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A photonic and phononic crystal (phoxonic crystal PxC) is a periodically patterned material that can at the same time localize optical and mechanical modes. Here we theoretically model one-dimensional PxC in diamond and find high quality mechanical resonances with very high frequencies > 10 GHz and optical properties comparable to those of PxC in other materials. The simultaneous confinement of photons and phonons leads to an optomechanical interaction that we calculate in a perturbation approach. The optomechanical coupling strengths reach values in the MHz range. We identify design rules to simultaneously achieve high optical and mechanical quality factors along with strong optomechanical coupling.
© 2014 Optical Society of America
1. Introduction
Recently there has been increasing interest in optomechanical interaction in massive systems at the micro- and nano-scale [1–4]. Strong coupling of optical and mechanical modes typically requires resonant enhancement in optical cavities, e.g. Fabry-Pérot type cavities where either an end mirror [5] or a membrane suspended inside the cavity [6] are acting as mechanical resonators. In another approach, whispering gallery mode resonators have been used where the optical mode can couple to flexural modes of the circular shaped resonator. Different cavity designs and materials have been realized e.g. microtoroids in silicon [7], microrings in aluminum nitride [8] and microdisks in silicon nitride [9]. Such cavities have been used to demonstrate strong coupling effects such as electromagnetically induced transparency, wavelength conversion and force measurements [9, 10].
Another very promising system are photonic and phononic crystal cavities, often called phoxonic crystal cavities (PxC). Here, refractive index structures at the scale of the optical wavelength (typical unit cell size ≈ 200 nm) confine light with small mode volumes and high quality factors. The periodic refractive index structures also lead to a periodic variation of acoustic properties along the PxC giving rise to the formation of a phononic band gap. Taking into account the longitudinal velocity of sound for typical materials ranging from vl = 7180 m/s (for silicon) to vl = 18075 m/s (for diamond) the unit cell size of photonic crystal cavities then defines the range of possible frequencies for mechanical oscillations typically some hundreds of MHz (for silicon) up to few ten GHz (for diamond). Combined with the optical properties of the PxC the simultaneous existence of photonic and phononic band gaps in one structure is possible [11, 12]. PxC cavities can at the same time localize high quality optical and mechanical modes. This has been studied for one-dimensional Bragg structures [13] and 2D cavities [14, 15]. The simultaneous confinement of photonic and phononic modes to a very small volume causes a strong optomechanical interaction. Various designs have been investigated so far with the use of different materials: 2D PxC made of indium phosphite have been used as suspended membranes [16,17] and experiments with a double layer of 2D PxC showed optical wavelength conversion [18]. Air slot designs have been proposed both in 2D PxC in gallium arsenide [19] and in 1D PxC in silicon [20, 21]. One-dimensional PxC cavities in silicon have successfully been laser-cooled into the quantum ground state [22] and the asymmetry of the mechanical sidebands in the quantum regime has been demonstrated [23]. Recently, one-dimensional PxC in silicon nitride have been used to demonstrate electromagnetically induced transparency [24].
An interesting material for optomechanical applications is single crystal diamond. Diamond shows good optical properties (low absorption) in the visible spectrum and an extraordinary high Young’s modulus along with high thermal conductivity. Experiments on mechanical resonators showed that the outstanding mechanical stiffness allows for high resonance frequencies even in simple resonator types like doubly clamped beams [25–27] and coupled-element resonators [28] in polycrystalline diamond as well as cantilevers [29–31] and nanodome resonators [32] in single crystal diamond. The fabrication of single crystal diamond PxC cavities would allow one to enter frequency ranges in the few ten GHz regime that no other massive mechanical resonator reached to date. Photonic crystal cavities in single crystal diamond have already been fabricated successfully and their optical properties have been investigated [33–35]. To our knowledge neither the mechanical properties of PxC in diamond nor the optomechanical coupling in such structures have been studied to date.
