Hybrid dielectric slot-plasmonic ring resonator for Purcell enhancement

We present a computational study of Purcell factor enhancement for a novel hybrid-plasmonic ring resonator using a novel implementation of the body-of-revolution (BOR) finite-difference time-domain (FDTD) method. In this hybrid structure, a dielectric slot ring is surrounded by a metallic ring such that a hybrid plasmonic mode is generated within two thin low-index gaps. The surrounding metallic ring decreases the binding loss for small ring radii, leading to high-quality factors and mode-field confinement. The hybrid resonator shows high quality-factor values above 103 and small mode volumes down to 10−3λn3 simultaneously, thus providing large Purcell factors (F p > 104). The distributed strong confinement within two gaps renders the proposed resonator useful for multi-emitter applications.


Introduction
Enhancing the spontaneous emission (SE) rate of optical emitters is essential for many applications.For example, realizing high brightness single photon sources for quantum applications [1] requires emitters with high SE rates.The SE rate is proportional to the local density of optical states (LDOS).The LDOS n(E, x) of a system is the number of electromagnetic states per unit frequency that photons of energy E can occupy, weighted by the spatial density of each available photon state at position x.Optical cavities can enhance LDOS by increasing temporal and spatial confinement of light; hence, they can also modify the SE rate of an emitter placed inside the cavity.The maximum enhancement of the LDOS at the resonance peak of a cavity mode is known as the Purcell effect and is given by the Purcell factor [2], where l n m is the wavelength of the mth resonance mode, n is the refractive index of the surrounding medium, Q is the quality factor (Q-factor) which quantifies the photons lifetime inside the cavity, and the effective mode volume V eff is the corresponding spatial confinement.Equation (1) assumes perfect spatial and polarization alignment of the emitter, which is typically achieved at a mode field antinode.
According to equation (1), the Purcell factor increases with high Q-factors and small mode volumes.Dielectric cavities have been especially useful in achieving high Purcell factors [3][4][5] as they can confine light in volumes approaching a single wavelength with very little loss.Plasmonic resonators are utilized in enhancing the emission of quantum emitters [6][7][8][9][10], as they can achieve much smaller mode volumes over broad bandwidths, though at the expense of low Q-factors.
The effective mode volume of dielectric resonators is commonly restricted by the diffraction limit, which places an upper bound on the associated Purcell factors.Among dielectric cavities, the dielectric slot geometries can confine light beyond the diffraction limit [11,12].Moreover, because of their low absorption loss, the spectrum linewidth of dielectric cavities is narrow in comparison to the broad spectrum of most optical emitters.Consequently, the coupling between optical emitters and dielectric cavities can be weak.Apart from dielectric resonators, one may utilize plasmonic resonators, whose high absorption loss prevents the achievement of high Q-factors and strong radiative emissions.
In most previous work on hybrid configurations of dielectric and plasmonic cavitites [13][14][15][16][17][18][19], a plasmonic nanoantenna is placed near a photonic crystal or disk resonator, resulting in a local intensity enhancement in the gap region between these two coupled systems.In such configurations, the Purcell factor enhancement is observed in a finite region between dielectric and plasmonic cavities, which leads to several drawbacks.These drawbacks include difficulty in tuning the cavities, challenges fabricating thin gaps and controlling the optical emitters within these gaps, and incompatibility with multi-emitter applications.
In this work, we present for the first time two types of hybrid photonic-plasmonic cavities that overcome these drawbacks.In addition, we use a novel numerical technique based on BOR-FDTD method to demonstrate the enhanced light-matter interaction.These new structures hybridize dielectric slot and plasmonic modes in a novel geometry: Instead of surrounding a localized nano-antenna, a metallic silver ring surrounds a dielectric ring resonator.The close proximity of the metallic to the dielectric ring would typically induce excessive loss and limit the quality factor of the device.However, since the hybrid plasmonic mode is strongly confined inside the gaps, the plasmon polaritons have only a small overlap with the silver ring.Moreover, the silver ring surrounding the ring resonator decreases the binding loss for small ring radii.As a result, the hybrid modes exhibit high Q-factors and small mode volumes.In addition, by adjusting the stripe width and gaps, the proposed device allows for a trade-off between mode volume and quality factor.This adjustment also allows for better control of the optical emitters as well as enhancement of the Purcell factor as compared to other proposed hybrid systems.Additionally, the high Purcell factors are achieved over a wide frequency range-from telecommunication to visible wavelengths.Owing to the distributed coupling of the dielectric and metallic rings, the proposed resonator can be utilized for multi-emitter applications.Previous researchers have also reported on Purcell enhancement in hybridized dielectric and plasmonic ring resonators [20].Compared to the structure presented in that work, the geometry of our proposed resonator is much simpler to fabricate and has a figure of merit (defined below) approximately 20 times larger.
Figures 1(a) and (b) show the top and cross-sectional views of the proposed resonators, respectively.The structures are designed based on silicon-on-insulator (SOI) technology.These structures consist of a high-index dielectric ring pair composed of a silicon core of dimension W c × H and a thin silicon stripe of dimension W str × H.The high-index pair is surrounded by a silver ring above a silica substrate.The ring radius R in is equal to the inner radius of the core, and g 1 and g 2 denote the first and second gaps, respectively.
Different types of hybrid plasmonic modes can be excited depending on the stripe and gap widths.We numerically examine the maximum Q-factor Q and minimum mode confinement V that can be achieved in each structure using a new BOR-FDTD method.In previous work, we generalized this method for the modal analysis of hybrid plasmonic annular resonators [21] and successfully used BOR-FDTD for the numerical analysis of corrugated waveguides [22].In this work, we use similar methods to show that the proposed hybrid structures have a higher Purcell enhancement than non-hybridized plasmonic and dielectric cavities, enabling a tunable trade-off between the Q-factor and the mode volume.

