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 New Journal of Physics The open–access journal for physics

Signal-to-noise ratio evaluation in resonant ring

metamaterial lenses for MRI applications
J M Algarin1 , M A Lopez, M J Freire and R Marques
Department of Electronics and Electromagnetism, Faculty of Physics,
University of Seville, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain
E-mail: jalgarin@us.es

New Journal of Physics 13 (2011) 115006 (12pp)

Received 30 June 2011
Published 8 November 2011
Online at http://www.njp.org/
doi:10.1088/1367-2630/13/11/115006

Abstract. In this paper, we present a method for the evaluation of the signal-
to-noise ratio in magnetic resonance imaging (MRI) coils loaded with resonant
ring metamaterial lenses, in the presence of a conducting phantom resembling
human tissue. The method accounts for the effects of the discrete and finite
structure of the metamaterial. Numerical computations are validated with
experimental results, including laboratory measurements and MRI experiments.

Contents

1. Introduction 1
2. Computation of the signal-to-noise ratio 3
3. Discussion 6
3.1. Validation of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Application of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. Conclusion 10
Acknowledgments 10
Appendix A 10
References 11

1. Introduction

Metamaterials are artificial structures that can be tailored in order to achieve a wide range of
electromagnetic properties that cannot be observed in nature, including simultaneously negative
1
Author to whom any correspondence should be addressed.

New Journal of Physics 13 (2011) 115006

1367-2630/11/115006+12$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
effective permittivity and permeability [1]. It is well known that a metamaterial slab with relative
permittivity εr and relative permeability µr , both equal to −1, behaves as a ‘super-lens’ with a
resolution [2] well beyond the diffraction limit of classical optics or sub-wavelength (sub-λ)
resolution. If the frequency of operation is sufficiently low, we are in the realm of quasi-statics,
and we only need a metamaterial slab with εr = −1 or µr = −1 (depending on the electric or
magnetic nature of the quasi-static field) in order to observe this effect [2]. The same effect can
be observed in two coupled magneto-inductive (MI) surfaces [3–5]. Effective electromagnetic
properties of metamaterials arise from their structure rather than from the nature of their
components, which often are conventional conductors and dielectrics. Many metamaterials
are made of the periodic repetition of some resonant elements with a sub-λ period, so that
some kind of homogenization can be made [6]. Due to the resonant nature of its constitutive
elements, the interesting properties of such metamaterials usually appear in a very narrow band
of frequencies. This narrow bandwidth is commonly considered as a major drawback for many
applications. However, it is not an important limitation for magnetic resonance imaging (MRI),
because MR images are acquired by measuring radio-frequency (RF) signals inside a relatively
narrow bandwidth of a few tens of kilohertz. Besides, due to the low frequency of operation of
most MRI systems (tens of MHz), designing metamaterials with a sub-λ period is an affordable
task. Thus, MRI appears as a natural application for resonant ring metamaterials.
As is well known, the generation of images in MRI is based on the detection of spatial
variations in the phase and frequency of the RF waves absorbed and emitted by the nuclear spins
of the imaged object. MRI detectors are conventional coils operating in the near-field region of
the RF magnetic field. Metamaterials offer the possibility of manipulating this magnetic near-
field in order to improve coil performance. The application of metamaterials in MRI has been
previously explored in several works making use of devices based on different types of resonant
elements, such as swiss-rolls [7–11], wires [12] and capacitively loaded split rings [13–16].
Many of these works have explored the sub-λ imaging ability of metamaterials with µr < 0.
A capacitively loaded split ring is a small open ring of copper that is loaded in the gap with a
chip capacitor. Diamagnetic properties of closed inductive loops were well known in the past
and Schelkunoff reported that this effect can be enhanced by adding a chip capacitor [17] to the
ring. Of course, this capacitor has to be non-magnetic for MRI applications. In previous works,
some of the present authors have analyzed the MRI applications of capacitively loaded split-
ring metamaterial slabs with effective permeability µr = −1. The ability of such metamaterials
to increase the sensitivity of surface coils [13, 14] and to improve the localization of the field of
view (FOV) of these coils [15] (a fact that may find applications in parallel MRI (pMRI)) were
analyzed. pMRI works by taking advantage of the spatially sensitive information inherent in a
receiving array of multiple surface coils in order to partially replace time-consuming spatial
encoding and thus to reduce the image acquisition time [19, 20]. Moreover, we have also
investigated the application of resonant ring slabs with effective permeability µr → 0 to locally
increase the signal-to-noise ratio (SNR) of surface coils [16]. Increasing the SNR of the image
is one of the main goals in conventional MRI, besides the reduction of the acquisition time.
In a previous work [14], some of the present authors presented a method for the evaluation
of the SNR of surface coils in the presence of metamaterial slabs of negative effective
permeability. This method was a first approach to the problem, since metamaterial slabs were
modeled as continuous slabs with an effective permeability extracted from a previously reported
homogenization procedure [18]. Therefore, this model did not take into account the effects of
the discrete and finite structure of the metamaterial. Moreover, other metamaterial structures,

