Chiral metamaterials: retrieval of the effective parameters with and

Chiral metamaterials: retrieval of the
effective parameters with and without
substrate
Rongkuo Zhao,1,2 Thomas Koschny,1,3 and Costas M. Soukoulis1,3,∗
1 Ames
Laboratory and Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA
2 Applied Optics Beijing Area Major Laboratory, Department of Physics, Beijing Normal
University, Beijing 100875, China
3 Institute of Electronic Structure and Laser, FORTH, and Department of Materials Science
and Technology, University of Crete, 71110 Heraklion, Crete, Greece
*soukoulis@ameslab.gov
Abstract:
After the prediction that strong enough optical activity
may result in negative refraction and negative reflection, more and more
artificial chiral metamaterials were designed and fabricated at difference
frequency ranges from microwaves to optical waves. Therefore, a simple
and robust method to retrieve the effective constitutive parameters for chiral
metamaterials is urgently needed. Here, we analyze the wave propagation in
chiral metamaterials and follow the regular retrieval procedure for ordinary
metamaterials and apply it in chiral metamaterial slabs. Then based on
the transfer matrix technique, the parameter retrieval is extended to treat
samples with not only the substrate but also the top layers. After the
parameter retrieval procedure, we take two examples to check our method
and study how the substrate influences on the thin chiral metamaterials
slabs. We find that the substrate may cause the homogeneous slab to be
inhomogeneous, i.e. the reflections in forward and backward directions
are different. However, the chiral metamaterial where the resonance element is embedded far away from the substrate is insensitive to the substrate.
© 2010 Optical Society of America
OCIS codes: (160.3918) Metamaterials; (160.1585) Chiral media; (160.4760) Optical properties.
References and links
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5 July 2010 / Vol. 18, No. 14 / OPTICS EXPRESS 14553
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(1990).) use the Drude-Born-Fedorov relations: D = ε (B+ β ∇×E), B = μ (H+ β ∇×H), where β characterizes
the strength of the chirality. They can be brought to the same form. The transformations between the parameters
of the two systems are given in Ref. 26.
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5 July 2010 / Vol. 18, No. 14 / OPTICS EXPRESS 14554
38. E. U. Condon,“Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432-457 (1937).
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1.
Introduction
Chiral metamaterials (CMMs), which are artificial materials that lack any planes of mirror symmetry, possess strong ability to rotate the plane of polarization of electromagnetic waves. This
ability is called optical activity which is of great interest to many areas of science, for example, analytical chemistry and molecular biology. Recently, it was predicted that strong enough
optical activity may result in negative refraction [1–6] and negative reflection [7]. After these
predictions [1–7], a lot of artificial CMMs were designed and fabricated at different frequency
ranges from microwaves to optical frequencies [8–22]. The basic physical quantities to characterize the optical properties of CMMs are the effective constitutive parameters. Therefore, a
simple and robust method to retrieve the effective parameters for CMMs is urgently needed.
Parameter retrieval [23–28] is a basic and important technique to obtain the electromagnetic properties of the effective media. The effective media are usually considered to be homogeneous when the size of structure is much smaller than the wavelength λ . The retrieval
parameters, ε and μ or n and Z, are well defined and can be determined from reflection and
transmission coefficients (S parameters) [23–28]. We follow the similar procedure [23] to do
the parameter retrieval for CMMs as it was done in ordinary metamaterials. The procedure of
the retrieval for free-standing CMM slabs was briefly shown in our previous paper [29]. However, in most of the experiments [17–22], the CMMs are fabricated on a substrate. Therefore, a
retrieval procedure for samples with substrates should be developed. This is an important development for obtaining the electromagnetic properties of CMMs from experimental data. Kwon
et al. [30] presented a retrieval procedure for CMMs with a semi-infinite substrate. However,
experimentally the thickness of the substrate is finite. Here, we present a simple and robust
retrieval method based on the transfer matrix technique. It can deal with samples not only with
substrates but also with top layers.
