1. Introduction

As first described by Purcell (see for example [1]), spontaneous emission must be viewed as a cooperative process involving an emitter and its surrounding electromagnetic environment. This has led to widely studied concepts for the enhancement of emission into a desired mode [2–4] or the suppression/inhibition of emission into undesired modes [5,6], with optical microcavities playing a central role. Control over spontaneous emission is the key enabler of several proposed ‘quantum’ light sources, such as the threshold-less laser [7] and the controlled source of single photons (i.e. the single photon source, SPS) [8,9].

Micro-scale confinement of light results in a set of sparsely distributed, small-volume cavity modes, such that the rate of emission into a single preferred mode can be greatly enhanced (i.e. the ‘Purcell effect’) [1]. However, control of emission through the Purcell effect alone requires that both the cavity and the emitter have a narrow linewidth. This requirement is effectively lifted in cases where the microcavity inhibits emission into other modes (e.g. the leakage into background radiation modes [5]). Most microcavities (e.g. microspheres, micro-pillars, microdisks, defects in 2-D photonic crystals) rely at least partially on total-internal-reflection-based confinement, and require additional engineering [10] to suppress emission into background radiation modes. A more ideal scenario is provided by defect micro-cavities formed within 3-D photonic crystals, which can, in principle, greatly suppress the local density of optical modes (LDOM) within one or more stop bands, thereby enabling essentially all spontaneous emission (over a broad wavelength range) to be directed into a single desired mode [6]. However, such cavities are challenging to implement at optical frequencies.

As is now well-known, 1-D photonic crystals (i.e. thin-film Bragg mirrors), under appropriate conditions [11], can exhibit an omnidirectional reflection band for light incident from a lower-index external medium. However, unlike 3-D photonic crystals, the 1-D omnidirectional reflector (ODR) supports in-plane ‘guided’ modes at all frequencies [12], including the spectral bands associated with external omnidirectional reflection. Nevertheless, for an emitter that is located in the external medium and at a sufficient distance to minimize coupling to the evanescent fields of these in-plane modes, it is possible for the 1-D ODR to facilitate significant inhibition of spontaneous emission [13,14].

Motivated by this, Xu et al. proposed and fabricated [15] a novel ‘onion’ microcavity with a spherical air core surrounded by ODR claddings, and subsequently analyzed its potential for the modification of spontaneous emission [16,17], showing that it might be possible to inhibit background emission by several orders of magnitude. This is an exciting property, which makes the onion resonator resemble the defect cavity within a 3-D photonic crystal. However, the onion resonator is also a challenging structure to fabricate, and moreover does not provide a convenient means for external coupling to the cavity modes of interest. Here, we demonstrate that buckled-dome microcavities [18] clad by ODRs can provide many of the same benefits, while offering much simpler fabrication and the possibility for straightforward and efficient coupling to the cavity modes by external fibers or free-space beams.

2. Background and overview

We have previously reported [18–20] experimental results on half-symmetric Fabry-Perot cavities fabricated by guided delamination-buckling within a thin film mirror stack. These cavities can be fabricated in large arrays, and the self-assembly nature of the fabrication process results in a very high defect-finesse. Reflectance-limited finesse and ‘textbook’ manifestations of Laguerre-Gaussian (LG) cavity modes have been achieved in both the 780 nm [19] and 1550 nm [20] wavelength ranges. Furthermore, cavity height (and thus resonance conditions) can be varied within a certain range through control of film stress and the choice of base diameter, and tuned on a finer scale through temperature control [20].

2.1. Representative experimental structure

As a test case for the modeling study, we focus on the small-mode-volume cavities from Ref [21]. A fundamental mode volume on the order of one cubic wavelength in the 1550 nm-range was demonstrated, and attributed in part to the high index contrast of the Si/SiO₂-based mirrors employed. Also of central importance to the present study, these mirrors provide a broad band of omnidirectional reflection for light incident from a lower index medium [11,22], such as from the air core of a buckled cavity.

