Nonlinear properties of split-ring resonators

Bingnan Wang1, Jiangfeng Zhou1, Thomas Koschny1,2 and

Costas M. Soukoulis1,2,∗

1 Ames Laboratory and Department of Physics and Astronomy, Iowa State
University, Ames, Iowa 50011, USA

2 Institute of Electronic Structure and Laser, FORTH, and Department of
Materials Science and Technology, University of Crete, 71110 Heraklion, Crete,
Greece

<EMAIL_ADDRESS>
###### Abstract

In this letter, the properties of split-ring resonators (SRRs) loaded with
high-Q capacitors and nonlinear varactors are theoretically analyzed and
experimentally measured. We demonstrate that the resonance frequency $f_{m}$
of the nonlinear SRRs can be tuned by increasing the incident power. $f_{m}$
moves to lower and higher frequencies for the SRR loaded with one varactor and
two back-to-back varactors, respectively. For high incident powers, we observe
bistable tunable metamaterials and hysteresis effects. Moreover, the coupling
between two nonlinear SRRs is also discussed.

OCIS codes: (350.4010) Microwaves; (999.9999) Metamaterials.

## References and links

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##  1 Introduction

Electromagnetic Metamaterials are periodically arranged artificial structures
that show peculiar behaviors such as negative refraction not seen in natural
materials[1, 2]. The photonic atoms, or the element structures of the
metamaterials, are typically much smaller in size than the working wavelengths
such that the metamaterials can be considered to be homogeneous and
macroscopic parameters such as electrical permittivity and magnetic
permeability can be used to describe the electromagnetic properties of the
metamaterials [3]. By carefully engineering the photonic atoms, both the
permittivity and permeability can be made negative such that negative
refraction can be achieved from the metamaterial [4, 5]. The most widely used
structures for this purpose is the composite of short metallic wires and
split-ring resonators (SRRs) [6, 7]. While the array of short wires gives a
negative permittivity in the wide frequency range below the effective plasma
frequency, the array of SRRs gives a negative permeability in the narrow
frequency range just above the resonance frequency so that the effective index
of refraction can be negative in a narrow frequency band. While most of the
research in this area is in a linear regime, where the electromagnetic
responses are independent of the external fields, some effort has been made to
study the nonlinear effects of the metamaterials, especially the nonlinear
tunability of the SRRs [8-12]. The SRRs are essentially LC resonators and the
resonance frequency is determined by the geometry of the rings. To tune the
magnetic responses of the SRRs, extra components or materials need to be
introduced into the SRRs.

In Ref. [13], ferroelectric films are added to the substrate of the SRRs and
the magnetic tunability is achieved by controlling the electric permittivity
of the ferroelectric films with the change of temperature. In Ref. [14], low-
doped semiconductors are photodoped within the slits of the SRRs and the
magnetic response is tuned by varying the conductivity of the semiconductors
with an external light source. In Ref. [15], ferrite rods are introduced to
ambient the SRR unit cells and the magnetic resonance is modulated by
magnetically tuning the inductance of the ferrite rods by an external magnetic
field. Compared to these methods, the use of varactors is more feasible in
microwave applications in that the tunability can either be realized by a
small DC bias voltage [9, 11, 16] or self-tuned by the intensity of the
applied electromagnetic fields without biasing [10, 17]. Tunable
metamaterials, based on the nonlinear SRRs with varactors, have been tested
experimentally in both transmission line form [11, 12] and bulk form [16].

In this letter, experiments are completed to analyze the properties of the
SRRs loaded with linear and nonlinear elements. The multivalue effect [10, 12]
of nonlinear SRRs with one varactor is analyzed. The nonlinear properties of
SRRs with one varactor and two varactors are compared. The coupling effect of
two nonlinear SRRs are also discussed.

##  2 Nonlinear properties of a single nonlinear SRR

The geometry of the SRRs is a single ring with the outer diameter of $7$ mm,
the inner diameter of $6$ mm, a slit width of $0.7$ mm, and fabricated on a PC
board substrate of $0.8$ mm thick. The resonance frequency of the SRR is
around 4 GHz. When a high Q capacitor with capacitance 2 pF is mounted into
the slit of the SRR, the resonance frequency is brought down to 0.9 GHz. This
makes the ratio between the wavelength and the SRR size as large as 50, which
is useful to miniaturize the size of potential microwave devices [18].