After the parameter retrieval, we give two chiral metamaterial examples: (1) twisted-crosses
and (2) four-folded rotated Ω-particles. For the twisted-crosses CMMs, the resonance elements
are very close to the substrate. The existence of the substrate, therefore, has a huge influence on
the thin CMM slab. It causes the resonance shifts to low frequencies and induces the homogeneous slab to be inhomogeneous. In order to correct the inhomogeneity, we need to introduce a
top layer, the same material as the substrate, to cover the CMM slab. The top layer will red-shift
the resonance further. For the four-folded rotated Ω-particles CMMs, the resonance elements
are embedded far away from the substrate. The influence of the substrate can be neglected. In
the meanwhile, profiting from the retrieval method and the precise simulation, we fit the retrieval results using the analytical formulas deriving from the Ω-particle resonator model. They
agree with each other very well.
The following sections are arranged as follows. In Sec. 2, we first obtain explicit analytic expressions of the retrieval parameters for free standing CMM slabs, then we extend this method
to deal with the samples with the substrate and top layers. In Sec. 3, we use the parameter retrieval methods to study two CMM examples and obtain the influence of substrate/top layer on
thin layer CMM slabs. Finally in Sec. 4, we present our conclusions.
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2.
2.1.
Retrieval method
Eigen waves in chiral materials
For reciprocal (Pasteur) chiral materials [31, 32], the strength of the cross coupling between the
magnetic and electric field is characterized by the parameter κ called chirality. The constitutive
relations in chiral medium are usually written as [33]:
E
D
ε0 ε
iκ /c0
,
(1)
=
H
B
−iκ /c0 μ0 μ
where ε0 and μ0 are the permittivity and permeability of vacuum, ε and μ are the relative
permittivity and permeability of the medium respectively. c0 is the speed of light in vacuum.
Assuming all fields to be plane waves, the source free Maxwell equations take the form:
k × H = −ω D, k × E = ω B,
(2)
where k and ω are the wavevector and the angular frequency of the plane wave in the chiral
medium. Inserting Eq. (1) into Eq. (2), we can get the eigenfunction of the electric field E
propagating in the chiral medium:
k × (k × E) = −k02 (ε μ − κ 2 )E − 2iκ k0 (k × E),
(3)
where k0 = ω /c0 is the free space wavevector. For simplicity but without loss of generality we
can assume k = kẑ and obtain the eigenvectors and eigenvalues as:
E± (z) = (x̂ ± iŷ)E0 eikz ,
(4)
k± = k0 (n ± κ ),
(5)
√
where n = ε μ . There are two eigenmodes in chiral medium. One is the right circular polarization (RCP/+) wave and the other is the left circular polarization (LCP/-) wave. Any wave
propagating though the medium can be uniquely decomposed into RCP and LCP and there is
no conversion between RCP and LCP as they propagate in the chiral medium. Then we can
define the index of refraction for RCP/LCP waves as:
n± = n ± κ .
(6)
If κ > n, n− will be negative. This constitutes an alternative approach to realize a negative
refractive index as proposed earlier by Pendry [1].
A chiral medium has two important properties. One is called optical activity which characterizes the rotation of the polarization plane of a linearly polarized light as it passes through a
chiral medium. Mathematically, it’s defined as the polarization azimuth rotation angle of elliptically polarized light:
1
θ = [arg(T+ ) − arg(T− )],
(7)
2
where T+ and T− are the transmission coefficients for RCP and LCP waves. The other property is circular dichromism which arises from the different absorption for RCP and LCP. It
characterizes the difference between the transmissions of two polarizations:
η=
1 −1 |T+ |2 − |T− |2
tan (
).
2
|T+ |2 + |T− |2
(8)
Artificial CMMs with large θ and small η are very important for applications.
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1
T ±
T±
Air
CMM
Air
n0, Z0
n±, Z
n0, Z0
R∓
R∓
z=0
z
z=d
Fig. 1. Schematics of the transmission and reflection coefficient of a standalone CMM slab.
2.2.
