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Abstract
In this study, we demonstrated an accurate estimation of the refractive index in nanoporous structures using a novel hybrid simulation method that combines electromagnetic field analysis and ray tracing based on wave optics and ray optics. A novel theoretical model for the refractive index in a nanoporous structure was developed using a simulation of the effective medium approximation (EMA) and screening effect with consideration of polarization dependence. The optical properties were then experimentally evaluated on aggregates of silica nanoparticles. Based on the measured properties, nanoparticles were predicted to have sufficient transparency for the optical cavity. A nanoporous microdisk laser was thus fabricated with laser dye to demonstrate that the nanoporous microdisk was capable of lasing. The microdisk functioned sufficiently as an optical cavity and had well-matched characteristics. This study provides a novel viewpoint of polarization dependence to estimate the refractive index of nanoporous materials by combining numerical simulations and experiments.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microcavity is a type of optical cavity that ranges in size from a few microns to several hundred microns [1–3]. The microcavity confines light with whispering gallery modes (WGMs), wherein light propagates at the circumference of the microcavity in total internal reflection, thereby resulting in a high quality factor (Q-factor) as high as $10^8$ [4,5]. Among microcavities, the planar ones confine most of the light near the edges and thus have a smaller mode volume than those of other shapes, making them suitable for relevant applications. Therefore, several studies have been reported for the use of such planar microcavities, such as in low-threshold lasers, frequency combs, signal processing, biosensors, and bioimaging, [6–11]. Microdisks are planar microcavities and are more applicable to integrated devices compared to the other planar shapes. Microdisks can be fabricated by a variety of methods that are classified into two main approaches: subtractive and additive. In the subtractive method, the photolithography method [12,13] and electron-beam lithography method [14,15] have been reported. These subtractive methods have advantages of simultaneous mass production with high precision. However, this method requires dedicated equipment for each process and places a high load on the material, such as in chemical etching and high-temperature processes. In contrast, we developed novel additive method called “inkjet printing method” [16,17]. This method is a simple ink-ejecting process under open air without any thermal treatment, and this method is suitable for on-site or on-demand fabrication owing to its additive process, so that only a single microdisk can be fabricated. The printed microdisk has an extremely smooth surface since the microdisk is formed via self-assembly with surface tension of the ink solution. A fully polymeric microdisk laser with an ultra-low threshold was reported [18], and a label-free bio-sensor was also reported [19]. This inkjet printing method can be applied not only to organic materials but also to inorganic materials using a solution of dispersed nanoparticles [20]. The printed microdisk with nanoparticle solution has a nanoporous structure because the inkjet printing method has no thermal treatment and thus keeps the pores in the aggregates of nanoparticles after evaporation of the solvent. Nanoporous structures are suitable for sensing applications such as gas sensing owing to their large specific surface area [21,22]. For example, a highly sensitive gas sensor composed of nanoparticles has been reported [23], demonstrating that a film of clustered nanoparticles has a very high sensitivity owing to the enlarged porosity caused by the clustered nanoporous structure. The combination of such nanoporous structures with microdisks is promising for more advanced applications owing to the strong light confinement of WGMs and the large specific surface area of the nanoporous structure. A direct and strong interaction between the confined light and the sample is expected by capturing the sample in the pores of the nanoporous microdisk. Designing the porosity of microdisks or filling nanoparticles into pores to change the refractive index is also envisioned to enhance the sensitivity and limit of detection. For such advanced applications, elucidating how light propagates in nanoporous microdisks is important. There are some reports on the relation between the nanoscale structure and the refractive index, called the effective medium approximation (EMA), based on the Clausius–Mossotti relation because they provide an effective refractive index or permittivity of the mixture [24–26]. However, EMA theories only define the refractive index of porous materials, and do not describe the light behavior in nanoporous structures. Therefore, comprehensively studying the refractive index of the nanoporous structure in consideration of light propagation is crucial.
