1. Introduction

Whispering-gallery mode (WGM) resonators promise to be used in a large variety of applications such as ultra-low threshold semiconductor micro-cavity-lasers or intra-cavity enhancement of nonlinear interaction. They have therefore continued to be in the centre of interest in research on optical micro-cavities for quite some time [1]. Indeed, disc shaped micro-cavities have been demonstrated to be a feasible system to achieve lasing in a wavelength sized dielectric cavity [2, 3]. WGM resonators are also very good candidates for realising strong coupling with applications in quantum networking, as they allow to simultaneously achieve small mode volumes and high quality factors (Q) [4, 5]. Most recently more unconventional applications of active or nonlinear WGM have been proposed such as an exploitation of the particular properties of cylindrical WGM modes as a new way to accelerate charged particles [6].

In a WGM laser (as in every other laser), the active medium nonlinearly interacts with the intra-cavity light fields. As these features directly depend on both, the dynamics of the active medium and the WGM dynamics it thus seems of increasing importance to study precisely this interplay, as well as the novel phenomena that we might expect as a result.

In this paper we present a new comprehensive theoretical approach and modelling framework for the study of the ultrafast nonlinear dynamics of 3D WGM lasers. Building on the principles discussed in [7] our computational method simulates the classical, three-dimensional (3D) evolution of the electromagnetic field which is coupled via a dipole Hamiltonian to the quantum mechanical evolution of a two level system. Thereby we take into account the circular polarised light transitions and dipole dynamics. This allows us to analyse the response of the pumped material on arbitrarily polarised light and avoid the unphysical breaking of the cavity mode circular symmetry. As we will show in the following, our approach opens the opportunity of exploring the ultrafast, sub-femtosecond nonlinear optical field dynamics of WGM lasers. As such time-scales are currently still beyond direct observation by experiment such information on the ultrafast dynamics of the optical field in WGM lasers will be vital e.g. for the assessment of the formation and the maintaining of coherence in and the use of WGM lasers in quantum networks.

2. Time-domain full vector Maxwell-Bloch modelling framework

The simulation framework is based on a semi-classical approach, where the microscopic dipole polarisation governed by quantum mechanics is coupled to the classical electromagnetic field (see Ref. [7]). The evolution of the electromagnetic field without free charges is described by the Maxwell curl equations,

These are then coupled with the microscopic polarisation P by the material relation

which takes into account the dipole moment d and the dipole density n _a. The dipole moment was set to a reasonable value which should model a quantum dot like material [8].

The dipole transition in these simulations is a σ type transition with a complex dipole moment d ∈ ℂ³. In order not to disregard this vital polarisation information and to investigate the associated dynamics we therefore generally have to assume complex-valued electromagnetic fields E,D,H ∈ ℂ³. The microscopic polarisation P and the population difference N on the other hand are real values (P,N ∈ ℝ). We derive the equations of motion of P and N on the basis of a dipole Hamiltonian of a quantum dot like two-level system with an energy separation of ΔE = ħω ₀ and the occupation probability of the upper level (ρ_bb , with N = 1-2ρ_bb ; for details, see Ref. [7]). In addition, we include a phenomenological polarisation damping term (decay constant γ _p) to model a Lorentzian line shape resonance [9]. It is one of the parameters determining the coupling of the cavity resonance to the gain medium. The frequency Ω² =${\omega }_0^2$ - ${\gamma }_{\mathrm{p}}^2$ is the resonance frequency of the Lorentzian. The resulting equations of motion of the polarisation P and population difference N then read:

Above, phenomenological terms model carrier injection in terms of a non-radiative pump rate Λ, and the non-radiative decay towards an equilibrium state N ₀ with a summarising decay constant γ _nr. Because of the interest in the ultrashort, transitional character of the system dynamics and our focus on a quantum dot like active material, carrier diffusion can safely be disregarded [10]. The simulations show that the non-radiative decay time has no significant influence on the particular femtosecond dynamics. Here it is included to model the basic threshold features of a laser, where the threshold inversion should lie above the transparency inversion (ρ_{bb, trans} = 0.5).

