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Abstract
Dielectric nanostructures exhibit low-loss electrical and magnetic resonance, making them ideal for quantum information processing. In this study, the periodic double-groove silicon nanodisk (DGSND) is used to support the anapole state. Based on the distribution properties of the electromagnetic field in anapole states, the anapoles are manipulated by cutting the dielectric metamaterial. Quantum dots (QDs) are used to stimulate the anapole and control the amplification of the photoluminescence signal within the QDs. By opening symmetrical holes in the long axis of the nanodisk in the dielectric metamaterial, the current distribution of Mie resonance can be adjusted. As a result, the toroidal dipole moment is altered, leading to an enhanced electric field (E-field) and Purcell factor. When the dielectric metamaterial is deposited on the Ag substrate separated by the silicon dioxide (SiO2) layer, the structure exhibits ultra-narrow perfect absorption with even higher E-field and Purcell factor enhancement compared to silicon (Si) nanodisks.
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1. Introduction
Controlling the spontaneous emission process has basic significance in enhancing light-matter interaction. The electric Purcell effect has gained significant attention as it manipulates the decay rate of quantum emitters through electrical transitions. Recently, an active optical nanoantenna has been suggested to improve and regulate the emission of quantum sources by resonant coupling with local modes [1]. Plasmon nanostructures have been widely used to regulate the spontaneous emission rate and the radiation efficiency of molecules [2–7]. The spontaneous emissivity of the molecules near the metal surface is generally enhanced due to the different environment compared to free space [8–10]. According to Purcell factor measurement, the enhancement is caused by the overlap between the frequency of the surface plasmon mode and the molecule emission spectrum. Therefore, metal nanostructures with surface plasmon resonance are preferred for effective coupling between radiation and quantum sources. Plasmon resonances, which are characterized by strong field enhancement near the metal-dielectric interface and high spatial localization of resonance modes [11], can improve spontaneous emissivity [11–13] and resonant fluorescence transfer [14], manipulate the directivity of emission [15], provide perfect absorption [16], and even achieve strong coupling [17]. Although great progress has been made in this field [18], plasmon systems have inherent disadvantages, such as high optical loss, which may inhibit spontaneous emission. In addition, due to the limited plasmon materials that exist in nature [18,19], the overlap between the surface plasmon mode frequency and the molecular emission spectrum is only suitable for a few narrow frequency bands, thus limiting the potential applications of plasmon materials. Metamaterials have been used to enhance the electrical response of materials in the spectrum, although they mostly depend on lossy metal structures [20–23]. Recently, their all-dielectric counterparts have been proven to be a good substitute for nano-photonics [24–26]. In recent years, high refractive index materials have been extensively utilized in nano-photonics and optics to create resonant all-dielectric nanostructures [27–29]. Moreover, they show great advantages in light scattering manipulation, light polarization control, Raman signal enhancement [30,31], nanoantenna emission directivity control [32–34], and achieving nonlinear optical effects [35–39]. Furthermore, there has been significant theoretical research on the interaction between quantum emitters and all-dielectric systems. This research has demonstrated the potential of all-dielectric systems in controlling emission modes and enhancing spontaneous emission. Recently, radiation-free excitation anapole has been demonstrated in single dielectric nanostructure and metamaterial via inducing electromagnetic multipole resonance [40–43]. The radiation modes of the electric dipole and toroidal dipole modes exhibit similarities: they undergo destructive interference, resulting in total scattering cancellation in the far-field area [44]. The anapole state enables the production of non-radiative light fields, resulting in highly confined electromagnetic fields inside limited regions. By stimulating dipoles, customized dielectric nanomaterials offer comparable opportunities to their metal equivalents. This new type of dielectric photon dark state in dielectric nanostructures has promising applications in local E-field enhancement and other optical sensing fields [45,46]. However, limited research has explored the interaction between QDs and dielectric structures [47–49].
In this study, a double-groove Si nanodisk supporting anapole excitation is designed and its optical properties are discussed. Subsequently, a QD emitter was placed in the center of the nanodisk, and the radiation power of the source was significantly enhanced with the support of the anapole. Changes in the nanodisk's geometry alter magnetic dipole moments and toroidal dipole moments, which in turn affects the anapole state and, in turn, the near-field energy concentration. In this study, we investigate different QD positions, directions of spontaneous emission enhancement, and local near-field enhancement of the anapole. Since the nanostructures can significantly improve the emission of near-field and single photons by utilizing a strong Purcell effect, they become a possible candidate for various optical elements and devices.