As shown in Fig. 1(a) (and detailed elsewhere [21]), the cavities considered have a base diameter of ~50 μm and a peak height of ~0.78 μm. The lower mirror is deposited on a standard silicon substrate (n_Si ~3.5) and the upper curved mirror is adjacent to the external environment (i.e. typically air, n ~1). Note that the cavities in Ref [21]. have 5-period mirrors, but we assume 4-period mirrors (with an additional a-Si ‘capping layer’ on the upper buckled mirror [18,20]) for the theoretical study below. Given typical absorption losses for our a-Si layers (deposited by pulsed-DC magnetron sputtering), the addition of the fifth period produces only marginal (or even negligible) increase in reflectance and finesse, but significantly degrades the potential transmittance ψ = T/(T + A) of the mirrors, and thus the out-coupling efficiency for an embedded emitter. We furthermore assume indices n_aSi ~3.7-0.0009i and n_SiO2 ~1.47 at 1550 nm wavelength, for the a-Si and SiO₂ layers, respectively. While our experimental process (employing a shared-use deposition system) is subject to some variability in film loss [21], these values are representative of the lowest-loss mirrors we have produced in our previous work (resulting in R~0.999, T~A, and ψ ~0.5 [23]). Finally, quarter-wave layers at 1550 nm wavelength, corresponding to nominal layer thicknesses ~105 nm and ~264 nm, respectively, were assumed in all theoretical treatments below.

 

Fig. 1 (a) Microscope image of a portion of a large array of 50-μm-base-diameter domes. (b) Typical wavelength scan showing a fundamental resonance (at ~1610 nm for this particular cavity) and several higher-order transverse resonances. The solid lines are Lorentzian fits, and mode-field-intensity images associated with each resonance are shown as insets. (c) Schematic 3-D cut-out view of a buckled dome cavity, showing the fundamental resonant mode (at ~1527 nm using the cavity parameters described in the main text) predicted by a finite-element numerical simulation. (d) Predicted reflectance versus incidence angle from air at 1550 nm wavelength, for the 4-period lower mirror of the cavities.

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Figure 1(b) shows a typical experimental transmission scan, which as discussed elsewhere [21] reveals good agreement with predictions from paraxial Gaussian beam optics for a spherical mirror cavity. The buckled mirror has effective radius of curvature ~200 μm, resulting in a transverse mode spacing Δλ_T ~22 nm for the half-symmetric cavity. The mirrors have normal-incidence reflectance R ~0.999, limited mainly by residual absorption in the a-Si layers, enabling a cavity finesse ~2-3x10³ [23] (also note that Q ~F for the fundamental spatial mode of interest here). A finite-element (COMSOL Multiphysics) simulation (see Fig. 1(c)) predicted V_M ~1.3λ³ at ~1527 nm, in good agreement with our previous results [21] and with the FDTD simulations described in Section 4. Finally, Fig. 1(d) shows transfer matrix simulations for a planar mirror with the thicknesses and indices mentioned above. At 1550 nm wavelength, the mirrors provide high omnidirectional reflectance for both polarizations, although there is some leakage for TM polarization at large angles. The omnidirectional reflection band spans the ~1300-1700 nm range [22].

2.2. High-level description of spontaneous emission processes

A buckled dome cavity with an embedded dipole emitter is depicted in cross-section in Fig. 2(a). Note that the electromagnetic modes available to the emitter can be categorized as follows: (A) The Laguerre-Gaussian (LG) modes of the air-core half-symmetric cavity; (B) the vacuum radiation modes; and (C) the cladding modes guided within the mirrors. In keeping with the discussion above, we made some key simplifying assumptions as follows. First, given the spectral separation of the LG modes, we assume that the emitter is spectrally aligned to only the fundamental cavity mode, and that coupling to higher order LG modes is negligible. Second, given the high-reflectance-omnidirectional-band of the cladding mirrors, the rate of emission into radiation modes is anticipated to be very low [16]. Thus, background emission is expected to be dominated by coupling to the cladding modes (i.e. light guided within the omnidirectional mirror layers [17]).

 

Fig. 2 (a) Schematic diagram of a dipole emitter (indicated by the horizontal block arrow) inside a buckled dome microcavity, showing 3 possible routes for spontaneous emission. Emission into a cavity mode (A) can be predicted by the Purcell factor, emission into free-space vacuum modes (B) is assumed to be negligible due to the omnidirectional nature of the cladding mirrors, and emission into cladding modes (C) can be treated approximately using a planar model. (b) Planar model used to estimate the rate of emission into cladding modes. The dipole is placed at some distance d_Z from the interface with the bottom mirror.