Another interesting property of diamond is that it can host various kinds of color centers [36] acting as single quantum emitters. The nitrogen vacancy (NV) color center, consisting of one substitutional nitrogen atom and an adjacent lattice vacancy, has been widely studied in the last years. Its interesting spin properties allow for applications as quantum bit [37]. Silicon vacancy (SiV) color centers consisting of a silicon atom surrounded by a split vacancy are among the brightest color centers to date with narrow linewidths down to 0.7 nm even at room temperature [38,39] and would thus be interesting for quantum cryptography applications [40]. Furthermore, evidence for an optically accessible electron spin of the SiV center has been discovered recently [41]. Color centers can be deterministically placed inside a PxC cavity using ion implantation [42, 43] and couple to the cavity field [33–35]. It would now be very interesting to not only couple a color center to an optical cavity mode but to also investigate the coupling between a color center and the mechanical cavity mode. It is known that a color center’s emission can shift in frequency if it is exposed to stress in the material [44–46]. Such an effect is also expected when a color center interacts with a localized mechanical mode, giving rise to phononic sidebands of the emission line. Phonon sidebands of a quantum emitter could e.g. be employed for optical cooling of massive structures, for investigating electron-phonon interactions [47, 48] or even a coupling of optical, mechanical and internal degrees of freedom in a coupled emitter-PxC system. Furthermore localized cavity phonons could be used to mechanically control the spin of NV color centers [49].
In the first section of this article we will determine the optical properties of one-dimensional PxC resonators in diamond. In a second step we investigate the mechanical properties of the same structure namely the formation of phononic bands and the properties of localized mechanical modes. We then calculate the optomechanical coupling in the one-dimensional PxC. We can identify two mechanisms accounting for the coupling, i.e. the moving boundaries and the photo-elastic effect. We also show a general design principle how the PxC’s properties can be improved by a modification of the cavity design.
As a general approach we only consider the fundamental optical mode and the lowest order fully symmetric (py = pz = 1) mechanical mode arising from the Γ-point because the optomechanical coupling is strongest for these modes as will be discussed in Sec. 4. In all simulations the material constants of single crystal diamond are used: refractive index n = 2.4, Young’s modulus E = 1035 GPa, Poisson’s ratio ν = 0.2, density of mass ρ = 3520 kg/m3, longitudinal velocity of sound vl = 18075 m/s and transversal velocity of sound vt = 11069 m/s. We consider the material to be homogeneous and isotropic.
2. Optical properties of the PxC
We investigate one-dimensional PxC cavities consisting of a diamond air bridge patterned with a regular line of air holes. In such a periodic pattern Maxwell’s equations can be written as an eigenfunction of the electric field E in a source-free linear dielectric ε:
This equation can be fulfilled by Bloch-modes that establish a photonic band structure inside the periodically patterned material. This band structure strongly depends on the geometric parameters, i.e. length a, width w, height h and hole radius r of the unit cell shown in Fig. 1(a).
Fig. 1 (a) Design of the unit cell with hole radius r, structure height h, width w and lattice constant a. (b),(c) Profile of the Ex- and Ey-component of the electromagnetic field of the first order optical mode.
When one of these parameters is varied along the beam, the band structure associated with each unit cell is changing over the structure length. Modes that are guided in the middle of the structure cannot propagate in the outer regions and are thus confined to the center of the PxC even without a complete photonic band gap [50].
It has been shown that quadratically decreasing hole diameters provide a harmonic confinement potential for photons. This leads to a Gaussian shaped envelope of the electric cavity field allowing for the smallest loss rate and thus the best mode confinement [51]. Cavity structures with a gap between the inner holes, such as investigated e.g. in [35], leading to an optical mode confinement in the resulting dielectric defect are not well suitable for optomechanical applications as they show smaller optical quality factors and no fully symmetric localized mechanical cavity mode. The hole cross-sections should ideally be circular to avoid additional losses at sharp edges [52]. PxC cavities are scalable and can match any desired wavelength. The design wavelength chosen here is λ = 740 nm to match the zero phonon emission line of SiV color centers. The PxC consists of a line of 40 holes equally spaced by a lattice constant a. Here we investigate a structure where the hole radii r decrease from the center towards the edges of the structure [53] according to r = r0(1 − Cx2) with x the hole position in units of the lattice constant a (x = ±0.5 a;±1.5 a;±2.5 a;...) and C a scalar determining how fast the radius is decreasing.
