- Email: [email protected]
Magnetoinductive waves in arrays of split-ring resonators
ARTICLE IN PRESS
Physica B 394 (2007) 180–183 www.elsevier.com/locate/physb
Magnetoinductive waves in arrays of split-ring resonators Ilya V. Shadrivova,, Alexander N. Reznika,b, Yuri S. Kivshara a
Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia b Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia
Abstract We study experimentally the propagation of linear magnetoinductive waves in arrays of split-ring resonators with different orientation and spacing. We summarize our experimental results characterizing the effect of coupling of individual elements in magnetic composite metamaterials. We observe the band broadening due to the excitation of magnetoinductive waves which should impose some limitations on the effective medium approximation used to analyze metamaterials. r 2007 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Bs; 33.15.Cr Keywords: Split-ring resonators; Metamaterials; Magnetic composites; Magnetoinductive waves
1. Introduction Recent advances in a design of composite materials indicates that the concept of left-handed materials suggested long time ago [1] can be realized for a novel type of artificial materials with the properties not available in nature [2]. Such materials have both negative dielectric permittivity and magnetic permeability over a finite range of frequencies and, thus, they exhibit nontrivial phenomena for the propagation of electromagnetic waves. In particular, imaging by flat ‘perfect’ lenses with a subwavelength resolution may become possible [3]. Currently, strong efforts are being made to design left-handed metamaterials for shorter wavelengths including terahertz and optical frequencies [4,5]. It has been well established that the composite metallic structures consisting of arrays of wires and split-ring resonators (SRRs) may possess left-handed properties in the microwave frequency range, and negative magnetic response becomes possible due to resonators with a strong magnetic momentum. Most of the currently demonstrated metamaterials are based on the composite structures with Corresponding author.
E-mail address: [email protected] (I.V. Shadrivov). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.12.038
periodically arranged split-ring resonators [2] or pairs of wires [5]. However, a standard theoretical approach for analyzing the properties of such composite metamaterials is based on the effective-medium approximation where a microstructured composite is treated as a homogeneous isotropic medium characterized by effective macroscopic parameters. This approximation is justified when the characteristic scale of the wavelength of the electromagnetic field is much larger than the period of the microstructured medium. Moreover, the effective medium approach is based on an averaging over the lattice of micro-elements, and usually it does not take into account any short-wavelength eigenmodes of the structure, which might exist due to periodic microscopic features of the composite. Within the static approximation, magnetic interaction of SRRs was investigated in a number of works, see, e.g., Refs. [6–12]. The eigenmodes in the material containing arrays of SRRs are usually referred to as magnetoinductive waves. In this paper, we study experimentally the propagation of magnetoinductive electromagnetic waves in arrays of split-ring resonators similar to those usually used for the fabrication of left-handed metamaterials operating for microwave frequencies (see inset in Fig. 1). In our structures, radius of the internal ring is 2.56 mm, the ring
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183 1
n-1
Reflection coefficient
n
a
0.9
181 n+1 Transverse waves
d 0.8
Propagation direction m
0.7
Longitudinal waves
0.6
RF input 0.5 2
2.1
2.2 2.3 Frequency (GHz)
2.4
Fig. 2. Schematic of the transverse and longitudinal orientation of SRRs in one-dimensional arrays. In the transverse orientation, the magnetic momentum of each SRR is perpendicular to the array axis, while for the longitudinal orientation the magnetic momentum is parallel to the array axis.
Fig. 1. Resonant reflection of incoming waves from a single split-ring resonator excited by a short antenna. Shown is the reflection coefficient vs. wave frequency. Inset gives a sketch of a mutual position of the antenna and SRR.
