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Comparative analysis of the transfer function of closed and open split-ring metamaterial slab lenses
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Metamaterials 5 (2011) 107–111
Comparative analysis of the transfer function of closed and open split-ring metamaterial slab lenses J.M. Algarín, M.J. Freire ∗ , R. Marqués Dept. of Electronics and Electromagnetics, University of Seville, Facultad de Fisica, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain Received 14 December 2010; received in revised form 23 May 2011; accepted 15 July 2011 Available online 22 July 2011
Abstract The transfer function of some in-plane periodic split-ring metamaterial slab lenses is analyzed for several configurations across the slab. The transfer function is obtained by means of full-wave electromagnetic computations using the simulation software CST Microwave Studio, and it is compared with the transfer function obtained analytically for a continuous slab of reference. Significant differences are found in the transfer function of the analyzed structures. The closest behavior to the continuous slab lens was found for the partially open structure comprising an integer number of periods across the slab. Experiments are provided which confirms the theoretical results. © 2011 Elsevier B.V. All rights reserved. Keywords: Metamaterials; Sub-wavelength imaging; Super-lens; Split-rings; Capacitively loaded rings; Magnetic resonanceimaging
1. Introduction One of the most interesting properties of metamaterial slabs with relative permittivity εr = − 1 and/or relative permeability μr = − 1 is the ability of these slabs to behave as “super-lenses” with sub-wavelength (sub-λ) resolution [1]. The key mechanism behind sub-λ resolution is the amplification, inside the slab, of the evanescent harmonics coming from the source. Therefore, for a slab of thickness d, the amplitude of these harmonics is restored at a distance 2d from the source plane [1]. Nevertheless, since losses in realistic slabs prevent the amplification for large thickness, in practice sub-λ resolution is only possible in the near field region [2,3]. In this region, we are in the realm of the
∗
Corresponding author. Tel.: +34 954556408; fax: +34 954239434. E-mail address: [email protected] (M.J. Freire).
1873-1988/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.metmat.2011.07.001
quasi-statics, where both the electric and the magnetic fields can be treated separately, and we only need a slab with εr = − 1 [1,4] or μr = − 1. The choice depends on the characteristics (electric or magnetic) of the source. Metamaterial slabs with negative μr for sub-λ imaging in the radio-frequency range have been reported in [5] using swiss-rolls and in [6] using capacitively loaded split rings. Split rings have the key advantage over swiss rolls of providing three-dimensional (3D) isotropy when they are arranged in a cubic lattice, which is essential in order to image 3D sources. Medical applications of such split rings lenses in magnetic resonance imaging (MRI) have been recently reported by some of the authors [6–8]. Whereas the configuration of a periodic and isotropic split-ring metamaterial slab lens is well defined in the plane of the slab, the optimal choice is not so apparent in the direction across the slab. Up to the date, most realizations of negative μr split rings metamaterial lenses included two top and bottom split-ring layers of square
108
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
(a)
Y Z
X
a
(b)
properties have been proposed and studied [10–15]. These devices take advantage of the magnetoinductive couplings between arrays of split-ring resonators, and have been analyzed in terms of the magneto-inductive waves [16] which can propagate along such arrays. In this paper, however, we will restrict ourselves to the analysis of metamaterial structures which try to mimic continuous media with negative permeability, and more specifically to the structures sketched in Fig. 1. Although magneto-inductive waves can also propagate in such structures, this fact does not invalidate the use of homogenization techniques, provided they are correctly applied [17]. 2. Analysis
(c)
For a metamaterial slab, the transfer function T(ω, k) is defined as the transmission coefficient for each evanescent harmonic with transverse wave-number k in the plane of the slab. For an homogeneous slab of magnetic permeability μ, and in the quasi-magnetostatic limit, the transfer function is given by (Ref. [3], Eqs. 5–70): 4μμ0 e−|k|d − (μ − μ0 )2 e−|k|d
Fig. 1. Sketch of the different split-ring structures analyzed: (a) closed, (b) one-side open ended, and (c) both-sides open ended.
