A circuit model for the hybrid resonance modes of paired SRR metamaterials

To better understand the resonance modes caused by the interelement couplings in the building block of metamaterials, we propose a circuit model for the hybrid resonance modes of paired split ring resonators. The model identifies the electromagnetic coupling between the paired rings by electric and magnetic coupling networks and well explains the variation of hybrid resonance modes with respect to the distance and the twist angle between the rings. The predictions of our model are further proved by experiments. ©2014 Optical Society of America OCIS codes: (160.0160) Materials; (160.3918) Metamaterials. References and links 1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). 2. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). 3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). 4. V. G. Veselago and E. E. Narimanov, “The left hand of brightness: past, present and future of negative index materials,” Nat. Mater. 5(10), 759–762 (2006). 5. M. J. Freire and R. Marques, “Planar magnetoinductive lens for threedimensional subwavelength imaging,” Appl. Phys. Lett. 86(18), 182505 (2005). 6. S. Engelbrecht, M. Wunderlich, A. M. Shuvaev, and A. Pimenov, “Colossal optical activity of split-ring resonator arrays for millimeter waves,” Appl. Phys. Lett. 97(8), 081116 (2010). 7. X. Xiong, W.-H. Sun, Y.-J. Bao, M. Wang, R.-W. Peng, C. Sun, X. Lu, J. Shao, Z.-F. Li, and N.-B. Ming, “Construction of a chiral metamaterial with a U-shaped resonator assembly,” Phys. Rev. B 81(7), 075119 (2010). 8. G. Dolling, M. Wegener, A. Schadle, S. Burger, and S. Linden, “Observation of magnetization waves in negative-index photonic metamaterials,” Appl. Phys. Lett. 89(23), 231118 (2006). 9. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97(24), 243902 (2006). 10. T. Q. Li, H. Liu, T. Li, S. M. Wang, F. M. Wang, R. X. Wu, P. Chen, S. N. Zhu, and X. Zhang, “Magnetic resonance hybridization and optical activity of microwaves in a chiral metamaterial,” Appl. Phys. Lett. 92(13), 131111 (2008). 11. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009). 12. A. Radkovskaya, O. Sydoruk, E. Tatartschuk, N. Gneiding, C. J. Stevens, D. J. Edwards, and E. Shamonina, “Dimer and polymer metamaterials with alternating electric and magnetic coupling,” Phys. Rev. B 84(12), 125121 (2011). 13. R. S. Penciu, K. Aydin, M. Kafesaki, T. Koschny, E. Ozbay, E. N. Economou, and C. M. Soukoulis, “Multi-gap individual and coupled split-ring resonator structures,” Opt. Express 16(22), 18131–18144 (2008). 14. A. N. Serdyukov, I. V. Semchenko, S. A. Tretyakov, and A. Sihvola, Electromagnetics of Bi-anisotropic Materials: Theory and Application (Gordon and Breach Science, 2001). 15. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). 16. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006). 17. S. V. Zhukovsky, C. Kremers, and D. N. Chigrin, “Plasmonic rod dimers as elementary planar chiral meta-atoms,” Opt. Lett. 36(12), 2278–2280 (2011). 18. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002). #199211 $15.00 USD Received 9 Oct 2013; revised 12 Dec 2013; accepted 13 Jan 2014; published 22 Jan 2014 (C) 2014 OSA 27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001920 | OPTICS EXPRESS 1920 19. L. Zhou and S. T. Chui, “Eigenmodes of metallic ring systems: a rigorous approach,” Phys. Rev. B 74(3), 035419 (2006). 20. R. Plonsey and R. E. Collin, Principles and Applications of Electromagnetic Fields (McGraw-Hill, 1961). 21. R. P. Feyman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964,), Vol. 2. 22. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76(7), 073101 (2007). 23. D. K. Ghodgaonkar, V. V. Varadan, and V. K. Varadan, “Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies,” IEEE Trans. Instrum. Meas. 39(2), 387–394 (1990).


