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Abstract
This work considers a perfect 3D omnidirectional photonic crystal; Spherical Bragg Resonators (SBR), for lasing applications. We use the recursive transfer matrix method to study scattering in an Er3+ doped SBR. We find the threshold gain factor for lasing by scanning poles and zeros of the S-matrix in the complex frequency plane. For a six Si/SiO2 bilayer SBR, the threshold gain factor corresponds to a dopant density of Er3+ of 5.63 × 1020 ions/cm3. We believe, our work is the first theoretical demonstration of the ability to engineer optical amplification and threshold gain for lasing in SBRs.
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1. Introduction
Photonic crystals are structures based on a periodic arrangement of dielectric materials. They have been widely used to confine and control light by introducing defects or light emitters inside the photonic bandgap [1,2]. The dimensions of a photonic crystal are linked to the number of directions in which the refractive index changes. Therefore, an array of concentric spheres constitutes a 3D photonic crystal [3–6]. Confinement of photons in the core is possible for a multilayer sphere array of alternating high and low refractive indexes with an optical thickness of the layers of about one-quarter wavelength. This structure is a Spherical Bragg Resonator (SBR), also known as the Bragg onion resonator [3,4,7]. The SBR has excellent potential as a tridimensional photonic crystal with a high-quality factor Q ${\sim} {10^6}$ and small modal volume V${\sim} {({\lambda /n} )^3}$ [8,9]. These resonators are of great interest for applications that demand a large spontaneous emission factor, like single-photon sources [10] or threshold-less lasers [11,12].
The light-scattering properties of SBRs have been studied theoretically with different numerical methods and matrix formulations [3,4,9]. Furthermore, an efficient numerical model called Spherical space Bessel-Legendre-Fourier has been developed for any lossy material or dielectric structure with arbitrary geometric [13]. This method provides the analyzed structure's resonance state frequencies and field profiles. However, the recursive transfer-matrix algorithm developed by A. Moroz [14] has been the most widely used in recent years. This algorithm is computationally attractive due to the simplicity of the implementation, reliability, and computational efficiency [15]. Furthermore, the method presents a general solution for fundamental cross-sections, orientation-averaged electric and magnetic field intensities and the radiative decay rate of a dipole placed at any arbitrary position in a multilayer sphere. The algorithm has successfully addressed different light scattering problems in multilayer spheres using proper combinations and manipulations of the transfer matrices [15–17].
On the other hand, incorporating a gain or active media in a resonator opens an avenue for improving existing applications like on-chip resonators for lasing [18] or sensing [19] and even new ones. In addition, the promise of using SBRs for spontaneous emission engineering [5,11] has also been suggested. However, a comprehensive study of the conditions to achieve such optical amplification by introducing a gain media at the core of these Bragg onion resonators are still missing.
This work presents a theoretical approach to the electromagnetic properties in Si/SiO2 SBRs with a variable number of layers. We estimated the multipole scattering efficiencies by a recursive transfer matrix method [14,15]. The Weierstrass factorization [20] of the S-matrix in the complex frequency plane is used to find the perfectly emitting (poles) and absorbing (zeros) structure modes. We exhaustively study how modifying the Er3+ ion concentration as a gain media in the SiO2 core affects the localization of the poles and zeros in the frequency plane. The relationship of Er3+ ion concentration and the structure modes are used to engineer optical amplification and threshold gain for lasing in SBRs.
These resonators can be fabricated with precise control of the spherical layers’ thickness and chemical composition by plasma-enhanced chemical vapor deposition (PECVD) technique and others [7,21,22]. Therefore, we believe this work can be a reference for the future fabrication of SBRs that can be used for threshold-less laser applications.
2. Theory of light scattering
The spherical Bragg resonator (SBR) studied is shown schematically in Fig. 1. This SBR consists of a low refractive index spherical core surrounded by alternating concentric dielectric shells with different refractive indices. The shells have an optical thickness of about one-quarter wavelength ($\lambda /4$) to produce constructive interference of waves reflected from the interfaces of each surface. This frequency range, where electromagnetic waves are not allowed to propagate outward from the resonator, is known as the band gap. As a result of the spherical symmetry of the SBR in the radial direction of light propagation, an omnidirectional band gap is created. The center of the bandgap, where the SBR acts as an omnidirectional Bragg reflector, is defined by the wavelength, ${\lambda _B}$. The system is illuminated with a plane electromagnetic wave, and the host medium is air. The gain medium is located in the SiO2 core of the onion resonator.
