Abstract

We experimentally studied the coupling between a double split ring resonator and a complementary split ring resonator. The greatest coupling occurs when the two resonators are separated by the average ring radius, and the dimensionless coupling is as large as 0.1, allowing a novel planar metamaterial based on this hybrid structure. The coupling strength can be varied up to a factor of 2 by changing the relative orientation of the split ring resonators. A 2×2 waveguide structure with −10 dB coupling factor can be achieved, and showing multi-mode plasmon-induced transparency. It can be considered one-dimensional metamaterials exhibiting negative permeability and permittivity simultaneously.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Recently artificial structures combining building blocks exhibiting electric and magnetic resonances emerge for realizing negative index material. [1,2] Among them, the split-ring resonators (SRR) and their counterpart, the complimentary split-ring resonators (CSRR) respectively providing negative permeability and permittivity within a narrow frequency window are widely used. [3–6] These resonators therefore enrich the engineering on planar microwave propagation [7], leading to miniature microwave circuits. [8] In addition to these first generation of meta-materials [5], a new class of meta-materials based on strongly coupled resonator emerges owing to their rich physics. As an example, it is found that strongly coupled resonators exhibit the interesting phenomena analogous to electromagnetically-induced transparency (EIT). [9] In the so-called plasmon-induced transparency(PIT), coherent interference between two strongly coupled resonators is enforced to induce transparency. [10–14] With the occurrence of PIT, a slow light behavior presents, promising vast applications. [15–17]

It has been proposed that many intriguing phenomena analogous to those in electronic systems such as Bloch oscillations [18], artificial gauge field [19,20] and nonlinear effects [21] can be realized in photonics by tailoring the photon hopping in coupled resonators. Recently, people have demonstrated artificial magnetic field for the hopping photon in two-dimension [22, 23], which leads to photonic topological insulators. [24–26] The importance of coupled SRRs can be envisaged in the viewing of the versatile optical properties empowered by the metamaterials constructed by strongly coupled resonators. Over the years intensive studies have explored how the coupling is governed by parameters such as spatial arrangement and orientation configurations. [27–29] Though much efforts have been put on this issue, unfortunately, previous works are mostly devoted to the same type of resonators. Indeed, “exotic” electronic behaviors typically occur in a lattice with diatomic basis, so the study of a “photonic diatomic basis” is mandatory. [30] Perhaps a structure combining a resonator and its complementary counterpart gets off to a good start because of their identical frequencies, mirror images in shape, and perfectly dual electromagnetic properties. As an example, a SRR/CSRR diatomic basis should provoke negative permeability and negative permittivity at the same frequency. We also like to note that previous experiments usually investigate a small quantity of samples, so very limited information on the coupling strength can be obtained. Here, by taking the advantage of a home-made moving platform, we in-situ change the relative position and orientation of a double split ring resonator (DSRR) and a CSRR to map out the shift of their resonances. In addition, hybrid DSRR/CSRR structure with strong inter-resonator coupling exhibit a multi-mode PIT effect and an efficient band-pass wave guiding.

2. Experimental methods

Coupled DSRRs (sample A), which is similar to common PIT device[14] were fabricated on 0.5 mm-thick glass substrate using standard photolithography and wet-etching of thermally evaporated 300 nm-thick Al film. The etching solution was diluted phorspher acid (phorspher acid: de-ionized water=4:1). The outer diameter of the DSRRs is 2r = 7.6 mm and the average ring radius ra = 3.2 mm. The inter-resonator (center-to-center) distance d varies from 7.7 to 8.2 mm. For probing the resonance frequency, a microstrip couples one of the DSRRs, which is referred as the “radiative” one. Fig. 1(a) illustrates an optical micrograph of a sample with “face-to-back” configuration. Here we refer the principle direction pointing the gap of the outer ring as the “face” side.

 figure: Fig. 1

Fig. 1 (a) A photograph of a “face-to-back” coupled DSRR sample. The scale bar is 3 mm. (b) The schematic of the moving platform for the coupled DSRR and CSRR. The metal and bare PCB regions are respectively drawn in yellow and blue colors. Our setup allows us to move the lower layer in respected to the upper layer. The two layers are firmly contacted to each other during the measurement. (c) Circuit model for capacitively and inductively coupled SRRs. (d) The transmission data of the “face-to-face” coupled DSRRs. d varies from 7.7 mm to 8.2 mm, with a step of 0.1 mm, corresponding to gap between DSRRs from 0.1 mm to 0.6 mm. Two distinct resonances appear owing to the large inter-resonator coupling. When d increases to 8.2 mm, the two resonance dips gradually merge into a single dip. (e) The frequency difference fHfL of two resonance dips in (d) as a function of d for the “face-to-face”(red symbol) and the “face-to-back”(blue symbol) samples. At d = 7.7 mm, the “face-to-face” configuration gives a larger fHfL, while at d = 8.2 mm, “face-to-back” configuration results in a larger fHfL.