3. Cold-cavity modes and quality factors

To elucidate the influence of the active spatio-temporal light field dynamics we contrast and supplement simulations based on the Maxwell-Bloch approach to cold-cavity calculations. The cold-cavity simulations do not include the equations that define the microscopic polarisation (Eq. 3, 4), but were otherwise (eg with respect to boundary conditions) identical and used to estimate the cold cavity resonances of the dielectric structure and their quality factors. The cold-cavity simulation was performed by employing a standard Finite-Difference Time-Domain (FDTD) algorithm on a regular 3 dimensional Cartesian grid with uniaxial perfectly matching layers (UPML) absorbing boundary condition [11]. The simulation grid, including the PMLs, was 200×200×100 cells with the dielectric cavity centred in the middle. The size of the grid cells was (15nm)³. The time step was chosen according to the Courant criterion. As initial condition, each computational run is seeded with a divergence-free electromagnetic pulse of small amplitude modelling a spontaneously emitted contribution of polarisation.

The simulations show that the cold cavity modes of the whispering gallery type in such cavities are almost independent of the pedestals, remains of the etching process which connect the discs to the substrates. They furthermore show the resonance frequencies and quality factors of a compatible and an incompatible mode in the microgear (see Fig. 3). The electric field anti nodes of the higher frequency, incompatible mode (shorter wavelength) are in the lower refractive index material (as modes in the air-bands of photonic crystals, see also [12]).

For direct comparison with experiment we chose an actual disc shaped micro-cavity that is very similar to the one investigated in Ref. [2]. It is made of dielectric material with refractive index of 3.48. The disc radius is 1.6μm and its thickness is 180nm. Embedded in the middle of the disc is a thin layer (60nm thick) of gain material. The Lorentzian resonance is tuned to match the resonance frequency of a cavity eigen-mode (in this microdisc ≈ 1.550nm).

 

Fig. 1. Discrete, spatially resolved Fourier transforms of the Cartesian electric and magnetic field components of the cold-cavity HEM₅₁₀ WGM. The contour lines show the logarithmic decay of the field.

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In the microdisc, the cold cavity mode with this resonance frequency belongs to a TE like (the electric field vector is predominantly in the disc plane) degenerate WGM with five fold azimuthal symmetry. The degeneracy is due to the azimuthal symmetry of the disc. The defining axial magnetic field component has one anti-node along both, the axial and radial directions (see Fig. 1). The quality factor Q was estimated (from the FDTD simulation) to be ≈ 800. The neighbouring high Q resonances are at 1.4 and 1.7nm. With the introduction of the gear-like corrugation around the circumference of the disc (see Fig. 2), this resonance is split up into two, as the continuous azimuthal degeneracy of the mode is removed (see also Ref. [3]). An FDTD simulation of a microgear with similar radius and thickness as the microdisc described above, but with a corrugation depth of 160nm revealed two resonances with diametrical quality factors as can be seen in Fig. 3. The higher Q resonance (the compatible mode) has a Q ≈ 770, almost the same as the degenerate mode in the disc without corrugation. The lower Q resonance features a much lower Q ≈ 130 and therefore called the incompatible mode. The higher Q mode (the compatible HEM510) was chosen to coincide with the resonance frequency of the Lorentzian transition (see Fig. 3) of the active medium. Table 1 summarises the material parameters which were used in the microdisc and microgear laser simulations, now solving the full, nonlinear set of Maxwell-Bloch equations including 3 and 4.

 

Fig. 2. Geometry of the cylindric resonator cavities used in the simulations. The 3D view on the whole dielectric cavity structure, including pedestal and the active region (yellow), is shown on the left. The right shows the corrugation parameters in the gear-like cavity.

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Fig. 3. The logarithmic resonance intensity (I) spectrum of the incompatible and compatible (insets from left to right) HEM₅₁₀ modes in the microgear. The Lorentzian line shape of the dipole resonance is included (dashed curve).

