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Abstract
Previous studies have shown that split-ring resonators (SRRs) can be utilized to achieve finely tuned nearest-neighbor coupling strengths in various one-dimensional hopping models. In our study, we present a systematic investigation of resonator coupling, providing a comprehensive quantitative description of the interaction between SRRs and complementary split-ring resonators (CSRRs) for any orientation combination. Our method includes an estimation of the coupling strength through a linear combination of periodic functions based on two orientation angles, with a sinusoidal expansion of up to the 3rd order, allowing for efficient and streamlined microwave structure design. Through our approach, we offer a satisfactory explanation of the band structure of SRR chains using a microwave-hopping model, which facilitates the exploration of exotic photonic band structures based on tight-binding theory.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic band engineering refers to the adjustment of the band gap in photonic crystals through composition control or layered material construction. Coupled-resonator optical waveguides (CROWs) differ from photonic crystals as they consist of periodic arrays of isolated structural elements, enabling manipulation of light propagation on a micro scale. [1,2] As energy transfers between adjacent unit cells, the tight-binding model used in electronic band engineering can be applied to this system. This analogy has led to proposals for exciting phenomena such as Bloch oscillations [3], artificial gauge fields [4,5], and nonlinear effects [6] for photons in coupled resonators. Recent advancements include the demonstration of a one-dimensional Su-Schrieffer-Heeger two-band topological model. [7] Artificial magnetic fields for hopping photons in two dimensions have also been achieved [8,9], resulting in photonic topological insulators. [10–12]
Metamaterials with unique wave-manipulating properties can be created by carefully adjusting the unit cell, also known as ‘artificial atoms’. For instance, a medium with negative permittivity and permeability simultaneously can result in negative refractive index. [13–15] Split-ring resonators (SRRs), which provide negative permeability within a narrow frequency range, are commonly used as metamaterial atoms. [16–18] As the counterpart, the complementary split-ring resonators (CSRRs) have an electric response similar to the magnetic response of SRRs. [19] Coupled SRRs have numerous applications, including the production of magnetoinductive waves [20], electroinductive waves [21], stereometamaterials [22,23], and slow and retarded waves [24–26] . The photonic band structure can be designed with a diatomic unit cell due to Brillouin zone folding. [27–29] The slow wave nature can be seen near the band edge created by the bandgap opening. In addition to the inter distance, the orientation of SRRs greatly impacts the inter-resonator coupling. [28,30–32] The importance of coupled SRRs can be envisaged in view of the versatile optical properties empowered by the metamaterials constructed by strongly coupled resonators.
Previous research on coupled SRRs has largely focused on the same type of resonators and achieved limited tunable ranges for the coupling strength by distance and orientation. [33] The coupling strength was mostly calculated using coupling capacitance and mutual inductance. [34–37] Our study, utilizing finite element calculations, demonstrates the benefits of coupling a SRR with a CSRR. By adjusting the orientation, the coupling strength can be adjusted from positive to negative, across zeros. The dimensionless coupling strength can reach up to 9% and 10% for printed circuit board(PCB) and sapphire substrates, respectively. Using a SRR/CSRR pair as a “photonic two-atom basis" can lead to more diverse band structures and cause negative permeability and negative permittivity at the same frequency, resulting in double negativity of propagating electromagnetic waves. [38–40]
2. Method: the microwave structures and resonance tuning
The designs of double split-ring resonators (DSRR) and complementary split-ring resonators (CSRR) are shown in Fig. 1(a). There are several reasons for choosing a double split ring structure instead of a single split ring counterpart in our work. Firstly, the presence of a inner ring allows the excited current to circulate effectively, ensuring a pronounced magnetic response of the structure. [41] Secondly, it has been observed that the single split ring is more affected by nearby conductors compared to the DSRR. The smaller change in resonance frequency makes it easier to design the DSRR-CSRR building block and ensures that they are closely resonant. Thirdly, the large capacitance in the small gap region between the rings in the DSRR significantly lowers the resonant frequency and concentrates the electric field. This provides a size advantage for the DSRR at the same operating frequency, facilitating the construction of large chains of resonators as the building block requires less occupation area. [42] The outer and inner radii of the resonators are denoted by $R_{\textrm {out}}$ and $R_{\textrm {in}}$, respectively. Microstrip width $w$, gap $g$, and ring split $s$ are designed to be identical.