width is 1.44 mm, and the same size for spacing between the rings and a slot size in each ring, namely 0.32 mm. To probe the electric field in the vicinity of a slot of the external ring of the SRR, we use a short electric antenna printed on the circuit board together with SRR (see inset in Fig. 1). The antenna is capacitively coupled to the resonator allowing us to excite the resonator or register the electric field strength in vicinity of the SRR gap. We connect the port of the vector network analyzer (VNA), Rohde and Schwarz model ZVB20, to the antenna and measure the reflection from the structure. The results of our measurements are shown in Fig. 1. Indeed, when the frequency of signal that we launch to the antenna coincides with the SRR resonant frequency, more energy is absorbed in SRR. We can then estimate the quality factor of the resonator with antenna, and we find it to be 50. The resonance frequency of a single isolated SRR is measured to be 2.22 GHz. 2. Experimental results We study experimentally the propagation of magnetoinductive waves in arrays of split-ring resonators of two geometries, transverse and longitudinal, as shown in Fig. 2. The wave propagation in such arrays is possible due to mutual coupling between the resonators. In the quasistatic regime, the coupling is mainly due to mutual inductance of the resonators. The terminology reflects the mutual orientation of magnetic momenta of SRRs with respect to the propagation direction. In the quasistatic limit, the longitudinal array supports forward waves, while the transverse structure supports backward waves only [7]. Such waves have been observed experimentally [7] in the case when the wavelength is about four orders of magnitude larger than the array spacing [7]. In contrast, here we study the magnetoinductive waves when the ratio of the resonant wavelength to the structure period is of the order of 10.
Fig. 3. Photograph of the transversely (top) and longitudinally (bottom) oriented one-dimensional arrays of SRRs.
One-dimensional SRR arrays with two orientations are shown in Fig. 3. Top figure represents the array with the transverse (or planar) geometry [7]. Bottom figure shows the longitudinal (or axial) orientation. The arrays contain 10 resonators identical to one studied in the reflection measurements mentioned above. Each resonator has an antenna imprinted next to it. We chose this configuration, instead of a free-standing single antenna, to perform measurements in order to avoid positioning inaccuracies. In our setup, the mutual position of antennas and SRRs remains identical with a high accuracy, allowing for almost identical coupling between resonators and antennas. Our additional measurements show that the coupling of antenna to the SRR changes drastically when the position of antenna is varied even by 10 mm near the SRR gap, indicating strong localization of the electric field in the gap of the SRR. First, we present our results for the transverse geometry. Period of the structure or SRR spacing is 12.5 mm. Output
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183
182 2.5
T3 T5 T 10
T3 T5 T 10
2.6
2.4
Frequency (GHz)
Frequency (GHz)
2.4 2.3
2.2
2.2
2 2.1 1.8 2 0.8
0.9 k(cm–1)
1
1.1
0
0.05
0
0.1
2
Fig. 4. Left: Dispersion of the waves propagating along an array in the transverse geometry (see Fig. 3, top). Thin horizontal line indicates the resonance frequency for a single SRR (spacing is 12.5 mm). Right: transmission of the array measured on the 3rd, 5th and 10th resonators.
of VNA is connected to the antenna next to the resonator at the edge of the structure, exciting this SRR. Due to interaction between the resonators, excitation is transferred to other resonators in the structure. We can measure the transmission between the first resonator and other resonators in the structure. An absolute value of the transmission coefficient between the first and, respectively, third, fifth, and then 10th resonator is shown in Fig. 4 (right). We notice that the transmission maximum is shifted towards high frequencies from the single SRR resonance. Measuring the phase of the transmitted signal, we find that for some range of frequencies it is a linear function of the coordinate. This indicates that in this structure we have a propagating wave, and we can determine its wavenumber. Dispersion of the waves generated in the SRR array in the frequency band where the phase change is linear is shown in Fig. 4 (left). We notice that the slope of the dispersion curve changes it sign indicating that we have either forward (positive slope) or backward (negative slope) waves. This result is in a sharp contrast to the previous results [7]. The difference may be due to significantly different ratio of the wavelength to the structural period. For the longitudinal orientation of SRRs (see Fig. 3, bottom), we can change the coupling between resonators significantly by changing the lattice period. For the transverse orientation, the lattice period is limited by the external diameter of the resonators, whereas for the longitudinal geometry, we can put resonators much closer, archiving much stronger mutual inductances between resonators. We perform measurements for two different values of spacing, 10 and 5 mm. Corresponding results are presented in Figs. 5 and 6. In both the cases we observe forward waves registered by a positive slope of the dispersion curve. Decreasing the
4
0
0.02
k(cm–1)
Transmission
0.04
0.06
0.08
Transmission
Fig. 5. Same as in Fig. 3 but for longitudinal geometry, SRR spacing of 10 mm.
