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Metamaterials and superresolution: From homogenization to rigorous approach
ARTICLE IN PRESS
Physica B 394 (2007) 163–166 www.elsevier.com/locate/physb
Metamaterials and superresolution: From homogenization to rigorous approach Andrey Lagarkov, Vladimir Kisel Institute for Theoretical and Applied Electromagnetics RAS, Izhorskaya 13/19, Moscow 125412, Russia
Abstract The paper touches upon the correspondence between the widely used homogenized approach and integral equation technique in simulating electromagnetic excitation of metamaterials. The attention is drawn to metamaterials with helix-type inclusions, their properties are briefly discussed. Special attention is paid to the thin metamaterial sheets as they are the primary candidates to manufacture the so-called ‘‘superlenses’’ that can reveal the resolution overcoming the well-known diffraction limit (about half a wavelength). Experimental results are compared to computer simulation using both homogenized model and a rigorous approach. A physical interpretation is suggested of the development of an image with superresolution in a real device. r 2007 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 41.20.Jb; 78.20.Ci; 42.30.d Keywords: Metamaterial; Superresolution; Composite; Modeling
1. Introduction Modern metamaterials with negative values of permittivity and permeability may be manufactured by using inclusions of different shapes. Here we would like to draw the reader’s attention to easy to manufacture helix-type inclusions which were investigated earlier [1] and to the experimental results [2], which have demonstrated the effect of superresolution predicted previously by Pendry [3] in Veselago’s lens [4], Fig. 1a. The experimental ‘‘superlens’’ was provided by a plate of a composite material filled with resonant elements—spirals with a small pitch and linear half-wave segments of copper wire (see Fig. 1b), excited by the magnetic and electric components, respectively, of a field generated by two linear wire radiators [2]. In Ref. [2] as well as in other works [5–7], the reasons were outlined for the restriction of the limiting attainable resolution of the system originating from introducing negative permittivity and permeability, i.e. from the homogenization principle. However, despite the perfect correspondence of the experimental results and theoretical Corresponding author. Tel: +7 495 4842644.
E-mail address: [email protected] (V. Kisel). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.12.020
considerations [2] and general progress in the investigation and understanding of the effects demonstrating unusual electromagnetic response of metamaterials, a lot of questions remain unanswered (Fig. 1). Particularly, the details of the field interaction with thin metamaterial plates consisting of very few rows of resonators are not yet clear, since widely used homogenized models cannot yield reliable results in this case. For example, it was shown in Ref. [8] that, in the case of composites containing extended resonant inclusions, the effective permittivity may be introduced only for sheet materials whose thickness exceeds some critical value and, generally speaking, the value of permittivity may differ depending on the experimental conditions (see also the discussion in Ref. [9]). The experimental determination of the effective parameters of composites containing resonant inclusions is based, as a rule, on the results of measurements under incidence of a wave with a quasi-plane front [10].
2. Numerical modeling We are going to demonstrate some characteristic features of electromagnetic wave propagation in a real
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
164
a
b
R L R
Propagating modes (rays)
L R L
Source
Image Wire
“Interface”: how to outline?
LHM Evanescent modes
Does it exist?
Source Correspondence -? Source
Image
Thin (≈λ/30)
Image(?)
What’s here?
Fig. 1. Homogenized left-handed medium plate and realistic metamaterial.