New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)

3
such as the MI lens reported by the authors [3–5], cannot be modeled by this approach. More
recently, a new approach was developed by some of the present authors in order to overcome
these drawbacks [21, 22]. This approach, however, was developed for metamaterials operating
in vacuum. Therefore, it cannot take into account the effect of human tissues, which is essential
for the evaluation of noise. Thus, it would be desirable to include in the analysis the presence
of conducting samples resembling human tissues. Such an analysis is developed in this paper,
which is organized as follows: in the next section, a method for the computation of the SNR is
developed. Next, the experimental validation of the predictions of such a method is presented,
and a comparison with the results from the continuous medium approach [14] is made. Then,
an example of the application of the proposed method to the optimal design of a metamaterial
MRI lens is presented. Finally, some conclusions are presented.

2. Computation of the signal-to-noise ratio

This section shows the details of the method developed for the computation of the SNR of
MRI receiving coils in the presence of realistic split-ring metamaterial slabs and conducting
samples resembling human tissues. For the sake of simplicity, the sample will be modeled as a
conducting half-space, with a conductivity value typical of human tissue. The analysis is divided
into two parts corresponding to the signal and the noise. According to reciprocity theorem, the
signal received by a coil from a magnetic dipole placed at certain point inside a sample is
proportional to the magnetic field per unit current (noted as B1 in the specialized literature)
produced by the coil at that point [23]. On the other hand, the MR noise is proportional to the
square root of the noise resistance R associated with the sample [24]. In our analysis, the coil
is assumed to be lossless, which means that both the coil losses and MRI system losses are
excluded from the analysis. Therefore, the computed noise will be a sort of intrinsic noise [25],
associated only with the presence of human tissues and metamaterial. Since we are interested in
comparison
√ of the SNR given by different configurations and since the SNR is proportional to
B1 / R, in our analysis we will compute and compare this quantity for different configurations.
Figure 1(a) shows a sketch of the configuration under analysis: a coil is placed at a distance
s from the surface of a split-ring slab of thickness d, which is placed on the interface of a
conducting half-space. The goal is to evaluate the SNR at a point located at a distance z inside
the sample.
For the analysis, the configuration shown in figure 1(a) is divided into two subsystems. The
first subsystem, or subsystem A, consists of the coil and the metamaterial slab as if they were
placed in vacuum. The second subsystem, or subsystem B, consists of only the conducting half-
space. In general, the metamaterial slab consists of a number N of rings which are periodically
arranged in a 3D lattice (see figure 1(b), as an example). In subsystem A, the field produced
by the coil at the exit of the metamaterial, as well as the resistance introduced by the lens,
Rlens , is calculated. This computation is made by following the method reported by us in [21],
where the matrix equation for the unknown currents in the rings and coil has been solved. It
may be worth noting that analysis of such a structure, with thousands of elements, cannot be
performed by means of the available commercial electromagnetic solvers based on standard
numerical methods. The matrix equation that has to be solved is Z · I = V , where Z is the
impedance matrix of the system with (N + 1) × (N + 1) elements, which include the N rings
of the metamaterial and the coil, I is the vector of unknown currents and V is the voltage
vector, which is set equal to 1 for the coil and to zero for the rings. The diagonal elements

New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)

4

Figure 1. (a) Sketch of the configuration under analysis: a coil is placed at a

distance s from the surface of a split-ring slab of thickness d, which is placed
at the interface of a conducting half-space. (b) Sketch of the three-dimensional
(3D) lattice of rings that constitute the split-ring slab. (c) Photograph of the
experimental setup.

