# Negative Refraction Using Frequency-Tuned Oxide Multilayer Structure

Authors: Yalin Lu; Gail J. Brown; Kitt Reinhardt
Journal: Advances in OptoElectronics
 (2008)

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## Abstract

An oxide-based multilayer structure was proposed to realize negative refraction. The multilayer composes of alternative layers having negative permittivity and negative permeability, respectively. In order to realize negative refraction, their dielectric and magnetic resonances of layers will be tuned to the frequency as close as possibly via changing their temperature, composition, structure, and so forth. Such oxide-based NIMs are attractive for their potential applications as optical super lenses, imagers, optical cloaking, sensors, and so forth, those are required with low-loss, low-cost,
and good fabrication flexibility.


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## Body

## 1. Introduction

Negative index materials (NIMs) have attracted extensive
attention in recent years since the first observation of negative refraction in
those artificially made metamaterials consisting of arrays of metallic rings/rods [1].
Majority of such NIMs use metallic split ring resonance (SRR), surface plasmonic
resonance, or a combination of magnetic and metallic materials with naturally
occurring negative permeability (μ) or
negative permittivity (ε). Very unique
properties of such metamaterials, including negative
refraction, have been previously demonstrated mainly for operation in microwave
frequencies. A new trend of current NIM research is toward the optical frequency.
For example, a lossless negative index slab material could act as a “superlens”
or a “perfect lens,” which could image object with a high resolution far below
the conventional optical diffraction limit, and are attractive to those applications
including data storage, medical imaging, biophysics, and so forth. In addition to those metamaterials, negative
refraction has also been found in other material formats including planar
transmission lines, photonic crystals, and so forth. However, all such NIMs use
complicated artificial structures requiring high-precision nanofabrication and
can hardly be rigorously considered as uniform materials, since the size of their
structural elements is usually greater than one tenth of the operation wavelength.
In addition, certain polarization and incident angle of an electromagnetic (EM)
field must be maintained in order to realize such negative refraction.Considering the fact that an
SRR in a metamaterial produces−εψ and −μψ, respectively [2], such a metamaterial will
be similar to a multilayer composed by alternating −εψ and −μψ  layers—the later will be
much easier to realize as a natural material. It is also well known that some
natural materials have inherent −εψ or −μ, which will be useful for multilayer NIM
fabrications. For example, certain ferrites have −μ in microwave frequencies, and
many conducting materials have −ε below the plasma frequency. Studies toward this
direction have been performed, mostly by simulations, in stacked
ferrite-dielectric waveguides, ferrite-metal, ferromagnetic-superconductor, and
ferrite-semiconductor multilayers [3]. However, the use of conductive materials
may excite surface waves at their interfaces, which in turn change their
electromagnetic characters, and more seriously, result in a significant high-loss
concern. Apparently, an NIM composing of nonconductive optical materials will
be promising.Availability of the
frequency tunability to an optical NIM will provide a much bigger freedom in
exploring its potential applications as sensors, modulators, switches, and so
forth. Variable electromagnetic response when changing a metamaterial’s
structure is not a practical way of frequency tuning. External tuning via filling
liquid crystals and photoconductive materials to metamaterials will work for
some frequencies. It is normally slow and has concern on both reliability and
stability of performances.This manuscript intends to discuss
one possible mechanism potential able to realize negative refraction using all
oxide-base materials in the format of multilayer. Such oxide-based negative
index multilayers  are
expected to potentially have low optical losses,
against to those existing designs using conductive materials, and are
potentially frequency-tunable if the NIM’s composing materials are
dielectrically tunable by external fields including electric, optical, and
microwave. Tunable dielectrics are easily realizable in many piezoelectric and
piezomagnetic complex oxides. Therefore, the impact of such NIMs to their
potential applications such as super lenses, optical cloaking, optoelectronics,
and sensing can be significant.