To study the nonlinear magnetic response of the SRR, a varactor is mounted
onto the slit of the SRR. The reflection is measured with a loop antenna and a
vector network analyzer. The small loop antenna is made by a semi-rigid
coaxial cable and the diameter of the loop is 8 mm. The antenna is kept 2 mm
away and on top of the SRR (see the inset of Fig. 1(a)). The measured
reflection at different input power levels is shown in Fig. 1(a). As the
incident power from the vector network analyzer increases, the resonance
frequency of the nonlinear SRRs moves to lower frequencies. We will able to
tune the resonance frequency of the SRR by $10\%$ by increasing the incident
power by $6$ dB. If the incident power increases more, one can see clearly
from Fig. 1(b) that the reflection coefficient starts showing jumps. In
addition, we have observed strong hysteresis and bistable behavior in the
nonlinear SRRs. (As one can see from Fig. 1(a), as the incident power
increases the resonance frequency shifts to lower frequencies and, in
addition, we observe a decrease in the quality factor of the nonlinear SRR.)

Fig. 1:  (a) The reflection of the SRR loaded with one varactor, measured by a
loop antenna. The curves show the measurement results at input power from -15
dBm to 9 dBm in 3 dBm steps. The inset shows the picture of the loop antenna
and the sample. (b) The hysteresis effect at high input power levels. The blue
curves are measured for forward sweep (the source frequency of the network
analyzer is scanned from low to high) and the red dashed curves are measured
for reverse sweep (frequency is scanned from high to low).

Below, we present some analytical arguments to explain the tunability and the
hysteresis effects of the nonlinear SRRs observed experimentally.

The varactor, mounted onto the slit of the SRR, is a Skyworks SMV1231-079
hyperabrupt tuning varactor. The nonlinear voltage-dependent depletion layer
capacitance $C(V_{D})$ ($V_{D}$ is the voltage across the diode) is described
as the following, provided by the manufacturer SPICE model.
$C(V_{D})=C_{0}\left(1-V_{D}/V_{p}\right)^{-M}$, where $C_{0}=2.2$ pF is the
DC rest capacitance, $V_{p}=1.5$ V is the intrinsic potential and $M=0.8$. The
dissipative current is given by
$I_{D}(V_{D})=I_{0}\left(e^{(V_{D}/V_{T})}-1\right)$. From
$C(V_{D})=\frac{\,dQ_{D}}{\,dV_{D}}$, we can determine the time-dependent
charge,
$Q_{D}=\frac{C_{0}V_{p}}{1-M}\left[1-\left(1-V_{D}/V_{p}\right)^{1-M}\right]$.
Assume $V_{D}<V_{p}$, then the voltage across the diode can be expressed by
the charge,
$V_{D}(q)=V_{p}\left[1-\left(1-q\frac{1-M}{V_{p}}\right)^{\frac{1}{1-M}}\right]$,
where the renormalized voltage is defined as $q=Q_{D}/C_{0}$.