composite and expose the aspects that turn out to be hidden when the effective parameters are used in Maxwell’s equations. This appeared to be possible by applying a rigorous approach (integral equation method) to obtain a full-wave solution for the electromagnetic fields. The system of integral equations of Pocklington’s type [11] was solved numerically for a detailed study of interaction between the electromagnetic wave and the composite material. Such a solution of the problem in a rigorous formulation enables one to reproduce all details of the electromagnetic processes occurring in the experimental setup, while completely taking into account the interaction between all wire resonators that make up the experimental plate. The equations are based on a thin-wire approximation with regard for the finite conductivity of the wire metal, ! Z 1 q2 eikr 0 ~i , ZI þ io m0 I ~ dl ¼ ~ vE v ~ v 2 0 k0 qvqv 4pr L P where L ¼ Li , Z ¼ ð1 þ iÞ=2pasd: Such thin-wire equations were extensively studied, and details regarding the kind of approximation, corresponding limitations, solution techniques, etc. can be found in numerous textbooks, see, for example, Refs. [12–14]. Each equation is set up relative to the linear current density I in wire elements Li of a composite; here, ~ v and v~0 are unit vectors of tangents to the wire axis at observation and integration points, respectively, r is the distance between ~i is the vector of intensity of incident these points, and E field of frequency o. The finite conductivity of the wire metal including the skin effect was taken into account by introducing its specific (per unit length) impedance Z (d is
the skin depth, and a and s denote the wire radius and conductivity of its material, respectively). The numerical algorithm to solve this integral equation was developed using the widely employed Galerkin method (i.e., the moment method with roof-top expansion and weighting functions) and was thoroughly tested [15]. The scattered field was calculated by the obtained currents using vector potentials. Discussed below are the results of numerical simulation obtained for a composite plate with a finite number of elements that corresponds to that of a real experimental sample [2]. The calculation results both reproduced the observed effect of superresolution in the presence of a complex composite medium, and made it possible to compare the phenomena occurring in real samples of composites (periodic systems of resonant elements) to phenomena occurring in homogeneous media with negative electrodynamic parameters (metasubstances) that exist only theoretically. It was demonstrated that a plate of a composite exhibits some properties typical of a plate of metasubstance. For example, a frequency band exists (as predicted by theory, it is located in the vicinity of and a little higher than the resonance frequency of inclusions) in which the effect of superresolution shows up (see Fig. 2a). For comparison, Fig. 2b gives analogous pattern in the absence of a plate, that is, an incident field pattern (all geometrical dimensions along the graph axes are given in electrical units, i.e., are multiplied by k ¼ 2p/l, where l is the wavelength in free space). By and large, the composite plate may be characterized as a device in which a backward wave exists, i.e., there is a zone of space in the vicinity of resonators in which the
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
phase and group velocities are opposite to each other. Fig. 3 shows the calculated vector of phase velocity of the total field (Ez-component) in the vicinity of the composite
Fig. 2. Excitation of the realistic metamaterial plate by a pair of filamentary sources separated by l/6.
a
Phase velocity Vph
165
plate, when a plane wave propagating in the direction of the X-axis is incident on this plate. Calculations were performed for plates with different width. In each case, the negative phase velocity region appeared to be about the same thickness, as can be seen from Figs. 3a and b. It is interesting to note that subwavelength images appear to be localized within regions with negative phase velocity (compare Figs. 2 and 3) unlike the case of the homogenized double-negative-medium plate. Then, we would like to suggest a physical interpretation for the development of an image with superresolution in a real device. Specifically, one can note that the fields of propagating harmonic waves and evanescent modes, which make up the spatial spectrum of radiation of the filamentary source, excite the resonators of the composite differently. This difference is largely due to the fact that, in the incident field of propagating harmonic (plane) waves, ~ and H ~ are in-phase, while the evanescent the vectors E modes are characterized by a phase shift of p/2 between the ~ and H. ~ Further, it is known that the phase and vectors E amplitude patterns of a system consisting of crossed magnetic and electric dipoles (models of resonators of the composite, see Fig. 4a) are defined by the relation of the phases of currents of these dipoles (curves 1 and 2, Figs. 4c and d correspond to the patterns of co-phased equivalent moments and p/2-shifted-phase moments correspondingly). Investigations reveal that the realization of suitable phase relations in the electric and magnetic resonators of the plate result in an interference of the radiated field with the incident wave field such that the propagating harmonic waves (in contrast to the evanescent modes) experience a significant attenuation. As a result, the preserved (and resonantly increased in the case of small losses) evanescent modes gives rise to the specific subwavelength maxima of the field intensity in the vicinity of the unilluminated face of the plate. Detailed graphs can be found in Ref. [15]. One can use these maxima for registering the location of sources with superresolution, as was done in the experiment.
b
Phase velocity Vph 2
2 Vph < 0
1
0
Plate
kx
kx
1
Plate
0
-1
-1
-2
-2 -1
0 ky
1
-3
-2
-1
0 ky
1
Fig. 3. Plane wave incidence on the metamaterial plates of different width.
