Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities

We describe the strong optomechanical dynamical interactions in ultrahigh-Q/V slot-type photonic crystal cavities. The dispersive coupling is based on a mode-gap photonic crystal cavities with light localization in an air mode with 0.02(lambda/n)3 modal volumes while preserving optical cavity Q up to 5 x 106. The mechanical mode is modeled to have fundamental resonance omega_m/2pi of 460 MHz and a quality factor Qm estimated at 12,000. For this slot-type optomechanical cavity, the dispersive coupling gom is numerically computed at up to 940 GHz/nm (Lom of 202 nm) for the fundamental optomechanical mode. Dynamical parametric oscillations for both cooling and amplification, in the resolved and unresolved sideband limit, are examined numerically, along with the displacement spectral density and cooling rates for the various operating parameters.


Introduction
It is well-known that light has mechanical effects [1] and its radiation forces can be used to manipulate small atoms and particles [2]. Nowadays, the effects of optical forces in various mechanical and optical structures and systems have attracted intense and increasing interest for investigation [3]. Especially, the field of cavity optomechanics develops very fast [4][5][6][7], with recent studies covering a vast span of fundamental physics and derived applications . In this field, the optomechanical coupling between the supported mechanical and optical cavity modes are of key importance due to its direct relevance to the generated optical forces, and one main goal of the developed techniques is to cool the targeted mechanical mode to its quantum mechanical ground state [10,20,24,27]. Several classes of cavity optomechanical systems have been explored. One of the initial efforts examines macroscopic movable mirrors in the Laser Interferometer Gravitational Wave Observatory (LIGO) project [29][30].
Based on the micro-and nano-fabrication techniques, optomechanical resonators such as mirror coated AFM-cantilevers [14], movable micromirrors [15][16], vibrating microtoroids [11,31] In this paper, we theoretically investigate the large dispersive optomechanical coupling between the mechanical and optical modes of a tuned air-slot mode-gap photonic crystal cavity [51]. First, the optical modes are shown to exhibit high optical quality factor (Q) with ultra-small modal volumes (V) [52-56], from three-dimensional finite-difference time-domain numerical simulations. The mechanical modes and properties are then modeled using finite element methods. Based on first-order perturbation theory [57][58] and parity considerations, the respective optomechanical modes are then examined numerically. The dynamical backaction of slot-type photonic crystal cavities are studied, including the optically-induced stiffening, optical cooling and amplification, and radio-frequency spectral densities, for various laser-cavity detuning, pump powers and other operating parameters. We also note that the slot-type photonic crystal cavity can operate in the resolved-sideband limit, which makes it possible to cool the mechanical motion to its quantum mechanical ground state.

Ultrahigh-Q/V cavity optical modes
The slot-type optomechanical cavity is based on the air-slot mode-gap optical cavities recently demonstrated experimentally for gradual width-modulated mode-gap cavities [51] or heterostructure lattices [54], and theoretical proposed earlier in Ref.  E-field and energy distribution of the first (above) and the second (below) cavity modes.
A scanning electron micrograph (SEM) image of the cavity is illustrated in Fig. 2 Fig. 1(c), the optical field is mainly distributed in cavity region, and the simulation results also show that the minimum number of lateral lattice rows next to the cavity to maintain the high Q is ~ three lateral lattice rows. We therefore designed each beam into three lines with eight holes in each line, l=8a.
Mode II

Cavity mechanical modes
The mechanical modes are examined numerically via finite-element-method (FEM) simulations (COMSOL Multiphysics) for the dynamical motion of the suspended beams. The cavity mechanical modes can be categorized into common and differential modes of in-plane and out-of-plane motion [48] as well as compression and twisting modes of the two beams. The displacement fields Q(r) of the first eight mechanical modes are shown in Fig. 3. In the numerical simulations, the beams are clamped at both ends using fixed boundary conditions at the two ends (x=±1.96um) of the beam, meanwhile limiting motion in the x-y plane (in boundary condition constraint of z=±110nm has a standard notion