Methods
Quasi normal modes (QNM) are the resonance modes of a dispersive cavity with complex eigenfrequencies.The QNMs are the solution to the non-Hermitian eigenvalue problem of source-free Maxwellʼs equations.These eigenvalues are complex; therefore, the imaginary portion of the eigenvalues is related to losses.Calculation of QNMs in plasmonic cavities is challenging, specifically because of the fine spatial and temporal resolution required to resolve subwavelength plasmonic modes.Exploiting cylindrical symmetry for analyzing circular geometries can efficiently reduce the complexities of analysis and computational costs.
Here, we use a BOR-FDTD for QNMs calculations and dynamic analysis of the proposed resonator.In rotationally symmetric geometries, the electromagnetic field components Ψ can be expressed as follows: where m determines the azimuthal mode number, A m is the complex amplitude of this mode, and the function F m (f) is a periodic function with a period of 2π/m.In BOR-FDTD, the azimuthal direction is treated analytically, wherein the spatial derivative f ¶ ¶ is replaced by ±m in Maxwellʼs equations.Discretization and analysis are then reduced to an arbitrary cross-section of the cavity on the f − z plane.Eigenvalues are thereafter calculated from transverse resonances.
To describe the dielectric constant of silver, we use a Drude model ˜( ) in which one may assume ( ) [23].To implement this dispersion model, we first introduce the auxiliary field for the Drude pole in the frequency domain according to Then, we perform the inverse Fourier transform of this auxiliary field in which the time derivative ¶ ¶t can be replaced by jω, i.e.
Uniform and extremely fine meshes with an increment of Δ = 1 nm are chosen to take the impact of plasmon polaritons into account.The resonator is driven by a z-directed short-duration modulated Gaussian source: where, (ρ s , z s ) is the position of the source, f c is the central frequency in the desired frequency range ( ) f f , min max which is chosen to be 300 THz, and the pulse width and delay are chosen to be t For a given m, we record the time evolution of the electric field components in the high-intensity region.Then using filter-diagonalization method (FDM), we perform a spectral analysis of this time-domain signal [24].Note that FDM extracts the eigenvalues by fitting the signal to a sum of exponentially damped sinusoids, i.e.
f k and  f k are the real and imaginary part of eigen-frequencies, Δt is the time step, K is the number of eigenvalues, and N t is the number of time samples.
The Q-factor measures the number of cycles that photons remain stored inside the cavity and is represented by To calculate the effective mode volume, we first construct the 3D field profile from the 2D field profile.This yields m jm We then calculate the effective mode volume from the volume integral of the magnitude of the electric field as follows: In the calculations above, we have normalized the effective mode volume to the cubic of resonance wavelength ( ) l n , i.e.