T3 T5 T 10
2.6
2.4 Frequency (GHz)
0.7
2.2
2
1.8 0
2
4 k(cm–1)
6
0
0.02
0.04 0.06 Transmission
0.08
0.1
Fig. 6. Same as in Fig. 3 but for longitudinal geometry, SRR spacing of 5 mm.
spacing between the resonators we increase the bandwidth and the resonances become visible (see Fig. 6) due to increased mutual coupling. The maximum of wave transmission moves towards higher frequencies, as we decrease the distance between the resonators. 3. Conclusions We have studied experimentally the propagation of linear transverse and longitudinal magnetoinductive waves in one-dimensional arrays of coupled split-ring resonators. We have shown that the array bandwidth changes dramatically as we vary the coupling coefficient between the resonators. Such magnetoinductive waves are not described by the continuous theory based on the effective-medium approximation, and they should be taken into
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183
account for describing the fundamental properties of the commonly used metamaterial composites. Acknowledgments We acknowledge a support of the Australian Research Council and RFBR Grant 06-02-16669. Alex Reznik thanks Nonlinear Physics Centre for a warm hospitality. References [1] V.G. Veselago, Usp. Fiz. Nauk 92 (1967) 517 (in Russian) [English translation: Veselago:1968-509:PUS]. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966.
183
[4] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, C.M. Soukoulis, Science 306 (2004) 1351. [5] V.M. Shalaev, W. Cai, U.K. Chettiar, H.K. Yuan, A.K. Sarychev, V.P. Drachev, A.V. Kildishev, Opt. Lett. 30 (2005) 3356. [6] M. Gorkunov, M. Lapine, E. Shamonina, K.H. Ringhofer, Eur. Phys. J. B 28 (2002) 263. [7] M.C.K. Wiltshire, E. Shamonina, I.R. Young, L. Solymar, Electron. Lett. 39 (2003) 215. [8] R.R.A. Syms, E. Shamonina, V. Kalinin, L. Solymar, J. Appl. Phys. 97 (2005) 64909. [9] E. Shamonina, V.A. Kalinin, K.H. Ringhofer, L. Solymar, J. Appl. Phys. 92 (2002) 6252. [10] E. Shamonina, L. Solymar, J. Phys. D-Appl. Phys. 37 (2004) 362. [11] R. Marques, F. Mesa, J. Martel, F. Medina, IEEE Trans. Antennas Propag. 51 (2003) 2572. [12] M.J. Freire, R. Marques, F. Medina, M.A.G. Laso, F. Martin, Appl. Phys. Lett. 85 (2004) 4439.
Physica B 394 (2007) 180–183 www.elsevier.com/locate/physb
Magnetoinductive waves in arrays of split-ring resonators Ilya V. Shadrivova,, Alexander N. Reznika,b, Yuri S. Kivshara a
Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia b Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia
Abstract We study experimentally the propagation of linear magnetoinductive waves in arrays of split-ring resonators with different orientation and spacing. We summarize our experimental results characterizing the effect of coupling of individual elements in magnetic composite metamaterials. We observe the band broadening due to the excitation of magnetoinductive waves which should impose some limitations on the effective medium approximation used to analyze metamaterials. r 2007 Elsevier B.V. All rights reserved. PACS: 41.20.Jb; 42.25.Bs; 33.15.Cr Keywords: Split-ring resonators; Metamaterials; Magnetic composites; Magnetoinductive waves
1. Introduction Recent advances in a design of composite materials indicates that the concept of left-handed materials suggested long time ago [1] can be realized for a novel type of artificial materials with the properties not available in nature [2]. Such materials have both negative dielectric permittivity and magnetic permeability over a finite range of frequencies and, thus, they exhibit nontrivial phenomena for the propagation of electromagnetic waves. In particular, imaging by flat ‘perfect’ lenses with a subwavelength resolution may become possible [3]. Currently, strong efforts are being made to design left-handed metamaterials for shorter wavelengths including terahertz and optical frequencies [4,5]. It has been well established that the composite metallic structures consisting of arrays of wires and split-ring resonators (SRRs) may possess left-handed properties in the microwave frequency range, and negative magnetic response becomes possible due to resonators with a strong magnetic momentum. Most of the currently demonstrated metamaterials are based on the composite structures with Corresponding author.