We simulate the optical properties of this structure with a finite-difference time-domain FDTD method (MEEP) [54]. The Ex and Ey components of the electrical field of the first order mode are shown in Figs. 1(b) and 1(c), respectively. It can be seen that the field is strongly localized in the center of the PxC.
The cavity mode can be described by the quality factor Q giving the temporal localization and the mode volume V quantifying the spatial localization. For coupling a cavity mode to a color center the ratio of Q/V should be maximized to achieve a high Purcell factor F determining the emission enhancement. We are thus optimizing the parameters to obtain a large Q/V ratio following the design rules described in [55]. The optimized cavity design is given by the parameters in Tab. 1. The hole radii range from rmin = 69 nm up to rmax = 94 nm. The mode has a quality factor Qopt = 1.4 × 107 and a mode volume of Vopt = 2 (λ/n)3. This result is comparable to simulation results on other cavity designs in diamond [56].
Table 1. Parameter values for the optimized photonic crystal cavity
3. Mechanical properties of the PxC
In analogy to the case of the optical mode the periodic pattern of the PxC cavity gives rise to a periodic potential for phonons. The mechanical displacement field u is a harmonic function with the mechanical eigenfrequency fmech. The equation of motion can be written as an eigenfunction:
Here C̱ is the fourth order elasticity tensor only being dependent on Young’s modulus E and Poisson’s ratio ν in an isotropic medium. ρ is the density of mass for the diamond material and fmech the frequency of the mechanical mode. ∇s = (∇ + ∇T)/2 is the symmetric gradient operator with ∇T the transposed differential operator and the colon denotes a double scalar product. The solution to Eq. (2) is a Bloch mode type for the mechanical displacement field u. This gives rise to a phononic band structure.The mechanical eigenfrequencies of the PxC are calculated with finite element (FEM) simulations (COMSOL Multiphysics). We perform mechanical eigenfrequency analysis on the PxC cavity structure modeled with an adaptive mesh. As a starting point we use the geometrical parameters corresponding to the optimized optical cavity mode shown in Sec. 2.
3.1. Phononic band structure
In a first step we simulate the phononic band structure of a unit cell with a radius of the air hole r = 94 nm corresponding to the holes in the center of the structure presented in Sec. 2. As shown in Fig. 2(a) there are various bands fulfilling different symmetry conditions. Simulations of the optomechanical coupling show that the coupling is strongest when the modes are completely symmetric (see Sec. 4). Therefore we will only focus on completely symmetric bands with parity py = pz = 1 here.
Fig. 2 (a) Phononic band structure for a PxC cavity with a unit cell length of a = 223 nm. The width w = 1.6 a and height h = 1.7 a are chosen according to the optimized optical cavity (see Tab. 1). The hole radius is r0 = 0.42 a corresponding to the holes in the center of the structure. The modes are sorted according to symmetry conditions where +(−) y/z denotes even (odd) mirror symmetry with respect to the y/z-direction. (b) Mechanical modes associated to the band edges of the second and third phononic band at the Γ-point G2 and G3 and of the first and second phononic bands at the X-point X1 and X2. The mechanical displacement is depicted in a color scale where red represents maximal elongation and blue unperturbed regions. (c) Frequencies of the band edges at the Γ-point for different hole radii r. (d) Frequencies of the band edges at the X-point for different hole radii r.
For this symmetry condition a complete band gap is formed in the frequency range 10 to 12 GHz. Figure 2(b) shows the modes associated with the lowest bands at the Γ- and the X-point (corresponding to the coordinates (0, 0) and (π/a, 0) in reciprocal space) named G2, G3, X1 and X2, respectively. The first band has no guided mode at the Γ-point. Considering the PxC cavity it would be favorable to avoid mechanical movement in the x-direction as there would be an adjacent unit cell forced to perform the mirror symmetric movement. Only the modes arising from the Γ-point provide an in-phase mechanical oscillation of neighboring unit cells. To achieve strong optomechanical coupling the spatial overlap of the mechanical vibration with the field maximum in the standing wave pattern of the optical mode should be high. The maximal elongation of the mechanical mode should thus ideally be oriented along the y-axis of each unit cell corresponding to the maximum of the electrical field of the optical mode. For these reasons we will use the mode G2 as starting point for localizing a mechanical mode in the PxC.
