1. 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 [4–7], with recent studies covering a vast span of fundamental physics and derived applications [8–28]. 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], and nano-membranes [21,32] have been examined recently. Radiation-pressure dynamic backaction could be observed in these geometries. In addition, another class of optomechanical devices utilizes optical gradient forces [33–38] based on near-field effects. Compared to radiation-pressure based optomechanical cavities, these devices can achieve wavelength-scale effective optomechanical coupling lengths due to the strong transverse evanescent-field coupling between the adjacent cavity elements [25,26,33–35,18,38]. Photonic crystal membranes can be very good candidate platform with great design flexibility [39–44], with photonic crystal cavities offering an ultrahigh optical quality factor with a small volume [45–47]. The internal optical intensity is very high and sensitive to the geometrical changes. However, to make these cavities support mechanical cavity modes with strong coupling with the optical modes, special design considerations are needed. Current reported geometries are either in-plane in side-by-side configuration [48,49] or vertically superimposed in face-to-face configuration [50]. Both configurations are recently examined experimentally. to be promising for cavity optomechanical operations.

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 [51,38]. 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.

2. Optomechanical slot-type cavity design

2.1 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 [38,51] or heterostructure lattices [54], and theoretical proposed earlier in Ref [47]. A non-terminated air-slot [55] is added to width-modulate line-defect photonic crystal cavities to create ultrasmall mode volume cavities. To better understand the various modes existing in the air-slot mode-gap cavities, the modes in the slotted photonic crystal waveguide with W1 line-defect width and their dispersion properties are first investigated and shown in Fig. 1(a) for the three localized waveguide modes. Mode I and II can be traced back to the W1 waveguide fundamental even mode and high-order odd mode respectively inside the photonic band gap, while mode III can be understood as arising from the second index-guided mode (as shown in Ref [59].) below the projected bulk modes. We produce the cavities by locally shifting the air holes away from the center of waveguide – thus the cavity mode resonances are created below the transmission band of the slotted waveguide. Two of the possible modes in the cavities are shown in Fig. 1(b). Confirmed from the mode frequency and symmetry, cavity mode I is due to the mode gap of slotted waveguide mode I [Fig. 1(b)] and is expected to have both high Q and sub-wavelength V. Cavity mode II [Fig. 1(c)] represents the mode with the same odd symmetry as mode II in slotted waveguide.

 

Fig. 1 (a) Photonic band structure of slotted PhCWG with s = 80nm. The blue dashed lines show the three modes in the slotted PhCWG. (b) H-field and energy distribution of waveguide modes I, II and III. (c) E-field and energy distribution of the first (above) and the second (below) cavity modes.

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The cavity is illustrated in Fig. 2 with a = 490nm, r = 0.34a, t = 0.449a, n_si = 3.48, s = 80nm, d_A = 0.0286a, d_B = 0.019a and d_C = 0.0095a. FDTD simulation is performed to numerically evaluate the properties of the cavity mode. For s = 80 nm, the air-slot mode-gap confined PCS photonic crystal nanocavity supports a high Q localized even mode [Fig. 1(c)] with Q factor up to 5 × 10⁶ and a mode volume V of 0.02 (λ/n_air)³ from numerical simulations [47,51]. 2D Fourier transform of the electric field shows few leaky components inside the light cone, supporting the high Q character of this air-confined mode. From 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.

 

Fig. 2 Illustration of air-slot mode-gap optomechanical silicon cavity, fabricated with electron-beam lithography [60,61]. The holes shifts are shown with d_A = 0.0286a (red), d_B = 0.019a (blue) and d_C = 0.0095a (green), where a is the crystal lattice constant, for increasing the intrinsic cavity Q.

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2.2 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 (the boundary condition constraint of z = ± 110nm has R_z = 0nm and R_x = 0nm, where R_z (R_x) is the deformation along z(x) axis), with silicon material properties: Young’s modulus E of 130GPa normal to [110] silicon crystallographic in-plane direction, thermal expansion coefficient α of 4.15 × 10⁻⁶K, specific heat capacity c of 703J/(kg⋅K), thermal conductivity κ of 156W/(m⋅K) and density ρ of 2330kg/m³. We choose the triangular mesh configuration, with an average mesh element volume of ~9 × 10⁻⁴μm³, with the eigenfrequency and modal analysis for the first eight mechanical modes [Fig. 3], with eigenfrequencies ranging from 460 MHz to 2.16 GHz.