T (ω, |k|) =
periodicity, with a cubic arrangement of split-rings in between [6]. Such configuration is shown in Fig. 1a for a “two periods thick” slab lens. However, other configurations can be chosen, with one or both external split-ring surfaces removed, as it is shown in Fig. 1b and c, respectively. Choosing one of these configurations may depend on the specific application of the device, although for general purpose applications the choice must rely on the similarities between the imaging properties of the device and the imaging properties of a theoretical continuous medium slab lens. Imaging capabilities of an infinite metamaterial slab are summarized by the transfer function, defined as the ratio between the field at the image plane and the field at the source plane. The transfer function of a continuous slab of negative μr can be obtained analytically, and the transfer function of the structure shown in Fig. 1a was already studied by some of the authors in [9], where the similarities and differences with the transfer function of a continuous media slab lens were also analyzed. In this paper, the transfer function of the three split-ring metamaterial slab lenses shown in Fig. 1 will be analyzed, and the results will be compared with the transfer function of a continuous negative μr slab lens in order to find the optimal configuration from the point of view of the resolution. Besides the aforementioned devices, other splitring metamaterial structures with remarkable imaging
where d is the thickness of the slab. For comparison purposes we will consider a slab of an effective medium with the magnetic permeability given by Eq. (13) of [17], where magneto-inductive effects are explicitly considered through the mutual inductances between nearest neighbors. The parameters used for the calculations are the same as in the experiments reported in [6], but with the resistance of the rings reduced by two orders of magnitude. This small resistance is introduced in order to avoid numerical instabilities. Fig. 2 shows a 3D plot of this transfer function for different frequencies and for values of the transverse wavenumber k ranging in the interval [0, 0.3π/a], where a is the periodicity of the split-ring lens. Two peaks appear in the plot at each frequency. These peaks form two crests in the 3D graphic, which correspond to surface waves which are the magnetic analogous of the surface plasmons in negative permittivity slabs [3]. A flat region of T 1 can be clearly seen between the crests in Fig. 2. This region defines the domain in the ω − k space where imaging is achieved [3]. Once the main characteristics of the transfer function for a continuous slab have been reviewed, we proceed with the analysis of the discrete split-ring slabs. Since the minimum resolution of a metamaterial slab cannot be made smaller than the slab thickness (see [3] and references therein), there is no reason to use more than two
(μ + μ0
)2 e|k|d
(1)
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
109
Fig. 2. Three-dimensional plot of the transfer function of an homogeneous slab corresponding to the homogenization of the split-ring lens reported in [6].
or three periods across the slab width in the implementation of a split-ring lens. From these considerations, in the present work we will consider lenses with just two or three periods across the slab depth. The considered structures with two periods across the slab depth are shown in Fig. 1. The transfer functions of these structures are computed using the simulation software CST Microwave Studio. Computations are implemented by considering just two unit cells across the slab width and imposing periodic boundary conditions in the transverse directions. Ring properties and periodicity are chosen as in [6], except for the resistance, which is set equal to zero in order to make the results of the analysis dependent only on the discrete effects and not on losses. The transmission coefficient for different kx evanescent harmonics (with ky = 0) are computed. These values of kx are chosen so that the corresponding wavelength λx = 2π/kx is an integer number n of the periodicity a, that is, λx = na. Fig. 3a–c shows the results of the computed transfer function for the structures of Fig. 1a–c, respectively, and for different values of n = 72, 36, 18, . . ., 4. A common feature of the results obtained for the three different structures is that the value of the transmission coefficient between the resonances decays with n, that is, for short wavelengths or high harmonics. Since this happens even if the structure is lossless, it can be concluded that the discrete structure imposes a low-pass filter behavior for the spatial frequencies. Moreover, the results of Fig. 3a show that there is a shift towards lower frequencies in the resonances corresponding to higher harmonics. Therefore,
Fig. 3. Amplitude of the transmission coefficient for various transverse wave-lengths expressed in number of unit cells. Each one of the figures corresponds to the structures shown in Fig. 1.
a flat region completely free of resonances is not found for the transfer function at any frequency. A similar shift but now towards higher frequencies can be observed in Fig. 3c. However, a pass-band free of resonances can be found in Fig. 3b, which corresponds to the structure of Fig. 1b. Therefore, it can be concluded that the one-side open ended structure of Fig. 1b is the structure which more closely resembles the behavior of an homogeneous slab. It is worth to note that this structure is the only one having exactly two periods across the slab width. Fig. 4a–c shows results similar to those shown in
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Fig. 4. Amplitude of the transmission coefficient for various transverse wave-lengths expressed in number of unit cells. Each one of the figures corresponds to the analysis of structures similar to that shown in Fig. 1 but with three unit cells in depth.