In this study, we have demonstrated an accurate estimation of the refractive index in a nanoporous structure using a novel hybrid simulation method combining electromagnetic field analysis and ray tracing based on wave optics and ray optics. First, a numerical simulation using the finite-difference time-domain (FDTD) method was conducted to compute the light propagation in a nanoporous structure to investigate the relation between the refractive index and porosity. The FDTD simulation method was performed using a novel approach that hybridizes wave optics and ray optics to analyze how light propagates in a nanoporous structure at the fine resolution of a nanoscale mesh. This novel method was highly extensive owing to the compatibility of nanoscale resolution and macroscale light propagation, but it is quite significant. A novel theoretical model of the refractive index of a nanoporous structure was developed from this simulation regarding the screening effect, which has an additional viewpoint of polarization dependence. Two different shapes of silica nanoparticles were then selected to experimentally realize and evaluate two different porosities of the nanoporous structures. The refractive indices and scattering coefficients were evaluated for the two shapes of silica nanoparticles. The porosity was estimated from the measured refractive index and a new model of the nanoporous material from the FDTD simulation. With the measured scattering coefficient and the calculated porosity, different shaped nanoparticles were both predicted to have sufficient transparency for the optical cavity. Finally, a microdisk laser was fabricated using the two shapes of nanoparticles with organic laser dye and acrylic oligomer to show that the nanoporous microdisk was sufficiently transparent and had a smooth surface for lasing action. This study provides a novel viewpoint of polarization dependence to estimate the refractive index of nanoporous materials by combining numerical simulations and experiments.
2. Numerical simulation of refractive index for nanoporous materials
Two-dimensional (2D) non-standard type FDTD method was performed for numerical simulation of refractive index of nanoporous material. Light propagation and refraction were simulated in a nanoporous material, which is a simple structure comprising air and dispersed nanoparticles. Nanoparticles of silica ($n=1.458$ at 600 nm [27]) with a diameter of 12 nm were set out with several different volume fractions from 0.1 to 0.74048. The volume fraction of 0.74048 was the densest ratio at the 3D packing of the solid sphere [28]. This porous material was placed below the right of the $14\,\mu \mathrm {m} \times 25\,\mu \mathrm {m}$ area, and all other spaces were air. A basic Gaussian beam light source with a 600 nm wavelength was placed at the left edge of the simulated area, and the beam was propagated into the 45$^{\circ }$ boundary of air and nanoporous material. The spot size of the light source was $2\,\mu \mathrm {m} \times 5\,\mu \mathrm {m}$ with a Gaussian distribution. The polarization of s or p had a transverse or parallel electrical field to the 2D simulated plane, respectively. Figure 1(a) shows the results of light propagation in the nanoporous material with a few refractions at the boundary. The angle of the refracted beam was calculated from the simulated profile of the intensity of the electrical field. From these results, the refractive index of the nanoporous material was calculated according to Snell’s law as a function of the volume fraction of the nanoparticles. Such a long range with fine mesh simulation was necessary to obtain an accurate refractive index because it was revealed that the calculated refractive index fluctuated within a short range, as shown in Fig. 1(b). Multiple reflections should influence the calculated refractive index in a short range as there are many nanoparticles that cause reflection. This influence was relatively large in short range but not in long range. At 0.5 volume fraction and s polarization, 13.13 $\mu$m or longer was necessary to accurately calculate the refractive index of the nanoporous material within a variation of 0.5 %. For other simulation conditions, a distance from 12.92 $\mu$m to 14.86 $\mu$m was required within 0.5 % variation. To reduce the computation cost, the mesh size of the fine resolution was changed from 0.5 nm to 20 nm at a volume fraction of 0.5, and p polarization. The exact refractive index was calculated only for a mesh size of $< 2\,\rm {nm}$, as shown in Fig. 1(c). In other words, the 2 nm mesh was fine enough to simulate nanoparticles of diameter 12 nm. The polarization of s showed an almost linear relation with the volume fraction, whereas p polarization showed a lower refractive index than s in Fig. 2(a). Both s and p polarization showed a similar trend in refractive index as the Bruggeman model [29], which is one of the popular equations of the EMA theory:
where $n_\textrm{BG}$ is the effective refractive index from the Bruggeman model, $n_i$ is the refractive index of medium $i$, and $\delta _i$ is the volume fraction of medium $i$ $(\sum _i\delta _i = 1)$. Critical differences in the calculated refractive index were between s and p polarization, although there were only changes in the polarization of the same nanoporous material. Specifically, at a volume fraction of 0.5, the refractive index for s was 1.24, for p it was 1.19, and it was 1.22 from the Bruggeman model, as depicted in Fig. 2(b). Focusing on these differences, another theoretical model for the refractive index of nanoporous materials was obtained regarding the screening effect [30]:
Fig. 1. Simulation results of light propagation in the nanoporous material. (a) Simulated model and propagated light. (b) Refractive index vs. distance of propagation at a volume fraction of 0.5 and s polarization. (c) Refractive index vs. mesh size at a volume fraction of 0.5 and p polarization.
