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We present a design methodology and analysis of a cavity optomechanical system in which a localized GHz frequency mechanical mode of a nanobeam resonator is evanescently coupled to a high quality factor (Q > 106) optical mode of a separate nanobeam optical cavity. Using separate nanobeams provides flexibility, enabling the independent design and optimization of the optics and mechanics of the system. In addition, the small gap (≈25 nm) between the two resonators gives rise to a slot mode effect that enables a large zero-point optomechanical coupling strength to be achieved, with g/2π > 300 kHz in a Si3N4 system at 980 nm and g/2π ≈ 900 kHz in a Si system at 1550 nm. The fact that large coupling strengths to GHz mechanical oscillators can be achieved in Si3N4 is important, as this material has a broad optical transparency window, which allows operation throughout the visible and near-infrared. As an application of this platform, we consider wide-band optical frequency conversion between 1300 nm and 980 nm, using two optical nanobeam cavities coupled on either side to the breathing mode of a mechanical nanobeam resonator.
© 2012 Optical Society of America
1. Introduction
Recent experiments using radiation pressure effects in cavity optomechanical systems [1], such as electromagnetically-induced transparency [2, 3] and ground state cooling [4, 5], as well as theoretical proposals for using such optomechanical systems in quantum information processing applications [6–10], generally rely upon the achievement of large zero-point optomechanical coupling rates and low optical and mechanical dissipation rates. They also typically require operation in the resolved sideband limit in which the mechanical resonance frequency must exceed the optical dissipation rate. One-dimensional optomechanical crystals [3, 8, 11], in which a periodic patterning is applied to a doubly clamped nanobeam in order to simultaneously localize GHz phonons and near-infrared (200 THz) photons, are one promising architecture for future work in quantum cavity optomechanics.
While the use of a single nanobeam in such work has many advantages, separation of the optical cavity and mechanical resonator can afford greater levels of flexibility. This can be useful in applications in which multiple optical modes are coupled to the same mechanical resonance, for example, to achieve wavelength conversion [6,7,12,13] or in converting traveling wave phonons to traveling wave photons [6]. One disadvantage of separating the mechanical resonator and optical cavity is that the optomechanical coupling rate generally tends to be smaller, given the reduced overlap between the mechanical and optical modes. To overcome this, one can focus on optical modes formed within the gap between the optical and mechanical resonator. Such air-slot type modes [14], which display a high electric field concentration in the air gap, have been used to achieve large optomechanical couplings [15–18], and sub-wavelength effective optomechanical lengths [19, 20]. In these demonstrations, however, the mechanical resonators and optical cavities were not separated, and mechanical resonance frequencies were typically < 150 MHz. For operation in the resolved sideband regime, higher frequencies are desirable, given that high-Q cavities in materials like Si and Si3N4 typically have optical dissipation rates that are > 100 MHz.
Here, we present the design of a cavity optomechanical system in which a GHz frequency localized mode in a mechanical nanobeam resonator is coupled to a localized 300 THz optical mode in a separate optical nanobeam resonator (Fig. 1). We show that the slot mode effect can increase optomechanical coupling rates with respect to those found in a single nanobeam optomechanical system, and consider applications of this approach to instances in which the separated mechanical and optical resonators are particularly advantageous, such as wide-band optical wavelength conversion [6]. We consider both Si3N4 and Si systems, although our primary focus is on Si3N4. In particular, the Si3N4 system has many potential advantages for applications in quantum cavity optomechanics: optical Q > 106 has been demonstrated over a wide range of wavelengths in the visible and near-infrared in Si3N4 [21–23]. This can allow compatibility with quantum systems based on trapped atoms, ions, and semiconductor quantum dots, whose relevant optical transitions are often below 1000 nm (where Si is opaque). Furthermore, these stoichiometric Si3N4 films under tensile stress have been shown to support high mechanical quality factors (≈ 106), both in commercial thin films [24] and in nanofabricated devices [25–27]. Finally, as Si3N4 does not exhibit the two-photon absorption and subsequent free-carrier absorption and dispersion that silicon does [28], higher optical powers may be usable in this system, potentially increasing the range of pump-enhanced optomechanical coupling values that can be achieved. Our geometry is predicted to support GHz mechanical frequencies, optical Q > 106, and a zero-point optomechanical coupling strength g/2π > 300 kHz. Such values should in principle enable strong radiation pressure driven phenomena.
