Negative magnetic permeability of split ring resonators in the visible light region

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1 Optics Communications 8 () 3 3.elsevier.com/locate/optcom Negative magnetic permeability of split ring resonators in the visible light region Atsushi Ishikaa a,b, Takuo Tanaka a, * a Nanophotonics Laboratory, RIKEN (The Institute of Physical and Chemical Research), - Hirosaa, Wako, Saitama 3-98, Japan b Department of Applied Physics, Osaka University, - Yamadaoka, Suita, Osaka -87, Japan Received July ; accepted 9 July Abstract Negative magnetic permeability of split ring resonator (SRR) is theoretically investigated in the visible light region. In our calculations, e considered the delay of the current inside the metal SRR in order to estimate the permeability of the SRR precisely. From the results, the silver SRR array ith small capacitance exhibits negative permeability in all the visible range. The dependence of the permeability on the filling factor of the SRR elements is also discussed. Ó Elsevier B.V. All rights reserved. PACS: 73..Mf; 78..Bh; 78..Ci Keyords: Negative permeability; Split ring resonator; Plasmonic nano structure; Metamaterial; Left-handed material. Introduction In, an extraordinary phenomenon termed perfect imaging or superlenses as predicted by Pendry []. He claimed that a metallic slab of negative refractive index acts as a lens to image the complete spatial components of an object into the other side. This imaging system exceeds the diffraction limit, and structural components smaller * Corresponding author. Tel.: ; fax: address: t-tanaka@riken.jp (T. Tanaka). than the incident avelength are imaged. The requirement for perfect imaging is negative value for the refractive index of the slab, or negative values for both permeability l and permittivity e [,3]. Hoever, in the optical frequency region, l of most materials in nature is approximately unity. Pendry [] has also indicated theoretically that an array of split ring resonators (SRRs) behaves as an artificially negative l material in a particular frequency region. After PendryÕs ork, Smith [ 7] made the array of SRRs and demonstrated experimentally that the SRR array orks as the negative l material in microave region. As far as e 3-8/$ - see front matter Ó Elsevier B.V. All rights reserved. doi:./j.optcom..7.7

2 A. Ishikaa, T. Tanaka / Optics Communications 8 () kno, this SRR array is the first negative l material. Recently experimental studies of magnetic properties of SRRs have been reported in the THz region [8], particularly in the 3 THz ( lm avelength) [9], and in the THz (3 lm avelength) []. On the other hand, theoretical studies of the SRRsÕ behavior have been done from THz to the near-infrared region [,] and no the interest of this field moves to the visible light range. In the visible light range, e must determine the effect of the dispersion of both the surface resistivity and the internal reactance of metals on the resonatorõs properties precisely in order to understand and estimate the magnetic responses of the SRRs. The message in this paper is that as the frequency increases, both surface resistivity and internal reactance of the metals increase. The increase of the surface resistivity results in the decrease of the Q-value of the SRR; the decrease of the Q-value degrades the tunable range of the permeability. The increase of the internal reactance results in the reduction of the resonant frequency of SRR. Hoever, e used the complete formula of the internal impedance, hich holds to the visible range, to estimate the dispersion of both surface resistivity and internal reactance accurately and e found that the silver SRR array realizes the negative l in all visible light range. We also determined the magnetic responses of the SRRs from THz to the visible light region. In addition, the dependence of the magnetic responses of SRRs on filling factor is also discussed.. Theory Fig. shos the SRR model used in our calculations. Fig. (a) depicts the structure of the SRR. The SRR consists of double rings ith a gap that governs the direction of an AC current around the to rings. When a time-varying external magnetic field H ext is applied to the SRR, an induced current J flos along the rings, and this induced current produces an internal magnetic field H int. The internal magnetic field H int is the origin of the magnetic response of the SRR. Fig. (b) and (c) shos the a c J d array of the SRRs and the definition of a plane conductor used in our calculations, respectively. The basic form of the effective permeability of the SRRs is derived from [,] and given as F x l eff ¼ l Re þ il Im ¼ ; ðþ x C gl g þ i ZðxÞx L g here F is the filling factor, x is the angular frequency, C g and L g are the geometrical capacitance and inductance, and Z(x) is the ring metal impedance. In the case of the SRR placed in vacuum, F, C g and L g are represented by F ¼ pr a ; r C g ¼ pr 3 e Hint Rs + ixs Hext J b pffiffiffiffiffiffiffiffiffiffiffiffi K t KðtÞ l z x Fig.. Models of split ring resonator (SRR) used in our calculations: (a) the element of the SRR; (b) the array of the SRR; (c) a plane conductor for the ring. R s, the surface resistivity; X s, the internal reactance. ; t ¼ d þ d ; ðþ ð3þ L g ¼ l pr ; ðþ l here r is the inner radius of the ring, is the idth of the ring, d is the distance beteen the to rings, a is the unit-cell dimension in the xy-plane, l is the distance beteen adjacent planes of the SRRs along the z-axis, K(t) is the complete elliptic integral of the first kind, and e and l are the permittivity and the permeability in vacuum. In Eq. (3), e used GuptaÕs formula [3] and PendryÕs recipe [] to derive the geometrical capacitance. This expression represents both the surface resistivity and the internal reactance of the SRRs precisely. y a a

