Planar magnetoinductive lens for three-dimensional subwavelength imaging

Planar magnetoinductive lens for three-dimensional subwavelength imaging M. J. Freirea兲and R. Marqués Departamento de Electrónica y Electromagnetismo, Facultad de Física, Universidad de Sevilla, Avenida Reina Mercedes s/n, E 41012 Seville, Spain 共Received 5 January 2005; accepted 15 March 2005; published online 28 April 2005兲 A planar near-field magnetoinductive lens operating in the microwave range is presented. The proposed device consists of two parallel planar arrays of metallic broadside coupled 共BC兲split-ring resonators 共SRRs兲, or BC-SRRs. The power coming from a point-like source located in front of the lens is focused into a receiver located in free space beyond the lens. This focus is clearly separated from the back side of the lens, and has a size that is an order of magnitude smaller than the free space wavelength of the incoming field. The imaging properties of the device relies mainly on the excitation of magnetoinductive surface waves on the BC-SRR arrays. By simply scaling the BC-SRRs’ size, as well as the arrays periodicity, the operation frequency of the device can be tuned in a wide frequency range. Thus, the proposed design is potentially useful for many applications ranging from megahertz to terahertz. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1922074兴 After the seminal papers of Veselago1and Pendry2on the imaging properties of left-handed slabs, there has been a great deal of interest on this subject, including the design of subdiffraction imaging devices.2In fact, lenses overcoming diffraction limits could find application not only in imaging devices, but also in the recording of information, microwave heating, and many other technological areas. However, it soon became apparent that losses and dispersion—both unavoidable in any practical left-handed material—would strongly limit the performance of these devices.3–5 In particular, losses will prevent subwavelength imaging at distances higher than a small number of wavelengths in any practical lens.4,5 Therefore, attention will soon focus in the near-field imaging of electromagnetic signals. In this way, a quasiperfect imaging in the quasielectrostatic limit by using a slab of negative dielectric permitivitty 共␧兲was proposed in Ref. 关2兴. Since many metals show this property at optical frequencies, thin metallic slabs could be good candidates for obtaining near-field subwavelength imaging at these frequencies. Quasiperfect near-field imaging can be also obtained by using magnetized ferrite slabs operating in the microwave range.6 Practical realizations of subdiffraction imaging devices are currently restricted to the microwave range.7,8 These realizations make use of planar circuit analogous of lefthanded media7as well as of very thin 共single-cell depth兲 highly anisotropic left-handed slabs in free space.8The first realization7does not seem to open the way to any possible generalization leading to three-dimensional practical lenses. The second one, apart from being highly anisotropic, does not enable the reproduction of images “of finite depth,”8 since the imaging is restricted to planes parallel to the lens. Microwave three-dimensional imaging was already reported in Ref. 9, although the size of the focus was not of subwavelength dimensions. Thus, it can be concluded that devices producing three-dimensional isotropic subwavelength imaging have not been previously reported. In this letter we present a planar magnetoinductive lens for near-field imaging in the microwave range. As in previous work,2,6–8 the key concept of the proposed imaging process is the amplification, inside the lens, of the evanescent Fourier harmonics 共FHs兲coming from the source. This amplification restores the amplitude of each FH at the image plane so that the source field is reproduced at this plane. In negative-␧imaging devices2and in magnetized ferrite slabs,6 this amplification is due to the excitation of surface waves 共surface plasmons in metals and magnetostatic surface waves in ferrites兲at the shadowed interface of the slab. These surface waves, which decay from this interface towards both the source and the image, are coupled to the decaying evanescent FHs coming from the source in such a way that the amplitude of these FHs is restored at the image plane.2,6 This mechanism is sketched in Fig. 1共a兲, where the decayinggrowing-decaying process is illustrated for a single FH. 