The proposed geometry, shown in Fig. 1, consists of two suspended parallel dielectric beams of refractive index nd, thickness t, widths WO and WM, and are separated by a gap of width wgap. Both beams are etched through to form quasi-periodic 1D structures in the z-direction. The ”optical” beam’s 1D lattice is modulated so that a quasi-TE (Ey = 0 across the center of the Si3N4 layer at y = 0) confined optical mode is supported at an optical frequency fo. Optical confinement along the beam is provided by a 1D photonic bandgap, while total internal reflection provides confinement in the x and y directions [11,29,30]. With a sufficiently small spacing wgap, the quasi-TE mode consists of an air-gap resonance as in [14], with a strongly concentrated, almost completely x-oriented electric field between the two beams. The ”mechanical” beam is designed to support a mechanical resonance of frequency fm, confined in z, that displaces the boundary B facing the optical resonator, thereby closing the gap and thus shifting the optical resonance frequency fo. It is important to stress that the optical cavity is actually formed by the ”optical” beam and the ”mechanical” beam’s closest arm (indicated in Fig. 1), so that optical resonances depend strongly on the gap width. Nevertheless, optical and mechanical resonances can be tuned with great independence, as we show in the following sections. Sections 2 and 3 present background information on the simulations performed and consider the case of a single Si3N4 nanobeam optomechanical cavity, as a baseline reference. Section 4 describes the design of the slot-mode-coupled optical and mechanical nanobeam resonators, emphasizing the design approach for achieving high-Q optical modes, GHz mechanical modes, and large optomechanical coupling. Finally, section 5 considers a few applications of this platform.
2. Background
The shift in the frequency fo of a particular optical resonance due to displacement of the nanostructure boundaries produced by a mechanical resonance at frequency fm is quantified by the optomechanical coupling parameter gOM = ∂ωo/∂x = ωo/LOM; here, x is the cavity boundary displacement, ωo = 2πfo, and LOM is an effective optomechanical interaction length [31]. The effective length LOM can be estimated via the perturbative expression [32]
Here, E and D are the modal electric and electric displacement fields, respectively, Δε = εdiel. − εair, , and εdiel.,air are the permittivities of the nanobeam material and air. The mass displacement due to the mechanical resonance is given by Q, and the normal surface displacement at the structure boundaries is |Q · n|, where n is the surface normal. The integral in the denominator is performed over the entire surface of the nanostructure.The optomechanical coupling parameter gOM can be converted into a pure coupling rate g between the optical and mechanical resonances: g = xzpf · gOM, where is the zero point fluctuation amplitude for mechanical displacement [6] and m is the motional mass of the mechanical resonance at frequency ωm. The motional mass can be obtained from the displacement Q and the nanobeam material density ρ by .
We point out that Eq. (1) only quantifies frequency changes due to boundary distortions. An additional, potentially important contribution to the optomechanical coupling comes from the photo-elastic effect, which corresponds to local, stress-induced changes of the refractive index. This contribution can be quite significant, and was in fact recently exploited to demonstrate, in combination with the moving-boundary contribution, zero-point optomechanical coupling rates in excess of 1 MHz in silicon nanobeam optomechanical crystals [33]. Giant enhancement of Brillouin scattering in suspended silicon waveguides, resulting from both photo-elastic and radiation pressure optomechanical coupling, has also been predicted [34]. In the present work, only moving-boundary contributions were taken into account.
Optical and mechanical resonator modes are obtained using the finite element method, as described in [15], and are used in the expressions above to yield g and gOM.
3. Si3N4 nanobeam
To gain some initial insight into the optomechanical rates achievable with a low index contrast nanobeam in a ’standard’ (i.e., single nanobeam) geometry, we start by investigating an optical cavity composed of a 1D array of air holes etched in a Si3N4 nanobeam, as show in Fig. 2(a) [35]. We seek to achieve the largest possible zero-point-motion coupling rate g and optomechanical coupling gOM. Our cavity was designed to support the optical mode at a wavelength λ = 960 nm shown in Fig. 2(b). The 980 nm wavelength band is technologically important due to the availability of triggered single photon sources based on InAs quantum dots [36] and silicon based single photon detectors which operate within this range [37]. Together with the optical mode, this cavity supports confined mechanical modes co-located with the optical resonance. The fundamental breathing mechanical mode at f = 3.2 GHz, shown in Fig. 2(e), with motional mass m = 570 fg, displays the highest zero-point-motion coupling rate g to the optical resonance in Fig. 2(b).
Fig. 2 (a) Si3N4 nanobeam optomechanical crystal. (b)–(d): first, second, and third order optical resonances. (e) Mechanical resonance.
To reach this design, the procedure outlined in Section 4.1 was initially employed to obtain a high Q optical mode. The resulting cavity displayed a quadratic variation of the lattice constant (the spacing between holes) from the center to the edges [29], and supported, together with the optical mode, a localized breathing mechanical mode. The cavity geometry was then optimized via a nonlinear minimization code to yield a maximized optomechanical coupling rate g/2π = 133.6 kHz (LOM = 5.1 μm), while maintaining a high optical quality factor Q > 1 × 106. The optimization was realized by allowing the aspect ratio of the elliptical holes (vertical axis over horizontal axis) to vary quadratically from the cavity center, then searching a 2D space of aspect ratios at the center and at the edge of the cavity. Reducing the aspect ratio at the cavity center proved to be beneficial in reducing the effective length LOM, by reducing the mechanical mode volume and promoting better overlap between the optical and mechanical modes. The Q factor was not strongly affected, most likely because the hole areas were maintained.