Parameter retrieval for CMM slabs without substrate
A slab of reciprocal (Pasteur) chiral material looks exactly identical for waves propagating
in forward or backward direction. However, for a normal incident circular polarized wave
E± (z) = (x̂ ± iŷ)E0 eikz along the z direction the reflected wave will be E± (z) = (x̂ ± iŷ)E0 e−ikz
along the −z direction. The y component is always 90◦ advanced(+) / retarded(-) with respect
to the x component when the wave vector changes to the opposite direction. Therefore, the
polarization of the reflected wave is reversed based on our current definition of handiness in
relation to the wave vector, i.e. the reflected wave of RCP (LCP) wave is LCP (RCP). Now we
consider a circularly polarized plane wave normally
incidents onto a CMM slab with thickness
d, refractive index n± , and impedance Z = μ /ε , as shown in Fig. 1 (note that the impedance,
Z’s, of RCP and LCP waves are identical [31]). The amplitude of the incident wave is assumed
to be unity; the amplitudes of the transmitted and reflected waves are T± and R∓ respectively;
those of the forward and backward propagating waves inside the slab are T± and R∓ . Applying
the condition of continuity of tangential electric and magnetic fields at z = 0 and z = d, we have
the following equations:
1 + R∓ = T± + R∓ ,
(9a)
T − R∓
,
1 − R∓ = ±
Z
T± eik± d + R∓ e−ik∓ d = T± ,
T± eik± d
− R∓ e−ik∓ d
Z
= T± ,
(9b)
(9c)
(9d)
where k± is the wave vector as defined by Eq. (5). Then eliminating T and R in Eqs. (9), we
obtain the transmission and reflection coefficients:
T± =
4Zeink0 d e±iκ k0 d
,
(1 + Z)2 − (1 − Z)2 e2ink0 d
(10a)
R∓ =
(1 − Z 2 )(e2ink0 d − 1)
.
(1 + Z)2 − (1 − Z)2 e2ink0 d
(10b)
Equation (10b) shows that R+ = R− . This can be understood intuitively as follows: We know
that the polarizations of RCP and LCP are reversed after being reflected. For the forward incident wave, the optical path of RCP (LCP) is n+ d (n− d), where d is the distance that the wave
has traveled; For the backward reflected wave, the optical path is n− d (n+ d). Therefore, the total optical paths for RCP and LCP are all n+ d + n− d. They pass through the same optical path
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for the reflected waves, so the loss should be the same for both of them. In addition, the same
impedance of the RCP and the LCP gives the same reflections for both circular polarizations
on the surface. Therefore, both the reflections of the single slab should be the same.
Inverting the above Eqs. (10), the impedance Z and refractive index n± are given below:
(1 + R)2 − T+ T−
,
(11a)
Z=±
(1 − R)2 − T+ T−
n± =
1
i
Z −1
{ln[ (1 −
R)] ± 2mπ },
k0 d
T±
Z +1
(11b)
where m is an integer determined by the branches. The correct sign of the square root in
Eq. (11a) and the correct branch of the logarithm in Eq. (11b) are selected according to the
following conditions:
Re(Z) ≥ 0, Im(n) ≥ 0,
(12)
which are required by energy conservation and causality [23]. Then the remaining retrieval parameters can be obtained with the following relations: n = (n+ + n− )/2, κ = (n+ − n− )/2, ε =
n/Z, and μ = nZ.
√
Actually, we have another route to get the retrieval results. If we redefine T = T+ T− and
R = R∓ , according to Eqs. (10), T and R can be written as:
T=
4Zeink0 d
,
(1 + Z)2 − (1 − Z)2 e2ink0 d
(13a)
R=
(1 − Z 2 )(e2ink0 d − 1)
.
(1 + Z)2 − (1 − Z)2 e2ink0 d
(13b)
These expressions are exactly the same as those for the ordinary nonchiral materials. Therefore, we can use the traditional retrieval procedure for the nonchiral metamaterials to get the
refraction index n, the impedance Z, the permeability ε and the permittivity μ [23]. Then from
Eqs. (10a), we can obtain
κ=
T+
|T+ |eiφ+
−i
−i
ln( ) =
ln(
),
2k0 d
T−
2k0 d
|T− |eiφ−
(14)
where φ± are the phases of T± . Then we immediately get the refraction indexes n± = n ± κ .