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4. Spatio-temporal dynamics of whispering-gallery mode lasers

In the following we will present some of the novel ultrafast optical field dynamics of WGM lasers. We will explicitly contrast the dynamics of a cylindrical WGM laser with the dynamics of an otherwise identical microgear WGM laser. The close similarities between the cold-cavity characteristics of microdiscs and microgears is in strong contrast to the dynamic characteristics revealed by the full laser simulations.

Table 1. The table summarises the two-level material parameters of the laser simulations; density (n _a); dipole moment (d); transition frequency (ω ₀); polarisation damping constant (γ _p); non-radiative decay constant (γ _nr); thermal equilibrium (ρ _bb,0) and initial occupation probability of the upper level (ρ _bb(t = 0)); pump strength (Λ).

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With the particular chosen pumping strength the overall averaged dynamics of the microdisc as well as the microgear lasers are characterised by an over-damped relaxation oscillation. However, as the graph in Fig. 4 shows, the nonlinear dynamics of the spatially averaged inversion inside the microdisc laser is characterised by an over-damped relaxation oscillation that is superimposed with a higher-frequency oscillation. This is in contrast to the relaxation oscillation dynamics of the simulated microgear which is shown in the graph in Fig. 5. The difference can be explained by the complex interplay of spatio-temporal light-field dynamics and the inversion which is effected by spatial hole burning. We note that Fig. 4 and Fig. 5 imply a lasing threshold being very close to transparency. This is a direct manifestation of the high Q-factor of the lasing mode in combination with a small volume leading to nearly threshold less lasing in microcavities [13, 14, 15].

 

Fig. 4. Snapshot of the spatial variation in the inversion probability of the dipoles in the microdisc cavity and the evolution of the averaged inversion. The end of the red line marks the time of the snapshot. Animations can be found at [16].

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Figures 4 and 5 also contain representative snapshots of the spatially resolved inversion. The remarkable difference is the clear visibility of the ten-fold symmetry of the in-resonance cold cavity WGM imprinted in the inversion profile of the microgear which is barely visible in the microdisc. Other features such as the inner ring only appear in the plain cylindrical cavity. The more pronounced ten-fold symmetrical pattern of maxima and minima in the case of the microgear laser (Fig. 5) is a first indication for a much more stable and localised HEM₅₁₀ WG like nonlinear material cavity “mode”, that allows such regular accumulation and removal of inversion.

 

Fig. 5. Snapshot of the spatial variation in the inversion probability of the dipoles in the microgear cavity and the evolution of the averaged inversion. The end of the red line marks the time of the snapshot. Animations can be found at [16].

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A closer inspection of the spatial dynamics during the first relaxation oscillation peak shows that the HEM₅₁₀ mode is excited in both cavities. It initially displays straight radially extending node lines with the mode itself oscillating in place. Soon after the relaxation oscillation swing, however, the ultrafast dynamics of the electric field in the microdisc laser shows a rotating HEM₅₁₀ like optical field pattern that resembles the characteristics of a vortex structure. The direction of the rotation depends on the initial conditions of the simulation. Figure 6(a) shows a sequence of six snap-shots that reveal the fan-like rotation of the optical field with the symmetry properties of the cold cavity WGM with the corresponding eigen-frequency. We link this rotation to spatial hole burning effects in the inversion density and an interplay of the different continuously degenerate HEM510 modes. These cause the electric field anti-nodes to start moving to regions with higher inversion that formerly had been occupied by the nodes of the mode. This ultra-fast rotation of the electric field is possible in a cylindrical cavity since all cold cavity modes have an azimuthal degeneracy by an arbitrary angle. It is this rotating electric field which leads to the smeared-out inversion profile as seen in Fig. 4.