The nanobeam also supports two additional high order modes, shown in Figs. 2(c) and 2(d), at wavelengths of 999 nm and 1036 nm, both with Q > 1 × 106, and which couple to the mechanical mode in Fig. 2(e) with g/2π = 55.2 kHz and g/2π = 17.5 kHz, respectively.
3.1. Wavelength conversion
A cavity optomechanical system supporting two optical resonances at different wavelengths and a shared mechanical resonance may be used for performing wavelength conversion of classical or quantum optical signals [6,7,12,13]. In this photon-phonon translator scheme, a signal tuned to one of the optical resonances is transduced to the shared mechanical resonance, and from the latter to a second optical resonance at a different wavelength. These conditions are achieved in the single nanobeam design above, where the three displayed optical resonances couple to the breathing mechanical mode. The wavelength separation between input and converted signals is ≈ 76 nm, given by the separation between the fundamental and third order optical modes. Wider separations can in principle be achieved in larger cavities, however, given the requirement of small optical and mechanical modal volumes for a large zero-point motion optomechanical coupling rate, g is expected to drop [38]. Ultimately, the achievable spectral separation in single nanobeam designs has an upper bound given by the achievable photonic bandgap width, which increases with the refractive index contrast. Nanobeams based on Si3N4, with a refractive index n ≈ 2.0, are considerably more limited than Si nanobeams (n ≈ 3.5).
4. Slot mode nanobeams
4.1. Optical cavity design
We start the design of a Si3N4 optical cavity for operation in the 980 nm band by considering the geometry in Fig. 3(a). This is a version of the geometry in Fig. 1 in which the mechanical resonator is stripped of its rightmost arm and connecting ribs (in what follows, we refer to the remaining arm of the mechanical resonator as the ’optical arm’). We first generate Bloch bands for photonic crystals with unit cells as highlighted in Fig. 3(a), with optical arm widths WA constant in z. As seen in Fig. 3(b), the fundamental TE Bloch band displays bandgaps at the X-point (bands for WA/WO = 0.25 and WA/WO = 0.65 are displayed), and no overlapping bands below the light-line in the spectral region near the lowest band-edge. Figure 3(c) shows the squared electric field amplitude for the Bloch modes at the Brillouin zone boundary, for the structure with WA = 0.65 · WO
Fig. 3 (a) Simplified nanobeam geometry with only the optical arm of the mechanical resonator. (b) Photonic bands for the TE mode (Ey = 0 at the y = 0 plane) of the unit cell indicated in (a), for WA/WO = 0.25 and WA/WO = 0.65. (c) Normalized electric field amplitude for the fundamental TE mode band at the X point (open circle in (b)). (d) Normalized frequency bands for the first four TE photonic Bloch modes at the X-point, and (e) normalized effective length for the fundamental TE mode, as functions of WA/WO.
Figure 3(d) shows the evolution of the first four TE bands at the X-point as a function of WA. The apparent reduction in bandgap width (i.e., between the first and second bands) with increasing WA is due to an effective decrease in the filling fraction, or the ratio between air and dielectric regions in the unit cell. The larger bandgaps achieved with smaller WA correspond to a large effective reflectivity and thus are better suited for strong spatial confinement. On the other hand, a wider WA is desirable for enhanced modal frequency shifts with gap width (i.e., optomechanical coupling). This is evident in Fig. 3(e), where LOM is plotted, assuming |Q · n| = 1 in Eq. (1), for the fundamental TE photonic crystal modes at the Γ-point, as a function of the WA/WO ratio. A minimized optomechanical length can be obtained for WA/WO ≈ 0.7, about 30 % lower than the maximum value within the plotted range. A trade-off between spatial confinement and optomechanical coupling must thus be achieved in a geometry where the mechanical beam width WA is gradually reduced away from the cavity center, until a ’nominal’ width is reached, for which the reflectivity is highest. We thus choose to let WA vary from 0.65 · WO at the cavity center to 0.25 · WO at the mirrors.
The photonic bands shown in Fig. 3(b) were obtained with wz = 0.35 · WO, wx = 0.71 · WO, and t = 0.75 · WO (wz and wx are the elliptical holes’ axes in the z and x directions, t is the Si3N4 thickness), which led to a large bandgap for WA = 0.25 · WO. We choose the Si3N4 thickness to be t = 350 nm and the gap to be wgap = 25 nm, and determine the remaining dimensions by regarding the cavity center, where WO = 0.65 · WA. A zero-finding routine determined the width WO and the corresponding cavity lattice constant a for the lowest X-point TE Bloch mode to be at a wavelength λ ≈ 980 nm. We point out that while such small gap width poses a great fabrication challenge, it may in principle be achieved as in [39], where the ≈ GPa Si3N4 film stress was harnessed to bring two nanobeams together as closely as 40 nm.
