We derive an analytical LC model from Maxwell's equations for the magnetic resonance of subwavelength ring resonators. Using the LC model, we revisit the scaling of split-ring resonators. Inspired by the LC model, we propose a hybrid metal-dielectric ring resonator mainly composed of high index dielectric material (e.g., TiO2) with some gaps filled with metal (e.g., Ag). The saturation frequency of magnetic response for the hybrid metal-dielectric ring resonator is much higher (up to the ultraviolet range) than that for split-ring resonators, and can be controlled by the metal fraction in the ring. The hybrid metal-dielectric ring resonator can also overcome the homogenization problem of all-dielectric magnetic resonators, and therefore can form homogenizable magnetic metamaterials at short wavelengths down to the ultraviolet range.
© 2013 OSA
The magnetic response of a nature material is usually very weak, especially at optical frequencies, where the magnetic permeability () is simply put as one. Non-unity or even zero or negative magnetic permeability is very charming, because it leads to unusual electromagnetic phenomena  and very exciting applications such as superlensing , cloaking [3,4], electromagnetic energy absorbing , tunneling , squeezing [6,7] and radiation engineering [8, 9]. In order to engineer the magnetic permeability, people developed magnetic metamaterials composed of subwavelength artificial magnetic resonators, which can be treated as effectively homogeneous media by homogenization. With the help of magnetic metamaterials, people have managed to achieve non-unity or even negative magnetic permeability from microwave frequencies to optical frequencies [10–29]. Optical magnetism can be realized through metallic structures supporting magnetic plasmonic modes (i.e., can interact with magnetic fields), such as split-ring resonators (SRRs) [12–14,24] and the like [15,17,19,23,26,29]. However, the operation of metallic structures at optical frequencies is limited by the kinetic energy of the electrons in the metal, leading to saturation of the magnetic response when we push the operating frequency deeper into the optical frequency by size scaling . All-dielectric magnetic resonators have no saturation effect and have a much lower loss than metallic structures. They can be applied in microwave or terahertz ranges where large permittivity of several tens of can be easily achieved, but they present a challenging issue in the infrared and visible frequency range due to a relatively low refractive index of materials in this range . Only recently has optical magnetism in the midinfrared been realized with tellurium dielectric cubic resonators . Optical magnetism in the visible range can also be realized using crystalline silicon, where a very specific fabrication method is needed in order to guarantee that the constituent silicon is crystalline silicon [33,34]. Nevertheless, optical magnetism or optical magnetic dipole responses are not sufficient for magnetic metamaterials, which can be homogenized according to the conventional definition for a metamaterial. To form a homogenizable magnetic metamaterial, they have to satisfy the homogenization requirement . If the refractive index of the constituent dielectric material is not large enough, the homogenization of all-dielectric resonators will be unreasonable. This limits the use of all-dielectric magnetic resonators as homogenizable magnetic metamaterials at short optical wavelengths due to the lack of dielectric materials of very high indices at short optical wavelength. Therefore, it is still necessary to develop elegant designs for realization of homogenizable magnetic metamaterials at short optical wavelengths, such as the blue, violet and ultraviolet regions.
In this paper, we aim to demonstrate a hybrid metal-dielectric ring resonator design for homogenizable magnetic metamaterials at short optical wavelengths down to the ultraviolet range. With this design, the saturation for magnetic response occurs at much higher frequencies (up to ultraviolet rang) as compared to SRR design while the resonators are of truly subwavelength size in contrast to all-dielectric design. We first use an LC model to describe magnetic ring resonators in section 2, where we classify ring resonators into three types: split-ring resonator, all-dielectric ring resonator and hybrid ring resonator. Then based on the LC model, we revisit the scaling of the split-ring resonator and analyze the origin of the saturation of the magnetic response in section 3. Unlike , where the kinetic inductance is derived from the kinetic energy of electrons, we derive the effective inductance from Maxwell's equations and the dispersive permittivity of a metal. From this, we find an effective inductance which is similar to for frequencies far below the plasma frequency of the metal, but significantly larger than for visible wavelengths, rendering the size scaling to short wavelengths even more difficult. Then in section 4 we propose a hybrid metal-dielectric ring resonator mainly composed of a high index dielectric material (e.g., TiO2) with some gaps filled with metal (e.g., Ag). Such a new magnetic metamaterial can operate at short wavelengths down to the ultraviolet range, and the operating principle is inspiring for more designs of short wavelength metamaterials. In section 5, we discuss the issue of effective medium homogenization. The homogenization problem of all-dielectric magnetic resonators is illustrated. The advantage of the hybrid ring resonator over all-dielectric magnetic resonators is interpreted on the homogenization issue.