Introduction
Metamaterials, whose electromagnetic responses can be tailored by engineering, provide a new way to manipulate the electromagnetic (EM) wave.Many abnormal phenomena, such as negative refraction, superlens and cloaking, have been predicted and demonstrated by using metamaterials [1][2][3][4][5].The building elements of metamaterials, i.e. the artificial "atoms", can be designed to yield desired electric and magnetic moments.As the "atoms" have subwavelength scale, the metamaterial is usually regarded as effective medium and its EM responses can be treated as the average responses of the "atoms".Up to now, many abnormal phenomena arising in metamaterials, such as negative refraction, have been successfully explained by the effective media theory (EMT), even though some "atoms" are not really electrically small.Nevertheless, there are still some phenomena like hybrid resonance and optical activity [6,7] cannot be clearly interpreted by EMT, which is known to be caused by the interaction between the elements within the unit cell, the interelement couplings.In fact, in some circumstances [8][9][10][11][12][13], the interelement couplings could be extremely significant enough to result in many special EM characteristics that do not exist in conventional metamaterials.The hybrid resonance is one that is usually considered originate from the very strong interactions [14] between the elements in unit cell.Hence, the hybridization model, originally used to describe the plasmon response of complex nanostructures [15,16], was applied to the metamaterial to understand the interelement coupling.A good example is the hybridization model by means of Lagrangian formalism [10,11] well describing the split of resonant frequencies of the paired split ring resonators (SRR) and its variation with respect to the relative position of the rings.Recently, chiral metamaterials attract lots of attentions [17] and the chirality may have some close relationship with the interelement coupling.Therefore, it is necessary to well appreciate and do some further studies for fully understanding the interelement coupling effects.
In this work, a circuit model is proposed to better understand the interelement coupling in the paired SRRs.The model identifies the electric and magnetic coupling using coupling capacitances and mutual inductance.With the derived eigenfrequencies of hybrid resonance modes, we explore the state of hybrid modes and the breakdown of the hybridization.The variation of resonances with respect to the electric and magnetic coupling coefficients is also discussed.The results reveal that the mutual inductance and the coupling capacitances can greatly change the resonance states, and the hybrid resonance mode does not necessarily need a very strong coupling.Our model can well explain the shift of the hybrid resonant frequencies with respect to the variation of the distance and twist angle of the two rings.All the predictions of our model are further demonstrated by experiments.