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Given these assumptions, our goal is to estimate the rate of emission into the cavity and cladding modes of the structure, which will in turn yield estimates of the cooperativity and the spontaneous emission coupling factor. Of course, the enhanced rate of emission into the fundamental cavity mode is the Purcell effect and can be estimated using well-known expressions [1]. To estimate emission into cladding modes, we adopt a planar model for the buckled cavity as depicted in Fig. 2(b) and described in greater detail in Section 3. The planar approximation is well-justified by the low aspect ratio of the buckled domes (i.e. peak height ~0.78 μm and base diameter ~50 μm). In other words, the electromagnetic environment for an emitter located near the center of the cavity is expected to be closely approximated by the planar environment shown in Fig. 2(b). 3-D numerical simulations were used to verify these approximate results, and are described in Section 4.

2.3. Definition of key parameters

It is worthwhile to briefly summarize a few pertinent quantities. First, note that we define the Purcell factor as F_P ≡ γ_C/γ₀, quantifying the rate of emission into a cavity mode of interest relative to the free-space emission rate. For an emitter located in air and at the field maximum, and spatially and spectrally aligned to the cavity mode, the ‘ideal’ Purcell factor is given by the well-known expression [1] F_P0 = {3/(4π²)}{Q_EFF/(V_M/λ³)}. For an emitter linewidth Δλ_EM less than the cavity mode linewidth, we can write Q_EFF ~Q_C, where Q_C is the cavity quality factor. On the other hand, for a broad linewidth emitter (but still aligned to the cavity mode), Q_EFF = (Q_C⁻¹ + Q_EM⁻¹)⁻¹, where Q_EM = λ/(Δλ_EM) is the effective quality factor of the emitter [16].

A related figure of merit is the single-atom cooperativity, which can be expressed [24]:

Another important parameter is the spontaneous emission coupling factor β, which is the fraction of photons extracted from the cavity mode of interest, often stated as β = F_P/(1 + F_P). However, the latter equation implicitly assumes γ_BG ~γ₀ and also neglects the possibility for loss of cavity-mode photons (e.g. due to scattering or absorption by mirror layers). A more general and correct expression is as follows [24]:

3. Quasi-analytical planar model

We first consider spontaneous emission by a dipole embedded within the planar cavity shown in Fig. 2(b). We restrict the analysis to the case of a horizontal dipole; this is the orientation expected to couple most efficiently to the fundamental LG cavity mode of interest in the actual curved-mirror cavity. We also assume the dipole to be embedded in air, at some distance d_Z from the bottom mirror interface. Rigorous treatment of spontaneous emission requires a quantum electrodynamics formalism, which accounts directly for the polarization of the emitter by the zero-point fluctuations of the vacuum field, summing over all possible modes [25]. However, in the weak-coupling limit (expected to be valid for most emitters in the cavities described here, given their moderate Q values [1]), spontaneous emission can also be treated classically. In the classical view, the spontaneous emission rate of the dipole is modified by its own reflected field, and can be quantified by the local density of modes (LDOM), in turn given by the imaginary part of the electromagnetic Green’s function at the dipole location inside the (generally) inhomogeneous environment. Multiple authors have derived the Green’s function for planar (slab) structures (see for example Neyts [26], and references therein), and used the results to assess LDOM and spontaneous emission modification. Here, we follow the formalism provided by Tomas [27,28], from which the spontaneous emission rate into TE- and TM-polarized modes (normalized to the free-space emission rate), for a horizontal (in-plane) oriented dipole, can be written [12,29]:

3.1. Dipole in a half-wavelength thick air cavity

Consider a dipole located inside the air cavity depicted in Fig. 2(b), with both the upper and lower mirrors terminated by quarter-wavelength, high-index layers adjacent to the air core. At the design wavelength of 1550 nm, the (vertical emitting) fundamental cavity mode corresponds to d_A ~λ/2. The standing-wave field profile (not shown) exhibits nodes at the mirror interfaces and a peak (anti-node) at the mid-point of the air cavity. Figure 3 shows results from numerical integration of Eq. (3), with the air core thickness set at d_A = 0.776 μm. In order to illustrate the nature of the modes that make the dominant contributions to the LDOM, plots of the integrand in the TE expression are shown in Fig. 3(a). For a dipole emitting at 1550 nm and located at the center of the cavity, the LDOM is dominated by the fundamental cavity mode, which occurs for near-normal incidence in this case. The four additional peaks in the LDOM spectrum are associated with the in-plane guided modes of the 4-period claddings (which can be viewed as 4-channel multilayer slab waveguides [30]), and are, relatively speaking, weakly coupled since the dipole location is ~λ/4 removed from the mirror interfaces. Analogous behavior occurs for the TM modes, as shown for one representative case in Fig. 3(a). The extra peak is due to coupling to a surface mode.