The scattering properties of Spherical Bragg Resonator (SBR) can be characterized by the S or T matrices [23]. In our case, we use a rigorous and computationally fast transfer-matrix method, and the well know relation between the $T$-matrix and the definition of the $S$-matrix [23] to obtain the appropriate expressions for the $S$-matrix coefficients:
where $\overline{\overline I} $ is the identity matrix.2.1 Decay rate
This transfer matrix can be used to estimate the Purcell factor ($P = $ Γ/Γ0) in multilayer spheres where the normalized radiative decay rate of a dipole placed at the core center is given by [14]:
In addition, the modal volume V of a multilayer sphere can be calculated from the quality factor $Q = {\omega _0}/\Delta \omega $ and Purcell factor [7,24].
where $\Delta \omega $ is the full width at half maximum of the normalized radiative decay, $\omega $0 and ${\lambda _0}$ are the mode frequency and wavelength respectively, and ${n_c}$ is the refractive index of the SBR’s core.2.2 Scattering, absorption, and extinction efficiencies
Our structures have a spherical geometry, which can be treated as the deconvolution of electric and magnetic multipoles of order l, in spherical coordinates. Therefore, the S-matrix is diagonalized. Consequently, S-matrix fully describes the scattering process for each multipole l and diagonal component, p. For our calculations, we apply the Weierstrass factorization to each component p (Transverse Electric and Transverse Magnetic) to decompose the $S$-matrix [20]:
The eigenvalues of the $S$-matrix correspond to two singularities: poles and zeros [25]. The poles represent the eigenmodes, while the zeros are associated with a perfect absorption [25]. The position of poles and zeros allows to describe the scattering properties of the system under consideration [20].
Also, the total scattering, absorption, and extinction efficiencies (denoted by the abbr. subscripts, sca, abs and ext, respectively) can be obtained from the S or T matrices:
2.3 Gain media and loss compensation
Many potential applications of optical metamaterials are limited by their intrinsic losses. Active media or gain materials are most commonly used, among other variants, to compensate for optical losses [26]. Thus, the use of semiconductor nanocrystals or doped dielectrics as active media can produce an optically or electrically activated population inversion. The stimulated emission of photons in coherence with the incident radiation will compensate for material losses and produce amplification. In the literature, the gain material is characterized by the material gain parameter g. This value determines the amount of gain material required for loss compensation. For example, the gain factor for loss compensation in silicon materials goes from 1 to 100 cm-1 [26] and directly leads to the dopant concentration required for supplying the gain using the gain cross-section data for the dopant. The total extinction coefficient ${\mathrm{\Omega }_{ext}} = \; {\mathrm{\Omega }_{sca}} + {\mathrm{\Omega }_{abs}}$ will define the capability of the stimulated emission to compensate for the intrinsic optical losses. The system will be operating in an under-compensated regime for ${\mathrm{\Omega }_{ext}} < 0$, a fully compensated regime for ${\mathrm{\Omega }_{ext}} = 0$, or an overcompensated regime for ${\mathrm{\Omega }_{ext}} > \; 0$ [23,26].
When there is no lasing, a gain media can be modeled based on classical electrodynamics without considering the quantum dynamics of the ground and excited states. In that case, the system can be assumed as a dielectric with a complex refractive index having a negative imaginary part [27,28]:
The parameter $\gamma $ (gain factor) represents the property of a material to amplify ($\gamma < 0$) or absorb ($\gamma > 0$) an electromagnetic field. Therefore, the material gain parameter g can be represented by $4\pi \gamma /\lambda \; [{c{m^{ - 1}}} ],$ where λ is the free-space wavelength [26]. Undercompensated systems behave like typical dielectric materials but with stronger electromagnetic fields surrounding the structure and narrower resonances. The system acts like an optical amplifier when overcompensated at frequencies outside its resonances. A laser-like behavior is observed when losses are fully compensated at frequencies around resonance [26,28]. Such model correctly predicts the threshold gain and the lasing generation wavelength. For describing the post-threshold operation, a nonlinear permittivity function must be considered [29].