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Other samples(B and C) were made on 1.6 mm-thick ordinary FR-4 printed circuit board(PCB), on which both sides covered by 36 μm-thick Cu. A home-made moving platform was built for continuously probing the inter-coupling between a DSRR and a CSRR (sample B). Fig. 1(b) schematically illustrates the platform setup, which combines two layers of PCB: On the upper layer, a “radiative” DSRR was coupled to a microstrip for probing resonances. A CSRR was made on the lower layer, which could be moved by a 3-axis stage with the position resolution of 0.1 mm. The CSRR could be “darken” when it was moved away from the microstrip. The two PCBs were stacked in resemblance a structure with a DSRR and a CSRR on the two sides of a single PCB. Both the DSRR and the CSRR have the same 2r = 7.6 mm and the average ring radius ra = 3.2 mm. The position of the CSRR can be described by a coordinate system, of which the origin is chosen at the center of DSRR. The 2.8 mm-wide microstrip is parallel to the x direction, centered at y = −5.4 mm.

To demonstrate the wave-guiding property of coupled resonators, we fabricated 3 samples(C1–C3), respectively employing short DSRR-, CSRR- and interleaved SRR-arrays to couple two microstrip transmission lines(TL). In C samples, center-to-center distance for DSRR (or CSRR) d = 10 mm and 2r = 7.6 mm. All samples were measured by a vector network analyzer(VNA, Agilent N5230A). For the measurement of S-parameters of sample C, ports not measured were terminated with a broadband 50 Ω load. Microwave simulations were performed by means of High Frequency Structure Simulator.

3. Results and discussions

3.1. Coupled DSRRs

The microwave transmission amplitude of sample A’s with the face-to-face configuration are illustrated in Fig. 1(d). When the inter-resonator distance d is 8.2 mm, there is only one resonance at f0 =3.0 GHz, which is close to the resonance frequency of a single DSRR. As d is progressively reduced, the single resonance dip splits and the splitting increases, exhibiting a typical PIT spectrum. By noting the lower and the higher resonant frequency respectively as fL and fH, the largest splitting fHfL = 200 MHz when d is the smallest, d = 7.7 mm(gap=0.1 mm between resonators). Figure 1(e) shows how the splitting fHfL changes on d. In general, the splitting reduces rapidly as d increases, though it depends on the relative orientation of the DSRRs. When d < 8.2 mm, the coupling is larger when the two DSRRs are face-to-face than face-to-back.

When two resonators are close, they are capacitively and inductively coupled as a dimer. It has been found that the global excitation modes present a lower resonant frequency(fL) for symmetric mode and higher frequency(fH) for anti-symmetric mode, and the splitting of resonance reflects the coupling strength. The splitting can be explained by the coupled-mode theory [31],

Ω=ω1+ω22±(ω1ω2)22+|κ|2,
in which κ is the coupling coefficient. Here Ω = 2πfL(H), and ωj(j = 1, 2) are intrinsic resonator frequencies. When ω1 = ω2 = 2πf0, the result becomes
Ω2π=f0(1±g).
g = |κ|/ω1 is the dimensionless coupling strength. In sample A with the face-to-face configuration, the largest dimensionless coupling strength achievable is 2g = (fHfL)/ f0 = 6.6%. When the two DSRRs are very close, the coupling strength for the face-to-back one is reduced by a factor of 2. Ideally, the circuit analysis reveals that the face side can be modeled as a capacitor, while the back-side can be viewed as an inductor. [32] Previous works have pointed out that the face-to-face configuration provides a large capacitive coupling, while the coupling for face-to-back one is rather small. [33] However, when the DSRRs are separated as far as 8.2 mm, the face-to-back configuration surpass the face-to-face one, probably due to parallel electromagnetic polarization in the SRRs.

3.2. Coupling between a DSRR and a CSRR

Next we discuss the hybrid dimer formed by a CSRR and a DSRR by presenting the data of sample B. Figure 2(a) illustrates the measured microwave transmission amplitude of sample B for various CSRR positions. When the SRRs are exactly aligned(x =y=0 mm), two resonances respectively at 2.92 GHz and 3.27 GHz can be found. The lower one is attributed to CSRR because it only appears when the CSRR is in proximity of the microstrip. This can be further confirmed that we only see the higher resonance when replacing the lower layer by a metal ground. Although having the same size, the SRRs do not produce the exact resonance frequencies because the CSRR is overlaid by PCBs from both sides.

 figure: Fig. 2

Fig. 2 (a) The transmission data of coupled DSRR/CSRR measurement for 3 different positions, (x /mm, y/mm)=(0,0), (0, −2), and (0, −6). At the last position, two resonance absorption are clear. When the coordinates move toward +y direction, the CSRR is moved away from the microstrip and the resonance of lower frequency, to which the CSRR corresponds is getting smeared. (b) A mapping result of fD. (c) fD and fC as a function of x at y = 0 mm. (d) The coupling strengths 2gC and 2gM as a function of x at y = 0 mm. γ = 0.2 is assumed for calculation of 2gM. The maximum coupling strength is at x = 3 mm.

