Three-dimensional superresolution in metamaterial slab lenses: Experiment and theory

Three-dimensional superresolution in metamaterial slab lenses: Experiment and theory F. Mesa Department de Física Aplicada I. Universidad de Sevilla, 41012-Sevilla, Spain M. J. Freire, R. Marqués, and J. D. Baena Department de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012-Sevilla, Spain 共Received 29 September 2005; published 27 December 2005兲 This paper presents a theoretical and experimental study on the viability of obtaining twoand threedimensional superresolution 共i.e., resolution overcoming the diffraction limit for all directions in space兲by means of metamaterial slab lenses. Although the source field cannot be actually reproduced at the back side of the lens with superresolution in all space directions, the matching capabilities of metamaterial slabs does make possible the detection of images with three-dimensional superresolution. This imaging takes place because of the coupling between the evanescent space harmonic components of the field generated at both the source and the detector. DOI: 10.1103/PhysRevB.72.235117 PACS number共s兲: 42.30.Wb, 41.20.Jb, 73.20.Mf, 78.20.Ci I. INTRODUCTION It is well known from the early works of Veselago1that a slab made of a left-handed medium will focus the electromagnetic energy coming from a point source to another point located at the opposite side of the slab. An experimental confirmation of this focusing of energy has been reported by Houck et al.2Subsequent works3–5 have shown that, under some circumstances, metamaterial lenses made of lefthanded media can produce images at certain planes with a resolution beyond the classical diffraction limit, or “superresolution imaging” 共SRI兲. This SRI has been attributed to an amplification, inside the lens, of the evanescent space Fourier harmonics 共SFHs兲coming from the source.3It has been also discussed6that this process gives rise to fields that decay exponentially from the lens toward the image, which causes superresolution to take place only in planes parallel to the slab interfaces. In the direction perpendicular to the lens, a strong decay of the field is observed, and thus, a threedimensional 共3D兲picture of the source cannot be directly obtained from the field distribution at the back side of the lens. In other words, superresolution in the transverse directions is obtained at the price of a drastic loss of resolution in the longitudinal direction. This fact has been experimentally corroborated by recently reported field measurements,4 where the field growth from the image plane to the “super lens” can be clearly appreciated. Other experimental results also lead to the same conclusion, showing that images “of finite depth”5cannot be directly obtained from field measurements in SRI experiments. Following the seminal paper of Pendry,3other superresolution devices have been proposed making use of slabs of negative permittivity,7,8 ferrite slabs,9coupled planar polariton-resonant metasurfaces,10 and a pair of coupled magnetoinductive surfaces.11 In spite of the different physical nature of their constitutive elements, all these structures present several common characteristics. In this way, the above well-defined family of imaging devices can be characterized by: 共i兲the presence of planar slabs, 共ii兲the distance between the source and the image planes is always 2d共where dis the slab width兲, and 共iii兲the imaging properties are based on the amplification of evanescent SFHs inside the slab. Consequently, the imaging properties of Pendry’s left-handed slab lens are also expected to be shared by these devices.6 Nevertheless, some experimental results recently reported by some of the authors11 have suggested the possibility of obtaining superresolution in the longitudinal direction in one of the devices mentioned above. In these experiments, a 3D map of a pointlike source 共i.e., a source of subwavelength size兲was obtained at the image side of the lens. In other words, three-dimensional superresolution imaging 共3D-SRI兲 was obtained. The aim of the present contribution is to provide the general theory underlying this 3D-SRI. It will be shown that 3D-SRI of pointlike sources is actually a general property of the aforementioned family of devices, provided that the appropriate detection procedure is followed. II. ANALYSIS The main difference between SRI and standard imaging