 

Fig. 3 Mechanical displacement profile of the first eight mechanical modes. Modes in color (a, c, h) are allowed by parity considerations to couple to the optical modes; modes in grayscale (b, d, e, f, g) are forbidden by parity for sizable optomechanical coupling.

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Only mechanical modes with parity p_x, = p_y = p_z = + 1 can coupled to the optical slot cavity modes due to the symmetry of the optical field, as described in Ref [43]. Among the first eight mechanical modes, five of them (illustrated in grayscale in Fig. 3) do not couple to the optical modes due to parity considerations – b, e and f do not have the right parity in the x direction while d, e and g modes cannot be excited because of asymmetry of the optical gradient force along the y direction. In this slot cavity, therefore, only the first, third and eighth mechanical modes (depicted in color) have strong dispersive coupling to the localized optical modes. These are in-plane differential modes with modal frequencies Ω_m/2πat 459MHz, 1.36GHz and 2.16GHz respectively for a suspended beam length L of 3.92um. The effective mass of each mode is computed from$m_{eff}={\displaystyle \int dV\frac{{(r-r_0)}^2}{{(r-r_0)}_m^2}\rho }$, integrated over the computational space with ρ defined as the mass density, r the position from a fixed origin r₀ and ${(r-r_0)}_m$defined as the maximum displacement. The effective mass of the first, third, and eighth mechanical modes are computed to be 200fg, 100fg, and 30fg respectively in our specific implementation with 3.92um beam length, width of 1.7um, and membrane thickness of 220nm.

There are a number of possible dissipative processes where mechanical vibrational energy is dissipated into heat, either inside the structure or via interaction with its surroundings. These processes include squeezed film damping due to air viscosity [62], clamping losses, internal viscous damping in the silicon structure, and thermoelastic damping. Thermoelastic losses often set a lower ballpark estimate of the attainable Q_m in a vibrating beam element, where Q_m,Zener of the fundamental mechanical mode is expressed by [63,64]: $Q_m{}_{,Zener}=\frac{c\rho }{E{\alpha }^{{}^2}{T}_R}\frac{1+{\left(\omega {\tau }_z\right)}^2}{\omega {\tau }_z}$ where T_R is the ambient reservoir temperature,τ_z is the thermal relaxation time defined by $\frac{b^2}{{\pi }^2\chi }$, $\chi =\frac{\kappa }{c\rho }$, and b is the width of the beam. With silicon material properties, T_R at 300K, and b at 1.7 μm, Q_m is found to be in the range of 12,000 for the fundamental mode, and 40,000 and 60,000 for the third and eighth mechanical modes respectively.

3. 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 displacement α, can be given by its Taylor expansion around ω_o(α). If we consider the first-order expansion, and also set ${\omega }_o\left(\alpha \right)={\omega }_o{|}_{\alpha ={\alpha }_0}$as the equilibrium resonance of the optical mode, then the first order g_om = dω_o/dα can be defined as optomechanical coupling rate. g_om also represents the differential frequency shift of the cavity resonance (ω_o ) with mechanical displacement (α) of the slot cavity beams. One can parameterize the interaction strength between optical and mechanical degrees-of-freedom by an effective coupling length L_om [42] described by: $L_{om}{}^{-1}\equiv \frac{1}{\omega }\frac{d\omega }{d\alpha }$, with a corresponding optomechanical coupling frequency g_om defined by g_om≡ω_o/L_om.