To produce an effective potential well for the fundamental TE mode at the X-point, the band edge must shift towards lower frequencies at increasing distances from the cavity center. This shift is opposite to what is obtained when WA is modulated in the desired way, and thus we choose to modulate the lattice constant along the cavity in order to produce the correct trend. Following the procedure described in [29], allowing both beam width WA and the lattice constant a to vary quadratically away from the cavity center (Figs. 4(a) and 4(b)), the desired trend for the lowest band edge is achieved (Fig. 4(c)), along with an approximately linear mirror strength (Fig. 4(d)). The mirror strength corresponds to the imaginary part of the Bloch wavenumber for bandgap frequencies at the X-point, k = π/a(1 + iγ). From 1D first order perturbation theory,
where ω1,2 are the dielectric and air band edges at the Brillouin zone boundary, and ω0 is the midgap frequency. Linear mirror strength profiles tend to produce optical modes with reduced spatial harmonics above the light line. This leads to reduced power leakage into the air, and thus higher quality factors [29, 40]. We point out that the quadratic lattice constant and beam width tapers employed here, though not optimal, produce sufficiently linear mirror strength profiles for high quality factors to be achieved.Fig. 4 (a) Photonic lattice constant, (b) optical arm (WA) over optical beam (WO) width ratio, (c) frequency (at the X-point) of the first two TE photonic bands, and (d) photonic mirror strength, as functions of the distance from the cavity center (z = 0). (e) Squared electric field amplitude for the photonic mode generated with the parameters in (a)–(d).
Figure 4(e) shows a cavity mode at λ = 976 nm, with mechanical beam width and lattice constant modulation as in Fig. 4(a). The field is strongly concentrated in air gap, as in Fig. 3(c), and the quality factor is Q = 12 × 106.
4.2. Mechanical resonator design
The optical resonator design dictates the geometry of the mechanical beam’s optical arm. The remaining parts of the mechanical resonator (Fig. 1) are designed around this constraint. While the secondary arm can in principle be chosen arbitrarily, for simplicity we let it be identical to the optical arm. The lattice formed by the connecting ribs are made to follow the optical lattice. This configuration minimizes deterioration of the optical quality factor due to scattering at the ribs. Rib widths have a strong and nontrivial effect on scattering, and were made small (≤ 50 nm) to limit Q degradation. More details are given in Section 4.4.
The mechanical resonator geometry studied here is similar to the optomechanical nanobeam crystal analyzed in [31]. While in that work a detailed account of the phononic crystal mechanical band structure is provided, here we focus only on the relevant mechanical bands for enhanced optomechanical coupling. We are interested in producing ’breathing’ mechanical modes which displace the nanobeam arms laterally, allowing the air gap width to change.
Phononic bands for the crystal shown in Fig. 5(a) are plotted in Fig. 5(b), for an arm width WA = 0.25 · WO. Bands A, B and C were obtained with symmetric boundary conditions for the displacement at the x′ = 0, y′ = 0 and z′ = 0 planes, so that no modes with other symmetries are displayed. We refer to ref. [31] for a more detailed account of the additional existing bands. The characteristic boundary displacements for each band are inset in Fig. 5(b). Near the Γ-point, the mechanical modes of band C display the desired lateral displacement pattern, necessary for the formation of breathing resonances.
Fig. 5 (a) Phononic crystal geometry. (b) Mechanical Bloch mode band structure for a phononic crystal with WA/WO = 0.25. The inset shapes show a top view of the unit cell, with exaggerated boundary displacements for modes on bands A, B and C. (c) Evolution of the Γ-point eigenfrequency for band C, as a function of the ratio between the optical arm (WA) and optical beam (WO) widths.
