeability, derived from magnetic resonance,
applies when Hrf lies in a plane perpendicular to MS , Ha ,
and the applied saturating field H. For certain BaM films
grown and measured in our laboratory, we find 4MS 4:5 kG, H 100 Oe, Ha 17 kG. We impose an external saturating field, H 1 kOe, to align the ferrite
magnetization parallel to the wire grid axis. These parameters result in a ferrimagnetic resonance frequency, fO 37:8 GHz. The index in the direction of propagation (x
p p
direction of Fig. 1) is nx "y z where z 1 lin 1 with being the volume fraction of ferrite.
(For thin BaM laminae x ! 1 y due to dynamic
demagnetization of Hrf .) Variations in the amplitude of
the external magnetic field H allows us to tune the FMR
frequency and, hence, the value of nx .
In our previous work as well as is noted in the literature
[17] we find that the wire grid must to be separated from
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PRL 99, 057202 (2007)
PHYSICAL REVIEW LETTERS
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3 AUGUST 2007
FIG. 1. A sketch of the ferrite (black) and dielectric (gray)
composite configuration with a square wire grid (black circles)
centered in the dielectric and grid size L. The arrows indicate
normal incident millimeter wave radiation. A 1 kOe external
field is applied out of the figure to align the ferrite magnetization
parallel to the wire axis and the wave polarization.
the ferrite. The wire current is extinguished if the wires are
embedded in a ferrite at frequencies where the ferrite
permeability is negative or the magnetic loss tangent is
large. Therefore, we offset the wire grid with Mylar dielectric. Ferrite and dielectric laminae are paired with pair
thickness equal to the grid spacing. (See Fig. 1.) We
arbitrarily limit the unit ferrite-dielectric pair spacing to
be approximately equal to or less than 1=10th the free
space wavelength. The Mylar has a low electrical loss
tangent and a dielectric constant of 3.0. The actual low
loss dielectric employed is not critical since the negative
permittivity below cutoff is dominated by the wire grid.
We have investigated many variations of ferrite and
dielectric thickness and grid spacing, but here we focus
on a sequence where the grid spacing (pair thickness) is
varied while keeping the ferrite and the dielectric lamina
thicknesses equal to one-half of the grid spacing. 1D
normal incidence transmission spectra were calculated by
Fourier analysis of the finite difference time domain
(FDTD) pulse response [18] of 39 BaM-Mylar pair composites at various wire grid spacings L. (See Fig. 1.)
Perfectly matched layer (PML) boundary conditions [19]
were imposed in the propagation direction and periodic
boundary conditions were set laterally. Our wires have
rectangular cross sections and are 1=48th the wire grid
lattice dimension in the (x) direction of propagation and
1=12th the grid size in the (z) direction.
In Fig. 2 we display a few selected transmission spectra
along with the average composite permeability from
Eq. (1). As we increase L the upper cutoff frequency is
reduced and the transmission peak above ferrimagnetic
resonance (FMR) remains close to 40 GHz in the < 0
region. At larger L a second peak appears below FMR.
[See Fig. 2(d).] The index at a transmission passband varies
nearly linearly with frequency across the peaks. [See
FIG. 2. A series of transmission spectra through 39 unit thickness of the BaM-Mylar-wire composite varying the spacing of
the square wire grid. As the spacing increases the high pass
cutoff is reduced in frequency. As the cutoff approaches the BaM
resonance a second peak appears below the FMR frequency (d).
The complex relative permeability of BaM at 50% loading is
shown for reference (a).
Fig. 3(a).] In Fig. 3(b) we plot n sampled at the center of
the transmission peaks versus L.
Negative index transmission peaks are seen above FMR
and positive index peaks are seen below FMR. The sign
and amplitude of the real index at the transmission peaks
were found from single frequency simulations by plotting
the amplitude of the rf E field in steady state versus time
and space and comparing the slopes of the phase progression inside and outside the composite. During the course of
the simulation this E field was sampled along an axis in the
direction of propagation centered on the composite. The
sign of the index was readily apparent from movie sequences of the simulation. The imaginary part of n was found by
fitting the exponential decay of the rms E-field amplitude
in the material. Fresnel oscillations were negligible due to
the attenuation and thickness of the composite.
The peak appearing at rf frequencies below ferromagnetic resonance and the wire grid cutoff was consistently
measured to have a positive index of refraction. This
implies that both the average " and are positive in this
frequency band. We believe that the large ferrite permeability 1 below resonance increases the selfinductance of the wires effectively reducing the wire grid
cutoff resulting in a positive index of refraction. This effect
has not been reported in previous studies.
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Fixing the wire grid size at L 648 m with the external field at 1 kOe, the composite index is nx 1:06.