Fig. 1. (a) The structure of a DSRR with add-on lumped LC circuit. (b) The oblique view of the proposed structure for investigating coupling between a DSRR and a CSRR. (c) The top view of the structure and the definition of orientation angles $\theta _D$ and $\theta _C$.
For investigating the coupling strength, DSRR and CSRR are positioned on opposite side of a substrate, with a horizontal distance $d$ between them, as illustrated in Fig. 1(b). Previous work has examined the dependence of coupling strength on $d$, and it has been found that the maximum coupling energy occurs when $d$ is approximately equal to the average ring radius (3.2 mm for the structure on PCB). [43] While studying the orientation at this particular distance would be interesting, it is crucial to consider the requirements for forming a chain of resonators for future applications. To minimize direct coupling between adjacent DSRRs in the chain while maintaining a sufficiently strong DSRR-CSRR coupling, we selected a distance $d$ satisfying the condition, $d/r_{\textrm {out}}=1.316$.
As illustrated in Fig. 1(c), the orientation configuration of two coupled SRRs is noted by $(\theta _D, \theta _C)$, which are the arguments of normal direction vectors when the displacement vector, pointing from DSRR center to CSRR center, lies in the x-direction. In our study, we considered two kinds of substrates: a low-loss printed circuit board RO4003C and sapphire. The former is commonly used for room-temperature applications, while the latter is commonly used in microwave circuits for quantum information processing. The thickness $h$ of RO4003C circuit board and sapphire substrate is assumed to be 1.6 mm and 0.5 mm, respectively. To match the typical operating frequencies in potential applications, we aimed to design the resonance frequencies to be around 3.5 GHz and 6 GHz for resonators on RO4003C and sapphire substrates, respectively. The detailed structure dimensions are listed in Table 1.
Table 1. Dimensions of the resonator design in unit of mm.
The resonance frequency of a microwave structure was determined through eigenmode solutions using high-frequency structure simulations (HFSS). The solutions provided a list of possible resonances in the desired frequency range, including unwanted eigenmodes due to the finite sizes of the substrate and solution boundary. From this set, we carefully selected the resonance that exhibited a sufficiently high Q-value and displayed reasonable field distributions as the desired resonance. In order to obtain the coupling strength between resonators, we continuously tuned the resonance frequency of one resonator, crossing the frequency “degenerate point” where the resonance frequencies are identical. To change the resonance frequency of the DSRR without altering its structure, we added a lumped capacitor or inductor to the resonators as a patch in the microwave structure, as shown in Fig. 1(a).
As seen in Fig. 2(a), the resonant frequency $\omega _D/2\pi$ is reduced from 4.0 GHz to 3.3 GHz as the add-on capacitance changes from $C_a=-150$ fF to 150 fF. The value of $C_a$ can be chosen to be negative if the lump element is made of ferroelectric material [44], field effect transistors [45] or 1D materials with strong electron interactions [46]. In Fig. 2(b), the resonance frequency changes from $\omega _D/2\pi$=5.014 GHz to 3.606 GHz as $L_a$ increases from 3 nH to 100 nH. The equivalent circuit for our structure is depicted in the inset of Fig. 2(b), inspired by the work of Baena et al., who provided a detailed analysis of the circuit model for SRRs. [34] $L_0$ is the inductance of the ring, and $C_0$ is the capacitance between the rings. The resonance frequency of the DSRR with an add-on capacitance $C_a$ can be written as,
Fig. 2. (a) The resonance frequency of the DSRR as a function of the add-on capacitance $C_a$. (b) The resonance frequency of the DSRR as a function of the add-on inductance $L_a$. The effective circuit of the modified DSRR is displayed in the inset. The blue box marks the equivalent circuit of a DSRR, while the red box marks the add-on element.