Figure 2(c) shows that both phononic modes at the Γ-point G2 and G3 shift to higher frequencies when the hole radii are decreased. This property can be used to confine a mechanical oscillation in the same way as the optical mode [14]. The phononic bands are only guiding a vibration of a certain frequency in a small region of space where the hole diameter is nearly constant. If the hole diameter decreases along the structure only modes with higher mechanical frequencies would be guided in the outer regions of the structure. Thus, the mode confined in the center region cannot propagate to the edges. Also the two lowest bands at the X-point X1 and X2 shift to higher frequencies when the hole radius is decreased as shown in Fig. 2(d). This means that the complete band gap is shifting to higher frequencies along with decreasing hole radii.
3.2. Localized mechanical modes
For the optimized optical structure described in Sec. 2 we can find over 20 different localized mechanical modes at resonance frequencies between 0.1 and 30 GHz. As the optomechanical coupling is highest for completely symmetric modes we will focus on the lowest order mode arising from the guided mode G2.
The mechanical quality factors are calculated following a procedure outlined in [57]. In order to introduce absorbing boundary conditions we add half-spheres to both ends of the structure. They are made of an artificially absorbing material where Young’s modulus Eabs = E (1 + ηi) and Poisson’s ratio νabs = ν(1 + ηi) have an additional imaginary part with η = 0.5. This approximative method can be used for comparing different parameter sets but the absolute values can deviate slightly for numerical reasons. For localized mechanical oscillations the effective mass meff is a measure for the spatial confinement of the mode and thus equivalent to the mode volume Vopt in the optical case. The effective mass is calculated as
Simulating a structure based on the geometric parameters listed in Tab. 1 we find the localized cavity mode shown in Fig. 3 corresponding to the guided mode G2 at a resonance frequency fmech = 12.4 GHz. The mechanical quality factor is Qmech = 1.4 × 106 and the effective mass meff = 143 fg.Fig. 3 Mechanical displacement field of the localized mechanical mode arising from the guided mode G2.
The mechanical quality factors could be further improved by introducing a so-called “radiation shield”, a two-dimensional mechanical band gap structure surrounding the PxC cavity drastically reducing radiation losses into the substrate [58].
The model presented here only considers radiation losses into the substrate. Additional losses arise from thermo-elastic damping [59], dissipation at the surface [29, 30] and internal dissipation in the material often described by tunneling states [26, 28]. There is evidence in literature that the diamond material quality and even the surface functionalization have a strong impact on measured mechanical quality factors [29]. To give a general figure of merit independent of the material quality we here neglect these internal losses and consider our calculated values for the mechanical quality factor as upper bound that would be lowered by imperfections in a real diamond material.
4. Optomechanical coupling
Optomechanical coupling arises from a mechanical shift Δα introducing a shift of the optical frequency Δωopt. As the mechanical shift is small compared to the structure size the resulting frequency shift can be treated as a perturbation of the optical mode. The first order correction of the shift in the optical resonance frequency ωopt is then
The optical frequency and the electric field E(0) are properties of the unperturbed PxC with the dielectric constant ε. The optomechanical coupling is thus caused by a variation of the dielectric function dε/dα due to the mechanical motion Δα. To quantify the coupling rate we consider two different effects where the mechanical oscillation introduces either a variation in the spatial distribution of the dielectric medium or a local variation in the absolute value of the dielectric constant caused by strain.As the spatial distribution of the dielectric changes over time the photonic potential will be varied and the coupling term can then be expressed as follows [60]:
Here n̂ is the surface normal vector, u the mechanical displacement field, Δε = εdiamond − εvacuum the index contrast and . Only the electric field component parallel to the surface E|| and the electric displacement field perpendicular to the surface D⊥ have to be considered. This contribution will be referred to as moving boundaries (MB) effect in the following. This effect is particularly strong when the electromagnetic field is localized near the surface and when the mechanical elongation is large.The other contribution to the optomechanical coupling is the photo-elastic (PE) effect. The mechanical oscillation can locally introduce a variation in the absolute value of the dielectric constant ε. The perturbation term can then be calculated as follows:
Here ε0 is the vacuum permittivity, n the refractive index of diamond, Sij the strain tensor and p11 = −0.277, p12 = 0.058 and p44 = −0.171 the photo-elastic constants of diamond [61,62]. The optomechanical coupling strength g0 is defined as product of the zero point fluctuation xZPF and the derivative of the optical resonance frequency ωopt at mechanical equilibrium α = α0: The zero point fluctuation xZPF of the mechanical oscillation is calculated as follows: The total optomechanical coupling is the sum of the individual contributions from the MB and the PE effect g0 = g0,MB + g0,PE. It is important to note that the individual contributions of both effects g0,MB and g0,PE can be in phase or out of phase leading to additive or subtractive contributions to the total optomechanical coupling [63].To quantify the optomechanical coupling we use the optical and mechanical simulation techniques presented in Secs. 2 and 3. For every structure, we determine the optical D- and E-field as well as the mechanical displacement field u. The optomechanical coupling can then be calculated from these field distributions as the combined effect of the MB and the PE effects.