2. Model description
The early artificial material exhibiting strong magnetic response is a type of ring resonator called split-ring resonator , which later appears in many variations, with the key mechanism unchanged. The unit cell () of the ring resonator metamaterial explored in the present paper is shown in Fig. 1. It is a two dimensional ring resonator with radius and ring thickness . Here the ring shape is fixed (as an example), but the size of the ring is determined by the value of (used as the size scaling parameter hereafter). The ring is composed of two types of materials with permittivity and , occupying two and two portions by angle (where ), respectively.
Based on the implication that the ring resonator has a well-defined magnetic dipole resonance, we can assume a uniform displacement current in the ring (i.e., for any angle position, the displacement current confined in the ring layer is the same). We also approximate the displacement current density at the middle (i.e., ) of the ring layer as the average displacement current density in the ring layer, so that we can express the displacement current as , where is the average displacement current density. Therefore the displacement current can be expressed as (assuming time-harmonic dependence )Eqs. (1) and (2):
3. Scaling of conventional split-ring resonator
To analyze the scaling problem of the conventional SRR at the optical wavelength, we use Ag as the metal material () and use air as the dielectric material (). The permittivity of Ag can be described by the Drude model to fit the experimental data  in the frequency range 150 – 700 THz (i.e., wavelength range 2μm – 430nm). The fit parameters are , and. The permittivity is dispersive, and thus is also dispersive. The permittivity has a negative real part and a positive imaginary part; therefore is a negative effective capacitance in parallel connection with a resistance, which is equivalent to a positive inductance in series connection with a resistance , i.e., , where [solid line in Fig. 2(a)] and . Then the effective magnetic permeability can be expressed as a resonant formFig. 2(b)] and the frequency line [dotted line in Fig. 2(b)]. For frequencies far below the plasma frequency of the metal (e.g., below 100 THz), can be approximated as . With this approximation, becomes the well-known kinetic inductance , which is non-dispersive as shown in Fig. 2(a) (dashed line), and . The resonance frequency can then be directly expressed as [dashed line in Fig. 2(b)], and the quality factor can be expressed as . For the 2D SRR shown in Fig. 1, when we scale all the geometrical parameters (), we see easily that (equal to ), and remain constant, while scales proportional to the scale factor as . For a silver SRR structure operating at optical frequencies, is comparable with or even larger than . Thus reducing by size scaling is not an effective method to push the resonance wavelength to the short optical wavelength. For frequencies not far from the plasma frequency of the metal, increases quickly when the wavelength decreases [cf. solid line in Fig. 2(a)], which makes it even more difficult to push the resonance wavelength to the shorter optical wavelength by size scaling. By comparing the dashed line with the intersection point denoted with a circle in Fig. 2(b), we can see that for the given structure parameters, the resonance frequency determined by the LC model with the approximation is overestimated by about 20 THz. For a resonator with higher resonance frequency, the overestimation would be even larger (as will be indicated below referring to Fig. 3). Some effort has been made to reduce by using a metal with a higher plasma frequency, e.g., aluminum . However, the loss of aluminum is much higher than silver, which means that although magnetic resonances at short wavelengths can be achieved, they are very weak. For SRRs, another problem during scaling is that although the quality factor decreases slowly, decreases very quickly (due to the decrease of and the increase of), which reduces the amplitude of the magnetic resonance. Hence, the conventional metallic SRRs are not suitable for magnetic metamaterials at short optical wavelengths.