Fig. 3a–c, but for structures with three unit cells across the slab. The results of Fig. 4 still make clearer that a pass-band free of resonances only exists for the one-side open ended, or “periodic”, structure. 3. Experimental results The results of the previous section strongly suggest that the one-side open ended, or “periodic”, structure of Fig. 1b is the choice which provides the minimal resolution. In order to check this hypothesis the images
Fig. 5. (a) Sketch of the experimental setup and (b) normalized measurements of the image of two small loop antennas separated 4, 5, 6 and 7 unit cells produced by a split-ring slab. The vertical dashed lines show the relative location of the antennas. Solid lines: calculations from the homogenization model. Red triangles and blue squares: measurements for the structures shown in Fig. 1a and b [8], respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
generated by the structures of Fig. 1a and b will be compared in an experiment (in fact, the first structure corresponds to the lens previously reported in [6] and [8]). In the experimental setup, a pair of small loop antennas of the same diameter as the rings of the lens are placed at a distance of the lens equal to its thickness. The transverse distance between the antennas is set equal to an integer multiple of the lens periodicity a. The “image”
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
is obtained at the exit interface of the lens by measuring the transmission coefficient between the antennas and a small loop probe. Measurements are carried out by means of an Agilent Technologies network analyzer E8363B. Fig. 5 shows the experimental results obtained for both structures and for two input antennas separated by 4, 5, 6, and 7 unit cells. The measurements for the lens of Fig. 1a (red triangles in Fig. 5) were done at the frequency of 63.87 MHz. This frequency was previously determined as the optimal one by means of an accurate model specifically designed for this structure [9]. Measurements for the lens of Fig. 1b were made at the frequency of 64.8 MHz, which corresponds to a frequency in the middle of the pass-band shown in Fig. 3b. The results provided by the homogenization model [17] are also shown (black solid line) in Fig. 5. The vertical dashed lines in the Figure show the relative location of the input antennas. Measurements show that the input antennas can be well distinguished by the closed structure of Fig. 1a when they are separated at least a distance of six unit cells, whereas they can be clearly distinguished by the “periodic” structure of Fig. 1b when they are separated by a distance of five unit cells. Even for a separation of four unit cells, the periodic structure still shows two peaks in the image. 4. Conclusion Three different possibilities for implementing a μr = − 1 quasi-magnetostatic metamaterial slab lens have been analyzed using the transfer function concept. The analysis shows that the transfer function of these structures usually present resonances which may distort the image due to the disproportionate excitation of some spatial harmonics. Only for one-side open ended, or “periodic”, structures such as that shown in Fig. 1b, the transfer function shows a frequency pass-band free of resonances. This result suggest that this configuration provides the minimal resolution. An experiment has been designed and carried out in order to check this conclusion. The experimental results clearly confirms our hypothesis. Acknowledgements This work has been supported by the Spanish Ministerio de Ciencia e Innovacion and European Union FEDER funds under projects Consolider-EMET
111
CSD2008-00066 and TEC2010-16948 (SEACAM), and by the Spanish Junta de Andalucia under project TIC06238 (METAMED). References [1] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett. 85 (2000) 3966. [2] R. Marques, M.J. Freire, J.D. Baena, Theory of three-dimensional subdiffraction imaging, Appl. Phys. Lett. 89 (2006) 211113. [3] R. Marques, F. Martin, M. Sorolla, in: Metamaterials with Negative Parameters: Theory and Microwave Applications, Wiley, Hoboken, New Jersey, 2008. [4] N. Fang, H. Lee, C. Sun, X. Zhang, Sub-diffraction-limited optical imaging with a silver superlens, Science 308 (2005) 534. [5] M.C.K. Wiltshire, J.B. Pendry, J.V. Hajnal, Sub-wavelength