From Eqs. (14), we have
φ+ − φ− + 2mπ
,
2k0 d
ln |T− | − ln |T+ |
,
Im(κ ) =
2k0 d
Re(κ ) =
(15a)
(15b)
where the integer m is determined by the condition of −π < φ+ − φ− + 2mπ < π for one unit
cell. We note that the real and imaginary parts of κ relate to the azimuth rotation angle θ and
the circular dichromism η respectively.
2.3.
Parameter retrieval for CMM slabs with substrate
The retrieval procedure discussed above identifies the scattering amplitudes of the CMM slab
with those of an equivalent homogeneous slab. The retrieved effective parameters of the CMM
are the parameters of this equivalent homogeneous slab and given as functions of the scattering
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matrix of the CMM. In this form, the retrieval procedure is only applicable if the CMM has the
same symmetry as the equivalent homogeneous slab. For instance, the reflection amplitudes for
a homogeneous chiral (Pasteur) slab are identical for incidence from the front or back interfaces.
For a CMM on top of a substrate (for an example see Fig. 2) the scattering amplitudes do
not necessarily have this property. In this case, we cannot directly apply the above retrieval
procedure to the measured or simulated scattering amplitudes of the sample to obtain effective
parameters.
In the experiments and simulations (for instance, the commercial software CST Microwave
Studio [34]), we can easily obtain the total scattering matrix Stotal of the CMM/substrate slab
which relates the incoming waves to the outgoing (scattered) waves. It is well known [35] that
any scattering matrix S corresponds to a transfer matrix M which relates the amplitudes of the
in- and outgoing waves in front of the slab to those behind the slab:
t − rr /t r /t r t
;
M
=
,
(16)
S=
t r
−r/t 1/t where t and r are the transmission and reflection amplitudes; the quantities without (with) prime
() refer to wave propagation in forward (backward) direction.
Now we consider a structure of substrate/CMM/top layer as an example and assume the positive (forward) direction to be from the top layer to substrate layer. Measurement or simulation
afford the total scattering matrix Stotal of the stack in terms of total transmission and reflection amplitudes. Using Eq. (16) we obtain the corresponding total transfer matrix Mtotal , which
factorizes into a product of the transfer matrices of the individual layers [35]:
Mtotal = Msubstrate MCMM Mtoplayer .
(17)
The homogeneous substrate and top layers are assumed to be known. Their transfer matrices,
Msubstrate and Mtoplayer , respectively, are just the well known transfer matix of a homogeneous
slab and can be calculated analytically (as the product of the transfer matix across the interface
from vacuum into the homogenous slab, the diagonal transfer matrix (phase & attennuation) for
the propagation within the homogeneous slab, and the inverse of the interface transfer matrix
again to step out of the slab back into vacuum):
1+Z 1−Z Z+1 Z−1 ink d
e 0
0
2Z
2Z
2
2
,
(18)
M = Z−1 Z+1
1−Z
1+Z
0
e−ink0 d
2Z
2Z
2
2
where the impedance of Z, refractive index of n, and the thickness of d are all known parameters
of the substrate or the top layer. Note that the transfer matrix of each layer is that of the slab
in free space, i.e. the interface of two media can be considered as two media separated by a
vacuum slab with zero thickness.
Solving the above equation [Eq. (17)], we can get the effective transfer matrix of the CMM
slab standing alone in free space:
−1
MCMM = M−1
substrate Mtotal Mtoplayer .
(19)
Having extracted the effective transfer matrix of the CMM slab alone, we can use the relation
[Eq. (16)] again to obtain the S parameters of the free standing CMM slab. Then the retrieval
procedure given in Sec. 2.2 can be applied to the isolated S parameters of the CMM slab. And
following this way, we can obtain the retrieval parameters of CMMs with substrate and/or the
top layer.
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3.
Examples of the CMM slabs with and without substrate
In this section, we will give two examples to demonstrate that a physically meaningful retrieval
can be obtained from scattering amplitudes measured for the combined system of CMM and
substrate/top layer and to show how the substrate influences the retrieval parameters of the
CMM slab. For this purpose, we choose the twisted-crosses CMMs which had been fabricated
in the optical region by Decker et al. [21] and the four-folded rotated Ω-particle CMMs which
is closest to the standard chiral model.