In Fig. 3, we show the scaling process of the SRR structures in terms of magnetic resonance frequency. From the simulation  results (solid lines), we can see the saturation phenomenon when the scaling factor is large. SRRs with saturate at about 400 THz, whereas SRRs with , and saturate at about 500 THz, 550 THz and 600 THz, respectively. This is because a larger results in a smaller, and thus a larger resonance frequency. Note that there are two assumptions [see the text before Eq. (1)] that lead to Eqs. (1) and (2), where we started the introduction of the LC model. The first assumption is that the displacement current in the ring is uniform. The second assumption is that the displacement current density at the middle of the ring layer may be taken as the average displacement current density in the ring layer. The first assumption mainly causes some quantitative error on the derived effective capacitance . This assumption is valid when all the materials forming the ring resonator are of large permittivity (with respect to the background), or the angle portion of the material of small permittivity is very small so that . When there is a material of small permittivity and its angle portion is not very small, the electric field in this material will not be well confined and will have a large radial component, and consequently the assumption of uniform displacement current is not valid in this material. This leads to underestimation of the effective capacitance, and thus overestimation of the resonance frequency in the LC model, as has been indicated in . The second assumption may also cause some quantitative error, and mainly affects the determination of the effective radius of the ring resonator. The displacement current density at is the average displacement current density in the ring layer. As implied by Eq. (2), we have taken as the effective radius . The effective radius may deviate a bit from. The possible value of is a value between (the inner edge of the ring layer) and (the outer edge of the ring layer). We find that the possible quantitative error (on the resonance frequency of the magnetic resonator) induced by the error on the effective radius (the deviation of from ) is much smaller than that induced by the error on the effective capacitance. Therefore, we carry out a quantitative correction only to the effective capacitance for the LC model, by simply adding a correction factor to as . The correction factor is larger for SRRs with larger . We use , ,and for SRRs with , , and , respectively. We can see from Fig. 3 that the resonance frequencies calculated using the LC model (dashed lines) match very well with the simulation results (solid lines). Without the correction to the effective capacitance, the LC model would give larger resonance frequencies (but still well describe the scaling process and the saturation effect). In fact, we do not intend to formulate/construct a very quantitatively accurate theoretic model, which requires lots of detailed factors to be involved at the cost of losing some degree of simplicity and physical intuitiveness. Our LC model aims at revealing the resonance conditions and key factors that influence the magnetic resonance, and thus inspiring new designs with specific features of performance (e.g., magnetic metamaterials at short wavelengths as described in section 4).
We can find again in Fig. 3 the different results given by the LC model with (red dashed line) and without (red dash-dotted line) the approximation . The LC model with the approximation overestimates the resonance frequencies. When the scaling factor is small (e.g., ), the overestimation is not apparent, which means that in this region the dispersive inductance can be well approximated by the non-dispersive inductance . However, when the scaling factor is larger, the overestimation is quite apparent, up to 60 THz. In this region, we have to use the dispersive inductance.