## 2. Oxide Multilayer NIMs

The mechanism to build the oxide
multilayer NIM to be discussed is a resonance-tuned multilayer (RTM) composing of alternative dielectric and magnetic
oxide layers having naturally occurring−εψ or −μ, respectively, and having their resonance
frequencies finely tuned to concurrence via both structural and compositional
engineering to both used materials.The RTM mechanism
considers the nonlattice-matched multilayer case and the two alternative oxide
layers as the multilayer have−μ  or −ε, respectively.  For example, the oxide multilayer can be
represented as (A/B)m, and
“A” has an intrinsic dielectric
resonance, and “B” represents a
material with an intrinsic magnetic resonance. The RTM approach will be used to
fine-tune the two resonance frequencies concurrence together, and realize the
negative refraction immediately after the merged resonance frequency.
Fine-tuning to each oxide material can be realized by changing composition, structure,
temperature, magnetic field, and so forth. Schematic of the RTM NIM is shown in
Figure 1. In Figure 1, only
two adjacent laminas  (one dielectric and the other
magnetic) having thickness of d1 and d2, respectively,
are shown. They will be the building block for further making into the
multilayer structure. The insets on the right side schematically show the
frequency spectra of both permittivity and permeability around the resonance
frequencies. Arrows indicate the directions of frequency tuning to make two resonance
frequencies concurrence. Figure 2 shows the schematic multilayer structure and
the concurrence-occurred frequency spectra showing the frequency region with
both negative permittivity and negative permeability.Figure 1
Schematic of a negative
permittivity and negative permeability double layer (the building block for
multilayer, the left side of the figure) and tuning of their resonance
frequencies to concurrence-occur to the same frequency (inset at the right
side).Figure 2
Schematic of the
frequency-tuned multilayer structure (left) and the concurrence-occurred
resonances on both permittivity and permeability (right). After the
concurrence-occurred resonance frequency, both negative permittivity and
negative permeability will occur, which indicates negative optical refraction.In order to achieve a concurrence
of both dielectric and magnetic resonances in dielectric and magnetic oxide
materials, different origins of dielectric and magnetic responses have to be
considered for different frequencies ranging from the UV/visible, far-infrared
(including the major terahertz band), to microwave. For the UV to visible
spectrum, for example, the dielectric response is primarily from the electronic
polarization modes and the magnetic response from the magnetic dipoles. In the
far-IR range, ionic polarization modes, and antiferromagnetic resonances (AFMRs)
are mainly responsible for dielectric and magnetic responses, respectively.
Similarly, for those microwave frequencies, electronic dipole reorientation
(domain wall moving) and ferromagnetic resonances (FMRs) are the main causes.Other than the resonance frequency of the
concerning materials, both degree of damping and amplitude of dielectric and
magnetic responses will also play the key rule. For example, in optical
frequencies, direct magnetic dipole response is orders of magnitude weaker than
electronic dipole response. The above-discussed SCS mechanism actually provides
an indirect coupling of the magnetic dipole response to a strong electronic
dipole resonance via the phonon resonance, yielding both negative permeability
and negative permittivity after the phonon resonance frequency.The dielectric dispersion due to the  electronic
dipole polarization can be fitted to the following empirical expression:(1)ε*(ω)−ε∞1ε0−ε∞1=∑i=1ngi[1(1+iωτi)],where ε∞1 is the permittivity due to the ionic polarization, ε0 is the low-frequency permittivity, ω is the angular frequency, and gi
is the distribution of
relaxation time (τi). On the other hand, the dielectric
dispersion due to the ionic polarization can be fitted to the following
four-parameter semiquantum model [4]: (2)ε*(ω)=ε∞2∏j=1nωjLO2−ω2+iγjLOωωjTO2−ω2+iγjTOω, where ε∞2 is the permittivity due to the electronic polarization,
n is the number of optical phonon mode, γ is the damping factor, and TO and LO represent transverse and longitudinal optical phonon modes,
respectively. The relationship between the ε* and IR
reflectivity R(ω) can be given by (3)R(ω)=|(ε*(ω)−1)(ε*(ω)+1)|.Equations (1), (2),
and (3) permit a direct identification of the dielectric spectra and resonance
frequencies via the experimental IR reflectivity measurement on a specific
dielectric material system. Contribution from domain wall relaxation is usually
masked by electronic and ionic relaxations, unless the temperature is close to
the material’s Curie temperature or phase transition temperature. It has been
usually reported at above 100 kHz frequencies, will be minor to those optical
frequencies of our current discussion interest.Magnetic
resonances can be identified from recording the complex permeability spectra on
those magnetic oxide materials of interests. Accordingly, the complex
permeability  can be expressed as a superposition of two contributions,μ(ω)=1+χ~dw(ω)+χ~rot(ω),
namely, the domain wall component χ~dw(ω) and the magnetic moment rotation χ~rot(ω): (4)χ~dw(ω)=χdwωdw2ωdw2−ω2+j(β/mdw)ω,χ~rot=(ωrot+jαω)χrotωrot(ωrot+jαω)2−ω2, where χdw and χrot are static (or low frequency) susceptibilities
for domain wall (DW) and magnetic moment rotation (ROT) motions, ωdw and ωrot are resonance frequencies of DW and ROT
components, β and α are corresponding damping factors, mdw
is the effective mass of
wall, and ω is the frequency of the driving ac magnetic field. Description of
magnetic moment motions will be highly dependent on materials. For simplicity,
we consider both AFMR and FMR cases which will be similar in theory.In a magnetic material having a uniaxial
anisotropy and the magnetic field is applied parallel to the easy axis, the
AFMR frequencyωAFMR at low temperature
(below Néel temperature) is given by  (5)2πωAFMRγ=2HEHA±H,(H<HSF),2πωAFMRγ=H2−2HEHA,(H>HSF),where γ(≡gμB/ℏ) is the
magnetomechanical ratio, HE
is the exchange field, HA
is the anisotropy field, and HSF is the spin-flop field. When
H is
applied perpendicular to the easy axis, then the ωAFMR is given by (6)2πωAFMRγ=H2+2HEHA.For
FMR, a simple expression about theωFMR is Kittel's equation (7)ωFMRγ=(HA,eff+HE)4πMs,where
HA,eff is the effective anisotropy field, and  4πMs the saturation
magnetization. For both AFMR and FMR cases and when for qualitative analyses,
it can be simplified to ω∝HAHE.Depending
on the material selection for such multilayer, the above equations provide the
way to measure their resonance frequencies via a variety of approaches
including optical reflection, time-domain terahertz (THz) transmission,
magnetic analysis, and so forth. The above equations also theoretically provide
the direction to realize the efficient frequency tuning in order to achieve
concurrence of both dielectric and magnetic resonance frequencies.