4.1. Results
4.1.1. Optimization of the optomechanical coupling
We calculate the coupling of the fundamental optical mode shown in Fig. 1 with various different mechanical modes. We find that the coupling is strongest for completely symmetric modes (py = pz = 1) arising from the Γ-point. This is due to the fact that coupling contributions from one side of the unit cell cancel with contributions from the second half if the movement is antisymmetric and the coupling of opposing sign. Therefore we will only consider the lowest order fully symmetric mode arising from the guided mode G2 that shows the strongest optomechanical coupling. Simulating a structure based on the geometric parameters listed in Tab. 1 we calculate the contributions to the optomechanical coupling g0,MB/2π = −7.15 × 104 Hz from the moving boundaries effect and g0,PE/2π = 1.16 × 106 Hz from the photo-elastic effect. As the contributions of the two effects are out of phase the total coupling strength is found to be g0/2π = 1.09 × 106 Hz. As general design rule for the optimization procedure it is important to maximize the combined effect of both contributions instead of the individual coupling strengths g0,MB and g0,PE. As the PE effect is predominant it would be favorable to increase this contribution further while suppressing the MB effect.
In the following optimization process we vary the geometric parameters of the structure with the goal to achieve a higher optomechanical coupling strength g0. The optimal parameters are a hole radius r0 = 96 nm, a scalar C = 0.0023, a unit cell length a = 223 nm along with a height h = 365 nm and width w = 335 nm. The optical and mechanical quality factors are then Qopt = 5.5 × 105 and Qmech = 4.4 × 105 with an optical mode volume of Vopt = 1 (λ/n)3 and an effective mechanical mass of meff = 112 fg. The individual coupling strengths are g0,MB/2π = −9.26 × 104 Hz from the MB effect and g0,PE/2π = 1.91 × 106 Hz from the PE effect which lead to an overall coupling of g0/2π = 1.82 × 106 Hz. As a preliminary result we find that the simple approach of solely optimizing the optomechanical coupling sacrifices both the optical and mechanical quality factors. It has to be noted that the optomechanical coupling only depends on the spatial localization of the optical and mechanical cavity modes and is not affected by the decrease in the temporal confinement associated with lower quality factors.
4.1.2. Design variation to increase performance
It turned out to be very difficult to achieve good values for both quality factors Qopt and Qmech and at the same time a strong optomechanical coupling g0. It would be a problem for future experiments requiring the resolved sideband regime if the quality factors are low even though the optomechanical coupling is strong. Therefore, a compromise between a good optomechanical interaction and high quality optical and mechanical modes has to be found even if that includes a slight decrease in the coupling strength.
To meet these requirements we propose a structure where we add a mirror section to the nanobeam cavity. This means that the hole radius is varied only to the K-th hole from the center and then kept constant in the mirror section as displayed in Fig. 4(a). The total number of holes remains unchanged compared to the original structure. The idea behind this design is that the outer holes were too small to provide a photonic and phononic band gap for the guided mode and thus should be enlarged to provide an efficient isolation to radiation into the substrate. Figures 4(b) and 4(c) show the impact of such a change in geometry on the optical and mechanical quality factors Qopt and Qmech, respectively. The parameters used for the structure are a = 223 nm, r0 = 96 nm, C = 0.0015, w = 335 nm and h = 366 nm. The optical mode volume Vopt = 2 (λ/n)3 as well as the effective mass meff = 143 fg remain stable with the variation of K.