Numerical analysis
We now calculate the resonance, the Q-factor, the effective mode volume, and the Purcell factor of the resonator for different structural parameters.We place the source at the center of the core.The fundamental TE-like mode is excited inside the core, and the surface plasmon mode (SPP) is excited at the air-metal interface by the evanescent tail of the dielectric mode.Data is recorded at sampling points inside the cavity.To observe a hybrid plasmonic mode, we record the time-signal data at gap regions where the field confinement of the hybrid mode is high.Figure 2 We then extract the QNMs from the time-domain signal using the FDM process as mentioned in the previous section.After the excitation source is deactivated and upon arriving a steady-state solution, the timedomain signal is processed using FDM.The results of FDM calculations are given in table 1.
To determine the hybrid-plasmonic mode among the eigen-frequencies reported in table 1, we have repeated the simulation with long-duration Gaussian sources whose central frequency are ¢ f k .This is used to obtain the electric field profile at the corresponding resonance frequency.The hybrid plasmonic mode is recognizable by strong confinement inside the gap regions.After investigation of various field profiles, the eigenfrequency ˜= ´-f j H z 2.7596 10 1.501 10 has the character of a hybrid-plasmonic mode.The Q-factor and mode volume of this mode amount to Q = 919, and . The in-phase coupling of the dielectric and SPP modes creates a hybrid-plasmonic mode that propagates between core and metal.Figure 2(b) shows the 2D electric field profile of the hybrid mode in the ring crosssection.Moreover, figure 2(c) shows the 1D field profile along the ρ-axis, for a constant z.For this hybrid mode, the enhanced ρ-polarized electric field due to its discontinuity at the interface between high-contrast-index dielectrics leads to nanoscale mode-field confinement inside gaps.
Note that in comparison to previously reported hybrid systems [13][14][15][16][17][18][19], we have incorporated an extra stripe ring between the core and the metallic ring.This stripe confines the hybrid mode into two gap regions.As depicted in figures 2(b) and (c), the field intensity is mainly localized inside the first gap.As a result, the concentration of the mode-field decreases near the metallic ring, and the Ohmic loss can be significantly  decreased compared to previous hybrid plasmonic configurations.As shown in subsequent sections, this new configuration shows high Q-factors while maintaining strong spatial confinement.The width of the stripe and second gap play an important role in the hybridization between dielectric and SPP modes.If the stripe width is large, a dielectric mode can be excited within the stripe.In this case, by coupling of dielectric modes inside the core and stripe, a gap mode is created inside the first gap.On the other hand, by overlapping the stripe and SPP modes, a hybrid-plasmonic mode is created inside the second gap.By in-phase coupling of first and second gap modes a new hybrid-plasmonic mode is excited which exists inside the gaps and the stripe regions.If the stripe width is chosen to be small enough, no dielectric mode can be supported inside the stripe, and the hybrid-plasmonic mode will be excited by in-phase coupling of the core and SPP modes.