If the DSRR has an add-on inductor $L_a$, the resonance frequency reads,
Here $\beta =L_a/L_0$ is the ratio of add-on inductance to the microstrip inductance of DSRR. The fitting result in Fig. 2(b) gives an estimation of the intrinsic inductance of the DSRR to be $L_0=9$ nH. The obtained $L_0$ and $C_0$ are in good agreement with the analytical calculations, which gives 14 nH and 0.78 pF, respectively, and consistent with the expected DSRR resonance frequency of about 3.55 GHz. [34,47]
3. Method: determination of coupling strength
The Hamiltonian of two coupled resonators can be represented as follows
where $\omega _{D,C}$ are the intrinsic frequencies of the DSRR and CSRR, respectively, and $g$ represents the coupling strength in frequency units. The eigenfrequencies can be expressed as,By modifying the resonant frequency of one resonator, e.g. DSRR with an add-on capacitor, we can replace $\omega _D$ with $\omega _D'$ using Eq. (1). In particular, the difference between the eigenfrequencies can be expressed as:
with its minimal value being $2|g|$ at the degenerate point ($\omega _D'=\omega _C$). The collective frequencies obtained from HFSS can be fit to this expression, as shown in Fig. 3. The results clearly indicate that $2|g|$ for $(\theta _D, \theta _C)=(\pi /2,2\pi /3)$ is significantly greater than that for $(\pi /2,4\pi /3)$. While our method is efficient, it does not provide information regarding the type of coupling present in the resonator dimer. Nevertheless, previous work has indicated that the inter-resonator coupling strength comprises both capacitive and inductive ones, with the ratio showing a significant dependence on the orientation. [43]Fig. 3. The eigenfrequencies of the coupled resonators with the orientation configurations of $(\theta _D, \theta _C)=(\pi /2,2\pi /3)$(blue), $(\pi /2,4\pi /3)$(red) and $(0,4\pi /3)$(green) on PCB (a) and sapphire (b). The solid curves are fitted to Eq. (3).
4. Results and discussions
The sign of $2g$ extracted from the anti-crossing structure is ambiguous, but it can be inferred that $g$ changes sign with respect to $\theta _C$ at $\theta _D=0$. As shown in Fig. 4(a), assuming all $g$ values to be positive results in a discontinuity in $g(\theta _C)$ curve at $\theta _C=\pi$ for $\theta _D=0$. However, a smooth dependence requires a sign change at $\theta _C=\pi$. Similar argument can be applied to the case in $g(\theta _D)$ curve at $\theta _C=\pi /2$ as shown in Fig. 4(b). By assigning the appropriate sign to the $g(\theta _C)$ curves, we could examine the parity of the $g\left (\theta _C\right )$ dependence. The data reveal that $g(-\theta _C)=g(2\pi -\theta _C)=-g(\theta _C)$ when subtracting a constant $g(0)$. In view of this, we could expand these smooth curves using a Fourier series,
It is worth noting that only sine functions are presented in the series, as the $g$ functions exhibit asymmetry with respect to $\theta _C$, except for a constant term $c_0$. This feature is also observed in the $g(\theta _D)$ curves in Fig. 4(b), that $g(-\theta _D)=g(2\pi -\theta _D)=-g(\theta _D)$ when subtracting a constant term $g(0)$. Furthermore, we have found that using $N=3$ in the Fourier series sufficiently captures the orientation dependencies.
Fig. 4. (a) The coupling strength as a function of $\theta _C$ when $\theta _D=0$, $\pi /4$ and $\pi /2$ of the structure on PCB. (b) The coupling strength as a function of $\theta _D$ when $\theta _C=0$, $\pi /4$ and $\pi /2$ of the structure on PCB. (c) The coupling strength as a function of $\theta _C$ when $\theta _D=0$, $\pi /4$ and $\pi /2$ of the structure on sapphire. (d) The coupling strength as a function of $\theta _D$ when $\theta _C=0$, $\pi /4$ and $\pi /2$ of the structure on sapphire.
In order to describe the coupling strength $g(\theta _D, \theta _C)$ in a general form, we consider the reflection symmetry of the coupled system in the $y$ direction. This symmetry implies that $|g|$ remains unchanged under a reflection in $y$. Therefore, as $y\rightarrow -y$, the angles $(\theta _D, \theta _C)$ change to $(-\theta _D, -\theta _C)$ and $g(-\theta _D, -\theta _C)=\pm g(\theta _D, \theta _C)$. Based on the observation that $g(3\pi /2, \pi /2)>0$ and $g(\pi /2, 3\pi /2)<0$ are in opposite signs, we conclude that the odd parity case must be considered. To expand the coupling strength to $N=2$ in a Fourier series, the possible odd parity terms are $\sin (\theta _{D})$, $\sin (\theta _{C})$, $\sin (2\theta _{D})$, $\sin (2\theta _{C})$, $\sin (2\theta _{D})\cos (2\theta _{C})$, and $\cos (2\theta _{D})\sin (2\theta _{C})$.