Fig. 4 (a) PxC structure with mirror section where the hole radii are only decreasing to the hole K = 10 and the outer holes are being kept constant. The hole radius in the mirror section is identical with the radius of the K-th hole. (b) Optical quality factor Qopt as a function of the number of holes K that have a non-constant radius. (c) Mechanical quality factor Qmech as a function of the number of holes K that have a non-constant radius. Note the logarithmic scale of the graph. The inset shows the frequency of the localized mechanical mode and the band gap of the mirror section for different values of K.
Figure 4(b) shows that the optical quality factor Qopt first increases as the length of the mirror section increases and then falls drastically for K ≤ 9. For K ≥ 10 the mirror section provides a better localization for the optical mode as compared to the holes with very small radii in the original structure. The decay in quality factor for K ≤ 9 is caused by the reduced depth of the potential confining the optical mode. The optical mode is thus leaking out and propagates along the nanobeam leading to a reduced optical quality factor Qopt.
As can be seen in Fig. 4(c) (note the logarithmic scale), the mechanical quality factor Qmech suddenly jumps two orders of magnitude as the cavity length decreases from K = 12 to K = 11. This behavior can be explained by having a look at the phononic band structure. The inset in Fig. 4(c) illustrates a schematic of the mechanical band gap shown in Fig. 2(a) for different values of K and thus different hole radii in the mirror section. It can be seen that for K ≥ 13 the frequency of the localized mechanical mode lies outside the band gap of the mirror section. In this regime the mirror section is not providing an effective isolation against radiation into the substrate. For K = 12 the resonant mode coincides with the lower edge of the mechanical band gap and thus the flat part of the first fully symmetric band as can be seen in Fig. 2(a). Flat parts in the band structure provide efficient guidance of modes and thus the localized mechanical mode is radiating strongly into the surrounding substrate leading to a low mechanical quality factor Qmech. For K ≤ 11 the outer holes are acting as band gap material strongly confining the mechanical resonance in the middle of the structure and suppressing radiation into the substrate.
To our surprise the optomechanical coupling is not affected at all by this change of the PxC design. This means that with the proposed variation of the cavity design we can improve both the optical and mechanical quality factors to Qopt = 2.4 × 107 and Qmech = 9.1 × 106 without decreasing the optomechanical coupling strength g0/2π = 1.5 × 106 Hz.
This adaptation of the structure geometry has an additional advantage when it comes to fabrication. The hole sizes in the simulated structures are so small, that the precision in matching desired hole diameters is poor and achieving steep vertical sidewalls becomes almost impossible. Fabricating a structure where the holes are larger and more homogeneously sized allows for a significantly higher precision.
4.1.3. Other cavity designs
Prior experiments on optomechanical effects in one-dimensional PxC cavities have been carried out with a ladder type of cavity [64]. As has been shown elsewhere optical cavity modes can reach higher optical quality factors when the holes are circular rather than rectangular [52]. As there is no well-established nano-patterning technique for diamond, fabrication imperfections will add up to such design-induced losses at sharp edges. Therefore we believe that any kind of ladder type structure in diamond would not show optimum performance in a real experiment.
A structure with smaller holes in the center and increasing hole radius towards the edges has been used for the demonstration of optomechanical coupling in Silicon PxC [23]. We performed simulations on a similar structure (referred to as structure 2 in the following) where the hole radii are quadratically increasing from the center of the structure and being kept constant in the outer regions. The lattice constant here is a2 = 226 nm to match the emission line of the SiV color center at 740 nm with radii ranging from rmin,2 = 0.19 a2 to rmax,2 = 0.37 a2. The structure height is h2 = 339 nm and the width w2 = 339 nm. The optical and mechanical cavity modes of this structure are shown in Fig. 5.
Fig. 5 (a) Profile of the Ey-component of the electromagnetic field of the first order optical cavity mode of structure 2. (b),(c) Mechanical displacement field of the fundamental mechanical cavity mode of structure 2.