Fig. 2. (a) Refractive index vs. volume fraction of nanoparticle for s and p polarization. The solid line represents the Bruggeman model as a reference. (b) Depicted plot at a volume fraction of 0.5.
3. Experimental evaluation on optical properties for nanoporous material
3.1 Refractive index
The refractive index was measured to estimate the porosity of the nanoporous structure. For the experimental evaluation of the refractive index of nanoporous materials, amorphous nanosilica sol was selected owing to the high transparency of silica. Two different shapes of nanoparticles in the sols were expected to show different nanoporous structures: one was spherical (MEK-ST-40, Nissan Chemical Corp.) and the other was non-spherical (MEK-ST-UP, Nissan Chemical Corp.). Thin films of nanoporous structures were fabricated using spin-coating to measure their refractive indices after solvent evaporation and nanosilica aggregation. The sol was diluted to 10 wt.% by the solvent methylethylketone (MEK) to fabricate a thin and homogeneous film needed to measure refractive index accurately. The rotation conditions of the spin coat were 200 rpm for 5 s at the 1st stage and 1900 rpm for 60 s at the 2nd stage. The nanoporous films were coated on $3\,\rm {cm} \times 3\,\rm {cm}$ acrylic substrates (acrylite S-001, Mitsubishi Chemical Corp.). Refractive index dispersion in the region of wavelength from 200 nm to 980 nm was measured using a spectroscopic ellipsometer (SE-2000, Semilab Japan KK) at an incident angle of 70$^{\circ }$. The spot size of measurement was 500 $\mu$m irradiated by a xenon lamp. Four measurements were taken at different positions on each film to evaluate the variation in one film. The model of fitting to the measured data was a combination of Sellmeier and Lorentz, represented as follows:
Fig. 3. Refractive index dispersion of nanoporous films composed of spherical and non-spherical silica nanoparticles. The orange line shows values at wavelength of 600 nm used to calculate the porosity.
Table 1. Fitted parameters for nanoporous film of silica with spherical and non-spherical shapes.
3.2 Scattering coefficient
The scattering coefficient represents the transparency of the nanoporous material and indicates whether the nanoporous material is capable of lasing. For the experimental evaluation of the scattering coefficient of the nanoparticles, the same nanosilica sols were measured using a spectrophotometer (V-630, JASCO Corp.). The sol was placed in a glass cell with a 1 cm light path, and the intensity of the transmitted light was measured in the wavelength region from 400 nm to 800 nm. To eliminate the effect of cell reflection and MEK absorption, the intensity spectrum of MEK in the cell was also measured, and the sol spectra were normalized by this spectrum of MEK. With this normalization, the scattering coefficient was calculated as a black line in Fig. 4. The scattering coefficients of the two shapes had the same tendency that fitted the Rayleigh scatter theory with the following scattering cross-section:
Fig. 4. Scattering coefficients of measured nanoparticle sols (solid black line), estimation of nanoporous microdisks (dotted black line), and Rayleigh model of sols (solid orange line) for (a) spherical shape and (b) non-spherical shape. (c) Absorption and scattering coefficient of oligomer mix with non-spherical nanosilica.