3. Results and discussion
The scattering properties of an SBR with the geometry shown in Fig. 1 were studied. The SBR considered for this study consists of an SiO2 core (${r_{core}} = \lambda /2{n_{core}} = 491.32\textrm{}nm$) surrounded by dielectric Bragg cladding bilayers. Each bilayer constitutes a layer of Si followed by a layer of SiO2. The layer nearest to the core is Si and has a thickness of ${L_{Si}} = \textrm{}101.71\textrm{}nm$, followed by a SiO2 layer with a thickness of ${L_{SiO2}} = 244.98\textrm{}nm$. This geometry was optimized to act as a Bragg reflector with a stopband center at ${\lambda _B} \approx 1415\; nm$. As we will see later, with this design, the dipolar resonance core mode in the SBR is around the maximum absorption of the erbium cross-section [30]. The calculation was performed using experimental scattering values reported for SiO2 and Si [31,32].
The Purcell factor was calculated from Eqs. (2) and (3) at the resonant wavelength (eigenfrequency) of $\lambda \approx 1536\; nm$ for different layer numbers. The results obtained are depicted in Fig. 2(a). As can be appreciated, it grows exponentially with increasing layers. This indicates that the normalized spontaneous emission from a dipole placed in the SBR's core can be significantly modified by varying the number of layers.
Fig. 2. Calculated (a) Purcell factor, (b) quality factor and (c) modal volume for a spherical Bragg resonators with a different number of layers at the resonant wavelength $\lambda \approx 1536\; nm$.
The modal volume was computed from Eq. (4) with the obtained values of the Purcell factor, and the quality factor $Q = {\omega _0}/\Delta \omega $. Figure 2 (b) and (c) show the quality factor Q and the cavity modal volume, V for SBRs with layer numbers ranging from 4 to 16. Analogous to the Purcell factor, the quality factor shows exponential growth with an increasing number of onion resonator layers. Cavity Q values above 104 are obtained for a relatively small number of layers (12). These values are comparable with results obtained for well-known whispering-gallery modes [33]. The modal volume (Fig. 2(b)) remains nearly constant for the range of layers studied, with a value in the order of ${({\lambda /{n_c}} )^3}$. Our analytically calculated values are in good agreement with those obtained from others numerical methods [34], as well as with those reported for other dielectric materials [17]. This reinforces our choice to use onion resonators based on Bragg reflections as omnidirectional photonic crystals with a high Q and a small modal volume in this study.
3.1 Scattering efficiencies and modal analysis
In the following sections, we will study and discuss the scattering properties of a 12-layer Si/SiO2 SBR. The electromagnetic scattering of the SBR was investigated numerically with the recursive algorithm described in [14,15]. The total scattering efficiency ${\mathrm{\Omega }_{sca}}$, total extinction efficiency ${\mathrm{\Omega }_{ext}}$ and total absorption efficiency ${\mathrm{\Omega }_{abs}}$ are calculated from equations (7–9) and the obtained T-matrix coefficients.
The total scattering efficiencies (${\mathrm{\Omega }_{sca}}$) computed considering the electric TE and magnetic TM multipole contributions, l, are exhibited in Fig. 3. Figures 3 (a) and (b) show the ${\mathrm{\Omega }_{sca}}$ for the electrical multipole contribution and the magnetic multipole contribution, respectively. The total scattering spectra (electrical and magnetic multipole contributions up order 5) is displayed in Fig. 3 (c). The peaks and dips represent the different resonant and anti-resonant modes of the SBR.
Fig. 3. Scattering efficiencies for a 12-layer spherical Bragg resonator for five multipoles (a) electric, (b) magnetic and (c) total modes.
These modes can be grouped into core modes and cladding modes. Core modes are of the Fabry-Perot type, and their angular quantum number l is relatively small (Eq. (1)). On the other hand, the cladding modes are similar to the whispering gallery modes in ring resonators and have larger angular quantum numbers l [7]:
For our SBR, the resonance frequencies in the cladding layers corresponded to $l = 3$ and 4 and those in the core to angular quantum number $l = 1$. Significant differences are found between the modes located in the core and the cladding layers. The scattering resonances of the cladding mode (TM5 and TE4) are more pronounced in amplitude than those of the core modes (TE1), as can be appreciated in Fig. 3.