2. Optical properties of DGSND
As shown in Fig. 1, the periodic double-groove Si nanodisk array includes a QD, Si, and SiO2. The yellow sphere represents the QD, the blue-green elliptical disk represents Si, and the gray cube represents SiO2. Figure 1(b) is a cross-sectional view of the DGSND in the x-y plane. The silicon disk's long axis R is 600 nm and short axis r is 400 nm. The radius c of the sphere enclosing the QD is 8 nm. The two slits have symmetrical geometric parameters: 160 nm in length a and 16 nm in width b. In Fig. 1(c), the period p of the unit cell is 700 nm. The thickness h1 of the SiO2 with a refractive index of 1.45 as the substrate is 100 nm, and the thickness h2 of the Si disk is 80 nm. The dispersion refractive index of Si is extracted from the experimental data [50].
Fig. 1. (a) Schematic illustration of the periodic DGSND array. (b) Overhead perspective of the fundamental building block. (c) The unit cell as seen from the side.
Numerical calculations are performed using the commercial software, COMSOL Multiphysics. To observe the optical properties of the DGSND, firstly, without QDs, the plane wave is incident in the z-direction and the E-field polarizes in the x-direction. The surrounding medium is air. In Fig. 2(e), there is a distinct dip in transmission at 972 nm. This transmission spectrum can be fitted with the Fano resonance model [51], and the full-width half maximum (FWHM) is approximately 4 nm. To study resonance in the dielectric metamaterial, the multipole decomposition in Cartesian coordinates is applied. Initially, the system's induced current distribution J at different frequencies is calculated. Next, the multipole moments are determined by the given formula [52]
Fig. 2. DGSND is excited by plane waves. (a) At 972 nm, the E- field is spread out along the middle cross-section plane of the x-y axis. The annular displacement current distribution is represented by the red arrow. (b) The H-field distribution at the x-z cross-section plane at a wavelength of 972 nm is depicted, with the red arrow indicating the distribution of magnetic lines. (c) The far-field scattering power of the DGSND can be expanded via multipole expansion: electric dipole (P), magnetic dipole (M), toroidal dipole (T), electric quadrupole (QE), and magnetic quadrupole (QM). (d) Related phase relationships of the DGSND. (e) The DGSND exhibits transmission (Tra), reflection (Ref), and absorption (Abs) spectra.
Equation (7) demonstrates that in the scattering process when the contributions of P and T moment are in opposite phases, the far-field radiation is reduced to zero.
When $\boldsymbol{P} ={-} ik\boldsymbol{T}$, the scattering of P and T can be eliminated, that is, the electric anapole state. As shown in Fig. 2(c), the scattering powers of the P (black curve) and the T (blue curve) cross at 973.3 nm, and according to Fig. 2(d), there is about π phase difference between P and ikT at the wavelength of 973.3 nm, indicating the destructive interference, so an anapole state is formed. The suggested configuration is obtained from a Si disk lacking slits, which sustains the anapole state. For the DGSND lattice, the periodic structure can produce excitation of lattice resonance, leading to anapole-anapole strong interaction and additional increase of the near-field energy, in addition to anapole excitation owing to electromagnetic multipole resonance of high refractive index dielectric [28]. The interactions between adjacent nanocubes can be regulated by parallel and vertical coupling concerning the polarization axis. However, due to the introduction of double slits in the Si disk, the distribution of displacement current and polarized charge are changed. The distribution of electric and magnetic fields at the resonant wavelength of 972 nm is illustrated in Fig. 2(a) and (b). There are two symmetric displacement current vortices in the x-y plane and two typical magnetic field concentrations in the y-z plane. Due to the structural destruction of the Si disk, the concentration of the E-field is primarily in the two slits, while the magnetic field remains confined to the Si. The maximum enhancement of the E-field is 69.2 at 972 nm, which is 5.77 times greater than that of the traditional Si disk structure [53].