The SRR can be modeled as an RLC circuit with external excitation and the
voltage equation can be expressed by
$-L\frac{\,dI}{\,dt}-R_{S}-V_{D}=\varepsilon(t)$, where $I$ is the current in
the resonator, $L$ is the inductance of the resonator determined by the ring
geometry, $R_{S}$ is the resistance, and $\varepsilon$ is the driven term
provided by the loop antenna in the experiment. For small excitations, $I_{D}$
can be neglected so the current can be estimated by $I\approx\,dQ_{D}/\,dt$.
The equation of motion is now
$\frac{\,d^{2}q}{\,dt^{2}}+\gamma\frac{\,dq}{\,dt}+\omega_{0}^{2}V_{D}=-\omega_{0}^{2}\varepsilon(t)$,
where $\omega_{0}^{2}=1/(LC_{0})$ and $\gamma=\omega_{0}^{2}R_{S}C_{0}$.
Expand the restoring term $V_{D}$ by the Taylor series for small oscillations
(the oscillation amplitude satisfies $(1-M)|q|<V_{p}$) and omit the higher
order terms, $V_{D}(q)\approx
q-\frac{M}{2V_{p}}q^{2}+\frac{M(2M-1)}{6V_{p}^{2}}q^{3}$. Assume harmonic
excitation so that $\varepsilon(t)=f\cos(\omega t)$, where $f$ is the
excitation amplitude and $\omega$ is the excitation frequency, the equation of
motion is further estimated by
$\frac{\,d^{2}q}{\,dt^{2}}+\gamma\frac{\,dq}{\,dt}+\omega_{0}^{2}q+\alpha
q^{2}+\beta q^{3}=-\omega_{0}^{2}f\cos(\omega t)$, where
$\alpha=-\frac{\omega_{0}^{2}M}{2V_{p}}$,
$\beta=\frac{\omega_{0}^{2}M(2M-1)}{6V_{p}^{2}}$. This is now a nonlinear
driven oscillator problem [19].

The driven frequency can be written as $\omega=\omega_{0}+\delta$. When
$\delta$ is small, the driven frequency is close to the resonance frequency.
Without the $q^{2}$ and $q^{3}$ term, the oscillator is linear and the
amplitude of oscillation, $b$, is given by
$b^{2}\left(\delta^{2}+\gamma^{2}/4\right)=\omega_{0}^{2}f^{2}/4$. The
nonlinear $q^{2}$ and $q^{3}$ terms make the eigen-frequency amplitude
dependent, $\omega_{0}\to\omega_{0}+\kappa b^{2}$, where
$\kappa=\frac{3\beta}{8\omega_{0}}-\frac{5\alpha^{2}}{12\omega_{0}^{3}}$. Then
$\delta\to\delta-\kappa b^{2}$, and the oscillation amplitude satisfies the
equation $b^{2}\left[\left(\delta-\kappa
b^{2}\right)^{2}+\gamma^{2}/4\right]=\omega_{0}^{2}f^{2}/4$. This is a cubic
equation about $q^{2}$ and the real roots give the amplitude of oscillations.
When the external excitation is small, the oscillation amplitude is also small
and the higher orders of $b$ may be neglected and the oscillation can be
considered to be linear. When the excitation power is larger, the curve is
distorted and the resonance shifts to a lower frequency, since, in our case,
$\kappa$ is negative. When the excitation power is large enough, there are
three real roots of $b^{2}$ and the curve is folded over, see Fig. 1(b). The
branch in the middle is unstable and the oscillation tends to go to the other
two branches. In experiment, the oscillation follows the lower branch until it
jumps to the higher branch for forward sweep, and follows the higher branch
until it jumps to the lower branch. So, the hysteresis effect is observed in
our experimental measurements. Note, when the voltage on the varactor is
larger than 0.5 V, the nonlinear DC dissipative current (
$I_{D}(V_{D})=I_{0}\left(e^{V/V_{T}}-1\right)$, where $V_{T}=k_{B}T/e$ is the
thermal voltage, $k_{B}$ is the Boltzmann constant, $T$ is the temperature,
and $e$ is the electron charge) sets in and this increases the loss on the SRR
and the reflection dip measured is not as strong as the small oscillation
case. See Fig. 1(a), the reflection minimum increases from around -40 dB to -3
dB when the input power is increased from -15 dBm to 9 dBm. This is not
covered in the simplified model of the nonlinear oscillator model.

Fig. 2: The oscillation amplitude vs. source frequency calculated from the
simplified nonlinear oscillator model. The curves from bottom to top
correspond to excitation power from low to high. The blue arrow shows the jump
for forward sweep of the top curve and the red arrow shows the jump for
reverse sweep of the top curve.