Z ii = R + jωL + 1/jωC correspond to the self-impedances of the rings and coil, where ω is the
frequency, R the resistance, L the self-inductance and C the capacitance (in the case of the coil,
R and C are taken to be equal to zero). All these parameters can be measured independently,
and are treated as external inputs. The non-diagonal elements Z i j = jωMi j depend on the
mutual inductances Mi j between the rings of the metamaterial and between the coil and
rings of the metamaterial. The mutual inductances between the rings of the metamaterial are
computed using the Neumann formula, which has been tested to provide enough accuracy for
our purposes. For the coil, a model based on two filaments [26] was used to account for wide
strips. The mutual inductances between the coil and rings of the metamaterial are calculated
using this model and the Neumann formula [26]. Symmetries are taken into account in order
to reduce the computation time. Once the matrix system is solved and all the currents are
obtained, the resistance introduced by the lens, Rlens , is given by the real part of the ratio
between the imposed voltage in the coil (1 V) and the computed current in this element I N +1 :
Rlens = Re(1/I N +1 ). Finally, the vector E
P N +1potential A(x, y, 0) at the exit of the metamaterial is
computed as the sum A(x, y, 0) = i=1 Ai (x, y, 0), where each summand is computed using
E E
standard electromagnetic formulae.
In our analysis, the field at the exit of the metamaterial slab in vacuum in subsystem A
is assumed to be the impinging field existing at the input interface of the conducting half-
space in subsystem B. This assumption implies that the mutual inductances between the rings
calculated in vacuum in subsystem A should be the same in the presence of the conducting
half-space. In order to validate this assumption, table 1 shows the mutual inductances obtained
for a pair of rings in axial and coplanar configurations, both in vacuum and in the presence of a
conducting sample. The dimensions of the rings correspond to the structure of the metamaterial
lens previously reported by us [13]. The same structure will be used in the next section for the
validation of the predictions of the method. The calculations of mutual inductances shown in
the table were obtained with a rigorous full-wave electromagnetic analysis using the commercial
electromagnetic solver CST Microwave Studio, which can be used for the analysis of the present
case with only two rings but not for the whole structure of thousands of rings, as has been
mentioned above. In the simulation, the conducting sample (σ = 1.6 S m−1 ) was finite but large

New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)

5

Table 1. Mutual inductances between two rings, each of external radius 6.02 mm
and width 2.17 mm and with their centers separated by 15 mm, in vacuum (Mv )
and in the presence of a conducting sample (Mc ) for both coplanar and coaxial
configurations. The conducting sample (σ = 1.6 S m−1 ) was placed at 1.5 mm
distance from the rings.
Case Mv (nH) Mc (nH)
Coplanar −0.170 −0.170 − j7.49 × 10−3
Coaxial 0.216 0.216 − j4.53 × 10−3

enough to avoid edge effects. The results in the table show that the real parts of the mutual
inductances obtained in vacuum and in the presence of the conducting sample are quite similar
and that the small imaginary part that appears in the presence of the conducting sample can be
neglected. Therefore, the main assumption of the present method is valid.
After the analysis in subsystem B, next the vector potential obtained at the exit of the
metamaterial in vacuum is decomposed into its spatial Fourier harmonics e E , k , 0) by
A(k x y
means of a fast-Fourier transform (FFT). Next, each harmonic is multiplied by a transmission
coefficient, T (k x , k y ), which accounts for the boundary conditions at the interface between
vacuum and the conducting half-space, so that the transmitted harmonics that propagate inside
the conducting half-space are given by
E , k , z) = T (k , k )e
E
y A(k x , k y , 0) e ,
− jk z z
A(k
e x y x (1)
q
where k z = ω2 µ0 ε − k x2 − k 2y and ε is a complex quantity accounting for the permittivity and
conductivity of the half-space, which resembles human tissue (Re(ε) ' 90ε0 , Im(ε) ' −σ/ω).
Once the vector potential is known inside the conducting half-space, the Fourier transform
of the magnetic field B ez (k x , k y , z) is calculated from the transverse components of the vector
potential as
ez (k x , k y , z) = jk y f
B A x (k x , k y , z) − jk x f
A y (k x , k y , z). (2)
Finally, the magnetic field BEz (x, y, z) is obtained by means of an inverse FFT, which provides
the signal for the calculation of the SNR.
The conducting half-space introduces in the coil an additional series resistance, Rcond , that,
from the point of view of the reciprocity theorem, is due to the power dissipated by the eddy
currents in the conducting half-space:
σ σ ω2
Z Z
Rcond = E 2
| E(x, y, z)| dτ = E
| A(x, y, z)|2 dτ. (3)
|I N +1 | 2 |I N +1 |2