2
3
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
166
b
c
Ce
Θ
a
E(Θ) 135
1
90
135
45
45 2 1
180
0 0.5 1 1.5 2
225 I2
Arg E(Θ)
2
Cm
I1
d
90
0 180
315
0 0 45 90 135180
225
315 270
270
d)
Fig. 4. Electromagnetic excitation of a couple of electric and magnetic resonators. (a) A couple of electric and magnetic resonators which models the unit cell of the composite. (b) Dipole electric and magnetic moments of the couple of resonators. (c) Far-field magnitude patterns of the couple of dipole moments. (d) Far-field phase patterns of the couple of dipole moments. Curves 1 (dashed lines) correspond to the case of co-phased moments while curves 2 (solid lines) correspond to p/2-shifted-phase moments thus depicting different electromagnetic responses of the couple due to excitation by propagating or evanescent components of the incident field.
Moreover, note that both the calculations and experiments with a thin plate of finite width revealed that there is no need to use a large plate for sub-wavelength imaging of sources. For example, if the plate thickness is about l/30 and the sources are spaced at a distance of l/6 from one another, their sharp separate images are obtained even for a plate width of l/4 to l/3; further increase in the plate width does not result in a marked improvement of the quality of image (resolution). This special property of a thin plane-parallel lens appears to be of particular interest, because conventional focusing devices must have lenses or mirrors that exceed significantly the wavelength in size.
focusing metamaterial structures are likely to be achieved through using full-wave solution of the electromagnetic boundary problem.
References [1] [2] [3] [4] [5]
3. Conclusion [6]
Thus, the data calculated by means of a rigorous technique were compared with the results of solutions of a similar problem in the universally accepted approximation based on the use of averaged constitutive parameters of Maxwell’s equations, namely, on the introduction of permittivity and permeability with negative values. Because the field of resonators extends significantly beyond the geometric bounds of the sample (including the zone of location of radiators and the zone of measurements), we feel there is no point in using the effective values of the parameters e and m in the considered case. Despite the usefulness of the models based on introducing effective e and m , further improvements in the design of thin-layer
[7] [8] [9] [10] [11] [12] [13] [14] [15]
A.N. Lagarkov, et al., Electromagnetics 17 (1997) 213. A.N. Lagarkov, V.N. Kissel, Phys. Rev. Lett. 92 (2004) 077401. J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. A.N. Lagarkov, V.N. Kissel, in: L. Hock, O.C. Kim, S. Matitsine, G.Y. Beng (Eds.), Proceedings of the International Conference on Materials for Advanced Applications, Symposium F: Electromagnetic Materials, World Scientific, Singapore, 2003, pp.145. D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, J.B. Pendry, Appl. Phys. Lett. 82 (2003) 1506. X.S. Rao, C.K. Ong, Phys. Rev. E 68 (2003) 067601. A.P. Vinogradov, D.P. Makhnovskii, K.N. Rozanov, J. Commun. Technol. Electron. 44 (1999) 317. L. Liu, S.M. Matitsine, Y.B. Gan, K.N. Rozanov, Electromagnetics 25 (2005) 69. D.R. Smith, S. Schultz, P. Markosˇ , C.M. Soukoulis, Phys. Rev. B 65 (2002) 195104. H.C. Pocklington, Proc. Camb. Philos. Soc. 9 (1897) 324. C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989. R. Mittra (Ed.), Computer Techniques for Electromagnetics, Pergamon Press, New York, 1973. J. Moore, R. Pizer, Moment Methods in Electromagnetics, Wiley, New York, 1984. V.N. Kissel, A.N. Lagarkov, Phys. Rev. B 72 (2005) 085111.