To investigate the difference between cladding mode and core mode resonances, we studied the spatial distribution of the normalized magnitude of total electric field ${|E |^2}/{|{{E_0}} |^2}$ and total magnetic field ${|H |^2}/{|{{H_0}} |^2}$ of a 12-layer SBR (Fig. 4). As can be noted, the field distribution of the core mode ($l = 1$, dipolar mode) TE1-TM1 are mostly confined within the SBR's core. Whereas the electromagnetic fields of the cladding mode ($l = 5$) TE5-TM5 are distributed among the cladding layers. In general, the core mode resonances exhibit more complex spectral features.
3.2 Gain
As discussed above, structures with a small modal volume and a large quality factor are ideally suited for applications that require a large Purcell factor. We previously estimated a modal volume of $V = 0.15{({{\lambda_0}/{n_c}} )^3}$ and a quality factor of $Q = \; $104 for a 12-layer SBR. This places our SBR as a potential candidate for threshold-less laser applications, as will be discussed next.
We investigated the total scattering efficiencies by adding a gain in the 12-layer SBR core. We used Er3+ (g∼1–100 cm−1) as a gain medium. In our model, a gain medium can be modeled by adding a negative imaginary part to the refractive index. Figure 5 shows the scattering efficiencies for a 12-layer SBR doped in the core with Er3+. The scattering efficiencies (${\mathrm{\Omega }_{sca}}$) are depicted in Fig. 5 (a). An enhancement of the scattering efficiency occurs at the core eigenfrequency $\lambda \approx 1536\; nm$ for a gain factor of $\gamma \approx 5.20 \times {10^{ - 5}}$. This significant increase in ${\mathrm{\Omega }_{sca}}$ was coupled with a reversal ${\mathrm{\Omega }_{ext}}$ and ${\mathrm{\Omega }_{abs}}$ to negative values as can be seen in Fig. 5 (c) and (d). It indicates that the active medium can compensate for radiative and dissipative losses. This is evidence of the suitability of the system to amplify the incoming radiation, creating an outgoing wave with more intensity than the incoming one.
Fig. 4. Spatial distribution of the normalized magnitude for (a) total electric field ${|E |^2}/{|{{E_0}} |^2}$ and (b) total magnetic field ${|H |^2}/{|{{H_0}} |^2}$ on the (x, y) plane for l = 1 at $\lambda \approx 1536\; nm$ and (c) total electric field ${|E |^2}/{|{{E_0}} |^2}$ and (d) total magnetic field ${|H |^2}/{|{{H_0}} |^2}$ on the (y, z) plane for l = 5 at $\lambda \approx 1794\; nm$ for a 12-layer spherical Bragg resonator.
Fig. 5. (a) Scattering, (c) extinction and (d) absorption efficiency dependency on gain factor for a 12-layer Er3+ core-doped Spherical Bragg Resonator. (b) FWHM (red square) and maximum scattering efficiency (black circles) dependency on gain factor.
The scattering intensity and full width half maximum (FWHM) were plotted as a function of the gain factor at the core eigenfrequency λ = 1536 nm in Fig. 5 (b). The relation between scattering intensity and FWHM exhibits a threshold-like gain factor of around $\gamma \approx 5.00 \times {10^{ - 5}}$, Fig. 5 (b). Below this threshold, the scattering emission intensity was very low, and the line width was wide. Above this threshold, the scattering increased suddenly, and the emission width decreased, indicating the transition of the system from an optical amplifier regime to lasing.
3.3 Pole zero gain threshold
We previously discussed that by studying the scattering efficiencies via the transfer matrix method it is possible to find the gain threshold for which both the net radiation and material losses in an SBR are compensated. Therefore, the lasing regime is achievable. Lasing is associated with the presence of a pole of the S-matrix in the complex frequency plane. If the gain increases, it shifts from the lower complex plane to the real frequency axis [20]. As a function of the total system loss, the distance from the pole to the real axis changes, which defines the mode's Q factor and the decay rate. Consequently, the laser requires the presence of an active medium and a pole to compensate for both material and radiative losses. This occurs for a high Purcell factor when the feedback regime is reached.
As calculated in section 3.1, TE1 mode has an eigenfrequency λ = 1536 nm. This resonance frequency coincides with the maximum of the Er3+ emission cross-section [30]. Figure 6 shows the complex S-matrix maps for the electric (c) and magnetic (d) components of the dipole mode. For reference, the scattering efficiencies are shown in Fig. 6 for TE1 (a) and TM1 (b). In the S-matrix maps for TE1 and TM1, we can follow the singularities, which correspond to the system's eigenvalues. A point-like singularity was found around our eigenfrequency of interest (λ = 1536 nm) for the TE1 mode, as seen in Fig. 7 (a). Therefore, the presented geometry will act as an omnidirectional resonant cavity for dopants with the same emission frequency as the singularity, such as Er3+.