3.1.
Retrieval results of the twisted-crosses CMM slabs with and without substrate
air/CMM/air
air/top layer/CMM/substrate/air
air/CMM/substrate/air
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 2. (a)-(c) are the Schematics of the twisted-crosses CMM slab. (a) free standing, (b)
with substrate and top layer, and (c) with substrate only. The amplititudes, (d)-(f), and
the phases, (g)-(i), are also shown correspondingly. The prime () denotes that the wave is
incident from the opposite direction.
The structure of the twisted-crosses CMM is shown in Fig. 2. It’s composed of two gold
crosses rotated against each other by an angle of 22.5◦ . Each arm of the crosses is a short
rectangular wire with the dimension of 25 nm × 56 nm × 315 nm. The separation between the
two crosses is 37.5 nm. The gold structure is embedded in a lossless dielectric with the refractive
index of n = 1.41 as shown in Fig. 2(a). The structure is fabricated on the glass substrate
(n=1.45) with the thickness of 500 nm as illustrated in Fig. 2(c); Optionally a thin glass layer
(n=1.45) with the thickness of 200 nm can be placed on top of the embedded gold structure,
ω2
see Fig. 2(b). The gold is modeled by a Drude model permittivity ε (ω ) = ε∞ − ω (ω +ip ωc ) with
plasma frequency ω p = 2π × 2159 THz, collision frequency ωc = 2π × 25 THz, and ε∞ = 9.07.
The lower two rows in Fig. 2 show the circular polarization scattering amplitudes (magnitudes and phases) of the CMM slab obtained from simulation. For the free standing slab (left
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air/CMM/air
air/top layer/CMM/substrate/air
air/CMM/substrate/air
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
(s)
(t)
(u)
Fig. 3. The retrieval parameters ε , μ , κ , n− , n+ , η , and θ for CMM slabs. (Left) freestanding, (Middle) with substrate and top layer, and (Right) with substrate only.These results were extracted using (19).
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column) the simulation directly yields the scattering amplitudes of the CMM slab; for the CMM
on substrate (right) and CMM on substrat with top layer (middle column) the shown CMM scattering amplitudes were extracted using Eq. (19) derived above. T−− (R+− ) and T++ (R−+ ) are
the transmission (reflection) of LCP and RCP, respectively. (In order to specify the polarization
conversion, for Tab and Rab , b refers to the incident wave and a to the transmitted/reflected
wave.) As can be seen in Figs. 2, the reflection amplitudes for RCP and LCP are exactly the
same, which confirms the previous analysis.
The scattering amplitudes of the homogeneous chiral slab are the same for propagation
though the slab in forward or backward direction. The simulated scattering amplitudes for the
free standing CMM shown in Figs. 2(a), 2(d), and 2(g) also have this property. We can directly
apply the retrieval procedure described in Sec. 2.2 and obtain the effective medium parameters.
The results are given in Figs. 3.
For the CMM on susbtrate no meaningful retrieval can be obtained by applying the retrieval
procedure described in Sec. 2.2 directly to the total scattering amplitudes; The presence of
the substrate leads to additional scattering not originating from the CMM. It also makes the
reflections in forward and backward direction unequal - an asymmetry which can never be
reproduced by the homogeneous chiral slab. Only after extracting the scattering amplitudes
of the CMM slab itself from the simualtion results using Eq. (19) derived in Sec. 2.3 we can
apply the standard retrieval procedure described in Sec. 2.2 to the extracted CMM scattering
amplitudes and obtain reasonable effective medium parameters for the CMM slab.
Even after extracting the CMM scattering amplitudes we still observe some weakly asymmetric reflections R+− = R+− in forward and backward direction for the CMM on substrate in
Figs. 2(f), 2(i). The extraction derived in Sec. 2.3 does only deal with the propagating modes
in the system. However, there is also interaction between the CMM slabs surface regions with
their environment mediated by evanescent modes. This interaction makes the surfaces of the
CMM slab on substrate unequal and prevents its approximation by an equivalent homogenenous effective medium slab.