4. Hybrid metal-dielectric ring resonator
In order to operate at a short wavelength down to the ultraviolet range, we propose a new type of ring resonator, of which is a high index dielectric material while is a metal (Ag). This is the case “” described in section 2. In this paper we refer to this new type of ring resonator as HRR, short for Hybrid Ring Resonator. Note that the Drude model fit used for SRRs in section 3 is not accurate at shorter wavelengths down to the ultraviolet range. Therefore, we use another Drude model to fit the experimental data of Ag  for the HRR in the frequency range 700 - 930 THz (i.e., wavelength range 430 – 323 nm). The fit parameters are , and. For demonstration, we assume , which is reasonable since low-loss dielectric materials with permittivity around this value at short wavelengths are available by using materials such as TiO2, SiC and ZnS from various deposition technologies. The effective capacitance is positive while is negative and equivalent to a positive inductance in series connection with a resistance , i.e., , where and . The effective permeability has the same form as Eq. (4)
The scaling of HRRs is represented in Fig. 4(a) for (black lines), (red lines), (green lines) and (blue lines) by using our LC model (dashed lines) and simulation (solid lines). For each scaling curve (i.e., constant ), we still observe the saturation phenomenon as expected by our LC model. However, the saturation frequencies are much higher than those for conventional SRRs (cf. Figure 3) mainly due to the reduction of . The saturation frequency can be pushed higher by reducing the metal fraction in the ring resonator (i.e., decreasing ). Therefore, we can achieve a strong magnetic response at a higher frequency by reducing the metal fraction. To demonstrate this, we keep the scaling factor constant as 100 nm [as indicated by the vertical dotted line in Fig. 4(a)] and change from to . The effective permeability spectra (retrieved from simulated S parameters ) are plotted in Fig. 4(b). By decreasing, the resonance frequency increases from 748 THz (i.e., 401 nm) to 899 THz (i.e., 334 nm). It is worthwhile to point out that according to the Drude model of Ag, the real part of the permittivity is only −0.55 at 899 THz (i.e., 334 nm) and is positive at frequencies higher than 928 THz (i.e., 323 nm). In fact, by decreasing , strong magnetic resonances can be achieved as long as the permittivity of the metal is still negative. We can also see from Fig. 4(b) that, as expected by our analysis with the LC model, the amplitude and quality factor of the magnetic resonances remain quite large, only decreasing gently while being pushed to higher frequencies.
Until now we have simulated HRRs using the Drude model in order to equally compare the simulation results with the LC model results (the Drude model was used in the LC model in order to give analytical expressions and evidently draw the mechanism/principle). The Drude model fit for the real part is quite reliable, though the fit for the imaginary part is less accurate. However, the imaginary part only affects the amplitude and quality factor of the resonances. We yet again check the performance of the HRR by simulation using experimental material data of Ag . The simulated effective permeability spectra are shown in Fig. 4(c). As expected, the simulation results in Fig. 4(b) correspond well with that in Fig. 4(c), except for the differences on the resonance amplitude and quality factor. For resonances at frequencies lower than 840 THz [blue and orange lines in Figs. 4(b) and 4(c)], the simulation using the Drude model results in smaller amplitudes and lower quality factors, while for resonances at frequencies lower than 840 THz [red and black lines in Figs. 4(b) and 4(c)], the simulation using the Drude model results in larger amplitudes and higher quality factors. This agrees with the fact that the imaginary part of the Drude model is larger than the experimental data at frequencies lower than 840 THz while smaller at frequencies higher than 840 THz.
5. Discussion on effective medium homogenization
As fundamental elements of a metamaterial which can be viewed as an effectively homogeneous medium, the subwavelength magnetic resonators should satisfy the homogenization requirements and . The two requirements are independent and have different physical significances. The first condition guarantees that inside the metamaterial the phase advance across the unit cell is not significant so that the spatial dispersion effect is negligible. The second condition guarantees that the electromagnetic wave shining onto the surface of the metamaterial sees a homogeneous interface. The first condition is usually stronger than the second condition because usually for a conventional material. However, the first condition will be a weaker condition when , which may occur quite often for a metamaterial. Thus we utilized both the normalized effective wavelength [where ] and the normalized vacuum wavelength to evaluate the reasonableness of the homogenization. The homogenization is more reasonable when the normalized wavelength is longer. A photonic band gap regime typically occurs around . Note that is the effective refractive index retrieved from the S parameters, assuming the metamaterial as an effectively homogeneous medium . In the magnetic resonance frequency region, the effective refractive index is strongly dispersive. changes quickly across the resonance frequency, from a larger value at the low-frequency side of the resonance to a smaller value (even to zero) at the high-frequency side of the resonance. Thus the normalized effective wavelength also changes quickly with the frequency in the resonance region.