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The DSRR and CSRR resonance frequencies fD and fC are mapped out in the range of 20.0 mm × 20.0 mm with a pitch of 1.0 mm. As illustrated in Fig. 2(b), fD displays the rotational symmetry and its minimum appears at d = 3.0 mm. Figure 2(c) illustrates the cross section of fD and fC mappings at y = 0 mm, and reveals two important findings: When DSRR and CSRR are separated by a distance of d = 3.0 mm, both resonance frequencies red-shifts mostly. When they are widely separated(d > 7.0 mm), or perfectly aligned(d = 0 mm), their frequencies are largest. Later we will discuss how the coupling strength between the SRRs affect the red-shifts. In regarding the movement in y-direction, fC red-shifts when CSRR moves toward the microstrip, that is confirmed by an experiment using a upper layer without a DSRR. The red-shift in fC owing to the impact of microstrip provides us a guidance how to strongly couple a CSRR to a microstrip, and is useful in designing sample C.

The red-shifted resonant frequencies go beyond the prediction of Eq. (1), which always describes the repulsion of the resonant frequencies when coupling presents. Nevertheless we can analyze this problem using an effective circuit model as that described in Appendix A. From the observation that both resonance frequencies reduce as the resonators are in proximity, we expect that the inter-coupling contains capacitance. One can easily judge the capacitance(C)-coupling strength by using the result that

α1+α2~1(Ω1Ω2ω1ω2)2~2(δω1ω1+δω2ω2).
Again ωj=1/CjLj and Ωj are intrinsic and shifted resonator frequencies, and δωj = Ωjωj. Cj and Lj are capacitance and inductance of the resonator. αj = Cp/Cj, in which Cp1=C11+C21+Cx1~Cx1. For extracting the mutual inductance(M)-coupling strengths, we can use the following expression by noting that βj = M/Lj,
Ω12Ω22=[(1α2)ω22(1α1)ω12]2+4γω12(α1β2)(ω22α1ω12β2)
Here γC1/C2 = α2/α1, and β1=γβ2ω12/ω22. Equation (3) provides a possible way to determine β2 if γ is known,
2ω12β2~α1(ω12+ω22)±4α12Δ4+γ1[4ΔΩ2(2Δ2α1ω12α2ω22)2]
with ΔΩ2(Ω12Ω22)/2 and Δ2(ω12ω22)/2. The above equations are approximated results. The exact values of α’s and β’s can also be found by numerical methods.

The dimensionless coupling strengths for C-coupling and M-coupling can be respectively defined by 2gC = (α1 + α2)/2 and 2gM=β1β2. As one can see in Eq. (2), when α’s and β’s are not large, the sum of the red-shifts is linearly proportional to 2gC. From above analysis, we conclude that the largest coupling in DSRR/CSRR hybrid dimer occurs at d = 3.0 mm ∼ ra, and the corresponding 2gC values are 2.0%. 2gM is also exactly calculated by assuming γ = 0.2. Their position dependences are illustrated in Fig. 2(d). We note that experimental 2gC is lessen by the impact of the lower PCB and metal stage in our platform. Simulations reveal that in an ideal single PCB case, a larger coupling 2gC = 7.7% is recovered

Here we try to discuss why the optimum coupling occurs at dra. When dr, the DSRR and CSRR can be respectively viewed as a single magnetic dipole and electric dipole. By assuming that the two dipoles always oscillates in-phase to reduce the energy, we found that the coupling energy reduces as d−2 when d is smaller than wavelength 2πc/ω. In the case d is much larger than the wavelength, the expression for radiation zone should be imposed and Ud−1.

When d < ∼ r, the structure of the resonators are important in determining the coupling energy. According to coupled-mode theory, the coupling energy between the normal modes of two structures separated by d⃗ can be estimated by the following integral on total space,

U(d)=dτE1(r1)E2(r2)~c1dτE1(r1)B1(r2).
Here is the volume element. E⃗j and B⃗j are the mode electric and magnetic fields produced by resonator j(= 1, 2). r⃗j are the position vectors of the element from the resonator centers, and d⃗ = r⃗2r⃗1. Ideally, two complementary structures produce dual electromagnetic fields. When their distance is small r⃗2r⃗1, E (or B) fields generated by two structures are always perpendicular to each other, so the coupling energy becomes minimal. Perhaps by taking the two trends into account, we should expect the maximum U and maximal coupling occurs at drra.

3.3. Orientation dependence

From the coupled DSRR case, we see that 2g for face-to-face configuration is roughly twice than that for face-to-back one. In light of this, we expect that the orientation of the SRRs may have a strong impact to the coupling strength. For systematical study, we parametrize the orientation alignment using two angles θD and θC, representing the angles between their principal (or “face”) directions to the center-to-center vector as illustrated in Fig. 3(a). We performed experiments using our movable system with selected orientation configurations and all showed that the condition dra gives the largest frequency shift in fD and fC. For convenience, we attribute the frequency results at d = 4.0 mm and d > 11.0 mm as the fully shifted (Ωj) and original frequencies(ωj) for the dimer.

 figure: Fig. 3

Fig. 3 (a) The definition of orientation parameters θD and θC. (b)(c) The dimensionless coupling constants 2gC and 2gM for coupled DSRR and CSRR in various orientation configurations. Again, γ = 0.2 is assumed. The configurations is presented in (θD, θC). In (a) the 1-layer and 2-layer simulation data are multiplied by factors of 0.38 and 0.5, respectively. In (c) the factors are 0.6 and 0.44. (d) The polar plot for 2gC (red symbol) and 2gM (blue symbol) at various of θC when θD = π/2 done by 1-layer simulation. The largest coupling 2gC occurs at θC ∼ 0 and the smallest at θCπ. The angle of the minimum 2gC and maximum 2gM are the same. The red curve is a function plot 2gC = 0.088 + 0.014 cos θC.