Circuit model
Consider that the metamaterial consists of broadside-coupled SRRs [18], whose unit cell is shown in Fig. 1(a).The two rings are supposed to be identical and the space between the rings is small so that the EM field in one ring could affect the other.The incident EM wave is assumed propagating along z-direction, normal to the SRR plane, and polarized along y-direction.For this configuration, the basic resonance mode of the rings will be activated when neglecting the interactions between the rings [19] and this mode is known as the LC-circuit resonance mode.However, if the interelement coupling is significant, the resonance modes will change a lot and some new modes will be created [11,12,18].Basically, there are two types of interelement couplings between the rings: the electric coupling and magnetic coupling.The magnetic coupling is due to the mutual inductance between the rings.The circulating current in one ring produces a magnetic field affecting the magnetic field distribution in the other ring.The two rings are magnetically linked.Also, the small space between the rings changes the distributions of the charges gathered in the slits of SRRs, which forms the mutual capacitance between the rings, making the two rings electrically coupled.When resonated, each ring can be modeled by an LRC circuit and the interactions between the rings are modeled by coupling networks.As shown in Fig. 1(b) where the stereo circuit is projected onto a plane, the whole circuit consists of two parts: the two LRC circuits for single-rings and the coupling networks connecting the two LRC circuits.The electric coupling network is comprised by capacitances C 1a , C 1b C 2a and C 2b .They respectively represent the mutual capacitances between the chargers accumulated at the split of two rings at resonance, i.e. the 4 ends of two rings which could be considered as the charge-holding objects [20].Although at resonance, the charges on SRRs are usually accumulated at the slits of the rings, it may distribute on the ring arms when the space between the rings is very small.Therefore, it would be better to consider the coupling capacitances as the capacitances between the arms.The magnetic coupling network is due to the mutual inductance M of the two rings.In the model, the resistors are mainly resulted from the metallic loss of the rings.For simplicity, here we neglect their effect since the resistors only change the quality factor with no effect on the resonant frequency.
where I 1 and I 2 are the currents in the two SRRs, and ω is the angular frequency of the EM wave.The coefficients a ij (i, j = 1, 2) are the functions of the coupling capacitances; they are and [ ] ( ) where c ij = C ij /C 0 (i = 1, 2; j = a, b) are the normalized capacitances.The capacitances and inductance are related to relative position and angle between the two rings.However, it is still challenge to get these coupling parameters by the shapes, relative position and angle of the rings.One simple way is to retrieve the parameters from resonant frequency under some circumstance, such as discussed as follows.The coefficients a 21 and a 12 are the same due to the reciprocity of circuit.The last term in the left-hand-side of the equations is the electromotive force for the mutual inductance and its sign is according to the linkage of the two rings [21].It is clear that coefficients a 21 (or a 12 ) and M, respectively, represent the electric and magnetic coupling between the rings.Coefficients a 11 and a 22 signify the load effect of the electric coupling networks to each ring, which is easy to understand if we set the current, for example, I 2 zero.The coefficients a 11 and a 22 are usually different except in some circumstance such as the two rings are in symmetrical arrangement, or the cross-end coupling capacitances can be neglected, or the difference among the cross-end coupling capacitances and the end-end coupling capacitances is very small.The load effect makes the two rings have different eigenfrequencies even though they are geometrically identical, which can be realized from Eq.
(1).By solving Eq. ( 1), the eigenfrequencies of the hybrid resonance modes of the paired SRRs are derived as following: ( ) ( ) where 0 is the eigenfrequency of single ring, m is the normalized mutual inductance M/L 0 , and shows the interaction or "hybridization" of the two rings produces two new resonances splitting from the basic resonance of single ring.
It would be very complicated to discuss the variation of eigenfrequencies with coefficients a ij and m in general.Therefore, we first consider the situation of symmetrical arrangement of the rings where the twist angle φ = 0° or 180°.In these two cases, the coupling capacitances are c 1a = c 1b = c 1 and c 2a = c 2b = c 2 .The coefficients a ij have simpler forms, which are expressed respectively as: Usually, capacitance c 1 is greater than c 2 in symmetrical configuration due to shorter end-end distance of the two rings [20].Hence, the eigenfrequencies of the system are simplified as: where upper and lower signs are respectively for φ = 0° and 180°.Specifically, if the capacitances are much smaller than C 0 and cross-end coupling capacitances are very small whose effects can be neglected, the coefficients a ij could be further simplified to in the first order approximation.And eigenfrequencies of the system become as follow: which are the same as those derived by Lagrangian formalism [11] in dipole-dipole interaction picture.
The states of the resonance can be reflected by rings' circulating current patterns and energy levels of the resonance.For twist angle φ = 0°, as an example, Eq. ( 1) gives the current ratio I 1 /I 2 as −1 at ω 01 and + 1 at ω 02 .Meanwhile, since the coefficients a ij and m are positive numbers, ω 01 is smaller than ω 0 .Thus, the lower frequency resonance will have the anti-symmetrical circulating currents and the higher frequency resonance has the symmetrical one, coinciding with the results in Ref [11].

