1. Introduction

Fiber-coupled photonic circuits are powerful tools in our information infrastructure. In order to leverage these circuits to control and analyze light in our environment, we need to control how they radiate and absorb radiation. Gratings are an established way of controlling how photonic circuits radiate. They are dispersive: tuning the wavelength of light incident on a grating changes the angle at which it is scattered. With a tunable laser gratings can thus steer light in one dimension [1, 2]. When incorporated with phase-shifters into an array, they can steer in two dimensions [3–8]. The size, weight, power, and cost of these integrated beam-steering systems is lower than the motorized optical gimbals currently used with free-space optical systems for lidar, optical wireless communication [9, 10], and free-space optical interconnects [11]. The growing presence of autonomous systems, such as self-driving cars, motivates the development of mass-manufacturable photonic systems. With low-power on-chip beam-steering, a host of remote sensing, communication, and display applications comes into reach.

With angular dispersion as the steering mechanism the etched grating’s period Λ fixes a relation between optical wavelength λ and scattering angle θ. In contrast, with the ability to tune the grating period Λ, a monochromatic beam can be formed and directed. Sound is a naturally tunable optical grating. An acoustic wave containing multiple wavelengths scatters light into multiple angles as realized in pioneering work on acousto-optic beam deflectors [12–14]. A progression to guided-wave, collinear systems [15, 16] and arrays [17] in Ti-diffused and proton-exchanged lithium niobate waveguides enabled large fields of view for monochromatic light. These low index-contrast lithium niobate waveguides limit integrability, resolution, and efficiency. We address these limitations by embracing high index-contrast, subwavelength-scale silicon waveguides to be incorporated into a dense phased array. These waveguides – engineered to guide both light and sound – have recently been shown to exhibit strong acousto-optic interactions between propagating waves with tailorable dispersion [18–20].

Here we develop the concept of an optomechanical antenna (OMA) and present a perturbative description of the coupling between guided and radiated light by sound analogous to cavity optomechanics and Brillouin scattering [18–23]. After illustrating the scattering process for a slab waveguide, we explore the optical and mechanical co-design of an OMA compatible with silicon photonics and practical for a phased array antenna. We will show that such an OMA can scatter light in millimeters with only hundreds of microwatts of mechanical power. We conclude with an outline of this device’s performance, an account of the effects of disorder, and an outlook on the new capabilities of this approach.

2. Mechanics as a dynamic grating

In an optomechanical antenna [Fig. 1(a)] a guided optical wave with electric field E exp(iβz − iωt) is scattered by a guided mechanical wave with displacement field u exp (−iKz − iΩt) into radiating light with electric field E_r exp (ik·r − iω_rt). Energy and momentum conservation for the counter-propagating, anti-Stokes process

 

Fig. 1 (a) A mechanical wave with wavevector K scatters a guided optical wave (β) into free-space (k) at an angle θ. A single antenna scatters light into a cone. Sweeping the mechanical frequency, and therefore K, steers the cone through a range of angles θ. (b) Incorporating these antennas into a phased array forms a beam in the far-field directed into angle ϕ. Antennas consist of partially released silicon ridge waveguides each of which supports guided optical and guided mechanical modes. The displacement field u that scatters guided light (color representing the electric displacement field D_x) out at θ = 60°. (c) Light from a phase-shifter/splitter network (not shown) counterpropagates with mechanical waves driven by a piezoelectric transducer. An interdigital transducer (gray) is patterned on a piezoelectric film (red) deposited on SOI (blue on white). The transducer injects mechanical waves with nanometer-scale displacement and milliwatt-level power into the silicon waveguides.

Download Full Size | PDF

Analogous to the treatment of interactions in cavity optomechanics and Brillouin scattering, we perturbatively compute radiation from an OMA. Mechanical deformations vary the dielectric permittivity ε → ε+δ_uε·u with photoelastic and moving-boundary contributions to the scattering given explicitly in appendix C. Making a first Born approximation (see appendix C) [24], we have

Coupling to radiation causes decay of the optical power $\mathrm{P}(z)$ in the waveguide. From perturbation theory we find that the matrix element coupling E and Er scales as u = max|u|. By Fermi’s Golden Rule, the radiated optical power per unit length is α = α_mu² and therefore proportional to power in the mechanical wave ${\mathrm{P}}_{\mathrm{m}}$. Neglecting mechanical decay,

The scattering rate α_m is the rate of conversion between guided and radiating optical fields per unit length per the square of the mechanical deformation amplitude. In contrast to static gratings where the scattering rate is fixed by fabrication, modulating the mechanical power modulates the effective aperture $L_{\text{eff}}={\alpha }_{\mathrm{m}}^{-1}{u}^{-2}$ of an OMA.

3. Radiation of a slab waveguide

We begin by analyzing a simple OMA: a 220 nm thick silicon slab waveguide suspended in air. A typical scattering process is plotted in Fig. 2(a) where an antisymmetric mechanical Lamb wave scatters a counter-propagating guided transverse-electric (TE) optical mode into free space at θ = 45°.

 

Fig. 2 (a) Scattering from the TE-polarized optical mode of a 220 nm thick suspended silicon slab is computed nonperturbatively and the electric displacement field D_y is plotted where the maximal displacement u is set to 30 nm for visibility of the radiated field. The single-sided scattering rate α is quadratic in u even for large displacements. (b) The displacement-normalized scattering rates for the slab and ridge OMA are plotted together for u = 1 nm. Slab rates plotted in black and gray are computed nonperturbatively in 2D (dotted) as well as perturbatively in 2D (dot-dash) and 3D (dashed). For the ridge OMA of section 4, radiation into the optical slab modes (green), air (blue), and the silicon handle (red) add to give the total scattering rate (yellow). (c) Normalizing the scattering rate by power, here 1 mW per antenna for a ridge OMA array and 1 mW per 1.41 µm for the slab, gives a practical measure of the field of view Δθ. Tight confinement of the optical and mechanical energy in the transverse direction for the ridge OMA leads to enhanced radiation for a range of angles compared to the slab case.