 

Fig. 3 (a) The LDOM distribution (i.e. the integrand of the TE expression in Eq. (3)) in transverse-wave-vector space is plotted for various dipole positions and emission wavelengths. The integrand of the TM expression for one representative case (λ = 1570 nm, d_Z = d_A/2) is also shown (dash-dot line). (b) The total relative emission rate (γ_TE + γ_TM)/γ₀ for a dipole emitting at λ = 1550 nm is plotted versus dipole location relative to the bottom mirror interface. The contributions from the cavity mode (i.e. 0 < k_// < k₀) and the cladding modes (i.e. k_// > k₀) are also shown separately.

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The dashed red line in Fig. 3(a) corresponds to the same emission wavelength but with the dipole located at the interface to the bottom mirror. Since the cavity mode exhibits a node at this location, its contribution to the LDOM vanishes. Due to the omnidirectional claddings, emission into radiation modes is also very weak, as apparent from the relatively low LDOM density in the k_// < k₀ region. However, emission into the cladding modes is much higher in this case, since the dipole overlaps more strongly with their evanescent fields extending into the air core. Finally, the dotted green line corresponds to a dipole located at the center of the cavity but emitting at a longer wavelength subject to cut-off of the fundamental cavity mode. As might be expected, the cavity mode contribution to the LDOM is greatly suppressed but the contributions from the cladding modes is nearly identical to that predicted for the resonant case. These simulations confirm that background emission is dominated by emission into the cladding modes. In other words, γ_BG ~γ_CLAD, where γ_CLAD is the spontaneous emission rate estimated by evaluating the integrals in Eq. (3) over the range k_// > k₀.

Figure 3(b) is a plot of the total relative spontaneous emission rate for the same cavity as in part a, with the wavelength set to 1550 nm and the dipole position varied. Contributions from core (0 < k_// < k₀) and cladding (k_// > k₀) modes are also plotted separately. The behavior is consistent with results predicted elsewhere for omnidirectional-clad planar cavities (e.g. see Fig. 7 in ref [29].). Emission into cladding modes dominates for small dipole-mirror spacing, but becomes negligible when d_Z > λ/4. For a dipole optimally located at the center of the air cavity (d_Z ~0.375 μm), the model predits γ_C/γ₀ ~2.7 (which is near the upper limit for a planar structure [7]) and γ_BG/γ₀ ~γ_CLAD/γ₀ ~0.031. A more conservative estimate can be obtained by evaluating the integrals in (3) over the entire range encompassing both radiation and cladding modes, but at an off-resonant wavelength (such as the λ = 1570 nm case depicted in Fig. 3(a)). This results in γ_BG/γ₀ ~0.04, thus I ~25. Using this latter value and neglecting mirror loss (i.e. assuming κ_out = κ), Eq. (2) then predicts β ~70 for the half-wavelength planar cavity. This is consistent with the value (β ~25) predicted by St. J. Russell et al. [13] for a water-filled cavity clad by similar mirrors. The higher value here is mainly due to the assumption of an air core, which results in stronger decay of the evanescent cladding mode fields away from the mirror interfaces (i.e. γ_CLAD is approximately a factor of 3 lower here).

We can now speculate on the highest potential cooperativity for emission into the fundamental mode of the buckled dome microcavities described above. For the case of 3-D confinement, the maximum rate of emission into the cavity mode is given by the Purcell factor for a narrow linewidth resonant emitter. Using V_M ~1.3λ³ and Q_C ~2x10³ (see Section 2.1), it follows that F_P ≡ γ_C/γ₀ = F_P0 ~120. Furthermore, in keeping with the arguments above, we postulate that the background inhibition factor can be extracted from the planar model and is thus given by I = γ₀/ γ_BG ~25. For the optimally positioned horizontal dipole, this suggests a potential for cooperativity as high as C = F_P·I/2 ~1500. While similarly high values of cooperativity might be possible for highly optimized fiber Fabry-Perot cavities [9], those projections were predicated on much higher quality factor (Q ~10⁶). Such high Q-factor places restrictions on the emitter linewidth, and complicates the spectral alignment of the emitter to the cavity mode.