Taking into account the Parseval theorem, the above expression remains as follows:
σ ω2
Z +∞ Z +∞ Z +∞
Rcond = dk x dk y dz|e E , k , z)|2 .
A(k (4)
x y
|I N +1 |2 −∞ −∞ 0

E , k , z) provides both the signal (2) and the additional noise (4) introduced
Therefore, e
A(k x y
by the conducting sample. With this procedure, both the magnetic field Bz (x, y, z) inside the
conducting half-space and the total resistance in the coil Rlens + Rcond can be calculated to finally

New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)

6

Figure 2. Axial field along the z-direction inside the conducting half-space.
The field is produced by a circular coil of 3 inches diameter placed at 15 mm
distance from a split-ring slab of thickness 3 cm and 18 × 18 × 2 unit cells with
periodicity 15 mm. The slab is separated by 4 mm from the input interface of
the conducting slab with conductivity σ = 1.6 S m−1 . Red: the discrete method;
green: the continuous approach; blue: CST simulation.

obtain the SNR as

|Bz (x, y, z)/I N +1 |
SNR(x, y, z) = √ . (5)
Rlens + Rcond
All the integrations in the method were numerically obtained by using Romberg’s method.

3. Discussion

3.1. Validation of the method

In the above section, we have shown a method for the computation of the SNR that takes into
account the effects of the discrete and finite structure of realistic resonant ring metamaterials.
In the present section, the predictions of this method, which will be termed the discrete method,
are compared with the predictions of the continuous medium approach reported by us in [14]
and with the experimental results for a real structure. The real configuration under analysis is
similar to that sketched in figure 1(a), and a photograph is shown in figure 1(c). It consists of
a circular coil of 3 inches diameter and strip width 1 cm that is placed at 15 mm distance from
the metamaterial lens reported by some of the present authors in [13], with this lens placed
at 4 mm distance from the input interface of a conducting sample. The sample consists of a
14 × 16 × 16 cm3 phantom filled with a water saline solution with conductivity σ = 1.6 S m−1 .
The lens consists of a 3D array of 27 × 27 × 3 cm3 containing 18 × 18 × 2 cubic cells with a
periodicity of 15 mm and a total number of capacitively loaded split rings of 2196 [13]. Each
ring contains a non-magnetic capacitor of the series ATC100B specially designed by American
Technical Ceramics Corp. (New York, USA) for MRI applications and manufactured with low
tolerance for our application. The rings were photoetched on an FR4 substrate. The red line
in figure 2 shows the calculation by the discrete method of the axial field Bz (0, 0, z) inside the

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7

Figure 3. Square root of the input resistance in the coil of the configuration
analyzed in figure 2. Red: the discrete method; green: the continuous approach;
blue: the CST simulation; black: the measurements with the real split-ring
slab and a 14 × 16 × 16 cm3 phantom filled with a water saline solution with
conductivity σ = 1.6 S m−1 .

conducting half-space (σ = 1.6 S m−1 ) which simulates the phantom. The computation has been
carried out at a frequency of 63.6 MHz, which corresponds to the Larmor frequency of the MRI
scanner used for the experiments. The green line in figure 2 shows the calculation given by the
continuous medium approach [14] by modeling the lens as an infinite continuous slab with the
same thickness (3 cm) along the z-direction as the real structure and the phantom as a conducting
half-space. In this approach, the permeability of this slab is modeled following equation (13)
in [18]. Finally, the blue line in figure 2 shows the field given by the electromagnetic solver
CST Microwave Studio when both the lens and the phantom are modelled with the same
parameters as those in the continuous medium approach but with finite dimensions of the real
structure.
The comparison of the green and blue curves in figure 2 shows that the continuous medium
approach provides the same results on the field for both the infinite and finite cases. The
results of the discrete method (red line) show a strong discrepancy with the results of the
continuous approach (green and blue lines) for distances of the order of unit cell length. The
same conclusion was previously reported in [21] for a similar configuration but in vacuum,
that is, in the absence of conducting samples. As was explained in [21], at distances smaller
than about one lattice constant, the near field of the individual rings dominates, so that at these
distances the total field is remarkably different from the field given by the continuous model.
This is a consequence of the discrete structure, which cannot be revealed by a homogenized
model. Once the magnetic field is computed, the second part of the analysis is calculation of the
resistance seen by the coil or the input resistance in the coil. Figure 3 shows the squared root of
the input resistance seen by the coil computed with the discrete model (red line), the continuous
approach (green line) and CST software (blue line) and finally the measurements (black line)
obtained with an Agilent PNA series E8363B Automatic Vector Network Analyzer.