3 3 A. Ishikaa, T. Tanaka / Optics Communications 8 () 3 3 In the optical frequency region, the conductivity of the metal is described as rðxþ ¼ x p e c ix ; ðþ here x p is the plasma frequency and c is the damping constant of the material. We used the empirical values x p =. s and c = 3.3 s for silver, x p = 3.8 s and c = 7. s for gold, and x p = 3. s and c =.9 s for copper []. To calculate the ring metal impedance, as shon in Fig. (c), the ring of the SRR as considered as a plane conductor hose thickness as greater than the penetration depth s hich is given by sðxþ ¼Re dðxþ ; ðþ i here d is the skin depth []. In the optical frequency region, the decrease in the penetration depth s is already saturated approximately at nm. Since the thickness of the ring in our model is not zero, the internal impedance must be considered. The internal impedance for a unit length and a unit idth of a plane conductor Z s (x) is defined rigorously as Z s ðxþ ¼ i rðxþdðxþ ¼ R sðxþþix s ðxþ. ð7þ The real and imaginary parts of Z s (x) are the surface resistivity R s and the internal reactance X s, respectively. By using Eq. (7), the ring metal impedance is described as ZðxÞ ¼ prz sðxþ. ð8þ Note that the imaginary unit in the numerator of Eq. (7) represents the delay of the current inside the metal, and this term is necessary to calculate the surface resistivity R s and the internal reactance X s accurately in the frequency region higher than THz. 3. Results Fig. shos the dispersion curves of the surface resistivity R s and the internal reactance X s of silver, gold, and copper. As the frequency increases, the surface resistivity saturates at the inherent frequency of each metal. The saturation value of silver is remarkably smaller than those of gold and copper. The internal reactance, on the other hand, does not saturate and moves aay from zero drastically as the frequency increases. As a result, in the optical frequency region, e found that the effect of the internal reactance on the resonant frequency of SRR as more dominant than that of the surface resistivity. In other ords, a large internal reactance reduces the resonant frequency that is determined by the geometrical capacitance and inductance of the structure, as pointed out in []. Fig. 3 shos the dependencies of the real and imaginary parts of the effective permeability l Re, l Im as the unit-cell dimension and the dimensions of the SRR uniformly decrease. We used silver as the material of the SRR, and e neglected the interference beteen the adjacent SRRs because a as much larger than d. As the dimensions of the SRR decreases, the magnetic response, hich is defined by the difference beteen maximum value of l Re and its minimum, decreases. The resonant frequency shifts to high value. Note that the increase of the resonant frequency is not linearly proportional to the dimensions of the SRR due to the increase of the internal reactance. - Xs : Ag : Au : Cu -. Fig.. Dispersion curves of the internal impedance of silver, gold, and copper. In the frequency region exceeding THz, the internal reactance is more dominant than the surface resistivity, and this internal reactance decreases the resonant frequency. Rs

4 A. Ishikaa, T. Tanaka / Optics Communications 8 () nm 8nm 3nm 9nm a r d l Fig. 3. Real and imaginary parts of the effective permeability of the silver SRRs as a function of the unit-cell dimension and the dimensions of the SRR. The labeling in each case indicates the unit-cell dimension a. The inset shos the corresponding dimensions of the SRR in nanometers. Fig. shos the frequency dependence of the minimum value of l Re for the SRR made of silver, gold, and copper. The l Re minimum frequency dependence of the unit-cell dimension a for the silver SRR is also shon. As the frequency increases, the minimum value of l Re approaches unity asymptotically. In the case of the SRR made of gold and copper, the minimum value of l Re becomes positive value in the frequency region above - - Visible Range 3 3 : Ag - : Au - : Cu - Fig.. Frequency dependence of the minimum value of l Re of the SRRs made from silver, gold, and copper. The l Re minimum frequency dependence of the unit-cell dimension a for the silver SRR is also shon. THz. On the other hand, only the silver SRR still exhibits negative l Re in the visible range. In Fig., e can see that there are to factors in the frequency dependence of the minimum of l Re, and e considered these factors by dividing the frequency region into to. The rapid increase of the minimum of l Re seen in the frequency region belo THz, is attributed primarily to the increase of the surface resistivity of the ring. In the frequency region above THz, the gradual increase is primarily due to the increase of the resistivity arising from the decrease in the idth of the ring. In contrast to the increase in the surface resistivity of the ring, the decrease of the ring idth does not result directly in the degradation of the tuning range of the permeability. We concluded that in the visible light region, the increase of the resistance as not dominant factor to realize the negative l. In order to describe the SRRsÕ behavior in terms of the filling factor F, e calculated the minimum value of l Re of the SRRs made of silver, gold, and copper by changing the filling factor. Fig. (a) (c) shos the results of the calculations for silver, gold, and copper, respectively. The frequency dependencies of the imaginary part of the effective permeability l hose real part has the minimum value are also shon. The minimum value of l Re dramatically changes according to the slight variation of the filling factor. As shon in Fig. (a), hen F is %, the minimum value of l Re is positive in the visible range. When F is %, the minimum of l Re becomes negative up to THz, hich is corresponding to nm in avelength. When F is 3%, the minimum of l Re becomes negative in all visible light range. It is evident that the filling factor is important for the realization of the negative l in the visible range. On the other hand, in the case of gold and copper, the negative l is never observed in the visible light range. As a result, e concluded that silver must be used for SRRsÕ material to obtain the negative l in the visible light region. Hoever, the l Re / l Im ratio of the silver SRRs is not large (.377), compared to those of less suitable metals like gold (.) and copper (.997), all at F = 3% and THz. This suggests the tradeoffs beteen the negative l and the propagation loss as pointed out by Dimmock [].