There are two possible solutions to this problem: First, we could follow the method introduced in Ref. 22 and approximate an isotropic reflection Rav as the geometric average,
(20)
Rav = RR ,
where, R (R ) is the reflection when the incident wave is from the top/vacuum (bottom/substrate)
side of the CMM slab. This method has been used successfully if the anisotropy was small
enough [27]. (Note that the inhomogenization is not inherent property of the twisted-crosses
CMM slab. It’s only the boundary effect of the thin layer sample. If a sufficiently transparent
sample goes to bulk material with infinity thickness, this boundary effect becomes negligible.)
Using this method we obtain the retrieval of the single sided CMM on sustrate shown in the
right column in Fig. 3.
Second, we could cover the top side of the CMM with a second layer of substrate. This layer
does not need to be very thick as the interaction via the evanescent modes with the surface layers
of the CMM decays quickly. As can be seen in Figs. 2(e), 2(h) this approach lead to perfectly
symmetric reflection amplitudes and allows to apply the standard retrieval procedure described
in Sec. 2.2 to the extracted CMM scattering amplitudes to obtain the effective parametes of the
CMM slab. The results for this case are show in the center column in Fig. 3.
Comparing the results with and without substrate, we can see that the substrate has two
main influences on the CMMs. One is lowering the frequency√of the resonance because the
substrate increases the capacity C of the resonator and ωR ∼ 1/ LC. The other is the substrate
may induce the homogeneous CMM to be inhomogeneous. For the structure with substrate
only, the two crosses are in the different environments. For the cross near the substrate, the
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evanescent field leak to the substrate while for the cross near the air, the field leaks to the
air, which makes the two substantially identical crosses unequal physically. Even the substrate
contribution has been deducted off mathematically by using Eq. (19), the CMM slab still shows
the inhomogeneous property (e.q. R+− = R+− ).
3.2.
Retrieval results of the four-folded rotated Ω-particle CMM slabs with and without substrate
Fig. 4. Schmetics of the four-folded rotated Ω-particle CMM. The inner and outer radius
of the loop are r = 5 μ m and R = 5.5 μ m. The length of each straight wire is L = 8.08 μ m.
d = 0.5 μ m. a = 32 μ m. b = 12 μ m. Each Ω-particle locates at the center of each quarter
of the unit cell. The silver Ω-particle is embeded in the polyimide with ε = 2.5 and loss
tangent δ = 0.03. The silver is charactored by the Drude model with the surface plasmon
ω p = 13.66 × 1015 rad/s and the collision frequency ωc = 2.73 × 1013 rad/s.
Not all the chiral metamaterials are so sensitive to the influence of the substrate. In the following, we will give such an example—four-folded rotated Ω-particle chiral metamaterials. The
structure and the dimensional parameters are shown in Fig. 4. The incident wave is perpendicular to the CMM slab.
The amplitude and phase of the reflection and transmission are shown in Figs. 5(a)–5(d).
The left is with 50 μ m substrate (n=1.27) only and the right is for the free standing sample.
There is a strong resonance at around 1.87 THz where both the amplitude and phase of the
transmissions of RCP and LCP are different, which indicates very strong optical activity and
circular dichromism. The interesting is that for the structure with substrate only, the extracted
reflections from different directions almost overlap with each other, e.g. R = R , even they have
a large difference before the extraction using Eq. (19). Comparing the results between with
[Fig. 5(a),5(c)] and without [Fig. 5(b), 5(d)] substrate, we see that the substrate almost has no
influence on the Ω-particle CMMs. That is because the resonance elements, the gap of the circle
magnetic resonator and the short wires for the electric resonance, are located in the middle of
the unit cell. They are far away from the unit cell surface, in other words, they are far away
from the substrate. The evanescent field leaked into the substrate is very little. Therefore, this
kind CMM is not sensitive to the existence of the substrate.