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8

Figure 4. Red: the discrete method; green: the continuous approach; blue: the
CST simulation; black: the measurements of the S21 coefficient between the coil
matched to 50  and a small probe.

Figure 2 shows good agreement between the results given by the discrete method and the
measurements in a wide range of frequencies. Next, figure 4 shows the results on the SNR at a
frequency of 63.6 MHz. These results correspond to the ratio between the field values shown in
figure 2 and the square root of the input resistance shown in figure 3. All the results in figure 4
are normalized to the maximum value obtained with the discrete method (red line) and are
shown in arbitrary units. The curve corresponding to the continuous approach for finite size
(blue line) is above the curve corresponding to the infinite case (green line). This is due to the
fact that the power dissipated by the eddy currents, and therefore the associated input resistance,
will always be higher in a semi-infinite sample than in a sample of finite size. The figure also
shows the results of the measurement of the SNR (crosses). These measurements correspond to
the transmission coefficient (S21 parameter in general transmission line theory [28]) measured
between the coil matched to 50  (the impedance of the feed) and a small probe placed inside
the phantom at different distances z, using the same network analyzer mentioned above. Under
these conditions, the S21 parameter is proportional to the SNR, as is easily demonstrated in an
appendix shown at the end of the paper. This type of measurement is an usual way to characterize
the performance of MRI coils in the laboratory previously to the test in the MRI systems. The
proportionality constant is obtained by previously fitting both the measured S21 and the SNR
calculated in the absence of the lens. The coil was matched to 50  by means of a simple
matching network consisting of a parallel capacitor and a series capacitor. The figure shows that
there is good agreement between the measurements (crosses) and the numerical results given by
the discrete method (red line) and that the results provided by the continuous approach disagree
both in magnitude and frequency with the measurements.

3.2. Application of the method

Once the method of analysis has been validated, we proceed to use this method for the
optimization of an arrangement of split-ring that can provide a good SNR and at the same time
New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)
9

Figure 5. (a) Photograph of a square coil of 12 cm length with an MI lens of

9 cm length placed over it. MR images of a 35 × 30 × 10 × cm3 agar phantom
(σ = 0.5 S m−1 ) obtained with (b) the coil and (c) the coil and MI lens.

enhancement of the localization of the FOV of a coil for application in pMRI. The main problem
associated with the use of split-ring slabs for pMRI applications is the high noise introduced by
the slabs [15]. The result of this optimization process is an arrangement that consists of two
parallel arrays of split rings, a structure that was previously studied by us and termed as MI
lens [3–5]. The configuration under analysis consists of a squared coil of 12 cm length and 1
cm strip width, and a pair of 2D arrays of 9 × 9 cm2 , which are parallel to the coil, with 6 × 6
split rings similar to that of the lens in [13]. The arrays are separated by 11 mm between them
and the closest array to the coil is placed at 6 mm distance from it by means of foam layers
(see the photograph in figure 5(a)). The coil was actively decoupled by a tuned trap circuit
including a PIN diode in transmission. The active decoupling for the loop was −25 dB with and
without the metamaterial slab. An MR experiment was carried out using this configuration with
a 35 × 30 × 10 × cm3 agar phantom with σ = 0.5 S m−1 . The MR experiment was carried out in
a 1.5 T Simphony MR system by Siemens (Siemens Medical Solutions, Erlangen, Germany) at
the Virgen Macarena’s University Hospital (Seville, Spain). Phantom images with and without
the MI lens were acquired using a FLASH sequence (TR/TE: 500/10 ms; FOV 22 × 22 cm3 ;
acquisition matrix: 128 × 128; flip angle: 60◦ ). Figure 5(b) shows an MR image obtained for the
agar phantom with the coil. Figure 5(c) shows the image obtained when the MI lens was placed
between the coil and the phantom. The images in figure 5 show that the FOV is much better
localized laterally with the MI lens.
Next, figure 6 shows the profiles of the SNR measured (black lines) in the phantom along
the axis of the coil (z-axis in the sketches shown in figures 5(b) and (c)) in the presence (black
solid line) and in the absence (black dashed line) of the MI lens, obtained from a series of
phantom measurements [27]. This figure also shows the computation of the SNR for both
situations by using the discrete method (red lines) and the measurements of the S21 (crosses)
in the phantom. The results show that there is good agreement between the simulation and the
measurements. The curves show that the SNR provided by the coil in combination with MI lens
is the same, for long distances, as that provided by the coil in the absence of MI lens, and that it
is even higher for shorter distances when using the lens. This proves that the MI lens can help
avoid the main problem associated with the use of metamaterial lenses with surface coils, that
is, the high noise introduced by metamaterial slabs [15].