The optical quality factor is Qopt,2 = 5 × 106 with a mode volume of Vopt,2 = 1 (λ/n)3. The mechanical mode has a resonance frequency of fmech,2 = 20.5 GHz, with a quality factor Qmech,2 = 4.3 × 106 and an effective mass meff,2 = 153 fg. We calculate a total optomechanical coupling rate g0,2/2π = 1.14 × 106 Hz consisting of the individual contributions g0,MB,2/2π = −8.73 × 104 Hz from the MB and g0,PE,2/2π = 1.23 × 106 Hz from the PE effect. Although the resulting values for fmech,2 and g0,2 look promising, the PxC’s performance crucially depends on a precise fabrication of the center region as both the optical and the mechanical quality factor critically depend on the hole placement in the mode confinement region. As structure 2 has smaller holes in the center of the structure the design parameters are much harder to reach in real structures as the aspect ratios of the holes are high (see also Sec. 4.1.2).
4.1.4. Fabrication tolerances and optical wavelength tuning
So far only ideal cavity structures have been considered. However, fabricated structures can deviate from the design geometry as the hole diameters are very small and diamond processing remains a challenge. Possible fabrication issues are imperfections in the placement and sizing of holes, not perfectly vertical sidewalls and losses due to surface roughness and absorption in the material. The latter two can be overcome by using high quality single crystal diamond substrates. In previous fabricated structures inclined sidewalls remained the biggest challenge causing a strong decrease in quality factor [33]. This can be improved by using dry etching techniques. Etched structures might reach quality factors that are only two orders of magnitude smaller than predicted by simulations, see e.g. [35]. This would allow for experimental optical quality factors in the order of 105 for the structure presented here. Simulations predict that mechanical quality factors decrease in a similar fashion as the optical quality factor when placement, sizing and sidewall steepness of the holes deviate from the ideal structure. A mechanical quality factor in the order of 104 can thus be expected for fabricated structures. It is important to note that the optomechanical coupling would not be affected by small optical or mechanical quality factors. It only depends on the spatial confinement of both cavity modes and is completely insensitive to changes in the temporal mode confinement caused by common fabrication imperfections.
The optical and mechanical resonance frequencies strongly depend on the geometrical parameters of the fabricated structures. On the one hand this is a challenge as it requires very precise patterning but on the other hand it can be used to tune the mode frequencies. It has been demonstrated that the optical cavity mode can be blueshifted into resonance with a color center emission line by homogeneously etching away diamond material [33,35]. A similar mechanism can be used for redshifting the cavity line when gas is condensed on the structure under cryogenic conditions [65]. Both techniques would at the same time shift the mechanical resonance frequency, that also depends on the geometric parameters of the structure.
5. Conclusion
In summary we showed that one-dimensional PxC structures in diamond allow the simultaneous localization of optical and mechanical modes. The localization of the optical mode with a wavelength of λ = 740 nm is based on the principle of the mode gap effect whereas the mechanical mode with a resonance frequency of fmech = 12.4 GHz is confined due to a complete band gap. The optical and mechanical quality factors are Qopt = 1.4 × 107 and Qmech = 1.4 × 106 respectively with an optical confinement Vopt = 2 (λ/n)3 and an effective mass meff = 143 fg. The optomechanical coupling reaches a strength of g0/2π > 1 MHz composed of the individual contributions of the photo-elastic effect and the moving boundaries effect. The performance of this structure can further be improved by introducing a mirror section at both ends of the cavity which provides a better confinement for both the optical and the mechanical mode. The quality factors can thus be improved to Qopt = 2.4 × 107 and Qmech = 9.1 × 106 respectively without decreasing the optomechanical coupling g0/2π = 1.5 MHz or the spatial confinement of the modes. To our knowledge this structure shows the highest mechanical resonance frequency in a massive system as well as the strongest optomechanical coupling in PxC cavities to date.
Acknowledgments
We thank M. Aspelmeyer and G. Cole for helpful discussions. L. Kipfstuhl acknowledges funding from the Stiftung der Deutschen Wirtschaft (SDW). This research has been partially funded by the European Community’s Seventh Framework Programme (FP7/2007–2013) under Grant Agreement No 611143 (DIADEMS).
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