## Abstract

A class of nano-scale wavelength-selective optical filters is proposed where the core of a metal-insulator-metal square ring is replaced with a split-ring core (SRC). The proposed resonator supports split-ring-resonator-like (SRR-like) resonant modes that are characteristics of the structure. These resonant modes are highly adjustable, via the gap size of the split-ring core, over a range of hundreds of nanometers. The proposed resonator can also incorporate tunable materials localized in the gap of the SRC or placed throughout the resonating path. By varying the refractive index (1 to 2) of the material in the gap of the SRC, first and second SRR-like modes can be tuned over ~200 and 300 nm, respectively. A circuit model based on transmission-line theory is proposed for the structure and used to derive the resonance conditions of the split-ring-resonator-like modes; the model compares favorably to the numerical results. The proposed resonator has the potential to be utilized effectively in integrated nano-scale optical switches and tunable filters.

©2013 Optical Society of America

## 1. Introduction

Metal-Insulator-Metal (MIM) waveguides are important metallic nanostructures used to guide surface plasmon polaritons (SPPs) at the nanoscale [1–4]. Strong confinement of light, no bend losses, and simple fabrication are examples of the interesting attributes of MIM waveguides [1–4]. These attributes have motivated researchers to design and investigate a variety of optical devices based on MIM waveguides. Among these, plasmonic resonant structures have been studied widely, particularly for optical signal processing and filtering applications [4–24]. For example, add/drop directional couplers [5,6,10], wavelength demultiplexing structures [20–24] and band-stop (-pass) filters [5–14,23] have been investigated.

According to previous studies, 2D MIM-based resonant structures suitable for filtering purposes can be mainly categorized as ring resonators [5–10,12], stubs [13–17] and slots [18–23]. Of particular interest in this paper are square-shaped ring resonators which have been investigated as efficient resonating structures to realize band-stop (-pass) or add/drop filters [5,6,10]. Similarly to rectangular-shaped MIM resonators, square-shaped resonators provide high coupling efficiency due to the long coupling section between the waveguide and the resonator compared with circular rings [5,6,10]. And, due to its symmetry, the odd resonances are degenerate leading to efficient application to add/drop filters [10]. Moreover, incorporating a nano-wall in the square ring allows efficient modification of the transmission spectrum of the filter either by changing the position or the width of the nano-wall [12].

In this paper, we propose a new structure combining SRRs [25–28] with MIM resonators for application to highly tunable optical filters. In our design, the core of the MIM square ring resonator is replaced with a split-ring core. We use the Finite Difference Time Domain (FDTD) method to investigate the resonant properties of the structure and we propose (and verify) a circuit model based on transmission-line theory [3,12,15,19,29] to predict the resonance behavior of the modes of interest. The results reveal that the proposed structure supports two sets of resonant modes. The first set is similar to those of the normal square ring MIM resonator, while the second set is SRR-like, specific to the new design, and highly tunable by the split-ring core gap. Also, the resonance wavelength of the first mode of the SRCRR is longer than that of the square ring of the same dimensions, demonstrating the potential of our new design to achieve resonators with smaller dimensions. Our ring resonator with a split-ring core (SRC) will henceforth be referred to as a SRCRR (split-ring core ring resonator).

## 2. Geometry and theoretical analysis

The schematic of the proposed SRCRR is shown in Fig. 1(a)
, from which it is noted that a SRC is used instead of a square core in the structure. Also, the gap of the SRC is highlighted in blue, where a tunable material might be incorporated. The structural parameters consist of the width of the MIM waveguides (*d*), the gap width between the bus waveguide and the resonator (*s*), the side length of the resonator (*L _{x}* =

*L*

_{y}=

*L*), the side lengths of the core (

*l*

_{x}=

*l*

_{y}=

*l*), the width of the SRC (

*w*), and the gap size of SRC (

*g*). The insulator is air and the complex permittivity of silver (

*ε*) is modelled by the Drude equation [4,30]. The structure is excited by the fundamental symmetric (relative to

_{m}*E*) SPP mode of the MIM waveguide, generated at the left end of the bus waveguide as shown (the complex propagation constant

_{y}*β*and field distribution of this mode can be obtained by solving an eigenvalue equation [31]). For all computations, the structural parameters of the filters are fixed to

*L*= 300 nm,

*l*= 200 nm,

*d*= 50 nm,

*s*= 20 nm and

*w*= 50 nm. Also, the SRR-like resonant modes are identified as TM

_{g}

*(*

_{k}*k*= 1, 2,...) modes, in which “

*g*” stands for the gap of SRC.