imaging at radio frequency, J. Phys.: Condens. Matter 18 (2006) L315. [6] M.J. Freire, R. Marques, L. Jelinek, Experimental demonstration of a μ = − 1 metamaterial lens for magnetic resonance imaging, Appl. Phys. Lett. 93 (2008) 231108. [7] M.J. Freire, L. Jelinek, R. Marques, M. Lapine, On the applications of μ = − 1 metamaterial lenses for magnetic resonance imaging, J. Magn. Res. 203 (2010) 81. [8] J.M. Algarin, M.J. Freire, M.A. Lopez, M. Lapine, P.M. Jakob, V.C. Behr, R. Marques, Analysis of the resolution of split-ring metamaterial lenses with application in parallel magnetic resonance imaging, Appl. Phys. Lett. 98 (2011) 014105. [9] L. Jelinek, R. Marques, M.J. Freire, Accurate modelling of split ring metamaterial lenses for magnetic resonance imaging applications, J. Appl. Phys. 105 (2009) 024907. [10] M.J. Freire, R. Marques, Planar magnetoinductive lens for threedimensional subwavelength imaging, Appl. Phys. Lett. 86 (2005) 182505. [11] M.J. Freire, R. Marques, Near-field imaging in the megahertz range by strongly coupled magnetoinductive surfaces: experiment and ab initio analysis, J. Appl. Phys. 100 (2006) 063105. [12] O. Sydoruk, E. Shamonina, L. Solymar, Tailoring of the subwavelength focus, Microw. Opt. Technol. Lett. 49 (2007) 2228–2231. [13] O. Sydoruk, M. Shamonin, A. Radkovskaya, O. Zhuromskyy, E. Shamonona, R. Trautner, C.J. Stevens, G. Faulkner, D.J. Edwards, L. Solymar, Mechanism of subwavelength imaging with bilayered magnetic metamaterials: theory and experiment, J. Appl. Phys. 101 (2007) 073903. [14] M.J. Freire, R. Marques, Optimizing the magnetoinductive lens: improvement, limits and possible applications, J. Appl. Phys. 103 (2008) 013115. [15] R.R. Syms, E. Shamonina, L. Solymar, Near-field image transfer by magneto-inductive arrays: a modal perspective, Metamaterials 5 (2011) 8–25. [16] E. Shamonina, V.A. Kalinin, K.H. Ringhofer, L. Solymar, Magnetoinductive waves in one, two, and three dimensions, J. Appl. Phys. 92 (2002) 6252–6261. [17] J.D. Baena, L. Jelinek, R. Marques, M.G. Silveirinha, Unified homogenization theory for magnetoinductive and electromagnetic waves in split ring metamaterials, Phys. Rev. A 78 (2008) 013842.
Metamaterials 5 (2011) 107–111
Comparative analysis of the transfer function of closed and open split-ring metamaterial slab lenses J.M. Algarín, M.J. Freire ∗ , R. Marqués Dept. of Electronics and Electromagnetics, University of Seville, Facultad de Fisica, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain Received 14 December 2010; received in revised form 23 May 2011; accepted 15 July 2011 Available online 22 July 2011
Abstract The transfer function of some in-plane periodic split-ring metamaterial slab lenses is analyzed for several configurations across the slab. The transfer function is obtained by means of full-wave electromagnetic computations using the simulation software CST Microwave Studio, and it is compared with the transfer function obtained analytically for a continuous slab of reference. Significant differences are found in the transfer function of the analyzed structures. The closest behavior to the continuous slab lens was found for the partially open structure comprising an integer number of periods across the slab. Experiments are provided which confirms the theoretical results. © 2011 Elsevier B.V. All rights reserved. Keywords: Metamaterials; Sub-wavelength imaging; Super-lens; Split-rings; Capacitively loaded rings; Magnetic resonanceimaging
1. Introduction One of the most interesting properties of metamaterial slabs with relative permittivity εr = − 1 and/or relative permeability μr = − 1 is the ability of these slabs to behave as “super-lenses” with sub-wavelength (sub-λ) resolution [1]. The key mechanism behind sub-λ resolution is the amplification, inside the slab, of the evanescent harmonics coming from the source. Therefore, for a slab of thickness d, the amplitude of these harmonics is restored at a distance 2d from the source plane [1]. Nevertheless, since losses in realistic slabs prevent the amplification for large thickness, in practice sub-λ resolution is only possible in the near field region [2,3]. In this region, we are in the realm of the
∗
Corresponding author. Tel.: +34 954556408; fax: +34 954239434. E-mail address: [email protected] (M.J. Freire).
