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Review paper - Epsilon-Near-Zero-Photonics
Project of course ’Recent Trends in Photonics’
Group members: Ziyue Zhang; Ruifeng Li
Abstract—Epsilon-Near-Zero(ENZ) material is a new class
of materials with a vanishing permittivity and has emerged
as an important field of research in recent years. As a
project in the course ’Recent Trends in Photonics’, this
review paper introduces the unique physical properties of
ENZ materials and show integrated device applications of
ENZ photonics.
Then applying Maxwell equations in a source-free area
leads to:
5×H=−
d
D=0
dt
d
B
dt
2
5 E=0
5×E=
(2)
I. I NTRODUCTION
Epsilon-near-zero (ENZ) photonics studies on
light–matter interactions in the presence of materials
with (effectively) near-zero permittivity. The
experimental realization of ENZ waveguide can
be traced back to around 2010 [1]. Although there
are many challenges to overcome in order to make
ENZ components active in the market, it has great
potential and has been an important field of research
in recent years. Integrated devices of ENZ photonics is
promising due to their unique properties, especially the
enhancement to nonlinear effects.
This review paper first introduces the basic physical
principles of ENZ materials and some realization methods , followed by introducing some novel features of
ENZ materials. And then some photonic devices based
on ENZ materials are shown. Some barriers and opportunities in the field of ENZ photonics are concluded in
the end.
II. T HEORY OF ENZ MATERIALS
A. basic concepts
Epsilon-Near-Zero(ENZ) materials have near-zero real
permittivity at given frequencies. The electric displacement field D as the product of permittivity tensor and
the electric field E is then near zero if we firstly neglect
the imaginary part of permittivity. A zero prmittivity
tensor also indicates that at a certain frequency the
polarisation fieldP is the opposite of the electric field
and leads to a zero D filed.
D = E = 0 (1 + χ)E = 0 (E + P)
(1)
The zero permittivity results in a curl-free Magnetic
field and a statistic electric field. This also implies that
the electric and magnetic fields decouples from each
other. If we assume the permeability µ of these materials
can be regarded as a constant, where µ = 1, which holds
for most natural materials in
p optical frequency. The
√
refractive index n = µ = µ(0 + i00 ) = n0 + in00 ,
where one prime stands for real part and double prime
is for imaginary part. It can be shown that a nearzero refractive
q index indicates that the real refractive
00
0
index n = µ2 and is in the scale of 0.5 for natural
metal ENZ points.
p µ Moreover, the impedance of ENZ
material Z =
diverges to infinity, which has a
huge mismatch with other media and results in poor
transmission coefficients.
However, for some artificial meta-materials the effective
permeability µ can also be tuned to near-zero with
some smart design. This kind of materials are called
epsilon–mu-near-zero (EMNZ), which is an important
subcategory of ENZ material with zero real refractive
index n0 = 0, thus also called Near-Zero-Index(NZI)
material. In NZI materials the effective wavelength of
the EM wave at the certain frequency inside the ENZ
material goes to infinity λef f = n0 λ = 0. This implies
the decoupling between temporal and spatial fields for
EMNZ materials. EMNZ materials can also overcome
the problemqof low transmission coefficient with their
ZEM N Z = µ→0
→0 ≈ Zvac .
Another important property of NZI materials is that
the wavevector |k| = 2π
λ n also goes to 0 due to the
near-zero refractive index. EM wave inside this kind of
materials tends to have an infinity phase-velocity and
0 group-velocity, given the expression vp = ωk and
vg = dω
dk . The frequency points where NZI occur is
therefore also called zero-group-velocity(ZGV) points.
Normally ZGV are accompanied with large groupvelocity-Dispersion(GVD), which is useful for dispersion compensation.