3.2. Dipole at the interface of the bottom mirror

While the value for C estimated in the previous section is compelling, it represents an ideal scenario in which a dipole emitter is trapped at the center of the air cavity. Techniques for isolating and trapping atoms have become quite sophisticated [31], but this is nevertheless a daunting challenge in practice. An alternative approach is to use a solid-state emitter embedded near the interface of one of the cavity mirrors [32,33]. Typically, the mirror in question is terminated by a low-index layer, so that the cavity field anti-node is aligned with the emitter position. Figure 4(a) shows a plot of the mode field intensity profile at resonance, for a planar cavity formed by combining a quarter-wave air layer and a quarter-wave ‘spacer’ layer (with index n_D = n_SiO2 for the particular example shown). Termination of the lower mirror by a low-index SiO₂ layer reduces its reflectance slightly, thereby reducing the cavity Q and finesse. To compensate this, such that Q_C is similar to the case above, we added an additional quarter-wave a-Si layer to the upper mirror (i.e. the a-Si capping layer was changed from a half-wave to a quarter-wave thickness, thus making it a 4.5-period mirror). The mirrors were otherwise assumed to have layer indices and thicknesses as described in Section 2.1. As expected, the field anti-node is aligned with the top surface of the spacer layer, so that a dipole placed there can couple efficiently to the cavity mode.

 

Fig. 4 (a) The fundamental mode intensity profile is plotted for a planar cavity comprising a quarter-wave air layer and a quarter-wave spacer layer with refractive index n_D. The multilayer structure, including 4.5- and 4-period upper and lower mirrors, respectively, is overlaid on the plot. A dipole emitter at the air-spacer interface is also depicted. (b) The predicted rate of emission into cladding modes is plotted versus spacer layer refractive index, for a horizontal monochromatic (λ = 1550 nm) dipole at the air-spacer interface, and with the spacer layer set to quarter-wavelength optical thickness in each case.

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Using the planar model described above, we estimated the emission rate into cladding modes as a function of the spacer refractive index, for a horizontal dipole located at the air-spacer interface. The results are shown in Fig. 3(b), and demonstrate that, for this more practical scenario, significant suppression of the background emission rate might still be possible. For example, simply using SiO₂ as the spacer layer still results in I ~2.7, corresponding to C ~160 for the cavity mode described above. Of course, this estimate again assumes a sufficiently narrow linewidth emitter, such that Q_EFF ~Q_C. A similar strategy for suppressing the emission into the cladding modes of an omnidirectional-clad hollow fiber was proposed by Bermel et al. [14].

4. Numerical modeling results

The quasi-analytical predictions from above clearly involve a few assumptions, particularly regarding the use of a planar model to approximate background emission for the real buckled dome cavities. To corroborate the results, a numerical study was conducted using a commercial FDTD software package (Lumerical). A 3-D structure was defined by using an experimental cross-sectional profile, fit to a Bessel function curve as predicted by elastic buckling theory [21], with peak buckle height 0.776 μm. Multilayer parameters, including thicknesses, indices, and loss were chosen as described in Section 2. A series of simulations were performed with increasing simulation times and decreasing mesh size, in order to confirm the convergence and accuracy of the results.

Figure 5(a) shows a typical fundamental mode-field profile extracted from the FDTD solution, which predicted V_M ~1.3λ³ and Q_C ~2000 at the fundamental mode resonant wavelength (λ ~1527.3 nm in this case). Figure 5(b) shows the relative emission rate predicted for an electric dipole oriented parallel to the plane of the substrate and located precisely at the center of the air cavity. Note that the software has a built-in capability to calculate dipole emission rates relative to the case of the dipole in vacuum. In very good agreement with the analytical approximations above, the predicted on-resonance emission rate is ~120 and the off-resonance emission rate saturates (for long wavelengths) to a value of ~0.04. As discussed in Sections 2 and 3, these correspond approximately to the Purcell factor (γ_C/γ₀ ~120) for the fundamental cavity mode and to the inhibition factor (I ~25) for background emission, thus confirming the estimate C ~1500 from Section 3.1.