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Figures 3(b) and 3(c) shows the deduced 2gC and 2gM for 10 major configurations using movable-stage experiment, and simulations for 1-layer and 2-layer PCB settings. Again, 2gC values obtained for the 3 methods follow the order that 1-layer value > 2-layer value > experiment value. Yet the data points agree well to each other if the later two are re-scaled by a common factor. In general, when θD is fixed, the coupling strength is smaller when θCπ. In addition, the maximal value can be as 2 times as the minimal value. Similar factor was found in the coupled DSRR case as mentioned before. If looking C-coupling strength only, we found the following order gC(π, 0) > gC(0, 0) = gC(π, π) > gC(0, π).

The above observation reveals that when θD = π/2, the coupling strength can vary greatly as θC changes. For a close look of θC dependence when θD = π/2, simulations with 1-layer PCB setting were performed. Presented as a polar plot in Fig. 3(d), the coupling strength 2gC shows a maximum value at θC = 0 and minimal one at θC = π. The data points can be roughly described by a function form of 2gC = 0.088 + 0.014 cos θC as marked by the red curve. The maximum and minimum of 2gM occur at θC = 3π/4 and 5π/4, respectively. Though the 2gM data cannot be described by a simple function.

3.4. One-dimensional array based on SRRs

Taking the advantage of strong coupling between resonators, we can build signal couplers between two TLs using hybrid DSRR/CSRR structure. Based on our orientation study, we chose the orientation (π/2, π/2) owing to the large coupling and good symmetry. Figures 4(a) and 4(b) respectively show the S-parameters S21, S31 and S41 of the coupler using 2 coupled DSRRs (C1) and 3 CSRRs (C2). Their coupling factor are very poor, except for C2, the coupling factors is ∼ −40 dB at the first resonance at 3.05 GHz but increases to −20 dB at second resonance 6.40 GHz. Unlike C1 and C2, coupled DSRR/CSRR (C3) provides a large coupling factor up to −10 dB in a wider frequency range from 2.6 − 3.3 GHz as illustrated in Fig. 4(c). At the second resonance of 6.0 GHz, it also maintains a −12 dB coupling factor. Furthermore, the coupling is symmetric in regarding to port 3 and 4. Intriguingly one can see the transmission coefficient shown in Fig. 5(a) and 5(b) exhibit a structure splits into 4 resonances dips, featuring the multi-mode PIT spectrum.

 figure: Fig. 4

Fig. 4 The S-parameters in inter-TL coupler experiment. Transmission lines coupled by 2 DSRRs(C1) (a), 3 CSRRs(C2) (b), and interleaved DSRRs and CSRRs(C3) (c). Because of the large d between DSRRs in C1 and CSRRs in C2, resonators are not coupled to each other. C3 structure gives a inter-TL coupling than the first two. In (c) the solid curves are experimental data and the dashed curves are simulation results. (d) The schematic of C3 and the measurement ports.

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 figure: Fig. 5

Fig. 5 (a)(b) Amplitude(a) and phase(b) of S21 for C3 near the resonance regime. It demonstrates a PIT spectrum with 5 modes; clear dips were found at 2740, 2865, 3027 and 3324 MHz, with an unclear one at 3220 MHz. (c) Circuit model for coupled SRR array.

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To understand the wave guiding property, we consider the following Lagrangian for a one-dimensional(1D) chain of N SRRs. Without loss of generality, pure C-coupling is under consideration,

=j=1NLjq˙j22(qiqx,j+qx,j1)22Cjj=1N1qx,j2Cx,j.
qj ’s and qx,j ’s are charge variables of SRRs and coupling capacitors. By noting that all DSRR(j=even) and all CSRR(j=odd) are identical and using the notation of α1=CpC11 and α2=CpC21, one gets
((1α1)ω12ω2α20α1(12α2)ω22ω2α10α2(12α1)ω12ω2)(q˜1q˜2q˜3)=0

Here qj (t) = jeiωt. In degenerate case ω1 = ω2, the 1D system forms a two-band structure with a dispersion relation,

ω=ω1(1α1α2)±α12+α22+2α1α2coskd.
When α1, α2 ≪ 1, each subband has a bandwidth about min (α1, α2) ω1, and centered at 1α1α2±(α12+α22)ω1.

The dips in Fig. 5(a) are roughly equal spaced by 150 MHz with a span of 584 MHz and centered at 3.03 GHz, showing a large coupling strength of α1α2 ∼ 2gC ∼ 10% between DSRR and CSRR. The experiment results clearly demonstrate that a hybrid structure with interleaved DSRRs and CSRRs allows in a high efficient band-pass waveguiding. In comparison to the arrays composed of single type of resonators which can only have one negativity, the hybrid structure has simultaneous negative permeability and permittivity. In addition, The DSRR/CSRR unit automatically forms a diatomic basis and paves the way to the photonic band structure with a phonon-like dispersion. We leave some of the properties for future study.