Modes from bands A and B stem from flexural modes of the connecting ribs - evident in the inset displacement plots - and display negligible lateral displacement. Band C has a minimum at kx = 0, which creates the conditions for the formation of a breathing mode phononic resonance. Indeed, a phononic cavity naturally arises from the quadratic spatial arm width (WA) variation, as evidenced in Fig. 5(c), where the Γ-point edge of band C is plotted as a function of WA/WO. As the arm width decreases from the cavity center towards the mirror regions, the band edge moves towards higher frequencies, placing the resonance frequency within the breathing mode phononic bandgap. Localized breathing modes as plotted in Figs. 6(a) and 6(b) result, with frequencies that decrease with increasing nanobeam widths WM, as shown in Fig. 6(d). It is also apparent that displacement profiles can also change considerably with WM. This is quantified in Fig. 6(e), where the normalized displacement
is plotted as a function of WM/WO. The integral is performed over the optical arm’s surface facing the optical beam, and so Dx is a measure of the uniformity of the optical arm’s displacement towards the optical beam. The evolution of the modal displacement pattern also results in a variation of the motional mass, as shown in Fig. 6(f).Fig. 6 (a) and (b): Mechanical beam breathing modes for (a) WM/WO = 2.6 and (b) WM/WO = 2.8. (c) Fundamental optical slot mode for the complete geometry. (d) Mechanical frequency, (e) normalized displacement Dx, (f) motional mass m, and (g) optomechanical coupling parameter gOM and zero-point motion optomechanical coupling rate g for the fundamental breathing mode as a function of WM/WO. As a function of gap width: (h) fundamental optical resonance wavelength, (i) quality factor, and (j) gOM and g of the fundamental mechanical breathing mode for WM/WO = 2.8, as in (b). Note that the geometry that produced (h)–(j) was optimized for a 25 nm gap. Improved values of gOM, g, and Q over those plotted can be obtained by optimization for different gap widths (see text). (k) Optical mode frequency and (l) optical quality factor as a function of rib width, for the geometry shown in (c), with a gap width of 25 nm and WM/WO = 2.8.
4.3. Optomechanical coupling
At this point, the optical and mechanical resonators designed in Sections 4.1 and 4.2 can be put together to form the composite optomechanical resonator shown in Fig. 6(c) (the fundamental optical mode of this structure, with electric field mostly concentrated in the gap between optical and mechanical resonators, is also shown). The optomechanical coupling parameter gOM and zero-point-motion coupling rate g between the optical resonance in Fig. 6 (c) and the breathing mode mechanical resonances (depicted in Figs. 6(a) or 6(b)) are plotted in Fig. 6(f), as a function of WM/WO. The maximum gOM and g values achieved for the WM/WO range shown are > 2× the value obtained with the optimized single nanobeam of Section 3, despite the larger motional masses (Fig. 6(f)). The strong influence of the gap width on the optomechanical coupling is evident in the fact that both g and gOM approximately follow the trend of Dx. In fact, the effect of increasing gap widths on the resonance wavelength is plotted in Fig. 6(h), displaying a fast red-shift for decreasing gaps. Because the cavity was initially designed to have a 25 nm gap, the quality factor decreases as this parameter is changed (Fig. 6(i)), and so do the optomechanical coupling gOM and the coupling rate g (Fig. 6(j)), despite the increase in the optical frequency fo (g ∝ xzpf ∝ fo, is the zero point fluctuation amplitude for mechanical displacement). For any fixed gap width, however, the lattice can be adjusted to yield a high quality factor. For instance, changing the lattice constant profile in Fig. 4(a) so it varies (quadratically) from 325 nm at z = 0 to 365 nm at the cavity edge gives an optical mode at wavelength λ = 986.4 nm with Q = 2.4 × 106 and gOM = 180 GHz/nm. In contrast, for the 25 nm gap design of Fig. 6, a 40 nm gap yields λ = 965.6 nm, Q < 5 × 105 and gOM = 150 GHz/nm (Figs. 6(h)–6(j)). This serves to show that our double nanobeam design affords great flexibility towards optimized performance under constraints such as resonance wavelength, quality factor, or gap width.
4.4. Effect of rib width
As mentioned earlier, the mechanical resonator ribs may degrade the optical quality factor significantly - for instance, the quality factor of 12 × 106 obtained for the simplified, ribless geometry shown in Fig. 4(e) drops to ≈ 2×106 when 50 nm ribs are introduced as in Fig. 6(c). To further illustrate this point, we plot in Figs. 6(k) and 6(l) the effect of varying rib width on the wavelength and optical quality factor of the mode for the 25 nm gap, WM/WO = 2.8 structure studied in the previous section. It is clear that with increasing rib width, the mode red shifts by an amount reaching > 0.5 % of its original value for widths > 80 nm. The quality factor varies in a non-trivial fashion with rib width, and decreases by more than an order of magnitude for widths > 80 nm. Because a more detailed analysis of this effect goes beyond the scope of this paper, we chose to work with a rib width of 50 nm, which is realistically realizable, and which yielded the highest quality factor.
4.5. Effect of index contrast
Equation 1 suggests that higher index contrasts generally yield higher optomechanical coupling. In addition, the electric field concentration in the slot region (∝ (Δn)2 [14]) is expected to add a significant contribution to gOM and g. Indeed, a design for operation at the 1550 nm band based on silicon (nSi ≈ 3.48), produced with the same procedure as above, yielded an zero-point-motion coupling rate g/2π ≈ 900 kHz for a mechanical resonance at fm = 1.38 GHz (an optimized single nanobeam design with similar geometrical parameters gave g/2π ≈ 400 kHz for a mechanical resonance at fm = 4.3 GHz). Table 1 shows the relevant parameters for this design. Here, the 1D lattice constant varied quadratically from 360 nm to 385 nm, and the optical beam width varied between from 325 nm at the cavity center to 150 nm at the edges.