processes is that, in the former, the information for the image formation is carried out by evanescent SFHs,3which can support subwavelength information, unlike the propagative SFHs responsible of the conventional imaging, which can only transport overwavelength information. Since evanescent fields cannot carry power and any image measurement requires some power transmission, it is apparent that the detection procedure has to somehow “create” a traveling power flux. This combined effect should then perturb, substantially, the fields around the detector in a way similar to that found in the tunneling effect. As is well known, the tunneling of power is due to the excitation of a pair of evanescent electromagnetic waves, whose interference gives rise to a nonvanishing flux of power. “Perfect tunneling” of power in a waveguide filled by a metamaterial has been recently reported by some of the authors.12 In this work it is shown that maximum tunneling of power in a setup with identical input and output waveguides is achieved when the output is placed at a distance from the input equal to that from the source to PHYSICAL REVIEW B 72, 235117 共2005兲 1098-0121/2005/72共23兲/235117共6兲/$23.00 ©2005 The American Physical Society235117-1 the image in a metamaterial superlens. This suggests that a similar effect could take place in a metamaterial superlens, provided that a detector identical to the source is used for the measurements. In the following, how to take advantage of such an effect in order to obtain 3D-SRI with metamaterial slabs will be shown. The present study starts with the canonical problem of the formation of images by a left-handed slab of thickness d characterized by ⑀ / ⑀ 0= ␮ / ␮ 0=−1+i ␦ , where ␦ Ⰶ1 accounts for the necessary losses factor to avoid the divergence of field integrals.13–15 In our study, and following a usual procedure in microwave SRI experiments, the source will be an antenna 共specifically, a loop antenna兲whose plane is located parallel to the slab interfaces as shown in Fig. 1. The field beyond the lens will be scanned by an output antenna that plays the role of detector. The source employed here is equivalent to an homogeneous surface distribution of magnetic dipoles given by Ms= 再 I0e−i ␻ t,inside the loop 0, outside, 冎 共1兲 where Idenotes the amplitude of the imposed time-harmonic current in the loop. The computation of the longitudinal magnetic field Hzat the image side of the slab 共z⬎⌬+d兲can be carried out by means of the following double-inverse Fourier transform: Hz共x,y,z兲=1 4 ␲ 2 冕冕 dkxdkyG ˜ 共kx,ky;z兲M ˜ s共kx,ky兲ei共kxx+kyy兲, 共2兲 where G ˜ 共kx,ky;z兲is the Fourier transform of the Green’s function of the structure under study and M ˜ s共kx,ky;z兲, the Fourier transform of the spatial surface distribution of magnetic dipoles. After applying the duality principle to expression 共6兲in Ref. 15 and taking ⌬=d/2, the Fourier transform of the Green’s function is found to be G ˜ 共kx,ky;z兲=−共kx 2+ky 2兲2 ␤ e−i ␤ 0共z+d兲 ␻ 兵共 ␤␮ 0− ␤ 0 ␮ 兲2ei ␤ d−共 ␤␮ 0+ ␤ 0 ␮ 兲2e−i ␤ d其, 共3兲 where ␤ =冑 ␻ 2␧ ␮ −kx 2−ky 2 ␤ 0=冑 ␻ 2␧0 ␮ 0−kx 2−ky 2. Since Ez=0 in the present case, all the remaining field components can be deduced from Hz.15 The numerical computation of 共2兲provides the map depicted in Fig. 2 for the magnitude of Hzin the 共x−z兲plane at the back side of the lens. It should be noted that, given that the operation frequency is 3 GHz 共␭0=100 mm兲and that d and ⌬are taken much less than the free-space wavelength, the ray model approximation cannot be employed here to obtain the fields in the considered region 共in other words, we are dealing with near fields and therefore in a SRI situation兲. According to previous discussions, Fig. 2 shows that the field magnitude has a strong decay along the zdirection, so that no information about the location of the source in the longitudinal direction can be extracted from the field distribution. A more detailed picture of the field distribution at two planes z=共2± ␦ 兲dis plotted in Fig. 3 共with ␦ set to 0.3兲and compared to the field amplitude near the source at z=± ␦ d共the field distributions at z= ␦ dand z=− ␦ dare identical兲.Asis expected from the properties of the metamaterial slab, the field distribution at the z=共2+ ␦ 兲dplane is almost identical to that at the z= ␦ dplane, thus confirming that the z= ␦ dand the z=共2+ ␦ 兲dplanes are, in fact, conjugate planes. On the contrary, the field distribution at the z=共2− ␦ 兲dplane substanFIG. 