Figures 5(e)–5(j) are the retrieval results of the optical parameters ε , μ , and κ . All these
curves behave well. We fit these curves using the analytical formulas given by Eqs. (29) derived
from the Ω-particle resonator model (see Appendix A). For with substrate only, the fitted parameters are as following: εb = 3.1738, μb = 0.9799, ω0 = 1.8651 THz, γ = 0.05519 ω0 , Ωε =
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air/CMM/substrate/air
air/CMM/air
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Fig. 5. (a)-(d) are the simulation results of the amplitude and the phase of the reflection and
transmission for the four-folded rotated Ω-particle CMM slabs. They are extracted using
(19). R is the reflection for the incident wave from the opposite direction. (e)-(j) show the
optical parameters ε , μ , and κ . Re/Im denotes the real/imaginary part of the value. The
subscripts s and f denotes the retrieval results from the numerical simulation and the fitting
results using Eqs. (29) respectively. The left is with substrate (n=1.27) only and the right is
for the free standing sample.
0.1537, Ωμ = 0.0627, and Ωκ = 0.0986. For without substrate, they are very close to those with
substrate. εb = 3.1736, μb = 0.9798, ω0 = 1.8713 THz, γ = 0.05463 ω0 , Ωε = 0.1560, Ωμ =
0.0625, and Ωκ = 0.0993. From these parameters, we can again see that the substrate has very
little influence on the four-folded rotated Ω-particle CMMs. The substrate only lowers the frequency of the resonance by 0.3%.
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In Figs. 5(e)–5(j), we can also see that the retrieval results from numerical simulations agree
very well with the analytical results given by Eqs. (29). It verifies that the ω dependences of
the optical parameters are valid in our previous work about realizing the repulsive Casimir
force [36] using chiral metamaterials. Note that from Eq. (25), the strength of electric polarization and magnetic polarization can be modified independently by the length of the straight
wires and the area of the circular loop respectively. Ωε > Ωμ here is just one case. But no matter
how to change the dimensional parameters, the relation Eq. (30) roughly always holds in such
designs composited by the passive material only.
4.
Conclusion
In summary, we have analyzed the wave propagation in chiral metamaterials and followed the
similar procedure that was done in ordinary metamaterials to do the parameter retrieval for
CMM slabs. Based on the transfer matrix technique, we extended the parameter retrieval technique to be able to treat samples with substrates and extra top layers. We studied the influence
of the substrate and top layers on the thin chiral metamaterial slabs. We found that the substrate
could lower the frequency of the resonance and induce the homogeneous chiral metamaterials to be inhomogeneous. The sensitivity of the influence relates to the strength of the leaked
evanescent field in the substrate. We also fitted the retrieval results using analytical expressions
derived from Ω-particle chiral model based on the effective LC circuit mode. We found they
agree with each other very well.
A.
Derive the forms of the constitutive parameters from the Ω-particle resonator model
Fig. 6. Schematics of the single right-handed (left) and left-handed (right) Ω-particle resonators.
Figure 6 gives the schematics of the Ω-particle resonator. The structure consists of an open
circular loop and two short wires. The area of the loop is S = π r2 and the length of each wire
is l. The wires are perpendicular to the loop and connected to the ends of the open loop. If the
Ω-particle is in a homogeneous external field, the driving electric potential can be written as:
U = 2lE0 ± μ0 SḢ0 ,
(21)
where ± signs correspond to right-handed and left-handed helix resonators, The hat-dot denotes
the first time derivative. Applying the effective RLC circuit model, we have
LI˙ +
q
+ RI = U,
C
I = q̇.
(22)
1
Let α = Ll , β = μ0 AL , γ = RL , and ω02 = LC
. Then assuming the external fields are plane waves,
−i
ω
t
, and considering stationary solutions, we get the following equation:
∼e
(−ω 2 − iωγ + ω02 )q = α E0 ± iωβ H0 ,
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(23)
Received 5 Apr 2010; revised 27 May 2010; accepted 4 Jun 2010; published 23 Jun 2010
5 July 2010 / Vol. 18, No. 14 / OPTICS EXPRESS 14565
then, we have
α
iωβ
E0 ±
H0 .