In the microgear laser, the spatial dynamics of the compatible, lasing HEM₅₁₀ mode is significantly more well-behaved. It is the higher loss of the incompatible cold-cavity mode in the gear geometry that prevents the nodes of the electric field to shift to areas of higher inversion (i.e. higher gain). Most notably, no vortex-like electric field structures such as in the microdisc laser were observed. Indeed, the time series of the axial electric field component in Fig. 6(b) shows the arrested HEM₅₁₀ mode fixed in place. In the case of the microdisc, the rotating field pattern smears out the inversion profile near the circumference of the disc. Due to the nonlin-earity, the mode profile in transitional phases does not necessarily represent an imprinted image in the inversion. In the microgear laser, the introduction of a corrugation around the discs circumference removes the azimuthal degeneracy and arrests the mode in place. Here, the mode profile in the transitional phase matches the spatial hole burning pattern in the inversion. This shows that the WGM is truly arrested in place.

Next to this rotation of the field pattern, the spatial dynamics of the inversion profile in the microdisc laser shows that it supports more than one mode (especially visible in the centre of the disc). These modes have nodes closer to the axis of the disc. Within the corresponding time frame the microgear however, supports only one lasing mode. Reasons for this different behaviour can be found in the spatial structure of the WGMs and their quality factors. A rule of thumb is that the quality factor is decreasing as the first radial antinode of a WGM is closing in on the axis of the cylindrical structure. Therefore the in-resonance WGM, due to a higher Q factor, will always start lasing sooner than other modes that have significant contributions closer to the central region of the cylindrical cavity. The in-resonance WGM will then consume from the energy along the circumference of the cavity, while the pump is building up excitonic energy in the middle of the disc.

 

Fig. 6. Ultrafast dynamic behaviour of the HEM₅₁₀ lasing mode in a disc (a) and gear (b). Shown is the azimuthal part of the electric field in the disc plane (x-y). The contour lines are even spaced on a logarithmic scale to enhance the visibility of the symmetric properties of the field pattern. The corresponding animations can be found at [16].

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In contrast to the microdisc, the in-resonance WGM in the gear is expanding towards the axis due to the fixed interaction of the dipoles with the WGM, thereby additionally consuming energy from the pumped dipoles nearer to the axis. This deformation from the cold cavity mode structure does not seem to occur in the microdisc laser where the optical field pattern is turning along the azimuthal direction, so that inversion is building up in a larger part of the central region than it happens in the gear.

At some later point in time, the inversion peak in the centre of the microdisc was large enough to also trigger the appearance of lasing modes which have their radial nodes closer to the axis of the disc than the in-resonance cold cavity WGM. This spatial deformation of the mode in the microgear and the lack of mode competition also results in a lower threshold inversion. Indeed, as table 1 indicates, in order to achieve continuous lasing operation the pump rate had to be chosen at a higher value in the microdisc than in the microgear laser.

5. Conclusion

We have presented the first full vectorial 3D simulations of a lasing process in microdisc and microgear cavities. As examples, cylindrical as well as gear-shaped microdiscs where analysed. Disc shaped dielectric cavities have low loss WGMs which are azimuthally degenerate. The shape, resonance frequency and quality factor of the WGMs is nearly independent of the pedestal or substrate arrangement. Cold cavity FDTD calculations show the occurrence of compatible and incompatible modes with diametrical quality factors in gear shaped cavities. They also revealed their eigen-frequencies and quality factors.

However, the full vectorial 3D Maxwell-Bloch simulations reveal intriguing dynamical features, in particular, during the transitional relaxation oscillation phase. Microdisc lasers with cylindrical resonator geometry display ultrafast rotating, vortex like optical field pattern which appear shortly after the first relaxation oscillation. The rotating field patterns are arrested by the introduction of a gear like corrugation thus removing the azimuthal degeneracy of the cold cavity modes.

Moreover, microgear cavities have a lower steady state inversion than the corresponding cylindrical cavities. They therefore require smaller pump rates and have lower steady state inversion. The lower quality factors of the incompatible mode and the modes which have their maximum in the centre of the disc in the microgear restrict lasing to the single high Q compatible resonance. The entire inversion is therefore used to sustain lasing of this mode as it does not need to be shared with any other modes as in the case of the microdisc.

Next to revealing these nonlinear characteristics of WGM lasers, our modelling framework provides a general basis for studying the ultrafast full vectorial, spatial dynamics of micro cavity lasers with complex structured resonators.