共Recently, a similar mechanism of amplification of evanescent modes has been reported in a pair of coupled resonator arrays.10兲In order to achieve ideal perfect imaging, this process should take place for all the FHs coming from the source. Since the lens operates at a single frequency, it implies that the dispersion relation for the surface waves at the shadowed interface of the lens 关the plane z=0 in Fig. 1共a兲兴 should be very flat, so that surface waves corresponding to whatever value of the wave vectors 共kx,ky兲can be simultaneously excited at the frequency of operation of the lens.2,6,10 This condition is actually fullfilled by the ideal surface plasmons supported by a negative-␧slab interface,2and by the magnetostatic surface waves supported by a magnetized ferrite slab interface.6 In the proposed device 共see Fig. 2兲, magnetoinductive surface waves are excited at the shadowed lens interface. Magnetoinductive waves were first reported by Shamonina et al.11 in both one-, two-, and three-dimensional arrays of split-ring resonators 共SRRs兲. These waves are produced by the inductive coupling between SRRs, due to the strong magnetic dipoles induced at each resonator near its resonance. Magnetoinductive waves behave as quasimagnetostatic sura兲Electronic mail: [email protected] APPLIED PHYSICS LETTERS 86, 182505 共2005兲 0003-6951/2005/86共18兲/182505/3/$22.50 © 2005 American Institute of Physics86, 182505-1 face waves in many aspects, as it was noted in Ref. 12. The dispersion relation for a planar square array of SRRs as those shown in Fig. 2 can be written as11,12 ␻ 0 2 ␻ 2=1+2M L关cos共kxa兲+ cos共kya兲兴,共1兲 where only the coupling between the nearest neighbors has been taken into account.12 In Eq. 共1兲,Land Mdenote the SRRs’ self-inductance and mutual inductance, respectively, and ais the array periodicity. The fractional bandwidth ⌬ ␻ / ␻ 0, predicted by Eq. 共1兲is given by ⌬ ␻ ␻ 0 ⬇4M L,共2兲 where ⌬ ␻ is the bandwidth of the magnetoinductive surface wave excitation. This bandwidth is very small, of the order of a 10% in the configuration shown in Fig. 2. Thus, the dispersion curve of the magnetoinductive waves is almost flat, which is required for subwavelength imaging. Since magnetoinductive waves are quasimagnetostatic waves, the magnetic field Hcan be obtained from the quasistatic magnetic potential ⌽mas H=−grad ⌽m. Since the magnetostatic potential satisfies Laplace’s equation in free space, the zdependence for the magnetoinductive waves that can be excited at the SRR square arrays is given by ⌽m共kx,ky,z兲=⌽m,0 exp关±k兩z−z0兩兴,共3兲 where ⌽m,0 is a constant, k=共kx 2+ky 2兲1/2, and z0denotes the position of each lens interface. According to Fig. 1, the plus sign in Eq. 共3兲corresponds to z0=−dand the minus sign corresponds to z0=0. It should be noted that the behavior of ⌽mat both sides of z=−dwould be nonphysical for an isolated SRR array, since the magnetostatic potential would grow to infinity at both sides of the array. However, the presence of the source at z=−3d/2, as well as that of the companion SRR array at z=0, allows for the excitation of this field configuration around the SRR array at z=−d. Regarding the behavior of ⌽mnear the SRR array, it differs from that of surface plasmons and ferrite magnetostatic potentials.2,6 On the one hand, the magnetostatic potential ⌽mshould experience a discontinuity at each lens interface equal to the mean magnetic dipole moment per unit area: ⌬⌽m=N具mz典共具mz典is the average magnetic dipole of the SRRs and Nis the number of SRRs per unit area兲. On the other hand, the normal component of the magnetic field 共Hz兲 should be continuous across the lens interfaces. The behavior of ⌽mfor the considered magnetoinductive surface waves is shown in Fig. 1共b兲. Although this behavior differs from that reported in Refs. 2 and 6 for surface plasmons and magnetostatic surface waves, it can still produce FH amplification, as sketched in Fig. 1共b兲关It is worth noting that if the magnitude of the magnetostatic potential were plotted, this graphic would be identical to Fig. 1共a兲.