Supposing the width of the second gap to be the structure consists of a pair of dielectric rings surrounded by a metallic ring.In this case, a dielectric slot mode will be generated inside the first gap so that no SPP mode can be excited at the stripe-metal interface.We note that the role of the metallic ring surrounding the dielectric resonator is to prevent the mode-field energy from radiation, particularly for small ring radii.Hence, it is required to keep the ring resonator small.
As discussed above, by tuning the dimensions of the stripe and the gaps, hybridization between dielectric and plasmonic modes can be controlled.To demonstrate this, we present a parameter analysis.Figure 3  over a wide frequency range of 127 THz.For g = 2 nm and m = 28, the The metallic ring surrounding the dielectric rings confines mode-field energy at small ring radii and consequently decreases the binding loss.Conversely, owing to the stripe layer between the core and metal, the overlap of the hybrid mode is decreased by the metal, leading to decreased Ohmic loss.As a result, the hybrid resonator shows high Q-factors at ring radii as small as 1 μm.However, the mode-field is strongly localized inside two thin gaps which leads to small mode volumes.As shown in figures 3(b) and (d), both the Q-factor and mode volume monotonically increase with increasing gap width.One reason for this behavior is the varying electric field distribution inside the resonator.With increasing the gap widths, the confinement decreases inside the gaps, while the field distribution simultaneously increases inside core and stripe.Because the confinement of energy decreases inside the second gap, the metallic ring decreases the overlap of the hybrid mode.Therefore, the Q-factor increases at the expense of reduced spatial confinement.Figures 4(a) and (b) show a quarter of the mode-field profile for g = 5 nm and g = 20 nm.As shown, the mode-field has been enhanced inside the core when the gap widths vary from 2 nm to 20 nm.
The width of the stripe is another critical parameter in tuning Q-factor and effective mode volume.Figure 5 depicts the modal properties as a function of the stripe width.Here, the gap widths are fixed at g 1 = g 2 = 5 nm.
The Q-factor decreases monotonically as the stripe width changes from 10 nm to 80 nm.On the other hand, a non-monotonic trend is observed for the effective mode volume with the changing stripe width.This behavior can be attributed to the varying coupling between dielectric and SPP modes.The Q-factors higher than 400 and normalized mode volumes in the range of The proposed hybrid plasmonic resonator can also be used for multi-emitter applications [25].Many localized intensity spots are created inside two gaps.For example, for m th resonance mode, the number of 2m optical emitter can be distributed inside the resonator.The in-phase emission of these multiple emitters can improve the efficiency of single photon sources. .By hybridization between the dielectric slot and plasmonic mode, strong mode-field confinements are achieved with two thin gaps.
In subsequent sections, a second design is considered in which the width of the second gap is chosen to be zero, i.e. g 2 = 0. Owing to the discontinuity of the TE-polarized electric field component at the silicon-air interfaces, a gap mode is created within the first gap.Here, no SPP mode can be supported at the stripe-metal    .By hybridization between the dielectric slot and plasmonic mode, strong mode-field confinements are achieved with the first gap.