3. Anapole states in DGSND excited by the QD
A QD emitter is located on the Si disk, and the QD emitter is modeled as an electric point dipole polarized along the main axis [54]. With the same parameters, only the spontaneous emission of QDs is relied upon to incident the DGSND. As can be seen from Fig. 3, the anapole state of THE DGSND is successfully excited. In Fig. 3(a), the scattered power of the P (black curve) and the T (blue curve) cross at 955 nm, while in Fig. 3(b), there is about π phase difference between P and ikT at the wavelength of 955 nm, forming an anapole state. The anapole state can be further identified from its electromagnetic field pattern. There are annular currents and annular magnetic moments, which conform to the distribution characteristics of the anapole state [55] .Fig. 3(c) and (d) have a similar distribution to Fig. 2(a) and (b). The E-field is localized in two gaps, while the H-field remains concentrated in the Si. However, the difference lies in the existence of electromagnetic hot spots in the region where the QDs are located, which is caused by the influence of the E-field in the position where the QDs are used as source emitters.
Fig. 3. DGSND is excited by the QD. (a) Far-field scattering power of the DGSND. (b) Related phase relationships of the DGSND. (c) At 955 nm, the E-field is distributed in the x-y middle cross-section plane, and the annular displacement current distribution is depicted by the red arrow. (d) Magnetic field distribution at the x-z cross-section plane at 955 nm, where the red arrow represents the magnetic field line distribution. (e)Enhancement of radiated, dissipated, and total power emitted relative to the power of the same dipole radiated in free space. (f) Details of the Purcell factor enhancement along the x-axis in the x-y plane.
QDs modeled as point dipoles produce a very wide spectrum [54], so that the emitted radiation couples to the modes in the DGSND, and some excitation modes are suitable for radiation, thus the DGSND can be used as nano-antenna. Due to material loss (non-radiative process), part of the total transmitted power ${P_{\textrm{tot}}}$ dissipates in the DGSND in the form of heat ${P_{\textbf{di}s}}$, while the rest propagates to the upper free space ${P_{\textrm{rad}}}$ in the form of radiation [54]. The power radiated, denoted as ${P_{\textrm{rad}}}$, is calculated by integrating the real component of the Poynting vector across the surface, where ${P_\textbf{{fs}}}$ represents the power transmitted by a solitary dipole in free space in the vacuum. The total power output of the dipole, ${P_{\textrm{tot}}}$, encompasses the power lost in the materials and the power transmitted into the upper space, represented as ${P_{\textrm{tot}}} = {P_{\textrm{rad}}} + {P_{\textbf{di}s}}$. Then by calculating the ratio of the radiated power of the QD in the DGSND and free space, the enhancement of the spontaneous emission is defined as the Purcell factor, i.e. Purcell factor =${P_{\textrm{rad}}}/{P_{\textrm{fs}}}$. The Purcell factor, which is based on physically significant and intuitive parameters, serves as an effective instrument for deciphering and managing the interaction between light and matter within resonators.
The calculated radiation powers ${P_{\textrm{rad}}}$, ${P_{\textbf{di}s}}$, and ${P_{\textrm{tot}}}$ of the x-directed QD dipole incident into the upper free space are shown in Fig. 3(e). The dipole is positioned above the center of the Si disk. These powers are normalized to ${P_\textbf{{fs}}}$. Due to the Purcell effect, the excitation of the anapole state of the dielectric metamaterial substrate enhances ${P_{\textrm{tot}}}$ by 120 times. At 971.5 nm near the anapole state wavelength, the QD radiation is maximized, and the Purcell factor is the strongest, while it does not have such characteristics in other frequency areas. 971.5 nm is exactly the position where the common peak values of the dipole moments in the multipole expansion are located. Meanwhile, at the same wavelength, the radiation emission and dissipation power are enhanced by 94 times (i.e., Purcell factor) and 25 times, respectively. This indicates that most of the total power transmitted is radiated into the free space above. To describe how the quantum spontaneous emission intensity will change after the introduction of a dielectric material, we investigate the quantum efficiency (${\eta _{\textrm{QE}}}$) enhancement, defined as ${\eta _{\textrm{QE}}} = {P_{\textrm{rad}}}/{P_{\textrm{tot}}}$ [56]. The results in Fig. 3(e) show that ${\eta _{\textrm{QE}}}$ is about 78%, indicating that DGSND has great advantages as an efficient nanoantenna.