 

Fig. 5 (a) Cross-sectional mode-field profile at the resonant wavelength λ ~1527.3 nm for a buckled cavity with ~λ/2 air-core thickness, as predicted by the FDTD-based solution. (b) Wavelength-dependent emission rate relative to the free-space emission rate, for a horizontally oriented dipole located in the middle of the air cavity. (c) Far-field projection of the power radiated from the top of the buckled cavity at the resonant wavelength. The plot is a 2-D representation (i.e. ‘overhead’ view, with the angle relative to normal indicated by the concentric circles in increments of 10 degrees) of the power distribution on a hemispherical surface of radius 1 m. (d) As in part (c), but for λ ~1531.1 nm.

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A rough estimate for the spontaneous emission coupling factor was made as follows. The angular distribution of power radiated through the upper buckled mirror of the cavity was solved using built-in ‘far-field projection’ capabilities of the software. As an example, the far-field intensity distribution at the fundamental resonance wavelength (~1527.3 nm) is plotted in Fig. 5(c). The plot is an ‘overhead’ view of the power distribution on a hemispherical surface of radius 1 m, with the concentric circles indicating the polar angle relative to the normal direction. Note that the fundamental Gaussian mode waist radius is w₀ ~2.47 μm for this cavity (from the FDTD results), corresponding to a predicted far-field divergence half-angle θ = λ/(πw₀) ~11.4 degrees. It is apparent from visual inspection of the plot that most of the radiated power lies within this range, and is thus coupled to the fundamental mode. By comparing the numerically integrated power to Gaussian-beam analytical predictions, we conservatively estimated that > 0.99 of the power emitted through the upper mirror is contained within the fundamental cavity mode. A similar analysis of the power radiated through the bottom mirror was in good agreement. This calculation neglects details such as the residual coupling to cladding modes (which is small, as discussed) and the angular dependence of the residual absorption by the mirror layers, which will tend to suppress far-field emission at high angles more than at normal incidence. Nevertheless, the result is consistent with the expectation of a near-unity value for β when mirror losses are neglected. In other words, assuming κ_out = κ and using C = 1500, Eq. (2) predicts β = 0.9997.

As a further illustration, Fig. 5(d) shows a similar far-field plot, but for a wavelength (λ ~1531 nm) slightly detuned from cavity resonance such that F_P ~1 (see Fig. 5(b)). Note that the emitted power is ~2 orders of magnitude less at this wavelength, as expected from the reduction in Purcell factor (and thus peak emission rate). In spite of this, even at this off-resonance wavelength, most of the out-coupled light is associated with the cavity mode. This is due to the strong suppression of radiation modes by the omnidirectional mirrors. From the discussion in Section 2.3 and neglecting mirror loss, we expect C = F_P·I/2 ~12.5 and β ~0.96 in this case. As above, we compared the numerically integrated power to the Gaussian beam prediction, and estimated that > 0.95 of the far-field radiated power is contained in the fundamental cavity mode at this wavelength. This provides further corroboration of the results in Section 3, and furthermore illustrates that, even without significant Purcell enhancement, a high spontaneous emission coupling factor can be attained when background emission is suppressed [16].

Finally, Fig. 6(a) shows the fundamental field profile predicted for a ‘spacer-embedded’ cavity, of the type discussed in Section 3.2. For this cavity, an additional SiO₂ (spacer) layer was added to the bottom mirror. Furthermore, the base diameter of the buckled mirror was set to 44 μm and the peak buckle height was set to 0.388 μm (i.e. a quarter-wave air layer at ~1550 nm wavelength). These values of base diameter and peak height were estimated based on the predictions of elastic buckling theory, as verified in our previous experimental work [20]. As discussed in Section 3.2, a 4.5-period upper mirror was used to compensate the reduced reflectance of the bottom mirror due to its low-index termination by the spacer layer. Also due to the addition of the spacer layer, the field penetration into the bottom mirror is somewhat increased (i.e. compare Figs. 5(a) and 6(a)). Thus, both Q_C and V_M are increased slightly compared to the non-spacer cavity above, but the ideal Purcell factor is similar in both cases. Figure 6(b) shows the predicted relative emission rate versus wavelength, for a horizontally aligned dipole located at the interface to the bottom mirror (i.e. at the top of the SiO₂ spacer layer, see Fig. 4(a)). Both the on-resonance ideal Purcell factor (F_P0 ~120) and the off-resonance inhibition factor (I ~1/0.3) are in good agreement with the analytical model from Section 3, confirming the potential for cooperativity C > 150. As above, these estimates assume a well-aligned emitter of sufficiently narrow linewidth to ensure Q_EFF ~Q_C.