4. Conclusion

In conclusion, we have demonstrated an efficient planar wave-guiding using hybrid structure of interleaved DSRRs and CSRRs at microwave frequency. It provides a −10 dB coupling between two TLs and a bandwidth of about 600 MHz. To understand the origin of the strong coupling between a DSRR and CSRR, we systematically investigated how the resonance frequencies of the two resonators change with their separation, and found that the coupling is greatest when dra. We also studied the coupling strengths when the two resonators are in various orientation, and found that a factor-of-2 reduction can be experimentally observed. Compared to chains built with single type of resonators, the overlapping structure yields a greater coupling between the resonators and greatly enlarges the transmission amplitude and bandwidth. When used in constructing large lattice structures, the DSRR/CSRR basis enables the engineering of the photonic band structure and could find vast applications. By utilizing state-of-the-art micro fabrication, one can reduce the resonator size and increase the operation frequency upto THz and optical one.

A. Circuit model for coupled SRRs

To analyze the observed shift in resonances, we used a model circuit as shown in Fig. 1(c). Each resonator is modeled by a LC circuit and they are coupled by a capacitance Cx and a mutual inductance M. The system Lagrangian reads,

=j=1,2Ljq˙j22+Mq˙1q˙2(q1qx)22C1(q2+qx)22C2qx2Cx.
Here q1 and q2 are charge variables producing current passing through inductor L1 and L2, respectively. qx is charge accumulate on the coupling capacitor Cx. The equations of motions for q1 and q2 are
d2dt2(q1q2)=(L1MML2)1(C11(1CpC11)CpC11C21CpC11C21C21(1CpC21))(q1q2)=11β1β2((1α1)ω12α1β1ω22(1α2)β1ω22+α2ω12(1α1)β2ω12+α1ω22(1α2)ω22α2β2ω12)(q1q2)

Here we used qxCp(C11q1C21q2) and Cp1C11+C21+Cx1. ωj=1/LjCj are the intrinsic resonance frequencies of the two SRRs. In the last equation, notations αjCpCj1 and βjMLj1 are used. The eigen-frequency Ω satisfies,

|(1α1)ω12α1β1ω22(1β1β2)Ω2(1α2)β1ω22+α2ω12(1α1)β2ω12+α1ω22(1α2)ω22α2β2ω12(1β1β2)Ω2|=0

For degenerate SRRs ω1 = ω2, the resonances are

Ω2ω12=(1α1+α22)±(α1α22)2+(α1β2)(α2β1).
Here higher order in α’s and β’s are omitted since the coupling strengths are usually much smaller than 1. One can clear see the frequency shift in pure capacitance(C)-coupling and mutual inductance(M)-coupling cases are different. For the former, Ω = ω1 and ω11α1α2, giving a repulsion of resonance ΔΩ = Ω1 − Ω2ω1(α1 + α2)/2 = 2ω1gC. For the later, Ω=ω11±β1β2, and repulsion of resonance ΔΩ~ω1β1β2=2ω1gM. If the SRR are identical, namely α1 = α2 and β1 = β2, the result can be simplified further as
Ω=ω11α1±|α1β1|

For non-degenerate case, a simple expression can be used to extract the C-coupling strengths only,

Ω1Ω2ω1ω2=1α1α21β1β2~112(α1+α2)=12gC
If the shift in resonance frequencies δωj are not large,
2gC~12(δω1ω1+δω2ω2).
The relative frequency shifts provides an easy way to estimate coupling strength 2gC.

Funding

Ministry of Science and Technology, Taiwan (102-2628-M-005-001-MY4, 106-2112-M-005-007).

Acknowledgments

We are grateful to the National Center for High-performance Computing for computer time and facilities. Fruitful discussions with C. C. Huang and C. S. Wu are acknowledged. This work is financially supported by the Ministry of Science and Technology, Taiwan under grant Nos. 102-2628-M-005-001-MY4 and 106-2112-M-005-007, and Research Center for Sustainable Energy and Nanotechnology, NCHU.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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19. R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011). [CrossRef]  

20. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012). [CrossRef]  

21. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013). [CrossRef]  

22. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011). [CrossRef]  

23. M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013). [CrossRef]  

24. R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012). [CrossRef]   [PubMed]  

25. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013). [CrossRef]   [PubMed]  

26. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013). [CrossRef]  

27. F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007). [CrossRef]  

28. 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, 125121 (2011). [CrossRef]  

29. E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012). [CrossRef]  

30. O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005). [CrossRef]  

31. H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE. 79, 1505–1518 (1991). [CrossRef]  

32. J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005). [CrossRef]  

33. 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, 18131–18144 (2008). [CrossRef]   [PubMed]  