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B. realisation mechanism
1) Natural ENZ materials: The easiest approach to
near-zero-epsilon is using those materials that can be
described by Drude Model:
ε(ω) = ε∞ −
ωp2
ω 2 + iγω
(3)
where ε∞ is the permittivity at infinity large frequency;
ωp is the electrostatic oscillation frequency (plasma free2
, N is the carrier density. When the
quency) ωp2 = εN0 m
e
incident frequency equals to the bulk plasmon frequency
√
of ω = ωo / ε∞ , the real part of epsilon becomes
zero,indicating the existence of ENZ points. This type
of ENZ behavior can be found over a wide frequency
range from the ultraviolet to the mid-IR range, and
even up to the THz regime. This broad band originates
from the wide range of N in different materials. Metals
are natural Drude-Model materials, but they have large
imaginary permittivity ,indicating too much loss for
most photonic systems. Thus the ENZ mode can only
exist when the metal layer is extremely thin (in d¡10nm
scale). Metals have ENZ points typically in the UV
band. ENZ points in metals are rather fixed because
their carrier density is difficult be tune.
Degenerately doped semiconductors such as fused silica, Cadmium oxide, and Indium oxide are also used
for ENZ materials, which are semiconductors with
high dopant density. These semiconductors have large
bandgaps and are transparent in visible range, and
are called Transparent conducting oxides ”TCOs” [2].
TCOs can be heavily doped and are able to be described
by Drude Model. In comparison with metals, Degenerately doped semiconductors have tunable carrier density
and low damping rate γ, leading to more flexible
ENZ points and lower loss(imaginary permittivity). The
ENZ points in Degenerately doped semiconductors are
normally in IR or MIR. However, the thickness of
such materials must be smaller than the scale of 1 100
micrometers to support ENZ mode due to loss. The
ENZ points in degenerately doped TCOs are observed
in experiments and widely used in integrated photonics.
Figure 1 [3] shows a comparison between theoretical
and experimental values for permittivity in a film of
ITO.
Figure 1: Linear relative permittivity of the ITO film
measured via spectroscopic ellipsometry and estimated
by the Drude model. [3]
There are studies focusing on the propagation modes in
Sub-wavelength thin film ENZ materials showing that
the ENZ mode is a part of long-range surface plasmonic
wave mode [4] [5]. The semicondutor type of ENZ
materials are easier to fabricate thanks to its reachable
thickness and the mature semiconductor chip industry.
2) artificial ENZ materials by media mixing: Another
way to obtain ENZ material is using multiple thin
layers of plasmonic materials and dielectrics. Materials
under long-wavelength approximation can be regarded
as homogeneous materials with a constant effective
permittivity εef f that can be determined by effective
medium theory(Maxwell–Garnett equation) [5]:
⊥,ef f =
dm m +dd d
dm +dd
1
k,ef f
=
dm /m +dd /d
dm +dd
where dm and dd are thickness of metal and dielectric layers and εm and εd are permittivity of metal
and dielectric layers. Dielectric materials with positive
permittivity combining with electron gas materials with
negative permittivity(when the frequency is below bulk
plasma frequency) then enables ENZ points at a certain
frequency in the artificial material.
This kind of ENZ materials are manufactured by stacking thin layers of dielectric, metal, semiconductors or
by inserting array of nanowires/nanonets into dielectric
host. They are mostly inhomogeneous because of the
broken symmetry compared with natural ENZ materials,
meaning that they have ENZ points with limited incident
angles.
The ENZ points are now much more tunable because
they are no longer determined by the carrier frequency
but the fraction of layer thicknesses. They can also
be easily fabricated by thin-film deposition techniques.
media-mixing ENZ materials are promising in nonlinear
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devices given these two advantages.
For example, 5 nm silver layers are stacked with 70 −
80 nm SiO2 layers in order to produce an ENZ point
around 800 − 900 nm wavelength [6].
3) artificial ENZ materials Operating at Cutoff Frequency: Metallic Hollow Waveguides with specific designs can also enable ENZ points in their spectrum.