3.1 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 $\Delta \alpha$characterizing the perturbation, the Hellman-Feynman theorem [65] provides an exact expression for the derivative of ω in the limit of infinitesimal Δα, $\frac{d\omega }{d\alpha }=-\frac{{\omega }^{(0)}}{2}\frac{〈E^{(0)}|\frac{d\epsilon }{d\alpha }|E^{(0)}〉}{〈E^{(0)}|\epsilon |E^{(0)}〉}$, 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], $〈E^{(0)}|\frac{d\epsilon }{d\alpha }|E^{(0)}〉={\displaystyle \int dA\frac{dh}{d\alpha }\left[\Delta {\epsilon }_{12}{\left|E_{\left|\right|}^{(0)}\right|}^2-\Delta ({\epsilon }_{12}^{-1}){\left|D_{\perp }^{(0)}\right|}^2\right]}$, for first-order perturbation of the cavity resonance. The integral is performed across the entire boundary surfaces of the optomechanical cavity, with h the displacement perpendicular to the unperturbed boundary surface, $\Delta {\epsilon }_{12}$defined as (ε₁ - ε₂) and $\Delta ({\epsilon }_{12}^{-1})$defined as (${\epsilon }_1^{-1}$ - ${\epsilon }_2^{-1}$). ${\left|E_{\left|\right|}^{(0)}\right|}^2$is the unperturbed E-field parallel to the boundary surface while ${\left|D_{\perp }^{(0)}\right|}^2$is the unperturbed electric displacement D normal to the boundary surface. From Ref [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:

3.2 Optomechanical coupling in slot-type optical cavities

Figure 4 shows the computed optomechanical coupling in the slot-type mode gap cavities, from first-order perturbation theory. As noted from parity consideration, here we show the optomechanical coupling strengths for the first optical mode to the allowed first (Ω_m,1) and second (Ω_m,3) mechanical modes, denoted as g_om(O₁-M₁) and g_om(O₁-M₂)respectively. We illustrate the coupling strengths for different slot gaps s of cavity, ranging from 40nm to 200nm. The electromagnetic field used is within a slot length l = 8a, since the cavity is confined by the PhCWG mode gap to a spatial localization of only several lattice constants a. As shown in Fig. 4, when the first optical mode is coupled with the second mechanical mode, the g_om is lower than that with the fundamental mechanical mode, which means the fundamental optical and mechanical modes provide the strongest dispersive coupling. The negative values depict as a decrease in optical resonance frequency for increasing slot widths s. For the fundamental mode, the dispersive coupling can go up to 940GHz/nm (or a coupling length of 202nm) for a slot width of 40nm. This strong optomechanical coupling is more than an order of magnitude larger than in earlier optomechanical implementations [21,25,48]. 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.

 

Fig. 4 (a) Computed optomechanical coupling rates of the fundamental optical mode with first (black solid squares) and second (red open circles) allowed mechanical modes, computed for the different slot gaps s. The inset panel is the corresponding coupling length. (b) Computed optomechanical coupling rates of the second optical mode coupled with first (black solid squares) and second (red open circles) allowed mechanical modes.

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4. Coupled mode theory

The coupled equations of motion for the optical and mechanical modes can be derived from a single Hamiltonian [66,8]:

 

Fig. 5 (a) Time-domain cavity amplitude a (solid blue line) and displacement x (dashed green line) of the first optical and first mechanical modes, with g_om of 940 GHz/nm, Ω_m/2π of 470 MHz, Q_m of 12,400, κ/2π of 425 GHz, and (1/τ_ex)/2π of 38 MHz. (b) Time-domain cavity amplitude for normalized detunings Δτ at −1, −0.25, 0, 0.25 and 1 (top to bottom).

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5. Displacement spectral density