New Journal of Physics 13 (2011) 115006 (http://www.njp.org/)

10

Figure 6. SNR along the z-direction for the configuration of figure 5(b) without
the MI lens (dashed lines) and for the configuration of figure 5(c) with the MI
lens (solid lines). Red lines: computations provided by the discrete method; black
lines: measurements from the MR images; crosses: measurements of the S21 .

4. Conclusion

A method has been developed for the computation of the SNR of MRI surface coils in the
presence of resonant ring metamaterial slabs and a conducting half-space resembling human
tissue. This method accounts for the effects of the discrete and finite structure of realistic
metamaterials. Numerical computations provided by this method have been compared with the
results provided by a continuous approach and with experimental results, thus making apparent
the agreement between the experiment and the predictions of the present method. This method
has also been used to optimize a resonant ring metamaterial structure in combination with a
surface coil in order to provide good SNR and pMRI capabilities. It has been found that the
best option consists of an MI lens slightly smaller than the area of the coil. This configuration
has been simulated and the theoretical predictions have been found to agree quite well with the
results of an MR experiment.

Acknowledgments

This work was supported by the Spanish Ministerio de Ciencia e Innovacion and European
Union FEDER funds under the projects Consolider-EMET CSD2008-00066 and TEC2010-
16948 (SEACAM) and by the Spanish Junta de Andalucia under the project TIC-06238
(METAMED). We also thank Dr Carlos Caparros, radiologist from Virgen Macarena’s
University Hospital in Seville, for his advice.

Appendix A

In this appendix, it is shown how the transmission coefficient, or the S21 parameter, between
a coil matched to 50  and a small loop probe is proportional to the SNR provided by the

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11
coil. The S21 parameter is defined, in general, in the transmission line theory [28] for two-port
networks as
V − 
S21 = 2+  + . (A.1)
V1 V2 =0
S21 is found by driving port 1 with an incident wave of voltage V1+ and measuring the reflected
wave amplitude, V2− , coming out of port 2; that is, S21 is the transmission coefficient from port
1 to port 2 when port 2 is terminated in a matched load (Z 0 = 50 ). The power associated with
the wave of voltage V1+ is P = (V1+ )2 /(2Z 0 ). Since, in our problem, the coil is also matched
to 50  (by means of a simple matching network consisting of a parallel capacitor and a
series capacitor), all this power is dissipated in the resistance R which loads the coil; that is,
P = R I 2 /2, where I is the amplitude of the current flowing in the coil and R comes from the
losses of the metamaterial and phantom in our experiment. Therefore, V1+ can be written as
p p p p
V1+ = 2Z 0 P = 2Z 0 R I 2 /2 = I Z 0 R. (A.2)
On the other hand, the voltage V2− in port 2 is the voltage induced by Faraday’s law in the small
loop probe, and this voltage can be written as
V2− = − jωBz S, (A.3)
where S is the area of the loop. Substituting the above expressions for V1+ and into theV2−
definition of the S21 parameter, we finally obtain
jωS Bz /I jωS
S21 = V2− /V1+ = − √ √ = − √ SNR, (A.4)
Z0 R Z0
which demonstrates that under the conditions of our experiment (i.e. a coil matched to 50  and
a loop probe of small area) S21 is proportional to the SNR.

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