Unlike dielectric waveguide with cutoff frequency
βcutof f = nmin k0 , metallic waveguides with perfect
electric conductivity(PEC)
boundaries have cutoff frep
π 2
k2 − ( w
) = 0, where w is the
quecy at β =
dimension of a 1D metallic hollow waveguide. EM
waves at the fundamental T E10 cutoff frequency of the
waveguide has zero propagation constant β 2 = k02 −
kx2 = 0 and can be regarded as having a zero effective
refractive index; a zero group velocity; a zero effective
permittivity. Therefore, a metallic hollow waveguide
can provide an ENZ point avoiding the requirement of
negative permittivity.
4) artificial ENZ materials Operating at the Centre of
the Brillouin Zone: The name ’Dirac Cone’ originates
from Semiconductor Quantum physics, where electrical
conduction can be described by the movement of charge
carriers which are massless fermions(in another word,
the effective mass of eletrons are 0 at Dirac points).
The Dirac points also indicates linear dispersion relation
between electron energy and the crystal momentum in
a 2D model . This concept can certainly be extended
to 2-D photonic crystals as ENZ points, with linear
dispersion relation between photon frequency and photon momentum in photonic crystals. When this Dirac
point is at the centre of the Brillouin zone where the
wavevector k =0, an isotropic(regardless of incident
angle in the 2-D plane) ENZ point is enabled in the
photonic crystal.
This structure is implemented in 2015 [7] with short
silicon pillars between parallel conductors. The photonic
dirac cone is shown in figure 2
III. NOVEL PHYSICAL EFFECTS OF ENZ MATERIALS
As shown in the previous section, ENZ materials have
many unique properties due to the singularity in its
parameter space. In this section several important novel
effects of ENZ materials are introduced.
A. normal emission
Any light wave incidents from an ENZ material should
be nearly normal to the surface after refracting due to
Snell’s Law with nEN Z ≈ 0, as shown in fig 3 1 on
the left. The right hand side of fig 3 demonstrates the
reciprocal effect: the coupling tolerance becomes very
critical because a near-zero refractive index makes most
wave incident on the ENZ material suffer from TIR.
Figure 3: normal refraction of ENZ material
nenz sin θenz = n sin θinc
(4)
This effect can be useful when designing for example
a surface emitting laser. This effect can also be used to
easily identify an ENZ material.
B. Transmission with Invariable Phase
An EM wave can maintain its phase when propagating
through an ENZ material simply because k = 2π
λ0 n = 0
and E = E0 ejk·r = E0 . This constant phase remains
all over the ENZ region, which can cause constructive or
destructive interference with other waves inside the ENZ
material. Thus, when there are multiple in-phase sources
inside the ENZ material, they will constructively interfere at everywhere. This effect is useful when designing
micro-antennas.
Figure 2: photonic Dirac Cone of the photonic crystal
structure [7]
1 this
is and original drawing and is not from any literature
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C. Field amplification
Considering the boundary conditions of Maxwell equation: n·D should be continuous on the boundary between
ENZ material and another dielectric material, where
n is the unitary normal vector. This means that the
normal component of electric field inside ENZ materials
can be amplified significantly due to the vanishing
|E
|
. However, this amplified
permittivity: |E⊥ | = 0,⊥
field can only exist as a surface wave mode because
of TIR, the proporgating mode is also called ’ENZmode’ or ’long-range surface wave mode’ . This effect is
usually implemented in sub-wavelength thin film ENZ
devices and is very important for enhancing non-linear
phenomenons.
the ENZ material due to Snell’s Law as explained in the
section [normal emission]. On the other hand, all light
waves will experience the same phase front diffusion
and enables phase modulation of emission by designing
the geometry of the exit ENZ surface as shown in figure
4.