To remove the nonlinear DC current and obtain a better self-tuning effect, a
new SRR is fabricated. The SRR ring is of the same geometry, but with another
identical cut on the other side of the ring and a varactor is mounted onto
each of the cuts, see inset in Fig. 3. The varactors are arranged back-to-back
such that no DC current can circulate in the SRR and the effective capacitance
characteristics $C(V)$ of the two varactors is now symmetric. This
configuration has the same effect of one heterostructure barrier varator (HBV)
diode [17]. The two varactors can now be regarded as two tunable capacitors
connected in a series. The total capacitance is smaller than a single varactor
at the same power level, so that the resonance frequency shifts to a higher
region (see Fig. 3). When the input power increases, the effective capacitance
decreases and the resonance frequency increases. Also see from Fig. 3 that the
resonance strength and the quality factor are almost the same for different
input power levels. The only problem of this configuration is that the
varactors discharge very slowly, due to the lack of a circulating current.

Fig. 3: The reflection of the SRR loaded with two back-to-back varactors,
measured by a loop antenna. The curves show the measurement results at power
levels from -14 dBm to 2 dBm, in 2 dBm steps. The inset shows the sample.

##  3 Mutual coupling between two nonlinear SRRs

To make nonlinear metamaterials out of the nonlinear SRRs, the mutual coupling
of the varactor loaded SRRs must to be studied. Since the mutual coupling of
two coplane SRRs are very weak, we studied the case of two parallel SRRs with
the same axis (solenoid case). The loop antenna is also parallel to the two
SRRs and has the same axis. The distance of the antenna and the first SRR is
fixed, and the second SRR is moved away from the first SRR. Reflection is
measured by the loop antenna for different distances between the two SRRs. In
Fig. 4(a), we present the frequency-dependence of the reflection coefficient
for two linear SRRs, as the distance between the two SRRs increases. As one
can see from Fig. 4(a), if the distance between the SRRs is large, only one
reflection resonance is observed and as the two SRRs move closer, the mutual
coupling becomes stronger and the reflection splits. The closer the two SRRs,
the wider the split. For two linear SRRs, the mutual coupling can be
calculated analytically with a simple LC model.

Fig. 4: The reflection of (a) two linear SRRs and (b) two nonlinear SRRs at
input power of $-15$ dBm. The legend shows the distance between the two SRRs.
The distance between the first SRR and the antenna is fixed.

In Fig. 4(b), we present the frequency-dependence of the reflection
coefficient for two nonlinear SRRs, as the distance between the two SRRs
increases. The incident power is $-15$ dBm, which is relatively low and the
results presented in Fig. 4(b) are almost equivalent to the linear SRRs
presented in Fig. 4(a), except that the reflection dip is not as deep as the
linear case, due to a higher loss in the varactor than the high-Q capacitor.
When the input power is low, the nonlinear SRRs behaves like the linear ones
and the splitting is nicely seen in the -15 dBm case of the 2d plot in Fig. 5.
When the input power gets higher, the resonance of the SRR gets broadened, due
to higher loss and the splitting becomes worse, as seen in Fig. 5. One can
clearly see from Fig. 5 that the hybridization gets very weak as the incident
power increases.

Fig. 5: The 2d plot of the reflection of two coupled nonlinear SRRs. The
x-axis shows the distance between the two SRRs and the y-axis shows the
frequency. The four figures from left to right display the result at input
power from -15 dBm to 0 dBm. The SRRs and the antenna are arranged such that
they have the same axis and the planes of the rings are parallel to each
other. The distance between the loop antenna and the first SRR is fixed and
the second SRR is movable along its axis, as seen from the inset.

##  4 Conclusion

In conclusion, we have demonstrated experimentally dynamic tunability,
hysteresis, and bistable behavior in nonlinear SRRs. The nonlinear SRR has a
typical SRR design and we have soldered in the gap of the SRR a commercial
varactor diode. Tunability of the resonance frequency was completed
dynamically by increasing the incident power of the vector network analyzer.
We have introduced different nonlinear designs and observed experimentally
that the resonance frequency can decrease or increase by increasing the
incident power. This way, we are able to change the sign of the nonlinearity.
A theoretical model was given that explained all the observed nonlinear
effects. Finally, we study the hybridization effects of the linear and
nonlinear SRR.

## Acknowledgments

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 office of Naval Research (Award No. N00014-07-1-D359), and
European Community FET project PHOME (Contract No. 213390).