1. Geometry of a metamaterial lens. The source is a loop antenna located at z=0. The detector is also an identical loop antenna. The lens is formed either by a left-handed metamaterial slab or by another planar device that produces SRI through amplification of evanescent harmonics. The output antenna can eventually be loaded with microwave resistors in order to minimize field perturbation. FIG. 2. 共Color online兲Mapoflog 10兩Hz兩corresponding to the field generated at the back side of the lens by a source loop antenna shown in Fig. 1. The slab is made of a left-handed medium of thickness d=4 mm with ⑀ / ⑀ 0= ␮ / ␮ 0=−1+i0.001, and is separated by a distance ⌬=d/2 from the source. The source is considered, for simplicity, a circular loop antenna of radius r1=5 mm. The wire radius is of 0.2 mm. The operation frequency is 3 GHz. MESA et al. PHYSICAL REVIEW B 72, 235117 共2005兲 235117-2 tially differs in magnitude from the field distribution at the z=− ␦ dplane, near the source. These facts corroborate the aforementioned discussions: the superresolution in the transverse directions 共the lateral dimensions of the source and the image are approximately one-tenth of the wavelength兲is compensated by an almost complete loss of resolution in the longitudinal direction. The problem of the measurement of the image in the SRI setup under study is next considered. In the microwave range, the image detection is performed by measuring the transmission coefficient between a source antenna and a receiving antenna, which is scanned in the image side of the lens.2,4,5 For simplicity, the receiving 共or output兲antenna is assumed to be identical to the input antenna employed as source. The transmission coefficient is measured by connecting the input antenna to a wave generator via a waveguide, and the output one to a detector through another identical waveguide 共more details of this measurement setup are reported in Ref. 11兲. In this approach, the metamaterial slab should be viewed as a matching device,16 whose transmission coefficient t共or S21 in the usual microwave terminology兲 is given by17 t=2Z12Z0 共Z11 +Z0兲共Z22 +Z0兲−Z12 2,共4兲 where Zij are the elements of the impedance matrix for the system formed by the two antennas and the left-handed slab, and Z0is the characteristic impedance of the input and output waveguide. In order to measure a superresolution image, the size of the loop antennas has to be smaller than the freespace wavelength, which additionally would make the real part of Zij 共the radiation resistance17兲be negligible with respect to its imaginary part; namely, Zij⯝−i ␻ Lij, where Lij is the inductance matrix of the system. The diagonal terms of the inductance matrix correspond to the inductances of a single-loop antenna faced to the left-handed slab. However, in the “perfect lens” configuration considered here, the slab does not affect the fields around the source3and, therefore, L11=L22 ⬅L, where Lis the self-inductance of the loops in free space. The nondiagonal terms of the inductance matrix account for the mutual inductance of the loop antennas in the presence of the slab, namely, L12 =L21⬅M. Thus, the transmission coefficient in 共4兲can be written as t=2i ␻ MZ0 ␻ 2共L2−M2兲+2i ␻ LZ0−Z0 2.共5兲 Since Ldoes not change with the position of the antennas, the spatial dependence in 共5兲comes only from M. The dependence of 兩t兩with the reactances X11=X22 =− ␻ Land X12 =X21=− ␻ Mdeduced from 共5兲is shown in Fig. 4. It can be seen how the maximum transmission 共兩t兩⬇1兲is achieved when X11⯝X12, except for very small values of the ratio X11/Z0. This effect is illustrated in Fig. 5, where the values of X12 and X11, which corresponds to a maximum of the transmission coefficient, are plotted. Therefore, except when ␻ L ⰆZ0, the transmission coefficient reaches its maximum at those points where M⬇L共兩t兩⬇1ifLⲏZ0兲. Since both the input and output antennas are identical, and the field at the source plane is reproduced at the image plane 共see Fig. 3兲, FIG. 3. Plot of the log10兩Hz兩at different z-planes of Fig. 1. The slab and the source loop antenna are as in Fig. 2. FIG. 4. Plot of the modulus of the transmission coefficient t =S21 as a function of the modulus of the reactances X11=X22= − ␻ Land X21=− ␻ M. FIG. 5. Plot of the values of the reactances X11=X22=− ␻ Land X112=X21=− ␻ Mfor which the modulus of the transmission coefficient 共5兲reaches a