E-mail address: [email protected] (I.V. Shadrivov). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.12.038
periodically arranged split-ring resonators [2] or pairs of wires [5]. However, a standard theoretical approach for analyzing the properties of such composite metamaterials is based on the effective-medium approximation where a microstructured composite is treated as a homogeneous isotropic medium characterized by effective macroscopic parameters. This approximation is justified when the characteristic scale of the wavelength of the electromagnetic field is much larger than the period of the microstructured medium. Moreover, the effective medium approach is based on an averaging over the lattice of micro-elements, and usually it does not take into account any short-wavelength eigenmodes of the structure, which might exist due to periodic microscopic features of the composite. Within the static approximation, magnetic interaction of SRRs was investigated in a number of works, see, e.g., Refs. [6–12]. The eigenmodes in the material containing arrays of SRRs are usually referred to as magnetoinductive waves. In this paper, we study experimentally the propagation of magnetoinductive electromagnetic waves in arrays of split-ring resonators similar to those usually used for the fabrication of left-handed metamaterials operating for microwave frequencies (see inset in Fig. 1). In our structures, radius of the internal ring is 2.56 mm, the ring
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183 1
n-1
Reflection coefficient
n
a
0.9
181 n+1 Transverse waves
d 0.8
Propagation direction m
0.7
Longitudinal waves
0.6
RF input 0.5 2
2.1
2.2 2.3 Frequency (GHz)
2.4
Fig. 2. Schematic of the transverse and longitudinal orientation of SRRs in one-dimensional arrays. In the transverse orientation, the magnetic momentum of each SRR is perpendicular to the array axis, while for the longitudinal orientation the magnetic momentum is parallel to the array axis.
Fig. 1. Resonant reflection of incoming waves from a single split-ring resonator excited by a short antenna. Shown is the reflection coefficient vs. wave frequency. Inset gives a sketch of a mutual position of the antenna and SRR.
width is 1.44 mm, and the same size for spacing between the rings and a slot size in each ring, namely 0.32 mm. To probe the electric field in the vicinity of a slot of the external ring of the SRR, we use a short electric antenna printed on the circuit board together with SRR (see inset in Fig. 1). The antenna is capacitively coupled to the resonator allowing us to excite the resonator or register the electric field strength in vicinity of the SRR gap. We connect the port of the vector network analyzer (VNA), Rohde and Schwarz model ZVB20, to the antenna and measure the reflection from the structure. The results of our measurements are shown in Fig. 1. Indeed, when the frequency of signal that we launch to the antenna coincides with the SRR resonant frequency, more energy is absorbed in SRR. We can then estimate the quality factor of the resonator with antenna, and we find it to be 50. The resonance frequency of a single isolated SRR is measured to be 2.22 GHz. 2. Experimental results We study experimentally the propagation of magnetoinductive waves in arrays of split-ring resonators of two geometries, transverse and longitudinal, as shown in Fig. 2. The wave propagation in such arrays is possible due to mutual coupling between the resonators. In the quasistatic regime, the coupling is mainly due to mutual inductance of the resonators. The terminology reflects the mutual orientation of magnetic momenta of SRRs with respect to the propagation direction. In the quasistatic limit, the longitudinal array supports forward waves, while the transverse structure supports backward waves only [7]. Such waves have been observed experimentally [7] in the case when the wavelength is about four orders of magnitude larger than the array spacing [7]. In contrast, here we study the magnetoinductive waves when the ratio of the resonant wavelength to the structure period is of the order of 10.
Fig. 3. Photograph of the transversely (top) and longitudinally (bottom) oriented one-dimensional arrays of SRRs.
One-dimensional SRR arrays with two orientations are shown in Fig. 3. Top figure represents the array with the transverse (or planar) geometry [7]. Bottom figure shows the longitudinal (or axial) orientation. The arrays contain 10 resonators identical to one studied in the reflection measurements mentioned above. Each resonator has an antenna imprinted next to it. We chose this configuration, instead of a free-standing single antenna, to perform measurements in order to avoid positioning inaccuracies. In our setup, the mutual position of antennas and SRRs remains identical with a high accuracy, allowing for almost identical coupling between resonators and antennas. Our additional measurements show that the coupling of antenna to the SRR changes drastically when the position of antenna is varied even by 10 mm near the SRR gap, indicating strong localization of the electric field in the gap of the SRR. First, we present our results for the transverse geometry. Period of the structure or SRR spacing is 12.5 mm. Output
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183
182 2.5
T3 T5 T 10
T3 T5 T 10
2.6
2.4
Frequency (GHz)
Frequency (GHz)
2.4 2.3
2.2
2.2
2 2.1 1.8 2 0.8
0.9 k(cm–1)
1
1.1
0
0.05
0
0.1
2
Fig. 4. Left: Dispersion of the waves propagating along an array in the transverse geometry (see Fig. 3, top). Thin horizontal line indicates the resonance frequency for a single SRR (spacing is 12.5 mm). Right: transmission of the array measured on the 3rd, 5th and 10th resonators.