Table 1. Silicon Based Optical Resonator Optomechanical Crystal Parameters. A slot width of 10 nm was assumed.
5. Applications
5.1. Wide spectral separation wavelength conversion
While a ’standard’ nanobeam geometry such as that studied in Section 3 already offers the necessary conditions for wavelength conversion, the spectral separation between optical resonances of different orders is limited to 70 nm. Even in silicon nanobeam designs, where the higher index contrast leads to a considerably wider spacing between resonances, a maximum separation of ≈ 100 nm is achievable [12]. As discussed in Section 3, the achievable wavelength separation in the single nanobeam design is ultimately given by the nanobeam photonic bandgap, which can be limited even for high contrasts such as in the Si case. The ability to perform wavelength conversion over considerably wider wavelength separations may be desirable for applications in which classical or quantum information transmitted over optical fibers at telecom wavelengths is originally produced in a completely separate wavelength band, for instance in the near-infrared or visible range. Such an achievement would be enabling for proposed hybrid quantum optical networks, whose nodes may be widely different from each other (trapped atoms [41], quantum dots [42], organic molecules [43], etc).
The physical separation between optical and mechanical resonances afforded through our double nanobeam optomechanical approach allows the formation of optomechanical resonators with widely spaced, spatially separate, and semi-independent optical resonances with appreciable optomechanical coupling to the same mechanical resonance. Such a resonator, schematically depicted in Fig. 7(a), consists of a central mechanical nanobeam laterally sandwiched by two optical nanobeams. Air gaps on the two sides of the mechanical resonator allow a single breathing mode resonance to couple efficiently to optical nanoslot modes on both sides. We demonstrate the feasibility of this scheme via an example in which optical resonances at wavelengths of 980 nm and 1310 nm are supported. Figures 7(b) and 7(d) respectively show the 1310 nm and 980 nm resonances of a double optical nanobeam resonator that shares the mechanical nanobeam resonance at fm = 1.37 GHz in Fig. 7(c). Both optical resonances display quality factors in excess of 106, which allows operation in the resolved sideband regime. The 980 nm wavelength optical cavity has exactly the same geometry parameters as that in Section 4.1. The 1310 nm cavity was designed with the same procedure as described in Section 4.1. Achieving high optical quality factors for both cavities required significant modifications to the original mechanical nanobeam geometry presented in Section 4.2. First, the nanobeam width WM had to be increased to 4 · WO, to minimize the interaction between the two optical resonators. This resulted in a lower resonance frequency, fm = 1.37 GHz. In addition, the 1310 nm resonator’s optical arm was made constant with WA = 0.25 ·WO, to prevent coupling of the confined 980 nm resonance to leaky resonances of the 1310 nm beam. This caused the optomechanical coupling gOM for the 1310 nm resonator to be ≈ 40 % lower than that obtained with modulation of the optical arm (gOM ≈ 112 GHz/nm). It is worth noting on the other hand that, despite the uniform arm width, gOM can still be reasonably high. This parameter, together with the optomechanical coupling rate g, effective optomechanical length LOM and motional mass m for the two resonators are given in Table 2.
Fig. 7 (a) Double optical cavity optomechanical crystal geometry. (b) 1310 nm optical mode. (c) Mechanical resonator breathing mode. (c) 980 nm optical mode. Green arrows indicate simultaneous coupling between the optical resonances and the mechanical mode.
Table 2. Double Optical Resonator Optomechanical Crystal Parameters. Both slots are assumed to be 25 nm wide.
As discussed in the Appendix, the conversion efficiency achievable in the scheme proposed in ref. [6] and demonstrated in ref. [12] depends on the cooperativities of optical cavities 1 and 2 (nc;1,2 is the average photon population in each cavity), and unity conversion requires C1 = C2 ≫ 1. The coupling rates calculated above are approximately a factor of three times smaller than those used in ref. [12]. These smaller coupling rates can be compensated by increasing the number of photons in the cavity (by a factor of nine, assuming similar optical and mechanical decay rates are achievable), which should be feasible in the Si3N4 system. We also note that the disparate coupling rates for the two cavities are not expected to influence the ability to achieve high efficiencies, as such differences may be compensated by proper balancing of the intra-cavity photon populations nc;1,2.