#### 2.1 Circuit model based on transmission-line theory

Equivalent circuit models are very useful as they provide insight on the main parameters affecting performance, and, conversely, they provide a means for rapidly estimating performance given the geometry and materials. Successful circuit models have been proposed to describe the resonance behavior of SRRs [25–28]. Also, based on microwave concepts, transmission line models have been successfully used to describe MIM-based wavelength-selective structures [3,12,15,22]. In this section, considering the symmetry properties of the square ring, we combine previous models for SRRs and MIM structures, and we propose a simple circuit model describing the resonance behavior of the SRR-like resonant modes of SRCRR structure. Before presenting the circuit model for SRR-like resonances, it is worth briefly discussing the effects of the SRC on the normal square ring MIM resonator modes, the TM_{1}, TM_{2c}, and TM_{2f} resonant modes [12].

Adding a SRC to the square ring structure reduces its symmetry group from *C*_{4v} to *C*_{1v} [32], affecting the degenerate modes. As a result, the degeneracy of the first resonance of the square ring will be lifted and only one of the degenerate modes will be excited; then, in the arrangement consisting of a SRCRR side-coupled to the bus waveguide, depending on the position of the gap, only one of them can be excited. In contrast, both the TM_{2c} and TM_{2f} modes (for which degeneracy is lifted by the corners of the square ring) can be excited.

Employing Ampere’s law, current patterns related to the TM_{1}, TM_{2c} and TM_{2f} modes can be predicted by considering the electric and magnetic field patterns of these modes. Peaks of the H-field profile of each mode and the corresponding E-field vectors are shown schematically in Fig. 1(b), where blue (red) circles indicate maxima (minima) of the magnetic field patterns, and black vectors represent the electric fields. As seen in Fig. 1(b), for the TM_{1} and TM_{2c} modes, the electric fields go through zero (and change direction) in the middle of the gap. In contrast, for the TM_{2f} mode, the electric field does not go through zero in the gap, resulting in the formation of displacement currents (*J*_{d} = -i*ω*** D**) therein; however, loop currents cannot completely form in the SRC due to the opposing direction of the electric fields in the SRC. Therefore, the TM

_{2f}mode will show a weak sensitivity to the gap of the SRC, while the TM

_{1}and TM

_{2c}modes will remain approximately unchanged. Conversely for the SRR-like (TM

_{g}

*) resonant modes, current loops can form in the SRC (as in conventional SRRs). This will create resonant modes possessing different characteristics in comparison with the modes of the square ring due to the concentration of electric fields in the gap of the SRC and the enhancement of the magnetic fields in the area bounded by the SRC.*

_{k}To find the resonance condition of the SRR-like modes, the impedance of the SRC should be added to the equivalent circuit of the square ring. The equivalent impedance of the SRC (*Z _{SRC}*) can be represented as [28,33]:

*R*=

*ρ l*/

_{eff}*w*,

*l*= 4(

_{eff}*l*-

*w*)-

*g*and

*ρ*= 1/

*σ*= 1/(-

*iω*(

*ε*

_{m}-

*ε*

_{0}));

*l*

_{eff}is effective length of the SRC and

*σ*is the ac conductivity.

*L*=

_{m}*μ*

_{0}(

*l*-

*w*) (

*l*-

*w*) /

*t*is the magnetic field inductance of the SRC (

*t*= 1 because the structure is 2D),

*L*

_{e}= -

*l*/ (

_{eff}*ω*

^{2}

*ε*

_{m}

*w*') is the electron self-inductance of the SRC [28, 33] and

*C*=

*ε*

_{0}

*w*'/

*g*is the capacitance of its gap;

*w*' =

*γ w*is the effective width of the SRC which is less than

*w*due to the skin depth and the asymmetry of the current distribution between the center and the external sides of SRC ($\gamma \text{\hspace{0.17em}}\approx \text{\hspace{0.17em}}0.6$) [28].