of VNA is connected to the antenna next to the resonator at the edge of the structure, exciting this SRR. Due to interaction between the resonators, excitation is transferred to other resonators in the structure. We can measure the transmission between the first resonator and other resonators in the structure. An absolute value of the transmission coefficient between the first and, respectively, third, fifth, and then 10th resonator is shown in Fig. 4 (right). We notice that the transmission maximum is shifted towards high frequencies from the single SRR resonance. Measuring the phase of the transmitted signal, we find that for some range of frequencies it is a linear function of the coordinate. This indicates that in this structure we have a propagating wave, and we can determine its wavenumber. Dispersion of the waves generated in the SRR array in the frequency band where the phase change is linear is shown in Fig. 4 (left). We notice that the slope of the dispersion curve changes it sign indicating that we have either forward (positive slope) or backward (negative slope) waves. This result is in a sharp contrast to the previous results [7]. The difference may be due to significantly different ratio of the wavelength to the structural period. For the longitudinal orientation of SRRs (see Fig. 3, bottom), we can change the coupling between resonators significantly by changing the lattice period. For the transverse orientation, the lattice period is limited by the external diameter of the resonators, whereas for the longitudinal geometry, we can put resonators much closer, archiving much stronger mutual inductances between resonators. We perform measurements for two different values of spacing, 10 and 5 mm. Corresponding results are presented in Figs. 5 and 6. In both the cases we observe forward waves registered by a positive slope of the dispersion curve. Decreasing the
4
0
0.02
k(cm–1)
Transmission
0.04
0.06
0.08
Transmission
Fig. 5. Same as in Fig. 3 but for longitudinal geometry, SRR spacing of 10 mm.
T3 T5 T 10
2.6
2.4 Frequency (GHz)
0.7
2.2
2
1.8 0
2
4 k(cm–1)
6
0
0.02
0.04 0.06 Transmission
0.08
0.1
Fig. 6. Same as in Fig. 3 but for longitudinal geometry, SRR spacing of 5 mm.
spacing between the resonators we increase the bandwidth and the resonances become visible (see Fig. 6) due to increased mutual coupling. The maximum of wave transmission moves towards higher frequencies, as we decrease the distance between the resonators. 3. Conclusions We have studied experimentally the propagation of linear transverse and longitudinal magnetoinductive waves in one-dimensional arrays of coupled split-ring resonators. We have shown that the array bandwidth changes dramatically as we vary the coupling coefficient between the resonators. Such magnetoinductive waves are not described by the continuous theory based on the effective-medium approximation, and they should be taken into
ARTICLE IN PRESS I.V. Shadrivov et al. / Physica B 394 (2007) 180–183
account for describing the fundamental properties of the commonly used metamaterial composites. Acknowledgments We acknowledge a support of the Australian Research Council and RFBR Grant 06-02-16669. Alex Reznik thanks Nonlinear Physics Centre for a warm hospitality. References [1] V.G. Veselago, Usp. Fiz. Nauk 92 (1967) 517 (in Russian) [English translation: Veselago:1968-509:PUS]. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966.
183
[4] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, C.M. Soukoulis, Science 306 (2004) 1351. [5] V.M. Shalaev, W. Cai, U.K. Chettiar, H.K. Yuan, A.K. Sarychev, V.P. Drachev, A.V. Kildishev, Opt. Lett. 30 (2005) 3356. [6] M. Gorkunov, M. Lapine, E. Shamonina, K.H. Ringhofer, Eur. Phys. J. B 28 (2002) 263. [7] M.C.K. Wiltshire, E. Shamonina, I.R. Young, L. Solymar, Electron. Lett. 39 (2003) 215. [8] R.R.A. Syms, E. Shamonina, V. Kalinin, L. Solymar, J. Appl. Phys. 97 (2005) 64909. [9] E. Shamonina, V.A. Kalinin, K.H. Ringhofer, L. Solymar, J. Appl. Phys. 92 (2002) 6252. [10] E. Shamonina, L. Solymar, J. Phys. D-Appl. Phys. 37 (2004) 362. [11] R. Marques, F. Mesa, J. Martel, F. Medina, IEEE Trans. Antennas Propag. 51 (2003) 2572. [12] M.J. Freire, R. Marques, F. Medina, M.A.G. Laso, F. Martin, Appl. Phys. Lett. 85 (2004) 4439.