Fig. 6. Scattering efficiencies of 12-layer Spherical Bragg Resonator for TE1 (a) and TM1 (b). Maps of the ${S_{pl}}$ matrix component in the complex frequency plane for TE1 (c) and TM1 (d).
Fig. 7. (a) Maps of the ${S_{pl}}$ matrix component in the complex frequency plane for TE1 in a 12-layer Er3+ core-doped Spherical Bragg Resonator. (b) Pole (blue) and zeros (orange) analysis of S-matrix scattering dependency on gain factor, gamma.
Looking closely at the TE1 singularity at $\omega /{\omega _0} \approx 0.65$, we can identify zeros and poles, Fig. 7 (a). The zeros correspond to perfect absorption, and poles represent the system eigenmodes. On the other hand, the addition of a gain media to the system will be reflected in a shift of the zeros and poles. Figure 7 (b) shows the imaginary parts of the pole and zeros when a scan over the gain factor is performed. Negative values of the pole's imaginary frequency are observed for small gains factor, while for gain factor values higher than $\gamma \approx 5.00 \times {10^{ - 5}}$, a positive pole's imaginary frequency was achieved. This value is the threshold gain for the amplification regime. Below this value, the scattering emission intensity is low, but above this value, the scattering emission is sharply turned-up at the eigenfrequency λ = 1536 nm. Therefore, $\gamma \approx 5.00 \times {10^{ - 5}}$ is the minimum gain factor (gain threshold) for lasing emission in a 12-layer Er3+ core doped Spherical Bragg Resonator.
The gain factor threshold for the lasing regime is highly dependent on the number of SBR layers. This is in agreement with the exponential growth of the Purcell factor observed previously. The gain threshold and Er3+ dopant concentration for lasing as a function of the number of layers in the SBR are shown in Fig. 8. The gain factor depends on the density of the dopant and its cross-section emission [35]. For the case of Er3+, whose emission cross-section is $\sigma = 7.27 \times {10^{ - 21}}\; c{m^2}$ at λ = 1536 nm [36], ${\sim} 1.50 \times {10^{22}}\; ions/c{m^3}$ are required to obtain laser emission in a 8-layer SBR structure. This value is slightly higher than the Er3+ dopant concentration in commercially available Er3+ doped silica materials. However, adding two more Si/SiO2 bilayer (12-layer) makes it possible to achieve a lasing regime with an Er3 + doped SiO2 core concentration of ${\sim} 5.63 \times {10^{20}}\; ions/c{m^3}$ since the threshold gain factor is reduced to $\gamma \approx 5.00 \times {10^{ - 5}}$.
Fig. 8. Gain threshold and Er3+ core-doped concentration for lasing in a Spherical Bragg Resonator as a function of the number of layers.
The dopant density of Er3+ in a 12-layer SBR structure is within the typical range [37]. Er3+-doped silica glass-based materials are commercially available with similar dopant densities, such as single and multi-mode optical fibers. Furthermore, SBRs have been fabricated with precise control of the layer's thickness and chemical composition by plasma enhanced chemical vapor deposition (PECVD) technique [7,21,22]. In fact, A. V. Medvedev et al. [21] have recently demonstrated the presence of a complete photonic band gap in a non-rare earth doped SBR structure.
We propose an SBR structure with a core made by melting a commercial Er3+ ion-doped silica optical fiber. The optical fiber tip radii can be controlled by the etching process. With a 980 nm diode laser, Er3+ ions can be excited through the silica fiber with the spherical tip. Depending on the Er3+ concentration in the silica doped microsphere, amplification or lasing of the scattered signal may be obtained. By placing a half-taper fiber close to the surface of the SBR, we can collect the scatter signal and analyze it with an optical spectrum analyzer.