References

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  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
    [Crossref] [PubMed]
  2. V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon. 1, 41–48 (2007).
    [Crossref]
  3. R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref] [PubMed]
  4. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
    [Crossref] [PubMed]
  5. C. Caloz and T. Itoh, Electromagnetic metamaterials: transmission line theory and microwave applications (John Wiley & Sons, 2005).
  6. F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
    [Crossref] [PubMed]
  7. F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
    [Crossref]
  8. M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
    [Crossref]
  9. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  10. Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
    [Crossref] [PubMed]
  11. M. Parvinnezhad Hokmabadi, E. Philip, E. Rivera, P. Kung, and S. M. Kim, “Plasmon-induced transparency by hybridizing concentric-twisted double split ring resonators,” Sci. Rep. 5, 15735 (2015).
    [Crossref] [PubMed]
  12. S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
    [Crossref] [PubMed]
  13. J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
    [Crossref]
  14. N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
    [Crossref]
  15. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
    [Crossref] [PubMed]
  16. F. Miyamaru, H. Morita, Y. Nishiyama, T. Nishida, T. Nakanishi, M. Kitano, and M. W. Takeda, “Ultrafast optical control of group delay of narrow-band terahertz waves,” Sci. Rep. 4, 4346 (2014).
    [Crossref] [PubMed]
  17. C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
    [Crossref] [PubMed]
  18. S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
    [Crossref]
  19. R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
    [Crossref]
  20. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
    [Crossref]
  21. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
    [Crossref]
  22. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
    [Crossref]
  23. M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
    [Crossref]
  24. R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
    [Crossref] [PubMed]
  25. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
    [Crossref] [PubMed]
  26. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
    [Crossref]
  27. F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
    [Crossref]
  28. 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, 125121 (2011).
    [Crossref]
  29. E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
    [Crossref]
  30. O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
    [Crossref]
  31. H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE. 79, 1505–1518 (1991).
    [Crossref]
  32. J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
    [Crossref]
  33. 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, 18131–18144 (2008).
    [Crossref] [PubMed]

2016 (1)

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

2015 (1)

M. Parvinnezhad Hokmabadi, E. Philip, E. Rivera, P. Kung, and S. M. Kim, “Plasmon-induced transparency by hybridizing concentric-twisted double split ring resonators,” Sci. Rep. 5, 15735 (2015).
[Crossref] [PubMed]

2014 (1)

F. Miyamaru, H. Morita, Y. Nishiyama, T. Nishida, T. Nakanishi, M. Kitano, and M. W. Takeda, “Ultrafast optical control of group delay of narrow-band terahertz waves,” Sci. Rep. 4, 4346 (2014).
[Crossref] [PubMed]

2013 (4)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

2012 (3)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
[Crossref]

2011 (5)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

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, 125121 (2011).
[Crossref]

C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

2010 (1)

S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

2009 (1)

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

2008 (2)

2007 (3)

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon. 1, 41–48 (2007).
[Crossref]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

2006 (1)

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

2005 (3)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
[Crossref]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[Crossref] [PubMed]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

2003 (1)

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

2001 (1)

R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

1991 (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE. 79, 1505–1518 (1991).
[Crossref]

Alexander, S.

S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

Aydin, K.

Baena, J. D.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Beruete, M.

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Bonache, J.

M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
[Crossref]

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic metamaterials: transmission line theory and microwave applications (John Wiley & Sons, 2005).

Cao, C.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Chen, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

Cong, L.

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Economou, E. N.

Edwards, D. J.

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, 125121 (2011).
[Crossref]

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[Crossref] [PubMed]

Falcone, F.

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

Fan, S.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

Faulkner, G.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

Fedotov, V. A.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

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[Crossref]

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S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
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[Crossref] [PubMed]

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Sydoruk, O.

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, 125121 (2011).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
[Crossref]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

Takeda, M. W.

F. Miyamaru, H. Morita, Y. Nishiyama, T. Nishida, T. Nakanishi, M. Kitano, and M. W. Takeda, “Ultrafast optical control of group delay of narrow-band terahertz waves,” Sci. Rep. 4, 4346 (2014).
[Crossref] [PubMed]

Tatartschuk, E.

E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
[Crossref]

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, 125121 (2011).
[Crossref]

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

Tomita, M.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

Tsai, D. P.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Tunnermann, A.

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

Umucalilar, R. O.

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Wan, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[Crossref] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Wu, C.

C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

Wu, J.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Xiao, B.

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

Xu, Q.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

Xu, W.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Yang, H.

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

Yu, Z.

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Zhang, Y.

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

Zheludev, N. I.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[Crossref] [PubMed]

Zhuromskyy, O.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
[Crossref]

Appl. Phys. Lett. (4)

F. Martin, J. Bonache, F. Falcone, M. Sorolla, and R. Marques, “Split ring resonator-based left-handed coplanar waveguide,” Appl. Phys. Lett. 83, 4652–4654 (2003).
[Crossref]

J. Wu, B. Jin, J. Wan, L. Liang, Y. Zhang, T. Jia, C. Cao, L. Kang, W. Xu, and J. Chen, “Superconducting terahertz metamaterials mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 99, 161113 (2011).
[Crossref]

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94, 211902 (2009).
[Crossref]