兴In Figs. 1共a兲and 1共b兲it can be seen how a single FH coming from the source 共at z=−3d/2兲is coupled to a surface wave excited at the shadowed lens interface 共at z=0兲, and how its amplitude is finally restored at the focus plane 共z=d/2兲. Figure 2 shows the practical device employed in this work. The lens consists of two parallel square arrays of SRRs separated a distance d共a layer of Rohacell™ foam was placed between the layers containing the arrays to achieve mechanical stability兲. Broadside coupled SRRs 共BC-SRRs兲 are used instead of conventional SRRs in order to avoid crosspolarization effects, and to reduce the electrical size of the resonators.13 The BC-SRRs are photoetched on a dielectric substrate of thickness h, as is shown in the inset of Fig. 2. Two square loop antennas are used as source and receiver, respectively. These square loops are also fabricated by photoetching a metallic pattern on a dielectric substrate. The size of these loops is taken as four times the unit cell area of the BC-SRRs array in order to average the detailed contribution to the total field coming from each individual resonator. The FIG. 2. Experimental setup for the measurement of the reported magnetoinductive microwave lens. The lens consists of two parallel square planar dielectric substrates with an area of 7⫻7cm 2, separated by a foam slab of thickness d=4 mm. A periodic two-dimensional array of BC-SRRs is photoetched on the two interfaces of each substrate. The square inset in the figure shows in detail the geometry of the BC-SRR, which has an outer radius r=2 mm and a ring width w=0.5 mm. The periodicity of the array is 5 mm. The dielectric substrate is alumina, which is commonly used in microwave circuits, with thickness h=0.254 mm and dielectric permittivity ␧r=10. Input antenna and receiving antenna are placed at the opposite sides of the lens. Both antennas are squared loops with an area of 1⫻1cm 2and are fabricated by photoetching a metallic pattern on an low permittivity substrate. The input antenna is fixed at a certain distance from the left interface of the lens, and the receiving antenna can be scanned along the Y-Z directions shown in the figure. FIG. 1. 共a兲Decaying-growing-decaying behavior of the electrostatic potential ⌿in the negative-permittivity 共␧兲lens. 共b兲Decaying-growing-decaying behavior of the magnetostatic potential ⌿min the magnetoinductive 共MI兲 lens. 182505-2 M. J. Freire and R. Marqués Appl. Phys. Lett. 86, 182505 共2005兲 input antenna 共the source兲is fixed at a distance d/2 from the lens; the output antenna is scanned in the y-zplane in order to search for the image of the input antenna. Figure 3共a兲 shows the measured transmission coefficient between both antennas when the lens is present, and Fig. 3共b兲shows the measurement when the lens is removed. Figure 3共a兲clearly shows that the transmitted power has a maximum at a distance d/2 of the shadowed interface of the lens, that is, at a distance 2dfrom the the input antenna, in agreement with the proposed theory. Nothing similar is observed when the lens is absent. The half-intensity width of the maximum is around 10 mm in the lateral direction 共yaxis兲, and less than 2 mm in the perpendicular one 共zaxis兲. These distances are smaller than the free space wavelength at the operating frequency 共about 100 mm兲by one order of magnitude. The intensity of the maximum is indeed remarkable, since it is one order of magnitude higher than the intensity at the same point when the lens is removed. A noticeable fact is that the reported maximum is not located at the lens shadowed interface but clearly removed to the neighborhood of z=d/2. Thus, a three-dimensional subwavelength image of the source is obtained. The slight asymmetries of the field patterns shown in