It is mentioned in the literature that to maximize the enhancement effect, the QD emitter is generally located at the electric hot spot and its dipole moment aligns with the local E-field [57]. Using dielectric metamaterial with the anapole to enhance the spontaneous emission of QDs also requires the spatial overlap between the emitter and the E-field hot spots. Therefore, the QD is placed at the position with the strongest E-field intensity to excite the maximum Purcell factor. Figure 3(f) shows that the Purcell factor of the dipole on the Si disk varies with the position on the x-axis, and the maximum value reaches 965 at the slit. Furthermore, the Purcell factor enhancement is symmetrical along the y-axis, the Purcell factor decreases as the distance between the dipole and the gap on the Si disk increases. Because of the excitation of the anapole state, which localizes near-field energy and hot spots in the slit area. The proximity of QDs to hotspot regions leads to an augmentation in the Purcell factor.
4. QD spontaneous emission regulated by the four-groove Si nanodisk (FGSND)
The above analysis shows that the anapole arises from the interaction between P and T. P can be regulated by opening two narrow slits in the x-direction. Changing the distribution of circulation by opening two square holes in the circulation area affects T, and further affects the anapole state and near-field energy accumulation. To test this hypothesis, two symmetrical rectangular holes are designed on the original DGSND. Figure 4(a) depicts a three-dimensional elementary diagram of the FGSND. The newly added holes are symmetric about the x-axis, with a length e of 180 nm and width f of 50 nm (as shown in Fig. 4(b)). The geometric parameters of the two holes are the same, and the spacing g = 210 nm.
Fig. 4. FGSND is excited by plane waves. (a) The proposed FGSND structure diagram with the unit cell. (b) At 948.1 nm, the E-field is distributed in the x-y middle cross-section plane, and the red arrow represents the annular displacement current distribution. (c) Far-field scattering power of the FGSND. (d) The FGSND exhibits transmission (Tra), reflection (Ref), and absorption (Abs) spectra.
To better understand the internal physical mechanism of rectangular holes, we analyze the optical characteristics of the FGSND under plane wave excitation. Figure 4(c) shows the complete multipole decomposition of the far-field scattering power of the structure is calculated. The current J distributes annularly in the nanocube and remains constant. After opening the hole, based on the law of conservation of current, the current density of the remaining Si nanostructure increases. An elliptical hole creates a circulating current with a higher density in the surrounding area, leading to a larger M distribution, leading to a stronger T. Two strong opposite parallel magnetic moments lead to an increase in magnetic quadrupole radiation. Therefore, in Fig. 4(c), a strong T (blue) and a strong QM (purple) are observed at 950.1 nm. Furthermore, the T contributes the most to the multipoles. And from the FGSND reflection spectrum in Fig. 4(d) very narrow linear resonance appears at 948.1 nm. The presence of holes changes the distribution of red arrows in the E-field diagrams of Fig. 4(b). The holes are surrounded by redistributed displacement currents, and the annular current density on both sides of the Si cube increases. The near-field energy of the two structures is strongly localized in the center, with a maximum E-field augmentation of 106 times.
Figure 5(a) and (b) show that when the QD is located on the x-axis, the distribution of the Purcell factor on the y-axis is symmetrical. At x = -78 nm (i.e., the slit area), the maximum Purcell factor is 2450, which is about 2.5 times higher than that of the DGSND. This is because of the stronger E-field of the FGSND compared to the DGSND. Since two holes are opened, the effect of the QD position on the y-axis is also studied. However, when the QD is located on the hole, the Purcell factor does not increase but reaches a minimum value of around 28. The Purcell factor is larger at the center and edge of the Si disk but does not exceed 210. The difference between the x-axis and y-axis arises from boundary conditions. The E-field can abruptly change in the direction of the E vector's oscillation, and the medium with the lower refractive index will have a greater E-field [42]. Therefore, in the slit of the x-direction, the E-field enhancement achieved the maximum. Similarly, the Purcell factor also shows the same trend in the x direction. Figure 5(d) and (e) show the Purcell factor enhancement as a function of the emitter position on the perpendicular z-axis. As seen above, the maximum E-field is localized in the gap of the DGSND, and the gap of the FGSND also enhances the maximum E-field, so the QD is placed at x = -78 nm and y = 0. In Fig. 5(d), it is obvious that emitters close to the nano-Si disk gap provide higher Purcell factor. Enhancements above 500 should be within 15 nm of the center. These results demonstrate an effective method for controlling spontaneous emission in QDs.