Upon further inspection of the $g(\theta _C)$ curves, we find that changes in $\theta _D$ only result in vertical shifts. Similar behavior is observed in the $g(\theta _D)$ curves at different $\theta _C$. This leads us to conclude that the cross terms are insignificant. Therefore, we can assume that the general form of the coupling strength has odd parity without cross terms, as described by the following equation:
For the special cases of $\theta _D = \pi /2$ and $\theta _C = \pi /2$, this equation simplifies to:
We can now obtain the coefficients unambiguously using Eqs. (6) and (7). The best-fitted coefficients, with $a_1$ set to 1, for $N=2$ and 3 are listed in Table 2. The general results using these fitting coefficients are plotted as solid ($N=3$) and dashed ($N=2$) curves in Fig. 4.Table 2. The coefficients for Fourier series. $g_0$ is defined such that $a_1=1$.
The change of sign in $g$ allows for tuning the resonators’ orientation angles to achieve zero coupling. As shown in Fig. 5, the boundary between positive and negative regions in $g/\omega$ represents the conditions for zero coupling, where $\omega =\omega _D=\omega _C$. The black points on the plot indicate configurations that result in the highest coupling strength, with $g/\omega =\pm 0.089$ and $\pm 0.101$ for PCB and sapphire substrates, respectively. The combination of a SRR and a CSRR in the microwave structure provides a broad range of tuning in $|g/\omega |$ from 0 to 0.1 when using a sapphire substrate.
Fig. 5. The scaled coupling strength $g/\omega$, in which $\omega =\omega _C=\omega _D$ as a function of $\theta _D$ and $\theta _C$ for structures on PCB (a) and on sapphire (b). The plots highlight the region where $g$ becomes negative and the conditions for $g=0$. The maximum coupling occurs at $\left (\theta _D, \theta _C\right )=\left (0.30\pi, 0.72\pi \right )$ and $\left (\theta _D, \theta _C\right )=\left (1.70\pi, 1.28\pi \right )$, as indicated by the black points. The maximum values are $\pm 0.089$ and $\pm 0.101$ in (a) and (b), respectively.
In Fig. 6, we present the field plot of the magnetic field component $H_z$ for 3 distinct coupling strengths associated with 3 orientation configurations at the eigenfrequencies, $\omega _+$ and $\omega _-$. Figure 6(a)-(d) are for the positive coupling strength of 210 MHz. Figure 6(e)-(h) are for the zero coupling strength ($|g|<1.8$ MHz). Figure 6(i)-(l) are for the negative coupling strength of -108 MHz. Generally speaking, we can find that the field at the lower frequency $\omega _-$ for positive and negative coupling cases have similar distributions but opposite sign in the magnetic field (for example, Fig. 6(c) and (k)).
Fig. 6. (a)-(d) The field plots of the magnetic field component $H_z$ for a positive coupling configuration $\left (\theta _D, \theta _C\right )=(\pi /2, 2\pi /3)$. (a) and (b) present $H_z$ distributions at $\omega _+$ for DSRR(top conductor layer) and CSRR(bottom conductor layer), respectively. (c) and (d) are $H_z$ distributions at $\omega _-$ for DSRR and CSRR respectively. (e)-(h) The field plots for the zero coupling configuration $(\pi /2, 4\pi /3)$. (i)-(l) The field plots for negative coupling configuration $(0, 4\pi /3)$
5. Applications: waveguide of the tailored orientation texture
The coupling formula can be used to create a linear chain of resonator dimers, each consisting of a DSRR and a CSRR. The structure is depicted in Fig. 7(a), with the DSRR and CSRR modeled as lattice sites A and B, respectively. The intra-cell coupling $t_1$ is directly given by Eq. (5), while the inter-cell coupling can be found by considering the system under a $\pi$-rotation about the $z$ axis:
Fig. 7. (a) A simple hopping model for 1D microwave propagation, where DSRR and CSRR are represented as lattice sites A and B, respectively. The red circles mark the unit cell of the “2-atom basis”. When the inversion symmetry is broken, the forward and backward hopping energies, $t_1$ and $t_2$ from A to B become unequal. (b) and (c) show the top and bottom views of the SRR array, respectively. Ports are indexed from 1 to 4. (d) The experimental S-parameter $|S_{41}|$ for $(\pi /2, \pi /2)$(blue) and $(\pi /2, 7\pi /6)$(red) samples. A band gap forms centered at 3.35 GHz in the later case. (e) The simulated S-parameter $|S_{31}|$ for $(\pi /2, \pi /2)$(blue) and $(\pi /4, 8\pi /9)$(red), with the later sample showing a large band gap.