Physica B 394 (2007) 163–166 www.elsevier.com/locate/physb
Metamaterials and superresolution: From homogenization to rigorous approach Andrey Lagarkov, Vladimir Kisel Institute for Theoretical and Applied Electromagnetics RAS, Izhorskaya 13/19, Moscow 125412, Russia
Abstract The paper touches upon the correspondence between the widely used homogenized approach and integral equation technique in simulating electromagnetic excitation of metamaterials. The attention is drawn to metamaterials with helix-type inclusions, their properties are briefly discussed. Special attention is paid to the thin metamaterial sheets as they are the primary candidates to manufacture the so-called ‘‘superlenses’’ that can reveal the resolution overcoming the well-known diffraction limit (about half a wavelength). Experimental results are compared to computer simulation using both homogenized model and a rigorous approach. A physical interpretation is suggested of the development of an image with superresolution in a real device. r 2007 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 41.20.Jb; 78.20.Ci; 42.30.d Keywords: Metamaterial; Superresolution; Composite; Modeling
1. Introduction Modern metamaterials with negative values of permittivity and permeability may be manufactured by using inclusions of different shapes. Here we would like to draw the reader’s attention to easy to manufacture helix-type inclusions which were investigated earlier [1] and to the experimental results [2], which have demonstrated the effect of superresolution predicted previously by Pendry [3] in Veselago’s lens [4], Fig. 1a. The experimental ‘‘superlens’’ was provided by a plate of a composite material filled with resonant elements—spirals with a small pitch and linear half-wave segments of copper wire (see Fig. 1b), excited by the magnetic and electric components, respectively, of a field generated by two linear wire radiators [2]. In Ref. [2] as well as in other works [5–7], the reasons were outlined for the restriction of the limiting attainable resolution of the system originating from introducing negative permittivity and permeability, i.e. from the homogenization principle. However, despite the perfect correspondence of the experimental results and theoretical Corresponding author. Tel: +7 495 4842644.
E-mail address: [email protected] (V. Kisel). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.12.020
considerations [2] and general progress in the investigation and understanding of the effects demonstrating unusual electromagnetic response of metamaterials, a lot of questions remain unanswered (Fig. 1). Particularly, the details of the field interaction with thin metamaterial plates consisting of very few rows of resonators are not yet clear, since widely used homogenized models cannot yield reliable results in this case. For example, it was shown in Ref. [8] that, in the case of composites containing extended resonant inclusions, the effective permittivity may be introduced only for sheet materials whose thickness exceeds some critical value and, generally speaking, the value of permittivity may differ depending on the experimental conditions (see also the discussion in Ref. [9]). The experimental determination of the effective parameters of composites containing resonant inclusions is based, as a rule, on the results of measurements under incidence of a wave with a quasi-plane front [10].
2. Numerical modeling We are going to demonstrate some characteristic features of electromagnetic wave propagation in a real
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
164
a
b
R L R
Propagating modes (rays)
L R L
Source
Image Wire
“Interface”: how to outline?
LHM Evanescent modes
Does it exist?
Source Correspondence -? Source
Image
Thin (≈λ/30)
Image(?)
What’s here?
Fig. 1. Homogenized left-handed medium plate and realistic metamaterial.