Simulations of SRR in symmetric configuration
Regarding the case of symmetric configuration, the variation of the hybrid resonance modes is due to the space between the rings [22].When two rings are far way, the interactions disappear and all the coefficients equal to zero.One resonance peak will occur in the transmission or reflection spectrum.Once two rings are close enough, the hybrid resonance occurs, two resonance peaks will appear and the shape of the resonance curve will vary with the distance between the rings.When the two rings are extremely close, capacitances c 1 and c 2 become huge and all magnetic flux in one ring goes through another, making a 11 ~1, a 21 ~0 and m~1.As a result, ω 01 tends to zero and ω 02 is finite, only one resonance peak will show in the transmission or reflection spectrum.
To intuitively demonstrate the above predictions, we simulated the transmission of an array of stacked U-shaped SRRs (uSRRs) and explored the variation of its resonant frequency by changing the space D between the two rings.The two SRRs are prepared on an FR4 (ε r = 2.5) substrate with thickness of 1.0 mm and stacked face-to-face with a dielectric slab in the middle.The resonance of uSRR is designed in microwave X-band and the dimension of uSRR is in its smallest size.The metallic strip is with side length L = 5 mm and width w = 1.8 mm.The lattice constant of the array is a = 13.2 mm [see Fig. 5(b)].Figure 2(a) plots the simulated results at twist angle φ = 0°.The simulations were performed by software CST microwave studio.The operating frequency range is 8-12GHz.We see the resonant frequency varies with the space D between the two rings.When D is large enough (over 10mm, nearly about λ/3 where λ is the wavelength corresponding to the central frequency), the system has only one resonant frequency and only one dip presents in the transmission [see the inset (I.3)].Without any EM interaction in the rings, each ring behaves like a free "atom" and the hybridization breaks down.When the two rings approach, two resonances appear.As shown in the inset (I.2), the basic resonance of single ring splits into two new resonances; one is lower than ω 0 and the other is higher than ω 0 .The current distribution at resonances is shown in Fig. 2(d).The currents are anti-parallel at lower resonant frequency but parallel at higher resonant frequency in good agreement with the prediction of our model.As D decreases, the difference between the two resonant frequencies increases.When two rings are very close (D = 0.05mm≈0.00167λ), the stacked SRR shows only one resonance again [see inset (I.1)].The resonant frequency is about 10.23 GHz, slightly greater than the resonant frequency of a single-ring 9.65 GHz, which indicates ω 01 goes to zero and leaves the higher frequency resonance in the transmission spectrum.Due to the very strong coupling, the currents in two rings are in the same direction and their distributions are almost the same [see Fig. 2(c)], therefore the two rings behave like a single ring.For the asymmetrical arrangement of SRRs (φ = 180°), the variation of the resonant frequencies with the distance D is similar to that shown in Fig. 2(a), however, the state of circulating currents are different, the currents are parallel at the lower energy level and anti-parallel at the higher energy level as shown in Fig. 2(e).In certain assumptions, we can even retrieve the coupling capacitances and mutual inductance from the resonant frequencies using Eqs.( 3) and ( 4).We assume when D is much smaller than the distance between the two arms of the single-ring, the normalized capacitance c 2 is much smaller than c 1 and thus can be neglected.This is because the mutual capacitance decreases rapidly with the increase of distance between the charged objects.Otherwise, the capacitances c 1 and c 2 are assumed to be equal.Figure 2(b) plots the retrieved coupling capacitance and mutual inductance at twist angle φ = 0° and 180°.We see the coupling capacitance and mutual inductance decrease rapidly with the increase of the distance between the rings.This variation is physically reasonable.With the increase of the space, less magnetic lines of one ring pass through the other ring, so the magnetic coupling decreases quickly.The electric coupling is always smaller than magnetic coupling since the mutual capacitance only works in a short distance.Comparing the two symmetric configurations, we see at twist angle φ = 180° the interactions between the rings are stronger.This is especially evident at the small space D. The reason is at angle φ = 180° the circulating currents excited by the incident wave are anti-parallel, then the magnetic interference between the rings causes the enhancement of circulating currents in the rings according to the Lenz's rule.This results in the increase of magnetic coupling.Because of the anti-parallel current distribution, the charges on the same side arms of the rings have opposite sign forming capacitors, which may be the cause of stronger electric coupling.The retrieved mutual capacitance and inductance indicate the hybrid resonance mode of the system is basically due to the magnetic interaction between the rings and it is not necessary to need a very strong coupling.