5.1 Optically-induced stiffening and effective damping rate

From the coupled equations, the x-dependent contribution to this adiabatic response provides an optical contribution to the stiffness of the spring-mass system. The corresponding change in spring constant leads to a frequency shift relative to the unperturbed mechanical oscillator eigenfrequency, or termed as optically-induced stiffening [4,48]. The non-adiabatic contribution in coupled equations is proportional to the velocity of the spring-mass system. The optical gradient force induced damping rate modifies the intrinsic mechanical resonator loss rate Γ_m, yielding an effective damping rate: ${\Gamma }_{eff}=\Gamma +{\Gamma }_m$where $\Gamma =-\frac{{\omega }_0}{2{\Omega }_m{L}_{om}^2{m}_{eff}}\left(\frac{2{\kappa }_{ex}}{{\kappa }^2+4{\Delta }^2}\right)\left[\frac{\kappa /2}{{\left(\Delta -{\Omega }_m\right)}^2+{\left(\kappa /2\right)}^2}-\frac{\kappa /2}{{\left(\Delta +{\Omega }_m\right)}^2+{\left(\kappa /2\right)}^2}\right]P$. We note that this is valid only in the weak retardation regime in which$\kappa >>{\Omega }_m$. We illustrate in Fig. 6 the corresponding frequency shifts and effective damping rate of the slot-type mode-gap cavity, for different input powers and normalized detuning (Δτ). With this classical model, the laser introduces a damping without introducing a modified Langevin force. This is a key feature and allows the enhanced damping to reduce the mechanical oscillator temperature, yielding as a final effective temperature T_eff for the mechanical mode under consideration: $T_{eff}\cong \frac{{\Gamma }_m}{{\Gamma }_{eff}}{T}_R$. As shown in Fig. 6, 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. 6 (a-c) Two-dimensional surface plots of the first optical – first mechanical mode linewidth (a) mechanical frequency (b) and effective temperature (c), for varying detunings and optical Q factors. A fixed pump power of 1pW is used, along with an effective mass of 200fg and a 300K bath temperature. The dashed white line denotes the condition for Ω_m = κ. (d-f) Example first optical – first mechanical mode linewidths (d), frequency shift (e) and effective temperature (f) with two input powers (P) and varying laser-cavity detuning. Otherwise indicated, the conditions are identical to panel (a), and with optical Q chosen at 5 × 10⁵.

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The spectral density of purely mechanical displacement in the oscillator is described as: $S_x(\Omega )=\frac{2{\Gamma }_m{k}_{B}T/m_x}{{\left({\Omega }_m^2-{\Omega }^2\right)}^2+{\left(\Omega {\Gamma }_m\right)}^2}$ without the optical stiffening and damping. Since the coupling will shift the oscillator frequency and damping, we can modify ${\Omega }_m$and ${\Gamma }_m$in the expression into ${\Omega }_m^{\text{'}}={\Omega }_m+\Delta {\Omega }_m$and${\Gamma }_m^{\text{'}}=\Gamma +{\Gamma }_m$. Figure 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.

 

Fig. 7 (a) Displacement spectral density of the first mechanical mode, with optical detuning from the first optical mode. With the input power increasing from 0 to 9.5uW, in addition to an observed optical stiffening, the amplitude decreases with a larger linewidth for a decrease in the effective temperature. The detuning Δτ is fixed at −0.25, for an optical Q of 5 × 10⁵, m_eff of 200fg, at 300K bath temperature. (b) Displacement spectral density of the first and second allowed mechanical modes with different detunings. The scale bar is in dB with units of m²/Hz (pump powers P₁ of 0.1uW and P₂ of 50uW used respectively in the modeling).

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As shown above, both cooling and amplification can be realized in the optomechanical cavity through the red- and blue-detuning to the cavity resonance. An important question is what limiting temperature is achievable with the optical gradient force backaction cooling technique as described above. Two theoretical papers [22,26] have extended the classical theory of radiation-pressure backaction cooling to the quantum regime and shown the close relationship that cavity backaction cooling has with the laser cooling of harmonically bound atoms and ions. The result can be simply divided by two conditions. In the unresolved side-band regime, $\kappa >>{\Omega }_m$, the ground state cooling is limited as: $n_f\approx \frac{\kappa }{4{\Omega }_m}>>1$, where n_f is the minimum phonon number. On the other hand, in the resolved side-band regime, ${\Omega }_m>>\kappa$, occupancies well below unity can be attained yielding: $n_f\approx \frac{{\kappa }^2}{16{\Omega }_m^2}<<1$. 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 × 10⁵ 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.

6. 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 modal volumes and cavity Qs up to 5 × 10⁶, we examined the optomechanical transduction of the various mechanical and optical modes for a dispersive coupling g_om up to 940 GHz/nm for the fundamental 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.