D. benefits to non-linear phenomenons
Nonlinear optics play an important role in the photonic
community. Nonlinear phenomenons becomes significant with two restrictions, namely phase-matching and
high incident intensity. Luckily ENZ benefits for both
aspects of nonlinear optics. Firstly, when there is phase
mismatch, there will be constructive interference in
the entire ENZ region among the nonlinear dipoles
which are in different locations, instead of destructive
interference in non-ENZ materials. Thus, ENZ material
tends to be ’phase-mismatch-free’. Secondly ,high input intensity is not needed to obtain the strong field
distribution required for the nonlinear processes due
to the unique electric field enhancement mechanism
introduced in the last section. Moreover, the group
velocity of EM waves in ENZ materials tends to be zero
in a certain frequency, which leads to long propagation
time inside ENZ material and earns more time for
nonlinear-optic interaction.
ENZ materials have enormous advantage in nonlinear
optics and is promising for integrated nonlinear devices
by enhancing effect on nonlinear effects. Using ENZ
materials to make frequency variation devices can obtain more outstanding functions.
Figure 4: an ENZ material can modulated the emitted
phase front [2]
B. All-Optical Switching
Based on its remarkable characteristics, ENZ materials
have shown potential in the following application fields.
In the integrated optical circuit, the optical logic gate is
the most basic element to realize the ultra-fast optical
network on the chip. This optical logic gate can be
realized by All-Optical Switching technology based on
ENZ material. The third-order optical nonlinear effect
induces a change of the optical refractive index which
can be directly used to realize the all-optical switch.
NRZ material has low linear permittivity in zero-epsilon
region. The relationship between refractive index and
light intensity can be approximately expressed as n =
∆n · I. The relationship between linear permittivity √ .
and the change in refractive index ∆n is ∆n = ∆
Therefore, the use of ENZ materials can not only make
the refractive index zero when the light intensity is
small, but also obtain a large gain rate, which means that
a small increase in light intensity can greatly change the
refractive index. Such characteristics can well realize
the distinction between the ’0’ state and the ’1’ state
of an optical switch. In addition, as mentioned above,
ENZ material itself is easy to achieve nonlinear effects,
which is also beneficial to the realization of all-optical
switches.
A. Directed Emission Devices
C. Nanoinsulators
For modern emitters, manipulation of the direction and
phase front of the emitted field are the goals that people
continue to pursue. On the one hand, the EM wave will
be in the normal direction of the surface when it leaves
Conventional circuits operating in the RF domain contain elements smaller than the operating wavelength,
based on which Engheta et al proposed a new concept ‘nanocircuit and nanoelements operating in the
IV. ENZ BASED INTEGRATED DEVICES
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optical domain’ in 2005 [8]. In such an optical circuit,
due to the change of optical characteristics in a high
frequency regime, the conducting current is no longer
the current flowing in the optical device. Instead, the
electric displacement vector D derived from Maxwell’s
equations can be regarded as a ’flowing optical current’.
In nanocircuits, sub-wavelength particles can be synthesized into nanoinductors, nanocapacitors and nanoresistors. The electric displacement vector in the ENZ
material can be regarded as zero, so the ENZ material
can be used as a nanoinsulator.
V. OPPORTUNITIES AND OBSTACLES
The development of ENZ materials still faces extreme
challenges. First, the inherent loss near the ENZ point
still needs to be reduced or eliminated. This inherent
loss will cause absorption and deformation of the signal
in the ENZ device, which makes the ENZ device unable
to be cascadely connected.
The second challenge is that currently ENZ can be only
implemented in a narrow bandwidth. This means that
WDM technology is difficult to use on ENZ devices,
and some nonlinear effects can not benefit from ENZ
materials.
Finally, the on-chip integration method is also a key
issue waiting to be solved. At this stage, there is no
complete industrial chain that can be used to realize the
integration of ENZ, which leads to its high cost.
VI. CONCLUSION
ENZ materials have great potential in photonic devices.
Content of this report is restricted by total words limitation but ENZ photonics have a much larger variety
of applications than what is demonstrated here. To
wrap up, several important features about ENZ materials
are explained in section III based on some simple
mathematical deduction in the theory section II. Some
modulation devices made use of ENZ materials are
listed. The limitations and expected future studies of
ENZ materials are shown.
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