Coupling factor and symmetry considerations
Cavity optomechanics involves the mutual coupling of two modes in the same spatially co-located oscillator: one optical (characterized by its optical eigenfrequency and electromagnetic fields) and one mechanical (characterized by its mechanical eigenfrequency and displacement fields) degrees-of freedom. The perturbed cavity optical resonance, modified by small displacement about equilibrium

Perturbation theory
Perturbation theory for Maxwell's equations with shifting material boundaries was used to calculate the coupling length L om [57,58]. With the parameter α Δ characterizing the perturbation, the Hellman-Feynman theorem [63] provides an exact expression for the derivative of ω in the limit of infinitesimal Δα, , where the terms with the (0) superscripts denote the unperturbed terms. With shifting material boundaries, the discontinuities in the E-field or the eigenoperator are overcome with anisotropic smoothening which gives the following expression for the integral in the numerator [57],  [43], one defines Q(r)=αq(r), where α is the largest displacement amplitude that occurs anywhere for the displacement field Q(r). From the perturbative formulation, one then obtains: , where n is the unit normal vector at the surface of the unperturbed cavity and the spatial r-dependence explicitly shown here. for a slot width of 40nm. These strong optomechanical coupling is more than an order of magnitude larger than in earlier optomechanical implementations. We also note that, since the electromagnetic field is negligible outside the cavity region of l =8a, the coupling length does not change much when l is longer than 8a.

Coupled mode theory
The coupled equations of motion for the optical and mechanical modes can be derived from a single . F L (t) is the thermal Langevin force. We illustrate the time-domain displacement x(t) and the normalized cavity amplitude of the first optical and first mechanical modes in Fig. 5(a). The cavity amplitude oscillates in-phase with the displacement within the mechanical frequency cycle as shown. In Fig. 5  . As shown in Fig. 5, as optical Q increases, at certain detuning the frequency shift becomes larger and the effective temperature is lowered, denoting the increased cooling rate. For a fixed optical Q in the unresolved sideband limit, there will be an optimal detuning where the linewidth reaches its largest value and the effective temperature is the lowest. In our case this optimal detuning Δτ is around -0.25 with an input power of 50pW and the effective temperature can be lower than 50K.  Fig. 7(a) shows the resulting displacement spectral density when the input power P changes from 0 to 6.9uW, and normalized detuning Δτ=-0.25 where the linewidth has the maximum value and the frequency shift is positive. With increasing input power, the peak value of the displacement spectral density goes down and the full-width at half-maximum becomes larger, which demonstrates an effective cooled temperature of the slot-type optomechanical oscillator. In Fig.   7(b) we show the optical stiffing and linewidth damping of the first two mechanical modes, for a span of detunings while maintaining a fixed input power. Note that the optical stiffening is not monotonic with increasing detuning. For a cavity decay κ/2π of 387 MHz, the optimal detuning is at Δτ of -0.43, for the largest optical gradient force stiffening. For the second allowed mode, in the region of normalized detuning from zero to -4, this stiffening is large which leads to a significantly suppressed spectral density. Moreover, note that in both Fig. 7(a) and 7(b), a large optical stiffening can be observed in the slot-type optomechanical cavity, where the optical stiffening can result in a modified mechanical frequency more than 1.86× the bare mechanical frequency. . Most of the present optomechanical cavities are in the unresolved sideband regime, either because low optical quality factor or low mechanical frequency, which limit the minimum phonon number higher than unity. However, since our ultrahigh-Q/V slot-type photonic crystal cavity has a high optical Q factor and higher mechanical frequency due to its small volume, it has significant potential to operate into the resolved sideband region. For example, for the first mechanical mode (Ω m /2π of 460 MHz), an optical Q of more than 5×10 5 will bring the optomechanical oscillator within the resolved sideband limit with a n f of 1 ×10 -3 , allowing the potential to cool the mechanical mode to its ground state.

Conclusion
We illustrate numerically the slot-type mode-gap photonic crystal cavities for strong optical gradient force interactions. With the simultaneous strong optical field localization in 0.02(λ/n) 3   modes. Temporal coupled oscillations between the optical and mechanical fields are examined, along with effects of large optically-induced stiffening, cooling and resulting displacement spectral densities, for the various operating regimes in the slot-type optomechanical cavities.