Special cases with inversion symmetry are $(\theta _D,\theta _C)=(\pi /2, \pi /2)$ and $(\pi /2, -\pi /2)$. In these cases, the hopping strengths satisfy $t_2=-t_1$ and result in a gapless spectrum. When inversion symmetry is broken, the ratio $|t_2/t_1| \neq 1$ and leads to a spectrum with a band gap. Examples are $t_2/t_1=0.022$ when $\left (\theta _D,\theta _C\right )=(\pi /2, 7\pi /6)$, and $t_2/t_1=-0.139$ when $\left (\theta _D,\theta _C\right )=(\pi /4, 8\pi /9)$ for RO substrate. The test samples of the resonator chain, consisting of 21 staggered resonators, are depicted in Fig. 7 (b) and (c). The chain is coupled to two microstrip transmission lines for microwave probing and the band structure is determined by measuring the S-parameters with a vector network analyzer (Agilent N5230A). We would like to note that the presence of microstrip lines inevitably introduces an impact on the resonator chains, and the coupling behavior may vary with different resonator orientations. [43,48] To mitigate this effect, we carefully chose the distance between the resonator and microstrip to ensure that it does not significantly affect the spectrum under investigation. S-parameter $|S_{41}|$ represents the transmission of microwaves through the chain, and provides information on the passband structure. As seen in Fig. 7(d), the reversion symmetric case $(\pi /2, \pi /2)$ has a single continuous pass band from 2.9 to 3.6 GHz, while the non-symmetric case $(\pi /2, 7\pi /6)$ has split passbands with a bandgap centered at 3.35 GHz. Figure 7(e) compares the simulated $|S_{31}|$ of the reversion symmetric case $(\pi /2, \pi /2)$ with the non-symmetric case $(\pi /2, 7\pi /6)$ for 10 staggered resonators. The latter case has larger values of $|t_1|$ and $|t_2|$, which results in a larger bandgap. The double negativity of the 1D chain is visually demonstrated in Fig. 8, where the wavefront moves in the opposite direction as the incident 3 GHz energy propagates through the 1D chain from port 1 to ports 3 and 4. This behavior is in contrast to wave propagation in transmission lines, where the wavefront movement aligns with the direction of energy propagation.
Fig. 8. The wavefront propagation in the SRR chain with $(\pi /2, \pi /2)$ excited from port 1. See Visualization 1.
In contrast to the maximum value of $t_1=-t_2=1.497 g_0$ at $(\pi /2, \pi /2)$, the maximum value of $t_1=t_2$ occurs when $(0.16\pi, 1.73\pi )$ with $t_1=0.874 g_0$. The equal hopping strengths ensures a reversal symmetric system, even though the geometric structure is not symmetrical. It is important to note that a change in the sign of $t_2$ in a linear chain of resonators will not result in a different spectrum as it only changes the sign of the eigenfunctions. However, in a circular chain with odd number cells $N$, a change in the sign of $t_2$ will result in a different spectrum. As illustrated in Supplement 1, the hopping direction rotates an angle of $\eta =\pi /N$ about $z$ axis as one moves from one lattice site (or resonator) to the next. This requires changing the inter-cell coupling strength to
The diagonalization of the hopping Hamiltonian reveals that samples with $t_1=t_2$ and have $N+1$ eigenfrequencies, which are doubly degenerate except for the highest and lowest ones. In contrast, samples with $t_1=-t_2$ have $N$ doubly degenerate eigenfrequencies, allowing one to distinguish the change in the sign of $t_2$ based on the eigenfrequency distribution, which is not possible in linear chains.