For SRRs and HRRs in the present paper, we plotted the minimum normalized effective wavelength (i.e., the worst homogenization condition in the resonance region) versus the scaling factor as dashed lines with solid symbols in Figs. 5(a) and 5(b). We also plotted the normalized vacuum wavelength as solid lines with solid symbols in Figs. 5(a) and 5(b). We found that both the normalized effective wavelength and the normalized vacuum wavelength increase monotonically when the resonators are scaled down (i.e., increases). When we scale down deep into the saturation region [see Fig. 3 for SRRs and Fig. 4(a) for HRRs, where the resonance wavelength decreases very slowly with ], both the normalized effective wavelength and the normalized vacuum wavelength increase nearly linearly with . It is worthwhile to point out that although the saturation effect for SRRs and HRRs is unfavorable for pushing the operating wavelength to the short end by geometrical size scaling, it is favorable for increasing the normalized wavelength (i.e., more suitable for homogenization) by just shrinking the structure. However, the strength of the magnetic resonance decreases as a trade-off when the normalized wavelength increases. We plotted the achievable minimum effective magnetic permeability versus as solid lines with hollowed symbols in Figs. 5(a) and 5(b). The achievable minimum effective permeability indicates the strength of the magnetic resonance. Note that the right y-axis for is in a reversed direction. We can observe the trade-off between the minimum effective permeability and the normalized wavelength (i.e., if longer normalized wavelength is expected, the minimum effective permeability achieved will be larger).
For all-dielectric magnetic resonators (e.g., the case “” described in section 2), there is no saturation effect, i.e., the resonance wavelength decreases linearly as the structure is scaled down (i.e., ). This means that, for all-dielectric magnetic resonators, the normalized vacuum wavelength is constant when we scale down the structure. Therefore, the normalized vacuum wavelength cannot be increased by shrinking the structure. In fact, the normalized vacuum wavelength is determined by the refractive index of the constituent material. For an all-dielectric magnetic resonator of a fixed size, it is well-known that the resonance wavelength increases with the refractive index of the constituent dielectric material. Thus the only effective way to increase the normalized vacuum wavelength of all-dielectric magnetic metamaterials is to utilize a dielectric material with a higher refractive index. Therefore, all-dielectric homogenizable metamaterials are challenging at short optical wavelengths due to the lack of dielectric materials of very high indices at short optical wavelengths. For all-dielectric resonators made of dielectric material with refractive index , it is very difficult to form a magnetic metamaterial with a regime where (the minimal requirement for possible homogenization) in the magnetic resonance region, although they can support well defined magnetic dipole resonances [33,34,40]. So far, no natural dielectric material with refractive index has been found in the NIR-vis-UV range, except crystalline silicon in the wavelength region 420~600nm (at wavelengths shorter than 420nm, the loss of crystalline silicon is too large). For all-dielectric magnetic metamaterials made of crystalline silicon, however, the normalized vacuum wavelength is still not large enough for good homogenization (which requires, for example, ). For our HRR design (using a dielectric material of refractive index of only 2.45), both the normalized vacuum wavelength and the normalized effective wavelength can easily get a value larger than 4 at optical frequencies [Fig. 5(b)].
We have proposed an LC model for ring resonators, based on which we summarized three cases of magnetic resonances. In the framework of our model, we revisited the scaling of conventional SRRs and proposed a hybrid metal-dielectric ring resonator with a much higher saturation frequency (up to the ultraviolet range) than that of SRRs and a truly subwavelength resonator size in contrast to all-dielectric magnetic resonators. The hybrid metal-dielectric ring resonators can form homogenizable magnetic metamaterials at short wavelengths, down to the ultraviolet range. Our model is physically intuitive and can provide some guidance for the design of magnetic metamaterials at optical frequencies. Our proposal for the hybrid ring resonator design is just a prototype, and variations and improvements are possible for better performance and ease of fabrication.
This work is partially supported by the National High Technology Research and Development Program (863 Program) of China (2012AA030402), the National Natural Science Foundation of China (60990322 and 61178062), the Swedish VR grant (621-2011-4620) and the Asian Office of Aerospace Research and Development (AOARD) (114045).
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