O. Sydoruk, O. Zhuromskyy, E. Shamonina, and L. Solymar, “Phonon-like dispersion curves of magnetoinductive waves,” Appl. Phys. Lett. 87, 072501 (2005).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

J. D. Baena, J. Bonache, F. Martin, R. M. Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garcia-Garcia, I. Gil, M. F. Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 (2005).
[Crossref]

J. Appl. Phys. (1)

E. Tatartschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111, 094904 (2012).
[Crossref]

J. Phys. B At. Mol. Opt. Phys. (1)

S. Alexander and N. Stefan, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B At. Mol. Opt. Phys. 43, 163001 (2010).
[Crossref]

Metamaterials (1)

M. Gil, J. Bonache, and F. Martin, “Metamaterial filters: A review,” Metamaterials 2, 186–197 (2008).
[Crossref]

Nature (1)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Nature Photon. (4)

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nature Photon. 7, 1001–1005 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, A. Tunnermann, S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomagnetic field and photonic landau levels in dielectric structures,” Nature Photon. 7, 153–158 (2013).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon. 6, 782–787 (2012).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photon. 1, 41–48 (2007).
[Crossref]

Nature Phys. (1)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nature Phys. 7, 907–912 (2011).
[Crossref]

Opt. Express (1)

Phys. Rev. A (1)

R. O. Umucalilar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Phys. Rev. B (1)

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, 125121 (2011).
[Crossref]

Phys. Rev. Lett. (5)

C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. 106, 107403 (2011).
[Crossref] [PubMed]

R. O. Umucalilar and I. Carusotto, “Fractional quantum hall states of photons in an array of dissipative coupled cavities,” Phys. Rev. Lett. 108, 206809 (2012).
[Crossref] [PubMed]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96, 123901 (2006).
[Crossref] [PubMed]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marques, F. Martin, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett. 93, 197401 (2004).
[Crossref] [PubMed]

Phys. Status Solidi B (1)

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[Crossref]

Proc. IEEE. (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE. 79, 1505–1518 (1991).
[Crossref]

Rev. Mod. Phys. (2)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Sci. Rep. (3)

F. Miyamaru, H. Morita, Y. Nishiyama, T. Nishida, T. Nakanishi, M. Kitano, and M. W. Takeda, “Ultrafast optical control of group delay of narrow-band terahertz waves,” Sci. Rep. 4, 4346 (2014).
[Crossref] [PubMed]

M. Parvinnezhad Hokmabadi, E. Philip, E. Rivera, P. Kung, and S. M. Kim, “Plasmon-induced transparency by hybridizing concentric-twisted double split ring resonators,” Sci. Rep. 5, 15735 (2015).
[Crossref] [PubMed]

S. Han, L. Cong, H. Lin, B. Xiao, H. Yang, and R. Singh, “Tunable electromagnetically induced transparency in coupled three-dimensional split-ring-resonator metamaterials,” Sci. Rep. 6, 20801 (2016).
[Crossref] [PubMed]

Science (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[Crossref] [PubMed]

Other (1)

C. Caloz and T. Itoh, Electromagnetic metamaterials: transmission line theory and microwave applications (John Wiley & Sons, 2005).

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Figures (5)

Fig. 1
Fig. 1 (a) A photograph of a “face-to-back” coupled DSRR sample. The scale bar is 3 mm. (b) The schematic of the moving platform for the coupled DSRR and CSRR. The metal and bare PCB regions are respectively drawn in yellow and blue colors. Our setup allows us to move the lower layer in respected to the upper layer. The two layers are firmly contacted to each other during the measurement. (c) Circuit model for capacitively and inductively coupled SRRs. (d) The transmission data of the “face-to-face” coupled DSRRs. d varies from 7.7 mm to 8.2 mm, with a step of 0.1 mm, corresponding to gap between DSRRs from 0.1 mm to 0.6 mm. Two distinct resonances appear owing to the large inter-resonator coupling. When d increases to 8.2 mm, the two resonance dips gradually merge into a single dip. (e) The frequency difference fHfL of two resonance dips in (d) as a function of d for the “face-to-face”(red symbol) and the “face-to-back”(blue symbol) samples. At d = 7.7 mm, the “face-to-face” configuration gives a larger fHfL, while at d = 8.2 mm, “face-to-back” configuration results in a larger fHfL.
Fig. 2
Fig. 2 (a) The transmission data of coupled DSRR/CSRR measurement for 3 different positions, (x /mm, y/mm)=(0,0), (0, −2), and (0, −6). At the last position, two resonance absorption are clear. When the coordinates move toward +y direction, the CSRR is moved away from the microstrip and the resonance of lower frequency, to which the CSRR corresponds is getting smeared. (b) A mapping result of fD. (c) fD and fC as a function of x at y = 0 mm. (d) The coupling strengths 2gC and 2gM as a function of x at y = 0 mm. γ = 0.2 is assumed for calculation of 2gM. The maximum coupling strength is at x = 3 mm.
Fig. 3
Fig. 3 (a) The definition of orientation parameters θD and θC. (b)(c) The dimensionless coupling constants 2gC and 2gM for coupled DSRR and CSRR in various orientation configurations. Again, γ = 0.2 is assumed. The configurations is presented in (θD, θC). In (a) the 1-layer and 2-layer simulation data are multiplied by factors of 0.38 and 0.5, respectively. In (c) the factors are 0.6 and 0.44. (d) The polar plot for 2gC (red symbol) and 2gM (blue symbol) at various of θC when θD = π/2 done by 1-layer simulation. The largest coupling 2gC occurs at θC ∼ 0 and the smallest at θCπ. The angle of the minimum 2gC and maximum 2gM are the same. The red curve is a function plot 2gC = 0.088 + 0.014 cos θC.
Fig. 4
Fig. 4 The S-parameters in inter-TL coupler experiment. Transmission lines coupled by 2 DSRRs(C1) (a), 3 CSRRs(C2) (b), and interleaved DSRRs and CSRRs(C3) (c). Because of the large d between DSRRs in C1 and CSRRs in C2, resonators are not coupled to each other. C3 structure gives a inter-TL coupling than the first two. In (c) the solid curves are experimental data and the dashed curves are simulation results. (d) The schematic of C3 and the measurement ports.
Fig. 5
Fig. 5 (a)(b) Amplitude(a) and phase(b) of S21 for C3 near the resonance regime. It demonstrates a PIT spectrum with 5 modes; clear dips were found at 2740, 2865, 3027 and 3324 MHz, with an unclear one at 3220 MHz. (c) Circuit model for coupled SRR array.