5.2. Strong optomechanical coupling
The observation of quantum behavior in cavity optomechanics relies on the possibility of strong coupling between optical and mechanical resonances. Ultimately, a regime is sought in which interactions at the single photon and phonon levels are observable. This single-photon strong coupling regime, in which a single phonon is able to shift an optical resonance by an extent comparable to the latter’s linewidth, is characterized by a coupling rate g comparable to both the optical resonance decay rate κ and the mechanical frequency Ω. Achieving single photon strong coupling at optical frequencies is challenging due to high optical losses. In [10], a scheme was proposed for producing a strongly enhanced, effective optomechanical coupling for quantum non-demolition (QND) photon detection and phonon number readout measurements. Such a QND measurement would extract information from the quantum system without disturbing its quantum state. In the setup of [10], an optomechanical crystal geometry supports two optical resonances split by a frequency 2J, comparable to the mechanical frequency Ω of an interacting mechanical resonance. The coherent interaction between photons and phonons is characterized by an effective coupling rate , where δΩ = 2J − Ω and g0 is the bare optomechanical coupling rate. In the limit 2J → Ω, the optical frequency shift is , where nb is the phonon number,which may be sufficiently large to produce a phonon number readout.
The flexibility of our double nanobeam geometry allows us to accommodate the optical mode splittings 2J ≈ Ω required in the scheme proposed in [10]. As shown in the inset of Fig. 8, an optomechanical resonator is formed by two identical optical beams sandwiching a mechanical resonator. The two individual optical resonances couple evanescently, forming a symmetric / anti-symmetric doublet whose spectral separation can be controlled by the mechanical beam width WM. The modal splitting 2J, shown in Fig. 8 for the same cavity parameters as in Section 3, indeed decreases with increasing WM, as the spatial overlap between the individual optical resonances diminishes. The main breathing mode mechanical frequency also decreases with WM, however remaining within the GHz range even for the largest mechanical beam widths plotted, so that the regime δΩ = 2J − Ω → 0 can be achieved. The coupling between the two modes can also be tuned by creating optical cavities with coinciding individual resonances, but different gap widths.
Fig. 8 Optical mode splitting for the symmetric double optical beam resonator in the inset. The dotted line in the inset indicates the symmetry plane that defines splits the original optical resonance into symmetric (blue) and anti-symmetric (red) modes. Parameters for the optical and mechanical resonators are as in Section 4, with WM/WO = 2.8.
5.3. Microwave to optical photon conversion
State-of-the art superconducting microwave circuits can be designed to provide controllable, coherent interactions between superconducting qubits and high quality microwave cavity resonators, and provide favorable conditions for the creation and manipulation of photonic states at microwave frequencies. In the context of a quantum network [41] where the nodes are connected via photonic links, however, low loss transmission of microwave photons over long distances becomes a challenge, as superconducting transmission lines would be required. The ability to convert between microwave and optical photons which can be transmitted via optical fibers thus constitutes an appealing solution to this problem.
The physical separation between optical and mechanical resonances in our geometry creates opportunities for the generation of microwave-to-optical frequency signal transducers. One possible geometry is illustrated in Fig. 9, where the far arm of the mechanical resonator incorporates a metallic strip, and, together with a neighboring ground plane, forms a capacitor that integrates a superconducting, planar circuit microwave resonator, similar to that demonstrated in [44,45]. The displacement of the mechanical beam affects the cavity resonance by causing a variation in the circuit capacitance. Coupling between the mechanical displacement and cavity capacitance is stronger for smaller gaps between the metallized beam and the ground plane. At the same time, the distance between the metallized mechanical beam arm and the optical resonator must be sufficient to prevent significant deterioration of the quality factor of the optical cavity, while keeping the superconducting microwave resonator sufficiently separated from strong optical fields is necessary to avoid heating and degradation of its performance. As shown in Section 3, this separation can be accomplished through increased mechanical beam widths. While this leads to decreasing mechanical frequencies for the fundamental breathing mode, as seen in the previous sections, mechanical frequencies in the GHz range are achievable.
Fig. 9 Schematic for possible optomechanical microwave-to-optical wavelength converter. The RF cavity resonance frequency is modulated by the gap capacitance Cg, which depends on the gap width, and therefore on the displacement of the mechanical resonator. The latter is coupled to the slot optical mode formed with the optical beam.
As is the case with optical-to-optical conversion, microwave-to-optical conversion under the protocol described in ref. [6] requires Cμwave = Coptical ≫ 1, that is, the cooperativity of the microwave and optical resonators (see Appendix) should be large and matched to each other. We anticipate that the design of the optical resonator will be similar to that described previously, while a full design of the microwave cavity is beyond the scope of this paper. Qualitatively, we note the conceptual similarity between the geometry shown in Fig. 9 and that studied experimentally in ref. [45], where a suspended Al nanobeam separated from a ground plane by a gap ≈ 20 nm was employed to perform vibration detection near the quantum limit, and for which subsequent experiments have shown radiation-pressure driven phenomena like electromagnetically-induced transparency [46]. The coupling between the microwave cavity and mechanical resonator is gμwave ∼ (∂Cg/∂wgap), where Cg is the coupling capacitance and wgap is the gap between the two elements. Reaching the regime Cμwave ≫ 1 with a microwave intracavity photon number consistent with recent experiments (up to 106 photons were used in ref. [46]) will thus require small gaps (10 nm gaps were reported in ref. [45]), along with low dissipation for both the microwave cavity and the mechanical resonator. Investigation of such suitable microwave cavity geometries is in progress.