1873-1988/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.metmat.2011.07.001
quasi-statics, where both the electric and the magnetic fields can be treated separately, and we only need a slab with εr = − 1 [1,4] or μr = − 1. The choice depends on the characteristics (electric or magnetic) of the source. Metamaterial slabs with negative μr for sub-λ imaging in the radio-frequency range have been reported in [5] using swiss-rolls and in [6] using capacitively loaded split rings. Split rings have the key advantage over swiss rolls of providing three-dimensional (3D) isotropy when they are arranged in a cubic lattice, which is essential in order to image 3D sources. Medical applications of such split rings lenses in magnetic resonance imaging (MRI) have been recently reported by some of the authors [6–8]. Whereas the configuration of a periodic and isotropic split-ring metamaterial slab lens is well defined in the plane of the slab, the optimal choice is not so apparent in the direction across the slab. Up to the date, most realizations of negative μr split rings metamaterial lenses included two top and bottom split-ring layers of square
108
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
(a)
Y Z
X
a
(b)
properties have been proposed and studied [10–15]. These devices take advantage of the magnetoinductive couplings between arrays of split-ring resonators, and have been analyzed in terms of the magneto-inductive waves [16] which can propagate along such arrays. In this paper, however, we will restrict ourselves to the analysis of metamaterial structures which try to mimic continuous media with negative permeability, and more specifically to the structures sketched in Fig. 1. Although magneto-inductive waves can also propagate in such structures, this fact does not invalidate the use of homogenization techniques, provided they are correctly applied [17]. 2. Analysis
(c)
For a metamaterial slab, the transfer function T(ω, k) is defined as the transmission coefficient for each evanescent harmonic with transverse wave-number k in the plane of the slab. For an homogeneous slab of magnetic permeability μ, and in the quasi-magnetostatic limit, the transfer function is given by (Ref. [3], Eqs. 5–70): 4μμ0 e−|k|d − (μ − μ0 )2 e−|k|d
Fig. 1. Sketch of the different split-ring structures analyzed: (a) closed, (b) one-side open ended, and (c) both-sides open ended.
T (ω, |k|) =
periodicity, with a cubic arrangement of split-rings in between [6]. Such configuration is shown in Fig. 1a for a “two periods thick” slab lens. However, other configurations can be chosen, with one or both external split-ring surfaces removed, as it is shown in Fig. 1b and c, respectively. Choosing one of these configurations may depend on the specific application of the device, although for general purpose applications the choice must rely on the similarities between the imaging properties of the device and the imaging properties of a theoretical continuous medium slab lens. Imaging capabilities of an infinite metamaterial slab are summarized by the transfer function, defined as the ratio between the field at the image plane and the field at the source plane. The transfer function of a continuous slab of negative μr can be obtained analytically, and the transfer function of the structure shown in Fig. 1a was already studied by some of the authors in [9], where the similarities and differences with the transfer function of a continuous media slab lens were also analyzed. In this paper, the transfer function of the three split-ring metamaterial slab lenses shown in Fig. 1 will be analyzed, and the results will be compared with the transfer function of a continuous negative μr slab lens in order to find the optimal configuration from the point of view of the resolution. Besides the aforementioned devices, other splitring metamaterial structures with remarkable imaging
where d is the thickness of the slab. For comparison purposes we will consider a slab of an effective medium with the magnetic permeability given by Eq. (13) of [17], where magneto-inductive effects are explicitly considered through the mutual inductances between nearest neighbors. The parameters used for the calculations are the same as in the experiments reported in [6], but with the resistance of the rings reduced by two orders of magnitude. This small resistance is introduced in order to avoid numerical instabilities. Fig. 2 shows a 3D plot of this transfer function for different frequencies and for values of the transverse wavenumber k ranging in the interval [0, 0.3π/a], where a is the periodicity of the split-ring lens. Two peaks appear in the plot at each frequency. These peaks form two crests in the 3D graphic, which correspond to surface waves which are the magnetic analogous of the surface plasmons in negative permittivity slabs [3]. A flat region of T 1 can be clearly seen between the crests in Fig. 2. This region defines the domain in the ω − k space where imaging is achieved [3]. Once the main characteristics of the transfer function for a continuous slab have been reviewed, we proceed with the analysis of the discrete split-ring slabs. Since the minimum resolution of a metamaterial slab cannot be made smaller than the slab thickness (see [3] and references therein), there is no reason to use more than two
(μ + μ0
)2 e|k|d
(1)
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
109
Fig. 2. Three-dimensional plot of the transfer function of an homogeneous slab corresponding to the homogenization of the split-ring lens reported in [6].