The scattering coefficients shown in Fig. 4 was at the sol with nanoparticles and the solvent; however, there should be nanoparticles and air at the nanoporous microdisk after solvent evaporation. Supposing the Rayleigh scatter, which had the same tendency as the experimental result, the actual scattering coefficient at the microdisk was assumed to be $n_2 =1$ of the refractive index contrast. In addition, at the nanoporous microdisk, the density $N$ of the nanoparticles was different from that of the sol. The microdisk density can be obtained from the results of the FDTD simulation and refractive index measurement, as calculated in subsection 3.1; 22.8 % and 41.8 % of porosity for spherical shapes and non-spherical shapes, respectively. With a nanoparticle radius of 6 nm, the densities of the nanoporous structure were calculated as $8.53\times 10^{17}\,/\rm {cm}^3$ and $6.43\times 10^{17}\,/\rm {cm}^3$ for spherical and non-spherical shapes, respectively. The scattering coefficient of the nanoporous microdisk should be 197 times higher than that of the spherical sol, and 346 times higher for the non-spherical shape. The dotted lines in Fig. 4 shows the estimated scattering coefficients at the nanoporous microdisk with these calibrations of x189 and x337 to experimental values. The Q-factor, defined by the scattering coefficient ($Q_\textrm{mat}$) was also presumed to be one maximal Q-factor limited by the material loss of the microdisk [32]:
where $n$ is the refractive index of the microdisk, and $\alpha (\lambda )$ is the scattering coefficient at wavelength $\lambda$. For the nanoporous microdisk, the $Q_\textrm{mat}$ was expected to be as high as $3.39\times 10^4$ at 600 nm with a measured refractive index and estimated scattering coefficient at the spherical shape, although this was a theoretical maximum. The same estimation for the non-spherical shape was $Q_\textrm{mat} = 1.81\times 10^4$ at 600 nm, which was slightly lower than that for the spherical shape. Consequently, these nanosilica sols with different shapes were both sufficiently transparent for the optical cavity.3.3 WGMs lasing
To demonstrate that nanoporous microdisks can be applied to optical microcavities, the microdisk structure was fabricated using the inkjet printing method. For inkjet printing, the same sols of silica nanoparticles were used, but the solvent was propyleneglycolmonomethylether (PGME) because MEK had a high vapor pressure unsuitable for a stable inkjet. Both spherical shape (PGM-ST, Nissan Chemical Corp.) and non-spherical shape (PGM-ST-UP, Nissan Chemical Corp.) were used to fabricate the microdisks. The spherical nanosilica sol was diluted by PGME to 15.5 wt.% that had the same concentration of the sol of the non-spherical shape; then, both sols were mixed with solution of acrylic-based oligomer resulting in 20.0 wt.% of total solute. The oligomer was added for structural stability of the nanoporous microdisk; simultaneously, the oligomer increases the refractive index of the microdisk to approximately 0.2 from that of only nanosilica. Thus, the fabricated microdisk was a kind of composite of nanosilica and oligomer. A laser dye (Rhodamine590 Chloride, Exciton Corp.) was mixed at 5.00 mmol/l into sols for lasing action in the orange region around a wavelength of 600 nm. An epoxy-based inkjet head (PIJ-60ASET, Cluster Technology Co., Ltd.) was positioned using a highly precise 3D robot (Shot mini 200$\Omega$, Musashi Engineering Inc.). A pulsed voltage of 21.00 V was applied to the piezoelectric actuator in the head with 300 Hz repetition frequency by using a dedicated driver (PIJD-1ASET, Cluster Technology Co., Ltd.) for the stable ejection of the ink solution. A vacuum pressure of $-1.5$ kPa was also applied to the inkjet head for stable conditions using an air compressor (ML-5000XII, Musashi Engineering Inc.). The droplet from the head flew through 1 mm onto the substrate, which was a film of fluorinated ethylene propylene (FEP) owing to its high water repellency, resulting in a small diameter of the printed microdisk. The nanosilica aggregates with oligomer were formed after the solvent of sol PGME was evaporated, and the formed nanoporous microdisk had no cracks and fine edges. The sizes of the microdisks of spherical or non-spherical nanoparticles were different because the states of ink ejection were slightly different. A little distortion of the microdisk’s circumference did not decrease lasing performance since this little distortion had large size scale compared to the wavelength of the light in visible region. Fig. 5(e) shows an image of nanoporous structure captured by a transmission electron microscope (TEM; HT7700, Hitachi High-Tech Corp.). The pores could be clearly seen in Fig. 5(e) resulted from aggregating nanoparticles. It was also obvious in Fig. 5(e) that nanoparticle and oligomer randomly aggregated, hence there were no effect by orientation. For the excitation of dye in the microdisk, a sub-nanosecond pulsed laser (PNP-M08010-120, Teem Photonics) was used with a pulse duration of $< 0.5$ ns and 20.0 Hz repetition rate. The wavelength was 1064 nm from the laser source and converted to 532 nm by second harmonic generation in a beta-bariumborate (BBO) crystal. The nanoporous microdisk was removed from the FEP using transparent scotch tape to confine light in WGMs. Figure 5 showed the removed microdisks measured for spectra. To obtain a clear signal of WGMs lasing, a pulse energy from 2 to 10 $\mu$J was necessary. The signal from the microdisk was collected using a 10x objective lens (Plan Fluor, NIKON), and the signal was led by an optical fiber (FG200UEA, Thorlabs Inc.) into a spectrometer (MS-7504, SOL instruments). The spot diameter measured by the optical fiber was approximately 10 $\mu$m in the view of the optical microscope, and the spot was carefully positioned at the edge of the microdisk because the WGMs were located near the cavity edge. A schematic of the experimental setup for laser irradiation is illustrated in Fig. 6, showing that this measurement system is the microspectroscopy type. Fig. 7(a) showed the measured spectrum on the nanoporous microdisk of spherical shape, clearly showing that the nanoporous microdisk was lasing in WGMs. Comb-like equidistant peaks were an evidence of WGMs. All peaks were fitted by the Lorentz function using the following formula to accurately measure the peak position of the mode:
where $f(x)$ is the Lorentz function of $x$, $a$ is the position parameter, $b$ is the width parameter, and $c$ is the height parameter. The most sharp peak and fitted curve are illustrated in Fig. 7(b). The result of fitting using this formula was needed to evaluate the following characteristics: effective refractive index and Q-factor. The free spectral range (FSR) in WGMs is represented by the effective refractive index using the following equation [33]: where $\Delta \lambda$ is the FSR, $\lambda$ is the resonance wavelength, $n_\textrm{eff}$ is the effective refractive index of the resonance mode, and $R$ is the radius of the microdisk. The FSR was obtained from the peaks shown in Fig. 7(a) represented as orange arrows. From the fitting results, the precise peak position was obtained as the position parameter $a$ so that the FSR was the difference between each adjacent $a$. Subsequently, $n_\textrm{eff}$ was calculated using Eq. (14), assuming that $\lambda$ was approximately 600 nm and $R$ was around 26 $\mu$m in the spherical shape and around 43 $\mu$m in the non-spherical shape, as shown in Fig. 5. Figure 7(c) showed the results of the calculation of $n_\textrm{eff}$ for microdisks fabricated by nanoparticles of spherical and non-spherical shapes. The error bar in Fig. 7(c) showed the first and third quartiles, and the plotted point was the median. In this wavelength region of lasing ($\sim 10$ nm), the refractive index dispersion was so small ($<0.0003$), so it could be ignored. In the comparison of the two shapes of nanoparticles, $n_\textrm{eff}$ showed a 0.04 difference, which was almost the same as that in Fig. 3 and mentioned in subsection 3.1. The calculated $n_\textrm{eff}$ were 1.52 for the spherical and 1.48 for non-spherical types, demonstrating slightly larger values compared to those in Fig. 3 as the oligomer filled some pores in the nanoporous microdisk. The Q-factor was also calculated from fitting result according to this formula: where $Q_\textrm{w}$ is the Q-factor calculated from the line width, $\lambda$ is the resonance wavelength of the mode, and $\delta \lambda$ is the full width at half maximum of the mode. The parameters in Eq. (13) can be applied to Eq. (15) as $\lambda = a$ and $\delta \lambda = 2b$. At the maximum values among the calculated Q-factor for all modes, $Q_\textrm{w} = 8.52\times 10^3$ for spherical shape and $Q_\textrm{w} = 9.34\times 10^3$ for non-spherical shape. These values were comparable to the estimation in subsection 3.2, and they showed a slightly lower value owing to other losses such as radiative loss. In addition, excitation density dependence was evaluated for lasing threshold of fabricated microdisk lasers. Figure 7(d) shows the excitation density dependence of microdisk laser of spherical nanosilica. As the lasing thresholds for microdisk lasers of spherical and non-spherical nanosilica, $18.5\,\mu \rm {J}/\rm {mm}^2$ and $17.9\,\mu \rm {J}/\rm {mm}^2$ were obtained by double linear fitting, respectively. The lasing capability of the nanoporous microdisk was proven with a well-matched effective refractive index and Q-factor. Furthermore, the performance of our novel model for refractive index on the nanoporous structure was proven by the agreement between the estimated and experimental Q-factors.Fig. 5. Optical microscope photograph of the microdisks removed from substrate by the transparent scotch tape, (a) spherical shape, (b) spherical shape at lasing, (c) non-spherical shape, (d) non-spherical shape at lasing. (e) TEM image of nanoporous structure.