6. Conclusion
We have presented a design methodology and analysis of an optomechanical crystal comprised of two laterally coupled 1D nanobeams which support spatially separate optical and mechanical resonances. This spatial separation enables independent design and flexible optimization of the optics and mechanics of the system, and can be of particular importance in applications requiring the coupling of multiple electromagnetic modes to a single localized mechanical resonance. In addition, the small gap (≈25 nm) separating the two nanobeams gives rise to a slot optical mode effect that enables a large zero-point optomechanical coupling strength to be achieved, even for nanobeam materials with small refractive index contrasts such as Si3N4. Our design predicts optical quality factors above 106 and mechanical frequencies above 1 GHz for nanobeams in both the Si3N4 and Si systems. While the optomechanical coupling strengths predicted here are moderately superior to those calculated and experimentally determined from single nanobeam geometries, optical and mechanical resonances can be designed with considerably more independence and control.
The large predicted optomechanical coupling strengths to GHz mechanical oscillators in Si3N4 are particularly attractive, as this material displays a broad optical transparency window, which allows operation throughout the visible and near-infrared. The material also offers a low intrinsic mechanical dissipation rate, and does not exhibit the two-photon absorption and subsequent free-carrier absorption and dispersion observed in silicon [28], potentially allowing higher powers to be employed and increasing the range of achievable pump-enhanced optome-chanical coupling values. All of these features provide good prospects for the realization of radiation-pressure mediated photon-phonon translation [6].
Appendix Optomechanical wavelength conversion
The wavelength conversion scheme proposed in Sections 3.1 and 5.1 involve two optical resonances at frequencies ωo,1 and ωo,2, both of which are coupled to a mechanical resonance at frequency ωm. Both optical cavities are prepared with strong pumps, red-detuned from the resonance centers by the mechanical mode frequency. For each optical resonance, this preparation gives rise to an effective optomechanical interaction for signals at the cavity center described by a beam-splitter type interaction Hamiltonian Hint,eff = hḠ (â†b̂ + âb̂†), where â and b̂ are the destruction operators for cavity photons at the center frequency and phonons in the mechanical resonator. The parameter G is the pump-enhanced optomechanical coupling, G = g|αss|, where αss is the square root of the steady-state photon population at the pump frequency, and g the bare-cavity optomechanical coupling rate. It is apparent from the effective Hamiltonian that the interaction allows the exchange of signal photons and phonons with a rate dictated by the intensity of the pump beam. In the wavelength conversion process, a signal photon at frequency ω1 is injected into the (prepared) cavity 1, and converted to a cavity phonon. The process is reversible, such that the newly created phonon may be subsequently transduced to a photon in (also prepared) cavity 2, at frequency ωo,2. This process can be described classically via the matrix equation [12]
obtained from the Heisenberg equations for the bosonic operators â1,2 (for photons in cavities 1 and 2) and b̂. To arrive at Eq. (A-1), we replaced â1,2 → α1,2e−i(ωc;1,2−Δt), β̂ → βe−iΔt, where Δ is the detuning of the optical signal from the cavity center frequency ωc;1,2, and the rotating wave approximation was used. Noise sources were completely disregarded. In eq (A-1), κ1,2 = κex;1,2 + κi;1,2 are the decay rates for optical modes 1 and 2, comprised of intrinsic (κi, associated with e.g., radiative losses), and extrinsic (κex, associated with coupling to external access channels) components; γi is the intrinsic decay rate for the mechanical mode; αin and is the input optical field. Finally, it was assumed that the detuning between the pump and signal beams (at the input and output optical cavities) was equal to the mechanical frequency, Δ = ωm. The field output from cavity 2 is simply , so Eq. (A-1) can be solved to give where γ = γOM;1 + γOM;2 + γi and (γOM;1,2 are the optical spring contributions to the total mechanical mode damping γ), and η1,2 = κex;1,2/2κ1,2.The conversion efficiency, then, is where are the cooperativities for optical cavities 1 and 2. It is apparent that, for η → 1, C1 = C2 ≫ 1 and η1,2 → 1 are sufficient. A more complete derivation of the wavelength conversion process, including treatment of noise, is given in [12].Acknowledgments
This work has been supported in part by the DARPA MESO program. We thank Vladimir Aksyuk and Jeff Hill for valuable discussions.
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