Simulations of SRR in asymmetric configuration
Now consider the variation of hybrid modes with rotating the twist angle φ as shown in Fig. 3(a).For arbitrary angle φ, the configuration of the ring is asymmetrical.At a fixed distance D, the coupling capacitances change a lot as rings rotating and the changes will alter the values of coupling coefficients.Figure 3(b) illustrates the variation of hybrid resonance modes frequencies with respect to electric and magnetic coupling coefficients.Here the load effect coefficients a 11 and a 22 are assumed to be different.We see the two resonance frequencies form two branches.The gap between the two branches becomes small when the difference of a and a 22 is small [Fig.3(c)].As a comparison, when a 11 and a 22 are the same [Fig.3(d)], the gap vanishes, which means there is a degenerated resonance mode when rotating the twist angle.This may happen when the distance between the two rings is relatively large.The differences among the cross-end coupling capacitances and the end-end coupling capacitances would be small enough leading to the load effect coefficients are almost the same.To verify this, we simulated transmissions of uSRR with a relatively larger space D = 2 mm when rotating the twist angle.Figure 4(a) plots the resonant frequency as a function of twist angle.Two resonance branches present; they tend to converge at first, and then shift away from each other.A degenerated mode happens at about twist angle 45°.In the figure, the arrows beside the resonant frequencies illustrate the direction of the circulating current at the resonant frequencies.We see symmetry of the circulating currents exchanges at about 45°. Figure 4(b) illustrates the details of the exchange of circulating current at resonance; the state with anti-parallel circulating currents changes from the lower resonant frequency to the higher one as twist angle increases.Similar results were reported in the twist SRR in THz frequency range [11].However, the avoided crossing of two resonance branches was found there, which is due to the small space between the rings that causes the coupling capacitances quite different.

Experimental verification
As an experimental demonstration for the prediction of our model, we fabricated an array of stacked uSRR.The array was prepared on FR4 substrate by photolithography.The geometric dimension of the unit cell was the same as that of our simulations.The sample is an array of 10 × 10 stacked uSRRs as shown in Fig. 5(b).The transmission measurements were performed using free space measurement technique [23].Figure 5(a) is the schematic of the experimental setup, which includes focus antennas and Agilent E8363A network analyzer.We first measured transmission of the sample at the twist angle φ = 0°.Figure 5(c) plots the measured resonant frequencies of two rings separated by different thickness FR4 slabs.The simulated resonant frequencies are also plotted in the figure as a reference.The measurements and the simulations are in good agreement.Figure 5(e) gives the measured typical transmission curves.When two rings are very close, for example, two rings are separated by 0.03 mm thick Kapton film (ε r = 3.9), only one resonant dip is shown.With the separating slab becomes thicker, two resonant dips present in the curve; for instant, separated by 0.25 mm thick FR4 slab, two resonance dip appears; one is strong at 10.1GHz and the other is weak at 9.57 GHz.The two dips become clear with almost the same strength as rings are spaced by a 2 mm thick FR4 slab.However, when the space between the two rings are large enough, for example separated by a 20 mm thick foam slab (ε r = 1.1), the transmission once again has only one sharp dip.Then, we measured the transmission when twist angle takes different values.Figure 5(d) plots the resonant frequencies against the twist angle.The resonant frequencies form two branches; they first tend to converge, cross at about 45°, and then shift away from each other, just the same with the simulations shown in Fig. 4(a).Figure 5(f) shows the measured transmissions at some typical twist angles.The results show the relative frequency change is more extensive than that reported in Ref [11].All the measurements are in good agreement with our theoretical predictions, indicating our circuit model well reflects the hybrid resonance modes existing in the paired SRR metamaterials.

Conclusion
In conclusion, we have developed a circuit model for the hybrid modes of paired split ring resonators.We identify the electromagnetic interactions between the rings and corresponding coupling coefficients, and derive the general form of the eigenfrequencies of the hybrid modes.This model provides a clear and intuitive picture for the hybrid resonance occurring in the paired SRRs and well explains the variation of the hybrid modes with respect to the spatial arrangement of the rings.Its predictions of the variations of the hybrid resonance modes with respect to the distance and twist angle are proved by the simulations and experiments.The way of circuit model offers a simple method to study interelement coupling effects of metamaterials and this method could be further used in exploring new phenomena related to the interelement interactions of metamaterials.Generally, two coupled rings can be modeled by the circuit shown in Fig. 6.Using Kirchhoff's current law at nodes a, b, c and d in the circuit above, we have equations for the current in each path:

Appendix
For the all loops of the coupling network, we have the following voltage equations according to Kirchhoff's voltage law, where j is the imaginary unit and ω is angular frequency: And for the loops of left and right rings, we have voltage equations as following:

Fig. 1 .
Fig. 1.(a) Schematic of unit cell of paired SRRs metamaterials; D is the vertical space between the two rings; φ represents the twist angle of the two rings.(b) Circuit model for the paired SRRs.The circuit is formed by two LRC circuits of each single-ring connected by the electric coupling network which consists of mutual capacitance C 1a , C 1b , C 2a , C 2b and the magnetic coupling network with mutual inductance M. The U-shaped shadows represent the two rings in panel (a) projected on the same plane.The variation of the angle φ and distance D will be reflected in the changes of the coupling mutual capacitances and inductance.Appling Kirchhoff's theorems on the circuit, we can obtain the following equations [see the Appendix], 1 1 2 0 1 11 12 2 1 0 0 0 2 2 1 0 2 22 21 1 2 0 0 0

Fig. 2 .
Fig. 2. (a) Resonant frequency changes with the space D between two rings.The inset is the transmission curve at D = 0.05, 2 and 12 mm, respectively.(b) The coupling capacitance ke = c 1 and the mutual inductance km = m retrieved from resonant frequencies.(c) Circulating current distributions at D = 0.05mm.The bottom image shows the circulating currents in the lower ring where the upper ring is hidden.(d)-(e) Circulating current distributions at resonance when two rings are in parallel and anti-parallel arrangement, respectively.The circulating currents are in opposite symmetry for these two arrangements.Higher energy resonance mode is represented by ω + and lower one by ω -.

Fig. 3 .
Fig. 3. Variation of hybrid resonant frequencies as a function of electric and magnetic coupling coefficients.(a) Schematic of rings rotation at fixed distance D. (b) Resonant frequencies vary with coefficients a 21 and m, where load effect coefficients a 11 and a 22 take 0.01 and 0.1, respectively.The two resonant frequencies form two branches of surface with a gap.(c) The gap between the two branches of resonance become small as the difference of a 11 and a 22 is small.Solid lines are the resonant frequencies for a 11 = 0.01 and a 22 = 0.1 and dashed line for a 11 = 0.08 and a 22 = 0.1.(d) A degenerated resonant frequency is shown as the load effect coefficients a 11 = a 22 (here the value is 0.1) are the same.This may happen when the distance between the ring is relative large and twist angle changes.

Fig. 5 .
Fig. 5. Experimental results for the paired SRRs metamaterial.(a) Schematic of experimental setup.It has two focus antennas and the sample is set in the between.(b) The image of the sample in experiments of which the top uSRR array are removed.(c) Measured resonant frequencies (triangles) and simulation ones (crosses) as a function of space D between the rings.The solid square is the resonant frequency for the rings separated by Kapton film.(d) Resonant frequencies measured by changing the twist angle.The space D is 2 mm.In the panels (c)-(d), ω + and ω -represent the higher energy and the lower energy resonance modes, respectively.(e) The typical transmission curves under different D. At D = 20 mm, the resonance dip shifts to 11.5 GHz because of the lower dielectric constant of the foam used.(f) The measured transmission curves at typical twist angles.

Fig. 6 .
Fig. 6.A general circuit of the paired SRR.The U-shaped shadows represent the SRR rings.L 0i , C 0i and R 0i (i=1, 2) are the intrinsic inductance, capacitance and resistance respectively.C ja and C jb (j=1, 2) are the mutual capacitances of the electric coupling network between the two rings, and M is the mutual inductance.I k (k=1, 2, …, 8) in the figure stands for the current in each path of the circuit.
and (8), we can obtain the current I 3 and I 6 as the functions of currents I 1 and I 2 .Then bring them back into Eq.(9) under the assumptions that the resistances R 01 and R 02 are zero.If the two rings are the same, we can have C 01 = C 02 = C 0 and L 01 = L 02 = L 0 .Then we can get the main equation Eq. (1) by rearranging the terms according to the currents I 1 and I 2 in the left and right rings, respectively.