Fig. 6. (a) Experimental setup to irradiate the pulsed laser on microdisk and to measure spectrum at the microdisk edge. (b) Focused view of the microdisk.
Fig. 7. (a) Spectrum from the nanoporous microdisk fabricated by spherical type. (b) Fitting result of mode with the highest Q-factor. (c) Calculated effective refractive indices from the spectra of microdisks of both spherical and non-spherical types. (d) Excitation density dependence of microdisk laser of spherical nanosilica.
4. Conclusion
In summary, we have developed an accurate estimation method for the refractive index in nanoporous structures by a novel hybrid simulation combining electromagnetic analysis and ray tracing. First, a numerical simulation was conducted to compute the light refraction in the nanoporous structure, revealing that the distance of light propagation is critical for accurately calculating the refractive index. A propagation distance of 13.13 $\mu$m or longer was necessary for accurate simulation at a volume fraction of 0.5, and s polarization. For this accurate calculation, a resolution of mesh size 2 nm or smaller was necessary to precisely simulate the nanoparticles with a diameter of 12 nm, although the model was extensive. A novel theoretical model of the refractive index in nanoporous materials was developed from the simulation as $q=f_\textrm{q}\left ( \delta _\textrm{a},\delta _\textrm{b} \right )$ regarding the screening effect and EMA theories. A 0.01 differences between EMA theory and the novel model was introduced by a viewpoint of polarization dependence. Two different shapes of silica nanoparticles were then measured for optical properties to evaluate the two different porosities of the nanoporous structure. The spherical shape aggregated more dense than the non-spherical shape, and their refractive indices of the spherical and non-spherical types were 1.33 and 1.22, respectively, indicating that their porosity were 22.8 % and 41.8 %, respectively from the novel model. The measured scattering coefficients at sols for the different shaped nanosilica were both fitted well to the Rayleigh scatter model; thus, the scattering coefficients at aggregates after solvent evaporation were estimated to be 189 times or 337 times higher than that at sols. The Q-factor was also estimated from the result of the scattering coefficient, which was sufficient for the optical cavity to be $3.39\times 10^4$ for the spherical shape and $1.81\times 10^4$ for a non-spherical shape at a wavelength of 600 nm. The nanoporous microdisk laser was thus fabricated with an organic laser dye and acrylic-based oligomer to demonstrate that the nanoporous microdisk was capable of lasing. The spectra were measured from the edge of the microdisk with optical excitation by a pulsed laser, and the spectra showed equidistant peaks as WGMs lasing. The value of $n_\textrm{eff}$ was calculated from the FSR of the measured spectra as 1.52 for the spherical type and 1.48 for the non-spherical type, demonstrating a well-matched and slightly larger value compared to the measured refractive index because the oligomer filled some pores for structural stability. The Q-factor from the line width of the spectra ($Q_\textrm{w}$) was also calculated to be as high as $Q_\textrm{w} = 8.52\times 10^3$ for spherical shape and $Q_\textrm{w} = 9.34\times 10^3$ for non-spherical shape. These $Q_\textrm{w}$ were comparable to the estimation from the scattering coefficient but slightly lower because of other losses such as radiative loss. The novel model of the refractive index in nanoporous materials is promising for advanced applications such as gas sensing by accurately estimating the refractive index.
Funding
Core Research for Evolutional Science and Technology (JPMJCR20T4); Japan Society for the Promotion of Science (JP18K14149, JP19KK0379, JP20J12903).
Acknowledgments
The authors would like to thank Nissan Chemical Corporation (Planning & Development Department, Performance Materials Division) for providing the sols (MEK-ST-40, MEK-ST-UP, PGM-ST, and PGM-ST-UP) and acrylic-based oligomer used in this study. This work was supported by JSPS KAKENHI Grant Numbers JP18K14149, JP19KK0379, JP20J12903, and JST, CREST Grant Number JPMJCR20T4, Japan.
Disclosures
The authors declare no conflicts of interest.
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.
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