or three periods across the slab width in the implementation of a split-ring lens. From these considerations, in the present work we will consider lenses with just two or three periods across the slab depth. The considered structures with two periods across the slab depth are shown in Fig. 1. The transfer functions of these structures are computed using the simulation software CST Microwave Studio. Computations are implemented by considering just two unit cells across the slab width and imposing periodic boundary conditions in the transverse directions. Ring properties and periodicity are chosen as in [6], except for the resistance, which is set equal to zero in order to make the results of the analysis dependent only on the discrete effects and not on losses. The transmission coefficient for different kx evanescent harmonics (with ky = 0) are computed. These values of kx are chosen so that the corresponding wavelength λx = 2π/kx is an integer number n of the periodicity a, that is, λx = na. Fig. 3a–c shows the results of the computed transfer function for the structures of Fig. 1a–c, respectively, and for different values of n = 72, 36, 18, . . ., 4. A common feature of the results obtained for the three different structures is that the value of the transmission coefficient between the resonances decays with n, that is, for short wavelengths or high harmonics. Since this happens even if the structure is lossless, it can be concluded that the discrete structure imposes a low-pass filter behavior for the spatial frequencies. Moreover, the results of Fig. 3a show that there is a shift towards lower frequencies in the resonances corresponding to higher harmonics. Therefore,
Fig. 3. Amplitude of the transmission coefficient for various transverse wave-lengths expressed in number of unit cells. Each one of the figures corresponds to the structures shown in Fig. 1.
a flat region completely free of resonances is not found for the transfer function at any frequency. A similar shift but now towards higher frequencies can be observed in Fig. 3c. However, a pass-band free of resonances can be found in Fig. 3b, which corresponds to the structure of Fig. 1b. Therefore, it can be concluded that the one-side open ended structure of Fig. 1b is the structure which more closely resembles the behavior of an homogeneous slab. It is worth to note that this structure is the only one having exactly two periods across the slab width. Fig. 4a–c shows results similar to those shown in
110
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
Fig. 4. Amplitude of the transmission coefficient for various transverse wave-lengths expressed in number of unit cells. Each one of the figures corresponds to the analysis of structures similar to that shown in Fig. 1 but with three unit cells in depth.
Fig. 3a–c, but for structures with three unit cells across the slab. The results of Fig. 4 still make clearer that a pass-band free of resonances only exists for the one-side open ended, or “periodic”, structure. 3. Experimental results The results of the previous section strongly suggest that the one-side open ended, or “periodic”, structure of Fig. 1b is the choice which provides the minimal resolution. In order to check this hypothesis the images
Fig. 5. (a) Sketch of the experimental setup and (b) normalized measurements of the image of two small loop antennas separated 4, 5, 6 and 7 unit cells produced by a split-ring slab. The vertical dashed lines show the relative location of the antennas. Solid lines: calculations from the homogenization model. Red triangles and blue squares: measurements for the structures shown in Fig. 1a and b [8], respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
generated by the structures of Fig. 1a and b will be compared in an experiment (in fact, the first structure corresponds to the lens previously reported in [6] and [8]). In the experimental setup, a pair of small loop antennas of the same diameter as the rings of the lens are placed at a distance of the lens equal to its thickness. The transverse distance between the antennas is set equal to an integer multiple of the lens periodicity a. The “image”
J.M. Algarín et al. / Metamaterials 5 (2011) 107–111
is obtained at the exit interface of the lens by measuring the transmission coefficient between the antennas and a small loop probe. Measurements are carried out by means of an Agilent Technologies network analyzer E8363B. Fig. 5 shows the experimental results obtained for both structures and for two input antennas separated by 4, 5, 6, and 7 unit cells. The measurements for the lens of Fig. 1a (red triangles in Fig. 5) were done at the frequency of 63.87 MHz. This frequency was previously determined as the optimal one by means of an accurate model specifically designed for this structure [9]. Measurements for the lens of Fig. 1b were made at the frequency of 64.8 MHz, which corresponds to a frequency in the middle of the pass-band shown in Fig. 3b. The results provided by the homogenization model [17] are also shown (black solid line) in Fig. 5. The vertical dashed lines in the Figure show the relative location of the input antennas. Measurements show that the input antennas can be well distinguished by the closed structure of Fig. 1a when they are separated at least a distance of six unit cells, whereas they can be clearly distinguished by the “periodic” structure of Fig. 1b when they are separated by a distance of five unit cells. Even for a separation of four unit cells, the periodic structure still shows two peaks in the image. 4. Conclusion Three different possibilities for implementing a μr = − 1 quasi-magnetostatic metamaterial slab lens have been analyzed using the transfer function concept. The analysis shows that the transfer function of these structures usually present resonances which may distort the image due to the disproportionate excitation of some spatial harmonics. Only for one-side open ended, or “periodic”, structures such as that shown in Fig. 1b, the transfer function shows a frequency pass-band free of resonances. This result suggest that this configuration provides the minimal resolution. An experiment has been designed and carried out in order to check this conclusion. The experimental results clearly confirms our hypothesis. Acknowledgements This work has been supported by the Spanish Ministerio de Ciencia e Innovacion and European Union FEDER funds under projects Consolider-EMET
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