 

Fig. 6 (a) Cross-sectional mode-field profile at the resonant wavelength λ ~1537.2 nm for a buckled cavity with ~λ/4 air-core thickness and ~λ/4 SiO₂ spacer layer, as predicted by the FDTD-based solution. (b) Wavelength-dependent emission rate relative to the free-space emission rate, for a horizontally oriented dipole located at the top of the spacer layer.

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5. Discussion and conclusions

We have demonstrated that buckled-dome microcavities clad by omnidirectional reflectors might enable very high cooperativities, even for moderate Q. Assuming parameters extracted from previously fabricated cavities with Q_C ~10³, we predict values as high as C ~1500 for an optimally located dipole emitter, due to significant inhibition of background emission by the omnidirectional claddings. Furthermore, assuming mirror losses can be reduced, near-unity spontaneous emission coupling factors might also be possible, even for relatively broadband emitters. It should be emphasized that precise positioning of an atomic emitter inside a buckled dome cavity remains a significant challenge. However, the monolithic nature of the fabrication process might lend itself to incorporation of nano-scale emitters, for example by patterned assembly [34] prior to the deposition and buckling of the upper mirror. Furthermore, in our previous work we have explored the fabrication of both nano-scale holes [19] and intersecting channels [21] as a means to provide open access for liquid- or gas-phase emitters. We hope to investigate these options in future work.

The benefits described, including the suppression of background emission over a wide spectral range and the possibility for mode volume approaching λ³, can be attributed (at least in part) to the high refractive index contrast provided by the a-Si/SiO₂-based mirrors. Of note, there is currently a significant worldwide effort aimed at the development of hydrogenated a-Si (a-Si:H) films for microphotonics. Optimized a-Si:H films can provide at least 3-4 orders of magnitude reduction in absorption loss at 1550 nm wavelength [35], compared to the typical value for the sputtered a-Si films described above. This could in turn dramatically improve both the attainable finesse and the out-coupling efficiency of the mirrors. Moreover, there is currently a strong motivation in the cavity QED literature to develop emitters and cavity devices for the so-called ‘telecom’ ranges, especially near the 1550 nm wavelength region best-suited to long-haul fiber transmission. Significant efforts towards the integration of 1550 nm range emitters (e.g. InAs/InP quantum dots [36,37], PbS quantum dots [38], and erbium [39]) with silicon-photonics devices are ongoing, and the present study is of direct relevance.

Nevertheless, cavity QED work has traditionally been conducted in the wavelength range below 1 μm, motivated by the wide variety of suitable emitters operating in this range (e.g. rubidium atoms and defect centers in diamond) and also by their compatibility with silicon-based single-photon detectors. TiO₂/SiO₂-based mirrors can be used in this range, and have sufficient index contrast to produce bands of omnidirectional reflection for incidence from air [11]. On the other hand, many studies have employed Ta₂O₅/SiO₂-based mirrors [32], which cannot provide omnidirectional reflection. Interestingly, for onion cavities Liang et al. [17] showed that even non-omnidirectional (SiN/SiO₂-based) mirrors could provide significant suppression of background emission under certain conditions. The degree of spontaneous emission modification possible in buckled cavities with lower-contrast mirrors is thus an interesting topic for future study.

Finally, it is worth mentioning that there is significant scope for optimization of the buckled-dome cavities. The present study was focused on a particular cavity geometry previously shown [21] to provide operation in a fundamental spatial mode regime, with correspondingly low mode volume. As discussed above, the moderate cavity Q is determined primarily by the residual absorption losses of the mirrors, and might be significantly increased through the use of a-Si:H. Furthermore, larger cavities can easily be fabricated [18], and could enable greater suppression of background emission for emitters located in the air core, due to the possibility for greater separation between the emitter and the mirror interfaces. The choice of a larger cavity would also imply higher V_M and Q_C (at relatively fixed F_P0 [9]), which would be desirable for certain emitters and applications.