Conclusion
In conclusion, we have presented two types of hybrid photonic-plasmonic ring resonators which are capable of achieving high Purcell factors.In the proposed hybrid resonators, an extra stripe is incorporated between dielectric core and metallic rings.The stripe ring decreases the overlap of the hybrid-plasmonic mode with the metal.Moreover, the surrounding metallic ring decreases the binding loss at small ring radii.This provides a trade-off between temporal and spatial confinement.Based on new BOR-FDTD calculations, each one of the introduced resonators shows Q-factors in the order of (10 3 − 10 4 ) and normalized mode volumes in the order of ( ) which lead to high Purcell factors in the range of (10 3 − 10 4 ).These characteristics may be useful for applications requiring strong light-matter interactions, such as single photon sources, optical forces in optical tweezers, light-matter entanglement, and bright multi-emitter sources.are achieved.

Figure 1 .
Figure 1.(a) Top view and (b) the cross-sectional view of the proposed photonic-plasmonic ring resonator.In this structure, a dielectric slot ring resonator is surrounded by a metallic ring.By coupling the dielectric slot and plasmonic modes, a hybrid mode is created within two gaps.
(a) shows a typical time evolution of E z (t) for mode number m = 32, where the dimensions of the structure are chosen as follows: W C = 200 nm, = W 40 nm str , H = 200 nm, g 1 = g 2 = 10 nm, and R in = 1 μ m.

Figure 2 .
Figure 2. (a) Typical time evolution of E z (t) at the center of the first gap g 1 .(b) The corresponding normalized mode field from a crosssectional view, and (c) 1D field profile along ρ-axis, in a constant z.The simulation parameters are chosen as follows: m = 32, W C = 200 nm, = W 40 nm str

3 are
shows the resonance frequency, the Q-factor, the effective mode volume, and the Purcell factor as a function of the gap widths.The results are shown for different azimuthal mode numbers.The first and the second gaps are assumed to be equal, i.e. g 1 = g 2 = g.Here, the other parameters are: = W 10 nm str and R in = 1 μm.As depicted in figure 3, the Q-factors larger than 1000 and normalized mode volumes in the range of achieved which leads to Purcell factors in the range of

Figure 3 . 3 are
Figure 3. (a) Resonance frequency (b) Q-factor (c) normalized mode volume and (d) Purcell factor as a function of the gap widths.The results are shown for different azimuthal mode numbers.The first and the second gaps are assumed to be equal, i.e. g 1 = g 2 = g, and the other parameters are: = W 10 nm str and R in = 1 μm.By increasing the gaps, the Q-factor and mode volume are monotonically increased, and the Purcell factor is monotonically decreased.

3 are
different azimuthal numbers.Moreover, the Purcell factors higher than F p 10 4 are obtained in a frequency range of 96 THz.As shown in figures 5(b) and (c), a desirable trade-off is achieved between Q-factor and mode volume.For = achieved at 226 THz, which leads to the Purcell factor of F p = 2.3 × 10 4 .Moreover, for = W 10 nm str , the Q-factor of 956 and normalized mode volume of 0.0062 are achieved at 310 THz, which leads to a Purcell factor of F p = 2.3 × 10 4 .Figure 4(c) shows a quarter of the mode-field in the top view for g = 5 nm, = W 40 nm str , and m = 40.To better understand the capability of the proposed resonator in enhancing light-matter interactions, we calculate a figure of merit (FOM), Q/V for different cases.Figures 6(a) and (b) show the phase diagram of Q and V as a function of gap and stripe widths, respectively.In both cases, FOMs greater than ( ) different azimuthal mode numbers.Moreover, an increasing monotonic behavior is achieved in the FOM of different gaps.Conversely, FOM for different stripe widths has a non-monotonic behavior.

Figure 5 .
Figure 5. (a) Resonance frequency (b) Q-factor (c) effective mode volume and (d) Purcell factor as a function of stripe width.Here, the gap widths are fixed at g 1 = g 2 = 5 nm.By increasing the stripe width, the Q-factor is monotonically decreased, and the mode volume and Purcell factor show non-monotonic behavior.

Figure 6 .
Figure 6.Phase diagram of Q and V (a) for different gap widths, g when the stripe width is fixed at = W 10 nm str and (b) for different stripe widths, W str , when the gap widths are fixed at g = 5 nm.In both cases, the FOMs greater than ( ) l -10 n 5 3are achieved.

Figure 8 .
Figure 8.(a) Resonance frequency (b) Q-factor (c) effective mode volume and (d) Purcell factor as a function of stripe width.The first gap width is chosen to be g 1 = 5 nm.By increasing the stripe width, the Q-factor is monotonically decreased, the mode volume is monotonically increased, and Purcell the factor is monotonically decreased.

Figure 10 .
Figure 10.Phase diagram of Q and V (a) for different gap widths, g and (b) for different stripe widths, W str .In both cases, the FOMs greater than ( ) l -10 n 4 3