composite and expose the aspects that turn out to be hidden when the effective parameters are used in Maxwell’s equations. This appeared to be possible by applying a rigorous approach (integral equation method) to obtain a full-wave solution for the electromagnetic fields. The system of integral equations of Pocklington’s type [11] was solved numerically for a detailed study of interaction between the electromagnetic wave and the composite material. Such a solution of the problem in a rigorous formulation enables one to reproduce all details of the electromagnetic processes occurring in the experimental setup, while completely taking into account the interaction between all wire resonators that make up the experimental plate. The equations are based on a thin-wire approximation with regard for the finite conductivity of the wire metal, ! Z 1 q2 eikr 0 ~i , ZI þ io m0 I ~ dl ¼ ~ vE v ~ v 2 0 k0 qvqv 4pr L P where L ¼ Li , Z ¼ ð1 þ iÞ=2pasd: Such thin-wire equations were extensively studied, and details regarding the kind of approximation, corresponding limitations, solution techniques, etc. can be found in numerous textbooks, see, for example, Refs. [12–14]. Each equation is set up relative to the linear current density I in wire elements Li of a composite; here, ~ v and v~0 are unit vectors of tangents to the wire axis at observation and integration points, respectively, r is the distance between ~i is the vector of intensity of incident these points, and E field of frequency o. The finite conductivity of the wire metal including the skin effect was taken into account by introducing its specific (per unit length) impedance Z (d is
the skin depth, and a and s denote the wire radius and conductivity of its material, respectively). The numerical algorithm to solve this integral equation was developed using the widely employed Galerkin method (i.e., the moment method with roof-top expansion and weighting functions) and was thoroughly tested [15]. The scattered field was calculated by the obtained currents using vector potentials. Discussed below are the results of numerical simulation obtained for a composite plate with a finite number of elements that corresponds to that of a real experimental sample [2]. The calculation results both reproduced the observed effect of superresolution in the presence of a complex composite medium, and made it possible to compare the phenomena occurring in real samples of composites (periodic systems of resonant elements) to phenomena occurring in homogeneous media with negative electrodynamic parameters (metasubstances) that exist only theoretically. It was demonstrated that a plate of a composite exhibits some properties typical of a plate of metasubstance. For example, a frequency band exists (as predicted by theory, it is located in the vicinity of and a little higher than the resonance frequency of inclusions) in which the effect of superresolution shows up (see Fig. 2a). For comparison, Fig. 2b gives analogous pattern in the absence of a plate, that is, an incident field pattern (all geometrical dimensions along the graph axes are given in electrical units, i.e., are multiplied by k ¼ 2p/l, where l is the wavelength in free space). By and large, the composite plate may be characterized as a device in which a backward wave exists, i.e., there is a zone of space in the vicinity of resonators in which the
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
phase and group velocities are opposite to each other. Fig. 3 shows the calculated vector of phase velocity of the total field (Ez-component) in the vicinity of the composite
Fig. 2. Excitation of the realistic metamaterial plate by a pair of filamentary sources separated by l/6.
a
Phase velocity Vph
165
plate, when a plane wave propagating in the direction of the X-axis is incident on this plate. Calculations were performed for plates with different width. In each case, the negative phase velocity region appeared to be about the same thickness, as can be seen from Figs. 3a and b. It is interesting to note that subwavelength images appear to be localized within regions with negative phase velocity (compare Figs. 2 and 3) unlike the case of the homogenized double-negative-medium plate. Then, we would like to suggest a physical interpretation for the development of an image with superresolution in a real device. Specifically, one can note that the fields of propagating harmonic waves and evanescent modes, which make up the spatial spectrum of radiation of the filamentary source, excite the resonators of the composite differently. This difference is largely due to the fact that, in the incident field of propagating harmonic (plane) waves, ~ and H ~ are in-phase, while the evanescent the vectors E modes are characterized by a phase shift of p/2 between the ~ and H. ~ Further, it is known that the phase and vectors E amplitude patterns of a system consisting of crossed magnetic and electric dipoles (models of resonators of the composite, see Fig. 4a) are defined by the relation of the phases of currents of these dipoles (curves 1 and 2, Figs. 4c and d correspond to the patterns of co-phased equivalent moments and p/2-shifted-phase moments correspondingly). Investigations reveal that the realization of suitable phase relations in the electric and magnetic resonators of the plate result in an interference of the radiated field with the incident wave field such that the propagating harmonic waves (in contrast to the evanescent modes) experience a significant attenuation. As a result, the preserved (and resonantly increased in the case of small losses) evanescent modes gives rise to the specific subwavelength maxima of the field intensity in the vicinity of the unilluminated face of the plate. Detailed graphs can be found in Ref. [15]. One can use these maxima for registering the location of sources with superresolution, as was done in the experiment.
b
Phase velocity Vph 2
2 Vph < 0
1
0
Plate
kx
kx
1
Plate
0
-1
-1
-2
-2 -1
0 ky
1
-3
-2
-1
0 ky
1
Fig. 3. Plane wave incidence on the metamaterial plates of different width.