Equations (16)

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Ω = ω 1 + ω 2 2 ± ( ω 1 ω 2 ) 2 2 + | κ | 2 ,
Ω 2 π = f 0 ( 1 ± g ) .
α 1 + α 2 ~ 1 ( Ω 1 Ω 2 ω 1 ω 2 ) 2 ~ 2 ( δ ω 1 ω 1 + δ ω 2 ω 2 ) .
Ω 1 2 Ω 2 2 = [ ( 1 α 2 ) ω 2 2 ( 1 α 1 ) ω 1 2 ] 2 + 4 γ ω 1 2 ( α 1 β 2 ) ( ω 2 2 α 1 ω 1 2 β 2 )
2 ω 1 2 β 2 ~ α 1 ( ω 1 2 + ω 2 2 ) ± 4 α 1 2 Δ 4 + γ 1 [ 4 Δ Ω 2 ( 2 Δ 2 α 1 ω 1 2 α 2 ω 2 2 ) 2 ]
U ( d ) = d τ E 1 ( r 1 ) E 2 ( r 2 ) ~ c 1 d τ E 1 ( r 1 ) B 1 ( r 2 ) .
= j = 1 N L j q ˙ j 2 2 ( q i q x , j + q x , j 1 ) 2 2 C j j = 1 N 1 q x , j 2 C x , j .
( ( 1 α 1 ) ω 1 2 ω 2 α 2 0 α 1 ( 1 2 α 2 ) ω 2 2 ω 2 α 1 0 α 2 ( 1 2 α 1 ) ω 1 2 ω 2 ) ( q ˜ 1 q ˜ 2 q ˜ 3 ) = 0
ω = ω 1 ( 1 α 1 α 2 ) ± α 1 2 + α 2 2 + 2 α 1 α 2 cos k d .
= j = 1 , 2 L j q ˙ j 2 2 + M q ˙ 1 q ˙ 2 ( q 1 q x ) 2 2 C 1 ( q 2 + q x ) 2 2 C 2 q x 2 C x .
d 2 d t 2 ( q 1 q 2 ) = ( L 1 M M L 2 ) 1 ( C 1 1 ( 1 C p C 1 1 ) C p C 1 1 C 2 1 C p C 1 1 C 2 1 C 2 1 ( 1 C p C 2 1 ) ) ( q 1 q 2 ) = 1 1 β 1 β 2 ( ( 1 α 1 ) ω 1 2 α 1 β 1 ω 2 2 ( 1 α 2 ) β 1 ω 2 2 + α 2 ω 1 2 ( 1 α 1 ) β 2 ω 1 2 + α 1 ω 2 2 ( 1 α 2 ) ω 2 2 α 2 β 2 ω 1 2 ) ( q 1 q 2 )
| ( 1 α 1 ) ω 1 2 α 1 β 1 ω 2 2 ( 1 β 1 β 2 ) Ω 2 ( 1 α 2 ) β 1 ω 2 2 + α 2 ω 1 2 ( 1 α 1 ) β 2 ω 1 2 + α 1 ω 2 2 ( 1 α 2 ) ω 2 2 α 2 β 2 ω 1 2 ( 1 β 1 β 2 ) Ω 2 | = 0
Ω 2 ω 1 2 = ( 1 α 1 + α 2 2 ) ± ( α 1 α 2 2 ) 2 + ( α 1 β 2 ) ( α 2 β 1 ) .
Ω = ω 1 1 α 1 ± | α 1 β 1 |
Ω 1 Ω 2 ω 1 ω 2 = 1 α 1 α 2 1 β 1 β 2 ~ 1 1 2 ( α 1 + α 2 ) = 1 2 g C
2 g C ~ 1 2 ( δ ω 1 ω 1 + δ ω 2 ω 2 ) .

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