## 3. Frequency Tuning

Tuning both dielectric and magnetic
resonance frequencies may follow a few different ways. In certain ferroelectric
materials, optical mode of the lattice (phonon) is dependent on both
temperature (T) and the material’s
Curie temperature (Tc), roughly
following an empirical relation: ω2∝(T−Tc) as an example. As an example, Figure 3 shows
the dependence of relative dielectric constant on the composition. The
dielectric maximum occurs at xBa∼0.7  [at where the
ferroelectric-paraelectric boundary (FPB) occurs], and it Curie temperature
changes from 415 K (xBa∼1) to about 40 K (xBa∼0)
almost linearly [5]. Similarly, tuning by changing both temperature and
material composition is also applicable to some magnetic oxide materials. Inset
in Figure 2 show a general frequency tuning schematic via changing the
temperature, in order to reach a concurrence at the same frequency for both
dielectric and magnetic resonances. Apparently, tuning their compositions to
reach a room temperature operation of the NIMs will be the most attractive,
simply because of the practicability for applications.Figure 3
Dependence of the dielectric
constant and Curie temperature of (Ba,Sr)TiO3 on its composition.
Inset shows a possible frequency tuning by changing the temperature for
reaching the concurrence of both dielectric and magnetic resonance frequencies.An
alternative way of possibly tuning the magnetic resonance frequency is by
changing the magnetic layer’s alignment inside the multilayer structure.
Figure4 shows the schematic of such a structure in which the FMR frequency will be
tuned by changing either the orientation angle θ or
the layer thickness, following the rewritten Kittel’s formula involving both
uniaxial anisotropy field (Hu)
and the additional shape anisotropy field (Hk,eff)
into the total effective anisotropy field (HA,eff)(8)HA,eff=Hu+Hk,effcosθ,2πωAFMRγ=4πMsHA,eff.Figure 4
Schematic of the
multilayer structure alignment required for possibly tuning the FMR resonance
frequency via changing the structural parameters—width, period,
duty cycle, and the orientation angle.In
Figure4, the main picture shows the parallel strait structure to be made in
the magnetic layer inside the superlattice, via selective etching the
as-fabricated magnetic layer or mask-shielding growth, and so forth. The inset
shows that when one measures the permeability horizontally, the result will be
actually tunable by changing the strait’s size (both period and width) or by
changing the angle respective to the easy axis of the magnetic material.  The multilayer structure itself to be
discussed in this theory actually provides the great easiness to reach such
frequency tuning for the magnetic part inside the multilayer. Structural
parameters that can be varied when designing include the layer thickness,
periodic duty cycle, and magnetic orientation, and so forth. An efficient way
to optimize material compositions and structural parameters may follow the
combinatorial strategy [6]. Effort toward efficient material selection,
simulation for frequency tuning, and the multilayer fabrication is ongoing and
will be published separately.


## 4. Conclusion

One
possible physical mechanism toward realizing negative refraction in a fully oxide-based
multilayer was theoretically discussed. Inside the multilayer structure, such
negative refraction may be achieved by tuning their resonances to the same
frequency via changing temperature, layer composition, structural parameter, and
so forth. Such oxide-based NIMs are attractive for their potential applications
as optical super lenses, imagers, sensors, and so forth, that are potential
with low-loss, cost-effectiveness, and good fabrication flexibility.



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*Source: 948614-2008-10-30.xml*