The general equivalent circuit model of the SRCRR is shown in Fig. 1(c), where the equivalent impedance of the SRC (Z_{SRC}) is added as a load to the transmission-line resonator representing the symmetric mode propagating in the square ring and possessing a characteristic impedance of *Z*_{0} = *βd*/*ωε*_{0} [3]. Due to symmetry, we have *Z*_{R} = *Z*_{L} = *Z* [29]:

*Z*

_{R}= Z

_{L}*, so to calculate the resonance wavelengths of the structure, minima in Im{

*Z*} are sought. It should be noted that the length of transmission-line resonator is modified as ${{L}^{\prime}}_{eff}={L}_{eff}+({l}_{y}\text{-}w)$which is due to the penetration of electric field into the SRC through its gap; the second term indicates the distance inside the SRC added to the optical length of the structure for the TM

_{g}

*modes.*

_{k}## 3. Results and discussions

The transmission spectra of the SRCRR (*g* = 40 nm) for the gap in horizontal (SRCRR^{h}) and vertical (SRCRR^{v}) arms of the SRC are compared in Fig. 2(a)
. Moreover, magnetic field patterns of the corresponding resonant modes are plotted below the spectra. For comparison, the resonance wavelength and magnetic field profile of the resonant modes of the square ring with the same dimensions are also plotted in Fig. 2(b).

As noted from Fig. 2, the SRCRR^{h} and SRCRR^{v} support SRR-like resonances (TM_{g}* _{k}* modes) with field patterns that are very different from the other modes which are characteristic of the square ring resonator. Both the TM

_{g1}and TM

_{g2}modes have concentrated magnetic fields in the area enclosed by the SRC (Fig. 2(a)). This concentration of the magnetic fields indicates that circulating currents have formed in the SRC. Hence, as discussed above, the SRC can be effectively used as an adjustable load to control the SRR-like resonances of the SRCRR. We also note that the TM

_{2c}and TM

_{2f}modes can also be excited in the SRCRR and that they are similar to those of the corresponding square ring (Fig. 2(b)). For the TM

_{1}mode, although both the SRCRR and square ring structures have a resonance around 1328 nm, their corresponding field patterns have different symmetries. This behavior, explained in the previous section, is due to symmetry reduction which lifts the degeneracy of the first resonance. In fact, based on the position of the gap, one mode of the degenerate pair would be excited. For the SRCRR

^{h}, the mode possessing an asymmetric field pattern with respect to the symmetry axis (the vertical bisector) is excited. In contrast, for the SRCRR

^{v}, the other mode of the degenerate pair is excited; this mode has a resonance wavelength of 1328 nm and its transmission drop is stronger than the mode in the previous case. A similar behavior is observed for the TM

_{g1}mode; placing the gap in a vertical arm of the SRC strongly affects this mode and lowers its coupling to the bus waveguide (Fig. 2(a)).

The sensitivity of the TM_{1} and TM_{g1} modes to the gap position in the SRC can be attributed to the concentration of their magnetic fields in the lower branch affecting coupling of light from the bus waveguide to the resonator and vice versa. This phenomenon can be explained by coupled-mode theory. Based on this theory, the transmission response of a resonator near resonance (*ω*_{0}) is expressible as [34]:

*τ*

_{0}is the decay rate due to loss, 1/

*τ*

_{e}is the decay rate into the bus waveguide, and 1/

*τ*

_{e1}is the decay rate in the forward direction towards the output. Since the resonant modes of the SRCRR are standing-wave modes, we have 1/

*τ*

_{e1}= 1/

*τ*

_{e}and the transmission on resonance becomes:

According to Eq. (4), decreasing *τ*_{e} reduces |*T*| decreases and higher drop can be achieved at the resonance. Therefore, for the SRCRR^{h} (SRCRR^{v}) structure, where the fields are more concentrated (lower *τ*_{e}), the TM_{g1} (TM_{1}) mode exhibits a larger drop on resonance. This is in accordance with the numerical results of Fig. 2(a). Also, it is seen that the TM_{1} (TM_{g1}) mode of the SRCRR^{h} is narrower (wider) than the TM_{1} (TM_{g1}) mode of the SRCRR^{v} structure, which is consistent with the expression for the external quality factor of the resonator: *Q*_{e} = *ω*_{0}*τ*_{e}/2. In what follows, the SRCRR^{h} is chosen for the computations because of the better performance of the TM_{g}* _{k}* modes and their sensitivity to the gap.

#### 3.1 Tunability of the resonances: Effects of the gap size of the split-ring core

As mentioned above, one motivation for the SRCRR is to provide a load for the square ring resonator that could be easily set to specific values commensurate with a desired filtering application. The gap of the SRC, which affects the capacitance in its equivalent impedance, plays an important role in this respect. Based on the theoretical discussion (subsection 2.1), it is expected that these SRR-like resonances will be more sensitive to the gap size of the SRC compared to other resonances. The resonant wavelengths of all modes computed for *g* varying from 10 to 60 nm are plotted in Fig. 3(a)
. It is seen that the TM_{1}, TM_{2c} and TM_{2f} modes change little in comparison to the TM_{g1} and TM_{g2} modes (SRR-like) as a function of gap size. The maximum variation in the resonance wavelengths of the TM_{1}, TM_{2c} and TM_{2f} modes is about 15, 7 and 40 nm, respectively. In contrast, the resonance wavelengths of the TM_{g1} and TM_{g2} modes vary by about 680 and 290 nm, respectively.