maximum. THREE-DIMENSIONAL SUPERRESOLUTION IN…PHYSICAL REVIEW B 72, 235117 共2005兲 235117-3 the condition M⬇Lis expected to be satisfied around the point 共x,y,z兲=共0,0,2d兲. In consequence, a maximum of the transmitted power should be detected around this particular point, and thus, it would appear as an “effective focusing point” of the perfect lens. In order to show the above expected effects in a practical situation, the magnitude of the transmission coefficient has been numerically computed for the configuration under study. The impedance matrix has been calculated by imposing known currents, Ii=1,0 A, in the loops, and then computing the corresponding self and mutual inductances. The computation of the self-inductance is an standard electromagnetic problem, and the mutual inductance is obtained after computing the flux of the magnetic field given by 共2兲 across the surface of the output antenna. The transmission coefficient is finally determined from 共4兲, assuming Z0 =50 ⍀. In Fig. 6, a map of the computed transmission coefficient for a system composed of two identical lossless loop antennas is shown. It can be observed that this figure substantially differs from Fig. 2; in particular, a clear maximum of 兩t兩can be observed in Fig. 6 in the neighborhood of the image at 共0,0,2d兲. These results clearly show the difference between the transmitted power and the field distribution in the absence of the output antenna, and also how 3D-SRI can be obtained when the appropriate detector is used. Let us now consider the measurement of the unperturbed field. For this purpose, the detector should be designed to affect the field distribution as little as possible. It could be closely achieved by loading the output antenna with an additional high resistance, which will significantly reduce the current induced at the output antenna, as well as its generated field. In Fig. 7 the computed values of the transmission coefficient along the z-axis of Fig. 1 are shown for different resistances, R, loading the output antenna. As is expected, the curve for R=0 ⍀shows a maximum near the location of the image, which thus appears as a “focusing point” of the lens, at z⬇2d.共The small difference between the actual location of this maximum with respect to z=2dcan be attributed to the not very high value of the ratio ␻ L/Z0⬇8 in the case under study兲. As is also expected, the curves for the highest values of Rresemble the behavior of 兩Hz兩in Fig. 2 for the unperturbed configuration. Another obvious strategy for reducing the perturbation of the field by the measurement is to reduce the radius of the output antenna of Fig. 1. This will reduce both L2共the inductance of the output antenna兲and Mwithout changing L1共the inductance of the input loop兲. In the limit when ␻ L2, ␻ M ⰆZ0,共4兲reduces to t⬇2i ␻ M i ␻ L1−Z0 .共6兲 Since both L1and Z0do not depend on the location of the output, the measured transmission coefficient turns out to be proportional to M, i.e. to the magnetic field Hzat the location of the output. Thus, the analyzed experimental setup is appropriate for the detection of the field of the unperturbed system formed by the input antenna and the lens. The magnitude of the transmission coefficient along the zaxis of Fig. 1 is plotted in Fig. 8 for several values of the radius of the output loop. The curves corresponding to r2艌4 mm clearly show that the location of the detected transmission maximum approaches the lens interface as the radius of the output loop is reduced. In the limit of small radius 共r2=1 mm兲, the previously predicted monotonic increase of the field toward the right-hand-side lens interface is observed. Figure 8 also shows that the sensitivity of the experimental setup to small differences between the shapes of the input and output antennas is not very high. As it can be readily seen from the curves corresponding to r2=5 and 4.5 mm, the transmission peak is located very close to 2dfor moderate deviations of r2 from its optimum value r2=r1. III. EXPERIMENT In order to show the application of the above theory to practical devices, the experimental demonstration of 3D-SRI recently reported by some of the authors11 will now be reexamined in light of the above considerations. Although the FIG. 6. 