−ω 2 − iωγ + ω02
−ω 2 − iωγ + ω02
q=
(24)
Therefore, the electric dipole, p = ql, and the magnetic dipole, m = ±IA = ±q̇A, can be written
as:
where, α A =
p
=
l
αl
E + ω 2 −±iωωβ
2 −iωγ H0 ,
ω02 −ω 2 −iωγ 0
0
(25a)
m
=
2 A
∓iωα A
E + ω 2 −ωωβ2 −i
H,
ω02 −ω 2 −iωγ 0
ωγ 0
0
(25b)
1
μ0 β l, l = l l̂0 , and A = Al̂0 ; l̂0
is the unit vector along the direction of the wires, i.e.,
the up direction. The electric polarization and the magnetic polarization are defined as P =
∑p
V0
and M = ∑Vm
respectively, where the summation is done in the unit cell and V0 denotes the vol0
ume of the unit cell. Assuming there are N resonators in one unit cell, the electric polarization
and the magnetic polarization can be written as:
P
=
M
=
±iωβ l
N
αl
N
V0 ω 2 −ω 2 −iωγ E + V0 ω 2 −ω 2 −iωγ H,
0
0
ω2β A
N
∓iωα A
N
V0 ω 2 −ω 2 −iωγ E + V0 ω 2 −ω 2 −iωγ H.
0
0
(26a)
(26b)
Here, the directions of l and A are merged into E0 and H0 to form the vectors of E and H.
Inserting Eqs. (26) to the relation D = ε0 E + P and B = μ0 H + μ0 M, we have
±iωβ lN/V0
α lN/V0
E+ 2
H,
2
2
ω0 − ω − iωγ
ω0 − ω 2 − iωγ
∓iμ0 ωα AN/V0
ω 2 μ0 β AN/V0
μ0 H+ 2
E+
H.
ω0 − ω 2 − iωγ
ω02 − ω 2 − iωγ
D = ε0 E+
(27a)
B =
(27b)
Comparing Eqs. (27) with Eq. (1), we can obtain the relative permittivity ε , permeability μ ,
and the chirality κ :
α lN/V0 ε0
ε = 1+ 2
,
(28a)
ω0 − ω 2 − iωγ
μ = 1+
κ=
ω 2 β AN/V0
,
ω02 − ω 2 − iωγ
±ωβ lc0 N/V0
±ω μ0 c0 α AN/V0
=
.
2
2
ω0 − ω − iωγ
ω02 − ω 2 − iωγ
(28b)
(28c)
Equations (28) are valid for the case that the metal chiral structure stands in free space. If it’s
merged into a background material with εb and μb , Eqs. (28) should be modified as follows:
ε = εb +
Ωε ω02
,
2
ω0 − ω 2 − iωγ
(29a)
μ = μb +
Ωμ ω 2
,
2
ω0 − ω 2 − iωγ
(29b)
κ=
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Ωκ ω0 ω
,
2
ω0 − ω 2 − iωγ
(29c)
Received 5 Apr 2010; revised 27 May 2010; accepted 4 Jun 2010; published 23 Jun 2010
5 July 2010 / Vol. 18, No. 14 / OPTICS EXPRESS 14566
where εb is usually larger than one; μb is usually very close to one. Ωε , Ωμ , and Ωκ are the
coefficients of the resonance terms in ε , μ , and κ , i.e. Ωε = V αεlNω 2 , Ωμ = βVAN
, and Ωκ =
0
0 0 0
β lc0 N
V0 ω0
= μ0Vc00ωα0AN , where c0 is the speed of light in vacuum. They describe the strength of the
resonance. The expressions of the frequency dependence of ε and μ given by Eqs. (29a) and
(29b) are the same as those in traditional metamaterials [37]. The linear ω dependence of κ is
the same as Condon model for the homogeneous chiral molecular media [38]. This is a general
feature of natural optically active materials [39].
Because α = Ll and β = μ0 AL , we then can obtain the following relation:
Ω2κ = Ωε Ωμ ,
(30)
which seems to limit the possibility of obtaining large Ωκ in these designs composited by the
passive materials only.
Acknowledgements
Work at Ames Laboratory was supported by the Department of Energy (Basic Energy Sciences)
under contract No. DE-AC02-07CH11358. This work was partially supported by the European
Community FET project PHOME (contract No. 213390). The author Rongkuo Zhao specially
acknowledges the China Scholarship Council (CSC) for financial support.
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