Thus, by means of the proposed structure resonant modes can easily be set to specific wavelengths without changes in any other dimension of the structure. The width of the SRC (*w*) is another parameter which can affect the capacitance of the gap and, consequently, shift the resonant wavelength of the TM_{g}* _{k}* modes; however, the effect of the gap size (

*g*) is much stronger. The high sensitivity of the SRR-like resonant modes to the gap size can also be understood by examining their electric fields. As shown in the inset of Fig. 3(a), the electric fields are highly localized and enhanced in the gap. This enhancement suggests that changing the refractive index in the gap would provide another means to control the filtering characteristics of the structure (as discussed in the next subsection).

#### 3.2 Tunability of the resonances: Variation of the refractive index in the gap of the SRC

One way of controlling resonant wavelengths is by varying the refractive index of the material(s) used to implement the resonators. Active control over this material property leads to active devices such as tunable filters or optical switches. Given the high sensitivity of the SRR-like (TM_{g}* _{k}*) modes to the gap of the SRC, changing the refractive index of the material therein should provide a degree of tunability. And, based on available tunable materials, optical switches or tunable filters can be achieved by changing the refractive index of the material in the gap by heating (

*e.g.*, polymers), by optical pumping (

*e.g.*, Kerr effect), by applying an electric field (

*e.g.*, liquid crystals), etc….

One way of filling the SRCRR with material is to completely fill the whole resonator (as in conventional MIM ring resonators). In this case, the sensitivity of the resonant modes of SRCRR and of a rectangular ring of the same size would be similar (for the latter, *Z*_{SRC} ~0 so the resonance condition of the SRCRR and a similar rectangular ring resonator are the same). However, this type of filling disturbs the coupling region between the resonator and bus waveguide (*i.e.*, the lower horizontal branch), such that the bus waveguide must be filled with the same material in order to maintain strong coupling [11,18]. An advantage of the SRCRR structure is that it allows partial-filling which prevents this disturbance.

We computed the resonant wavelength of the SRCRR modes as the refractive index of the material in the gap of the SRC (blue region of Fig. 1(a)) varies from 1 to 2, as plotted in Fig. 3(b). For the TM_{1g} and TM_{2g} modes, maximum red shifts of about 313 and 202 nm are predicted, respectively, whereas for the TM_{1}, TM_{2c}, and TM_{2f} modes maximum red shifts of 9, 4, and 29 nm, are predicted respectively. The bulk sensitivity of the TM_{1g} and TM_{2g} modes, computed as the slope of curves (resonance wavelength *versus n*), is also plotted in Fig. 3(b); it is seen that the TM_{g1} mode is more sensitive to variations in *n* for *n* > 1.15, while the TM_{g2} mode is more sensitive for *n* < 1.15. Thus, by partially filling the resonator (*i.e.*, the gap region) one can tune the TM_{g}* _{k}* modes while keeping the other modes unchanged.

Another filling scheme consists of filling the whole area inside the SRC, including the gap, with material (complete-filling). The resonant wavelength shifts of the TM_{g}* _{k}* modes for this type of filling were computed and compared to the shifts achieved by filling the gap only (partial filling). For each type of filling, a polymer was assumed as the tunable material. Assuming polymer (n ~1.45) and applying an index change of 0.02 (for a ~100 °C change in temperature) we compute for the TM

_{g2}mode in the SRCRR

^{h}(g = 60 nm), structure wavelength shifts of 5 and 14 nm for partial-filling and complete-filling, respectively. For the TM

_{g1}mode the wavelength shifts were 7 and 26 nm for partial-filling and complete-filling respectively.

## 4. Conclusion

A new type of MIM-based optical nano-filter, combining square-ring and split-ring resonators, termed the *split-ring core ring resonator* (SRCRR) is proposed and investigated theoretically. The proposed structure can support resonant modes of the square ring as well as new SRR-like modes characteristic of the structure which are highly tunable. These SRR-like resonances can be efficiently manipulated by changing the gap size of the split-ring core or by varying the refractive index of a material incorporated in its gap. A circuit model based on transmission-line theory is proposed for these modes and validated by comparison against results obtained numerically. The SRCRR offers great flexibility for the design of nano-scale tunable optical components.

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