2
3
ARTICLE IN PRESS A. Lagarkov, V. Kisel / Physica B 394 (2007) 163–166
166
b
c
Ce
Θ
a
E(Θ) 135
1
90
135
45
45 2 1
180
0 0.5 1 1.5 2
225 I2
Arg E(Θ)
2
Cm
I1
d
90
0 180
315
0 0 45 90 135180
225
315 270
270
d)
Fig. 4. Electromagnetic excitation of a couple of electric and magnetic resonators. (a) A couple of electric and magnetic resonators which models the unit cell of the composite. (b) Dipole electric and magnetic moments of the couple of resonators. (c) Far-field magnitude patterns of the couple of dipole moments. (d) Far-field phase patterns of the couple of dipole moments. Curves 1 (dashed lines) correspond to the case of co-phased moments while curves 2 (solid lines) correspond to p/2-shifted-phase moments thus depicting different electromagnetic responses of the couple due to excitation by propagating or evanescent components of the incident field.
Moreover, note that both the calculations and experiments with a thin plate of finite width revealed that there is no need to use a large plate for sub-wavelength imaging of sources. For example, if the plate thickness is about l/30 and the sources are spaced at a distance of l/6 from one another, their sharp separate images are obtained even for a plate width of l/4 to l/3; further increase in the plate width does not result in a marked improvement of the quality of image (resolution). This special property of a thin plane-parallel lens appears to be of particular interest, because conventional focusing devices must have lenses or mirrors that exceed significantly the wavelength in size.
focusing metamaterial structures are likely to be achieved through using full-wave solution of the electromagnetic boundary problem.
References [1] [2] [3] [4] [5]
3. Conclusion [6]
Thus, the data calculated by means of a rigorous technique were compared with the results of solutions of a similar problem in the universally accepted approximation based on the use of averaged constitutive parameters of Maxwell’s equations, namely, on the introduction of permittivity and permeability with negative values. Because the field of resonators extends significantly beyond the geometric bounds of the sample (including the zone of location of radiators and the zone of measurements), we feel there is no point in using the effective values of the parameters e and m in the considered case. Despite the usefulness of the models based on introducing effective e and m , further improvements in the design of thin-layer
[7] [8] [9] [10] [11] [12] [13] [14] [15]
A.N. Lagarkov, et al., Electromagnetics 17 (1997) 213. A.N. Lagarkov, V.N. Kissel, Phys. Rev. Lett. 92 (2004) 077401. J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. A.N. Lagarkov, V.N. Kissel, in: L. Hock, O.C. Kim, S. Matitsine, G.Y. Beng (Eds.), Proceedings of the International Conference on Materials for Advanced Applications, Symposium F: Electromagnetic Materials, World Scientific, Singapore, 2003, pp.145. D.R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S.A. Ramakrishna, J.B. Pendry, Appl. Phys. Lett. 82 (2003) 1506. X.S. Rao, C.K. Ong, Phys. Rev. E 68 (2003) 067601. A.P. Vinogradov, D.P. Makhnovskii, K.N. Rozanov, J. Commun. Technol. Electron. 44 (1999) 317. L. Liu, S.M. Matitsine, Y.B. Gan, K.N. Rozanov, Electromagnetics 25 (2005) 69. D.R. Smith, S. Schultz, P. Markosˇ , C.M. Soukoulis, Phys. Rev. B 65 (2002) 195104. H.C. Pocklington, Proc. Camb. Philos. Soc. 9 (1897) 324. C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989. R. Mittra (Ed.), Computer Techniques for Electromagnetics, Pergamon Press, New York, 1973. J. Moore, R. Pizer, Moment Methods in Electromagnetics, Wiley, New York, 1984. V.N. Kissel, A.N. Lagarkov, Phys. Rev. B 72 (2005) 085111.