We study the 2D response of a rectangular slab of this
material. The simulation space is bounded by PML boundary conditions in the x and z directions and periodic
boundary conditions in the y direction (Fig. 1). In Fig. 4
a line source is positioned close to the front surface of the
39 0:648 mm 25:3 mm thick composite slab. A homogeneous isotropic solid with the index n 1:06
would focus the line source radiation inside the material
as well as on the far side [5,20]. Our extremely anisotropic
material does not focus the source. Instead a chevron
radiation pattern is seen in the solid. The wave fronts in
the material propagate toward the source as is expected
when n < 0. Replacing the composite with a homogeneous
anisotropic slab with horizontal index of nx 1 and
p p
vertical index of nz "y x i yields a similar
phase propagation pattern.
Moving the line source of Fig. 4 farther from the slab or
impinging plane waves at modest angles of incidence
results in the usual negative refraction with refractive angle
slightly less than that expected for an isotropic sample with
the same negative index. For larger angles of incidence
FIG. 3. (a) The frequency dependence of the complex index
for the case of a 648 m grid spacing with a 1 kOe field bias.
The transmission spectrum is seen in Fig. 2. (b) A plot of the
index of refraction taken at the center of the transmission peaks
as seen in Fig. 2. At larger grid spacings a peak with n > 0
appears at frequencies below ferrimagnetic resonance. (c) The
variation of the index of refraction at 40 GHz for the 648 m
composite with a small (75 Oe) modulation of the 1 kOe
external field.
The transmission peak frequency can be shifted by
variation of the external magnetic field. For small field
variations the transmission peak can be scanned through
a fixed frequency. In Fig. 3(c) we show tuning of the
648 m composite index at 40 GHz by varying the
1 kOe external alignment field by 75 Oe. Over this range
of alignment fields the index varies nearly linearly by
40%. Larger fields will shift the transmission peak beyond the frequency of operation and extinguish the transmission. Thus the external field can be used to both tune the
center frequency of the < 0 passband and the value of n
at fixed frequency.
FIG. 4 (color). A two-dimensional FDTD simulation of the
BaM-Mylar-wire composite studied in Figs. 2 and 3. A 40 GHz
line source is positioned close to the composite slab. The color
intensity scale is saturated to better visualize the 40 GHz E field
amplitude inside the composite. The index in the horizontal
direction in the composite is n 1:06. The inset plots the
refracted angle versus the angle of incidence when plane waves
are incident on the same composite. The refraction is limited to
41
consistent with the chevron angle observed when the line
source is close to the material. The solid line shows the refraction
expected for a homogeneous isotropic material of index 1:06.
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PHYSICAL REVIEW LETTERS
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and ferrite components [21]. The interaction of the ferrite
and the wire self-field is not easily predicted and results in
the novel appearance of a positive as well as a negative
index transmission peaks below the grid cutoff frequency.
F. J. R. gratefully acknowledges prior collaborations
with his NRL colleagues Peter Loschialpo, Douglas
Smith, and Donald Forester, on unpublished studies of
magnetic NIMs. This work was performed under a
DARPA NIM contract, Valerie Browning, program
manager.
FIG. 5 (color). The composite of Fig. 4 is cut to form a
concave lens and the line source is moved away from the front
surface. Negative refraction at the concave face results in the
formation of a focus.
refraction greater departures from Snell’s law are observed.
(See inset of Fig. 4.) For the 648 m composite of Fig. 4
we find in the inset that the angle of refraction for plane
waves asymptotically approaches 41
. This is consistent
with the 41
chevron angle noted in Fig. 4.
In Fig. 5 we have sculpted the 648 m grid composite
block of Fig. 4 into a concave cylindrical lens. The 40 GHz
line source was moved approximately 2.6 cm from the
front surface of the lens. Negative refraction at the concave
face focuses the radiation behind the composite lens as is
expected for an isotropic negative index concave lens.
Modulating the 1 kOe external alignment field shifts the
index at a fixed frequency, linearly changing the focal
length of the lens from 1.21 cm at 925 Oe to 0.84 at
1075 Oe. A larger field increment moves the negative index
peak beyond the operation frequency extinguishing the
transmission.
Oriented ferrite/dielectric/wire composites display
negative refraction even in the presence of extreme anisotropy in the index of refraction. Although, planar slabs of
our composite do not focus close by divergent sources, they
can negatively refract radiation impinging at modest angles
of incidence with deflections comparable to that expected
for homogeneous NIM media. At greater angles of incidence the refraction asymptotically approaches the chevron angle seen for near source radiation. The index and
transmission frequency of these materials can be tuned by
modulation of the amplitude or orientation of the external
alignment field. The wire cutoff frequency is very sensitive
to the wire local environment. Further reduction of losses
in this composite will require greater separation of the wire
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