Fig. 5. FGSND is excited by the QD. (a, b) Details of the QD movement along the x-axis and the Purcell factor enhancement in the x-y plane. (c, d) Details of the QD movement along the z-axis and the Purcell factor enhancement in the x-z plane.
Given that the Purcell factor attains its maximum value in the gap region, it remains constant in this specific area (i.e., x = -78 nm, y = 0, and z = 0) for the sake of comparison. Figure 6 shows the relationship between the Purcell factor and the hole size. In Fig. 6(a), as the length e of the rectangular hole increases, the spectrum blueshifts and the Purcell factor initially increases and then decreases. When the e is 180 nm, the maximum Purcell factor is 2450. Additionally, when the e = 180 nm, the hole width f is adjusted. In Fig. 6(b), a width of 50 nm yielded a maximum enhancement factor of approximately 2450.
Fig. 6. FGSND is excited by the QD, and the QD is fixed in this area (i.e., x = -78 nm, y = 0, and z = 0), Purcell factor with different geometric parameters of modified Si disk. (a) The rectangular slit length e (fixed width f = 50 nm). (b) Rectangular slit width f (fixed length e = 180 nm).
By observing Fig. 2(c), it can be found that the current distribution on both sides of the DGSND structure along the y-axis direction is circular. For better consistency, the two rectangular holes on the y-axis are replaced with two circular holes. Figure 7(a) depicts a three-dimensional schematic diagram of the FGSND with circular holes (CH-FGSND). Essentially, two circular holes are opened on each side of the DGSND. The radius L of circular holes is 65 nm, and the distance D between two circular holes is 210 nm. The other dimensions remain unchanged for comparison. The diagram on the right in Fig. 7(a) illustrates the excitation of the M and T modes within the unit cell, representing the design concept of the CH-FGSND. By changing the distribution of J in CH-FGSND, it regulates M and T.
Fig. 7. CH-FGSND is excited by plane waves. (a) The proposed CH-FGSND structure diagram, and the excitability diagram of T and M in the unit cell, with J representing circular current distributions. (b) Transmission (Tra), reflection (Ref), and absorption (Abs) spectra of the CH-FGSND. (c) At 946 nm, the E-field is distributed in the x-y middle cross-section plane, and the red arrow represents the annular displacement current distribution.
Firstly, the optical characteristics of the CH-FGSND under plane wave excitation were analyzed. The transmission, reflection, and absorption spectra show that a narrow linear resonance at 946 nm in Fig. 7(b). Similar to FGSND, due to the presence of holes, the E-field diagram in Fig. 7(c) demonstrates that the near-field energy is focused in the slits, the greatest E-field augmentation reaches a factor of 131. This supports the previous conclusion that circular holes result in larger E-field enhancement and narrower linewidth due to the distribution of displacement current.
Figure 8 illustrates the corresponding maximum amplification of the E-field with different hole radii. Figure 8(a) illustrates that as L changes from 45 nm to 75 nm in 5 nm intervals, the resonance wavelength gradually blueshifts, and the E-field in the gap region initially grows and subsequently falls. E-field enhancement of 133 times is achieved when L is 65 nm. Next, considering the impact of the CH-FGSND on the spontaneous emission of QDs, QDs are still placed at the gap of x = -78 nm, y = 0, and z = 0. In Fig. 8(b), the distribution of the Purcell factor is shown for different L values. As L increases from 55 nm to 75 nm, the peak position shifts to shorter wavelengths. The Purcell factor reaches its maximum value of 3560 when L is 65 nm.
Fig. 8. (a) CH-FGSND is excited by plane waves, the E-field enhancement, and the resonant wavelength corresponding to various L. (b) CH-FGSND is excited by the QD, and the QD is fixed in this area (i.e., x = -78 nm, y = 0, and z = 0), Purcell factor with different L of the CH-FGSND.