The spontaneous emission of rare earth atoms and quantum dots (QDs) can be suppressed or enhanced by different classes of photonic structures such as woodpiles, concentric cylinders, spherical Bragg resonators, or inverse opals [11,37]. In our case, the high-quality factor Q and small modal volume V of our SBR significantly influence the emission rate of Er3+ atoms in the core. To demonstrate that, we will consider an Er3+ emitter located in the center of our structure. As the maximal optical emission of Er3+ atoms match with the single mode of the cavity (resonant condition), the emission rate (Γ/Γ0) is enhanced through the Purcell effect [12,38]. The lifetime of the Er3+ which is the inverse of the emission rate and, in the cavity, contrasts with its value in free space. This modification can be estimated from the Purcell factor of the cavity (P) and is given by the following relation [39,40]:
Shorter lifetimes raise spontaneous photons, increasing stimulated emission and amplifying spontaneous emissions [26]. In the case of our 12-layer Er3+ core doped SBR, the Er3+ lifetime decreased from ${\tau _r} \approx 4\; ms$ in free space [41] to ${\tau _{SBR\_E{r^{3 + }}}} \approx 127\; ns\; $ seconds in the cavity core. We have considered that the decay rate is purely radiative (${\tau _{nr}} \approx 0$), as previously reported theotrically and experimentally in Er3+-doped silica glass with a dopant concentration of ${\sim} 5.00 \times {10^{20}}\; ions/c{m^3}$ [41]. The enhancement of the emission rate of an Er3+ emitter (Γ/Γ0${\sim} \; {10^4}$) situated at the core, further confirms the suitability of our proposed SBR geometry for light sources, amplifiers, and lasers.
In contrast, for off-resonance, significant inhibition of the spontaneous emission of an Er3+ emitter is produced [12]. Such conditions can be achieved by increasing (or decreasing) the core radius ${r_{core}}\; $ by a factor of $({2s + 1} )/2$, were s is an integer value, resulting in cavity modes different from the Er3+ emission wavelength. For example, considering a 12 layers-SBR with a radius of ${r_{core}}/2$, the lifetime of an Er3+ atom will be longer (${\tau _{SBR\_E{r^{3 + }}}} \approx 25\; s$) than the free space value, inhibiting its spontaneous emission from the SBR core, by a factor of Γ/Γ0${\sim} \; {10^{ - 4}}$. Its effect is essential for optical quantum information processing applications [42–44]. Similar values of enhancement and suppression of the spontaneous emission of a dipole emitter embedded in a spherical Bragg resonator have been reported previously [11].
We have found a gain threshold using two different methods: scattering efficiencies and band-with, and pole and zero in the S-matrix map. The pole-zero method provides a better understanding of the amplification regime and confirms the lasing condition. In addition, we discuss the potential of the SBR for light emission control.
4. Conclusions
We have investigated the scattering properties of Si/SiO2 Spherical Bragg Resonators doped in the core with Er3+. The SBR geometry proposed in this work was optimized based on experimentally obtained refractive index values and dopant concentrations. Our onion resonator has a modal volume of $V = 0.15{({{\lambda_0}/{n_c}} )^3}$ and a quality factor of $Q = \; $104 for 6 Si/SiO2 bilayers highlighting its potential for spontaneous emission engineering. Using the transfer matrix T for the scattering in Er3+-doped Spherical Bragg Resonators, we found an enhancement in the scattering efficiency (${\mathrm{\Omega }_{sca}}$) for the dipolar core mode (TE1) at λ = 1536 nm and a dopant gain $\gamma \approx 5.20 \times {10^{ - 5}}$. Negative ${\mathrm{\Omega }_{ext}}$ and ${\mathrm{\Omega }_{abs}}$ were observed for these values, indicating a compensation of radiative and dissipative losses. We find the threshold gain factor for lasing by scanning poles and zeros of the S-matrix in the complex frequency plane. Six Si/SiO2 bilayers (or 12 layers) were found to be sufficient to reduce the threshold gain factor to $\gamma \approx 5.00 \times {10^{ - 5}}$. This value corresponds to a dopant density of Er3+ of $5.63 \times {10^{20}}\; ions/c{m^3}$. Materials with such concentrations are commercially available. On the other hand, the plasma-enhanced chemical vapour deposition (PECVD) technique has been used to obtain core-shell type structures. Therefore, we expect this work to pave the way for future experimental demonstrations of threshold-less lasers based on Spherical Bragg Resonators.
Funding
Natural Sciences and Engineering Research Council of Canada (Discovery grants, Strategic Research grants).
Acknowledgments
R. Kashyap acknowledges financial support for the Canadian NSERC Discovery grants and Strategic Research grants programs.
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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