Fig. 3 can be attributed to tolerances in the lens fabrication process, or to the positioning device used for scanning the output antenna. In summary, subwavelength microwave threedimensional imaging by a flat magnetoinductive microwave lens has been shown. The reported image has a subwavelength size, which is in agreement with the “perfect lens” theory,2and is clearly removed from the edge of the lens. An advantage of the reported device over previous near-field lens designs is the simplicity of its manufacturing process, which consists mainly of etching two planar metallic patterns on a conventional circuit board by means of standard microwave circuit manufacturing techniques. For these reasons, we feel that the reported concepts can substantially improve present approaches to electromagnetic imaging devices. Since BC-SRRs are metallic resonators with a linear response, whose frequency of resonance can be tuned by simply scaling their size, the operation frequency of the proposed device can be also tuned in a broad frequency range. This frequency range will be limited mainly by the usefulness of the SRR topology as a practical magnetic resonator design. Since this usefulness seems to have been actually demonstrated up to the terahertz range,14 the limits of application of the proposed design would also extend at least to these frequencies. In addition to its theoretical interest, devices based on the reported concepts can find practical applications in many technological areas, such as microwave imaging or heating for industrial and health applications. This work has been supported by DGI, Ministerio de Educación y Ciencia 共SPAIN兲, under project contract no. TEC2004-04249-C02-02. 1V. G. Veselago, Usp. Fiz. Nauk 92, 517 共1967兲关Sov. Phys. Usp. 10,509 共1968兲兴. 2J. B. Pendry, Phys. Rev. Lett. 85, 3966 共2000兲. 3N. García and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 共2002兲. 4D. R. Smith, D. Schurig, M. Rosebluth, S. Schultz, S. Anantha- Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 共2003兲. 5R. Marques and J. Baena, Microwave Opt. Technol. Lett. 41,290共2004兲. 6R. Marques, F. Mesa, and F. Medina, Appl. Phys. Lett. 86, 023505 共2005兲. 7A. Grbic and G. Eleftheriades, Phys. Rev. Lett. 92, 117403 共2004兲. 8A. N. Lagarkov and V. N. Kissel, Phys. Rev. Lett. 92, 077401 共2004兲. 9A. A. Houck, J. B. Brock, and I. L. Chuang, Phys. Rev. Lett. 90, 137401 共2003兲. 10S. Maslowski, S. Tretyakov, and P. Alitalo, J. Appl. Phys. 96, 1293 共2004兲. 11E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, J. Appl. Phys. 92,6252共2002兲. 12M. J. Freire, R. Marques, and F. Medina, Appl. Phys. Lett. 85,4439 共2004兲. 13R. Marqués, F. Medina, and R. Rafii-El-Idrissi, Phys. Rev. B 65, 14440 共2002兲. 14S. Linden, C. Enrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, Science 306, 1351 共2004兲. FIG. 3. 共Color online兲Experimental results obtained with the setup shown in Fig. 2. The data are referred to the magnitude of the transmission coefficient between the input antenna and the receiving antenna with the lens 共a兲 and without it 共b兲. The measurements were carried out by using a Vector Network Analyzer HP 8510 B. The input antenna is fixed at a distance of 2 mm from the left interface of the lens 共the location of the left interface corresponds to z=−3d/2 in Fig. 1共b兲and its operating frequency is 3.23 GHz. The receiving antenna is scanned along the Y-Zdirections shown at the right side of the lens in Fig. 2. Note that in 共a兲the image is formed at the point y=0 mm, z=2.25 mm 关z=d/2 in Fig. 1共b兲兴; that is at a distance of 2.25 mm from the right interface of the lens. This distance agrees very approximately with the half-width of the lens; i.e., with the sum of the foam half-width 共2mm兲and the microwave board thickness 共0.254 mm兲. Therefore, the image formation obeys the proposed theory. Note that in 共b兲the magnitude of the transmission coefficient in the point y=0 mm, z =2.25 mm in the absence of the lens is one order of magnitude lower than in the same point when the lens is present. 182505-3 M. J. Freire and R. Marqués Appl. Phys. Lett. 86, 182505 共2005兲