The same method can be applied to find the electrodeless state of the plasmon structure. Surface plasmon mode is an electromagnetic excitation that concentrates the electromagnetic field at a sub-wavelength scale by coupling with collective electron oscillation at the metal-dielectric interface [58,59]. Therefore, combining the electrodeless state and nanoscale confinement of the electromagnetic field can further enhance the electromagnetic near field. We designed a sandwich structure (Fig. 9(a)). The thickness of the Ag layer is 100 nm, the thickness of the SiO2 spacer layer is 20 nm, and the QDs are placed at x = -78 nm, y = 0, and z = 0. Other geometric parameters are similar to the structure of the CH-FGSND. The structure diagram is shown in Fig. 9(a), and the relative dielectric constant of Ag adopts the Johnson mode. Thanks to the mirror effect [60], the CH-FGSND with Ag substrate has a larger average E-field enhancement factor and a narrower linewidth. Moreover, the CH-FGSND with Ag substrate not only concentrates electromagnetic fields but also improves the spontaneous emission of QDs.
Fig. 9. (a) Diagram illustrating the periodic structure of the CH-FGSND with an Ag substrate. (b, c) The E-field distribution in the x-y plane and the H-field distribution in the y-z plane at 805 nm. (d) The spectra of CH-FGSND with an Ag substrate include transmission (Tra), reflection (Red), and absorption (Abs). (e) The periodic CH-FGSND with Ag substrate is excited by the QD, and the QD is fixed in this area (i.e., x = -78 nsm, y = 0, and z = 0), Enhancement of ${P_{\textrm{tot}}}$, ${P_{\textrm{rad}}}$, and ${P_\textbf{{dis}}}$ relative to the same dipole in free space.
As shown in Fig. 9(b), a diagram illustrating the distribution of the electric field in the x-y plane is generated at the resonance wavelength. The E-field is limited to the gap, while the H-field is predominantly present in the dielectric layer. This is a distinctive feature of localized surface plasmon resonance. Since the thickness of the bottom silver film is greater than the skin depth, the transmission channel of the proposed device will be blocked. In Fig. 9(d), the minimum reflection leads to the maximum absorption, and there is an obvious resonance at the wavelength of 805 nm, corresponding to an absorption rate of 99.7% and a FWHM of about 1.6 nm. In addition, based on the anapole state, the |E|/|E0| of the CH-FGSND with Ag substrate is 2.2 times greater than the Ag/SiO2/Ag slotted structure [53], this suggests that anapole-based dielectric metamaterials can form ultra-narrow perfect absorption and also have superior E-field enhancement effect. In addition, the results in Fig. 9(e) show that the Purcell factor is 5029 and ${\eta _{\textrm{QE}}}$ is about 90%, which is higher than the DGSND structure.
Metal plasmon resonance modes can be used in a variety of ways to increase the electromagnetic field, including single nanoparticles, nano-dimer, and nanoarray. Similarly, there are multiple ways to improve the Purcell factor by metal plasmon resonance mode, such as plasma nano-dimers [61] and nano-gap created by gold nanorods on metal thin films [62]. These methods can achieve small mode volume and improve spontaneous emissivity. Our designed structures, compared to other proposed structures, are superior to some Si nano-disk structures and comparable to plasmon enhancement in metal structures. Moreover, our structures can combine these two enhancements. In Table 1, we compare our structures with previous metal or dielectric structures to highlight their advantages.
Table 1. Comparison of |E|/|E0| and Purcell factor of various structures proposed in previous works
5. Conclusion
To summarize, we successfully excite the anapole state through the spontaneous radiation of QDs and also proved that the anapole state can influence the QDs to achieve high Purcell factors. Next, different structures were designed to enhance the E-field and Purcell factor of the anapole state. Firstly, the optical properties of these structures excited by plane waves are analyzed, and then the influence of different dipole positions of QDs in nanodisks is studied. By optimizing the dipole position and manipulating the anapole state, the spontaneous emission of all-dielectric nanostructures is maximized. By including the Ag substrate beneath the SiO2 and utilizing the mirror effect, the E-field is enhanced to 153, the Purcell factor is 5029, and 90% of the high ${\eta _{\textrm{QE}}}$ is achieved. This research will pave the way for stronger Purcell effects from emission sources in Si Mie resonators. The study's findings pave the way for effective nano-light sources built on resonant structures that facilitate Mie modes. The specific realization of optimized nanostructures will provide better control over light-matter interactions, which has the potential for innovative applications in novel light sources.
Funding
National Natural Science Foundation of China (12174351).
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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