In order to test our theory, we selected a circular chain with $N=5$, as depicted in Fig. 9(a) and (b). Two samples were chosen, one with all positive hopping strengths and the other with alternating-sign strengths. The positive strength sample had orientations $\left (\theta _D, \theta _C\right )=(1.089\pi, 0.778\pi )$ and hopping strengths $\left (t_1, t_2 \right )=(0.88, 1.30)g_0$, while the alternating-sign sample had orientations $\left (\theta _D, \theta _C\right )=(0.422\pi, 0.617\pi )$ and strengths $\left (t_1, t_2 \right )=(2.03, -2.20)g_0$. The experimental results of the transmission amplitude $|S_{31}|$ and $|S_{41}|$ for the positive strength sample are shown in Fig. 9 (c), which only exhibits 4 major peaks, instead of 6. It appears that two non-degenerate resonances in the sample with positive hopping strengths did not appear in the transmission. However, Fig. 9 (d) shows the results for the alternating-sign sample, which clearly displays all 5 peaks as predicted. These findings are further confirmed by simulations, as shown in Fig. 9 (e) and (f).
Fig. 9. The top and bottom views of a circular chain of SRRs are depicted in (a) and (b) respectively. (c) The experimental results for S-parameters $|S_{31}|$(blue) and $|S_{41}|$(red) for $\left (\theta _D, \theta _C\right )=(1.089\pi, 0.778\pi )$ exhibit 4 resonance transmission peaks, marked with green diamonds.(d) The experimental $|S_{31}|$(blue) and $|S_{41}|$(red) for $\left (\theta _D, \theta _C\right )=(0.422\pi, 0.617\pi )$ display 5 resonance transmission peaks. The corresponding simulated results for the two configurations are shown in (e) and (f) respectively.
6. Conclusion
To summarize, we have studied the interaction between resonators and provided a comprehensive analysis of the orientation-dependent coupling strength of a DSRR and a CSRR. Our sinusoidal approximation up to the third order allows for the design of desired orientation textures with dimensionless coupling strength up to 0.1. We have shown applications of this system in both linear and circular chains of coupled resonators as a waveguide. The unit cell, comprising a DSRR and a CSRR, allows for double negativity and reversed wavefront propagation. The intra-cell coupling strength $t_1$ and inter-cell coupling strength $t_2$ can be calculated from the orientation configurations. A gapless waveguide can be achieved when $|t_2/t_1|\sim 1$, and a gapped transport is observed when $|t_2/t_1|\neq 1$. The eigenfrequency distribution distinguishes the change in the sign of the inter-cell hopping strength in circular chains with odd number $N$. The chain with alternating-sign hopping strengths exhibits all $N$ doubly degenerate resonances, while the chain with all-positive hopping strengths only shows $N-1$ doubly degenerate resonances. Our results provide a foundation for designing coupled SRR systems to simulate other hopping models in one and two dimensions.
Appendix
The modified resonance frequency of the circuit in Fig. 2(b) can be determined using its ac impedance
in whichHere $x=\omega ^2 C_0L_0$, $\alpha =C_a/C_0$, and $\beta =L_a/L_0$. It is worth noting that in the absence of add-on elements, the impedance is given by $Z_2=2/{i\omega C_0}$, which corresponds to $\alpha = 0$ and $\beta = \infty$. Notably, setting $\beta = 0$ would lead to a short circuit between the outer and inner split rings. After simplification, the impedance reads,
When no add-on elements are present ($\alpha =0$ and $\beta =\infty$),The resonance condition $Z=0$ is met when $x=4$, resulting in the intrinsic DSRR frequency
In general cases, the resonance condition $Z=0$ is given by,
When the add-on element is a capacitor, $\beta =\infty$, and the resonance condition requires, When the add-on element is an inductor, $\alpha =0$, and the resonance condition requires, with solutionThe positive sign should be chosen for a reasonable solution, as the negative solution yields $x \sim 1/{\beta }$, which becomes 0 when $\beta$ tends to infinity.
Funding
National Science and Technology Council (110-2112-M-005-001, 111-2112-M-005-010).
Acknowledgments
Fruitful discussions with R. K. Lee and C. S. Wu are acknowledged. We are grateful to the computing support from National Center for High-performance Computing.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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