共Color online兲Map of the magnitude in decibels of the transmission coefficient, 兩S21兩, between the input and output antennas of Fig. 1. The structural parameters are as in Fig. 2. The output antenna is identical to the input one. FIG. 7. Magnitude of the computed transmission coefficient along the Zaxis between the input and output antennas of Fig. 1 for different resistances loading the output antenna. The structural parameters are as in Fig. 2. MESA et al. PHYSICAL REVIEW B 72, 235117 共2005兲 235117-4 underlying physics of the device analyzed there is not exactly the same as that of the left-handed perfect lens,11 both systems show the same process of amplification of evanescent modes and are equivalent in practice for the present purposes. Thus, experimental results plotted in Fig. 9 show the location of the maximum of the transmission coefficient along the zaxis of the magnetoinductive lens11 for different values of the resistance loading the output loop. The experimental setup used to obtain these results is described in detail in Ref. 11. The only difference is that the output antenna is now loaded by different microwave resistors to obtain the results shown in Fig. 9. These results show the same type of behavior as that reported in Fig. 7; namely, the location of a maximum for the magnitude of the transmission coefficient in the neighborhood of z⬇2dfor identical output and input antennas, and a similar displacement of the maximum of the transmission coefficient as the radius of the output antenna decreases. The agreement between the theory and the experiments can be considered as a validation of the present theory. It also shows that the ideas developed in the previous analysis are very general, and that, actually, they are applicable to any superresolution lens based on the amplification and amplitude restoration of FHs along the device. IV. CONCLUSIONS Superresolution in metamaterial superlenses has been investigated. As is well known, this imaging is primarily due to the amplification of evanescent modes inside the lens. As a consequence, superresolution in the planes parallel to the slab is unavoidably compensated by a drastic loss of resolution in the direction perpendicular to the slab. However, since evanescent modes do not carry power, any physical detection 共namely, a measurement兲of the image at the back side of the lens will require the existence of some amount of transmitted power from the source to the detector, which can significantly affect the original field distribution. If this last effect is taken into account, the secondary fields generated by the presence of the detector should be considered. Following this approach, it has been shown that the transmission coefficient between the source and the detector can be increased considerably. Provided that the appropriate detector is employed 共typically, a lossless output antenna identical to the input antenna兲, this maximum will occur in a neighborhood of the image, thus resulting in superresolution imaging also in the direction perpendicular to the slab. However, it should be emphasized that for obtaining this 3D-SRI, some previous knowledge of the source is necessary in order to design the detector properly. Thus, a general conclusion arises from the analysis: Superresolution in metamaterial superlenses is always incomplete; if the distance from the source to the lens is known, the shape and characteristics of the source can be recovered without uncertainty from the analysis of the field distribution at the image side of the lens. Conversely, if the shape and characteristics of the source are known, it is possible to design an appropriate detector in order to localize the source spatially from the values of the transmission coefficient between the source and the detector. However, it is impossible to determine, simultaneously, by means of a metamaterial superlens, the location, shape, and characteristics of an unknown source with a resolution overcoming the diffraction limit. We feel that the reported analysis and experiments, as well as the conclusions arising from them, will be of importance in the design of metamaterial superresolution devices. ACKNOWLEDGMENT This work has been supported by the Spanish Ministry of Education and Science by project Contract No. TEC200404249-C02-02. FIG. 8. Magnitude of the computed transmission coefficient along the zaxis between the input and output antennas of Fig. 1 for several values of the radius of the detector 共r2兲. The structural parameters are as in Fig. 2. FIG. 9. 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