5 3 A. Ishikaa, T. Tanaka / Optics Communications 8 () a Ag b Au - c Cu : F=% : F=7% : F=% : F=3% - : F=% : F=7% : F=% : F=3% : F=% : F=7% : F=% : F=3% Visible Range - Fig.. Frequency dependencies of the minimum value of l Re of the SRRs made of: (a) silver; (b) gold; (c) copper according to the filling factor: F = %, 7%, % and 3%. The frequency dependencies of the imaginary part of the effective permeability l Re hose real part has the minimum value are also shon. Only the silver SRR exhibits negative l in the visible range: 7 THz Discussion As seen in Eq. (), the small geometrical capacitance C g gives the high resonant frequency of SRR, and the decrease of SRRsÕ size is the straightforard method to obtain small C g. Hoever, as the size of SRR decreases, the actual capacitance of SRR becomes larger than our estimation because GuptaÕs formula is valid only hen the thickness of the ring is negligible. Therefore, e must decrease the geometrical capacitance of SRR in order to obtain the resonance of SRR and negative l in the visible region. To reduce the geometrical capacitance of the structure, employing an SRR consisting of a single ring is effective. In addition, dividing the ring along the circuit can also reduce the capacitance of the circuit [7]. Fortunately, recent progress in photolithography and to-photon micro-nanofabrication techniques [8,9] enables us to fabricate various resonant nano-scale structures ith high accuracy.. Conclusion We demonstrated that the array of SRRs realizes negative permeability in the visible light region. In our theoretical analysis, e derived the effects of the surface resistivity and the internal reactance on the magnetic responses by considering the delay of the current in the conductor. Our results reveal that the surface resistivity effect is small, because the surface resistivity is already saturated at the frequency loer than THz. In addition, e claimed that reducing the geometrical capacitance and using silver for SRR are necessary to realize the negative l in the visible light range. References [] J.B. Pendry, Phys. Rev. Lett. 8 () 39. [] V.G. Veselago, Sov. Phy. Usp. (98) 9. [3] Pendry also claimed that if the polarization of the incident field is P-polarized light and the thickness of the metallic slab is thin (several tens nm), the slab enhances the evanescent field and the slab focuses the image of object even if the permeability of the slab is positive.

6 A. Ishikaa, T. Tanaka / Optics Communications 8 () [] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Steart, IEEE Trans. Microave Theor. Tech. 7 (999) 7. [] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 8 () 8. [] R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz, Appl. Phys. Lett. 78 () 89. [7] R.A. Shelby, D.R. Smith, S. Shultz, Science 9 () 77. [8] T.J. Yen, W.J. Padilla, N. Fang, D.C. Vier, D.R. Smith, J.B. Pendry, D.N. Basov, X. Zhang, Science 33 () 9. [9] A.-C. Hsu, Y.-K. Cheng, K.-H. Chen, J.-L. Chern, S.-C. Wu, C.-F. Chen, H. Chang, Y.-H. Lien, J.-T. Shy, Jpn. J. Appl. Phys. 3 () L7. [] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, C.M. Soukoulis, Science 3 () 3. [] N.-C. Panoiu, R.M. Osgood Jr., Phys. Rev. E 8 (3). [] S. OÕBrien, J.B. Pendry, J. Phys. Condens. Matter () 383. [3] K.C. Gupta, R. Garg, I. Bahl, P. Bhartia, Microstrip Lines and Slotlines, second ed., Artech House, Boston, 99. [] P.B. Johnson, R.W. Christy, Phys. Rev. B (97) 37. [] S. Ramo, J.R. Whinnery, T.V. Duzer, Fields and Waves in Communication Electronics, third ed., Wiley, Ne York, 993. [] J.O. Dimmock, Opt. Exp. (3) 397. [7] S. OÕBrien, D. McPeake, S.A. Ramakrishna, J.B. Pendry, Phys. Rev. B 9 (). [8] S. Kaata, H.-B. Sun, T. Tanaka, K. Takada, Nature (London) () 97. [9] J. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, S. Kaata, Appl. Phys. Lett. 8 ().

GHz magnetic response of split ring resonators

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