Download Full Size | PDF

The slab waveguide is simple enough to admit to an analytical approach. A full coupled-mode description for roughness-induced scattering has been developed and is applicable to optomechanical scattering [25, 26]. We take a numerical approach easily extended to arbitrary geometries in which finite-element-method solutions of the uncoupled equations drive the inhomogeneous Eq. (3).

The scattering rates plotted in Fig. 2(b) show that nanometer-scale oscillations yield millimeter-scale effective apertures. Light which propagates with an effective index n_eff = 2.8 scatters out symmetrically above and below the waveguide in the phase-matched region when K/2π is between 1.2 µm⁻¹ (θ = 0°) and 2.5 µm⁻¹ (θ = 180°). The moving-boundary term dominates the optomechanical interaction while the photoelastic contribution is 50× smaller for this OMA (see appendix B.3). The interaction between TE guided light and Lamb waves is captured by an optomechanically-induced polarization current along $\widehat{y}$

Since the mechanical frequencies are much smaller than the optical frequency, they can be treated quasi-statically. Rather than the perturbative approach, the waveguide can be statically deformed by u and solved for the radiating field by frequency or time-domain methods. The quasi-static, nonperturbative approach agrees well with perturbative calculations and results from literature (see Appendix A).

While easily compared to scattering rates for etched gratings, the displacement-normalized α_m hides an important aspect of optomechanical antenna performance: the mechanical power. The fixed-power antenna functions [Fig. 2(c)] fall rapidly at higher angles θ since K and therefore ${\mathrm{P}}_{\mathrm{m}}\propto {\Omega }^2$ increase. In comparison to OMAs employing surface acoustic waves in lithium niobate [17], suspended structures are compliant and tightly confine the mechanical energy of their modes. This reduces the mechanical power ${\mathrm{P}}_{\mathrm{m}}$ needed to achieve a given scattering rate α_m by orders of magnitude.

4. Optomechanical antennas for silicon photonics

Having explored a two-dimensional optomechanical antenna, we add transverse structure to our calculations to yield a design for an OMA practical for an array and compatible with microelectronics manufacturing. In doing so, the mechanical waves of the slab are replaced by a multi-moded mechanical response of the core and socket of the waveguide in Fig. 1(b). Mixing between core and socket modes becomes an important feature of the antenna’s optomechanical response.

The OMA we describe is designed for 220 nm silicon-on-insulator (SOI) common in silicon photonics. A ridge waveguide is defined by a 170 nm partial etch, leaving a compliant 50 nm socket connecting the 450 nm wide optical core to the substrate. The ridge is partially released leaving a 1.33 µm wide suspended region. This width allows for subwavelength 1.41 µm pitched arrays with a transverse field of view Δϕ of up to 67°.

This OMA supports both guided optical and guided mechanical modes. At 193 THz optical waves are confined to the waveguide core and have an effective index n_eff = 2.33, well above the slab modes of the 50 nm SOI stack (hatched in Fig. 3(a)). The effective index sets the range of wavevectors K ∈ 2π × [0.86, 2.15] µm⁻¹ that phase-match β to free-space k. For these wavevectors, Lamb-like flexural modes of the waveguide have lower phase velocities than any other mechanical excitation in the system. They comprise bands that fall below the cone of surface and bulk waves of the 50 nm SOI stack, represented by the hatched region in Fig. 3. Consequently, they do not suffer mechanical radiation losses in the absence of disorder [20, 27].

 

Fig. 3 The ridge OMA guides light and sound supporting a range of mechanical modes over which the 193 THz optical mode (dot on black band of (a) E_y inset) can be scattered from 0° to 90° (shaded in blue). The mechanical power needed to achieve max |u_z| = 1 nm oscillations of the core is plotted above the bands of (b). Inset plots of the displacement show the low K behavior (red) where the mode resembles the fundamental Lamb mode of a uniform, clamped membrane. The first excited symmetric mode in the shaded region (black dot) is the mode of interest with scattering rates plotted in Fig. 2. After the anticrossing the mode is expelled from the core into the sockets (blue dot). In (a), the antisymmetric (with respect to y-reflection) and symmetric bands are plotted in black and red, respectively. In (b), symmetric and antisymmetric bands are plotted in black and red, respectively. In both band diagrams, the continuum of radiation modes of the 50 nm SOI stack are hatched.

Download Full Size | PDF

Mechanical modes of the structure can be understood in terms of waves in the sockets and waves in the core. Only motion of the core causes the antenna to radiate, directing our attention to the first excited symmetric band of Fig. 3(b). The fast band of the core mixes with the slow bands of the socket giving rise to avoided crossings at K/2π of 0.8 µm⁻¹ and 1.5 µm⁻¹. Below and above the avoided crossings, the sockets are mechanically decoupled by the core. This leads to nearly degenerate symmetric (black) and antisymmetric (red) bands. The motion of the core is suppressed in these degenerate bands, and therefore there is no optomechanical coupling.

The scattering rates are computed perturbatively and plotted in Fig. 2(b) alongside the results for a slab. The scattering rate α is peaked at θ ≈ 60° and falls off near the avoided crossings. Since the scattering rates are normalized by the maximum displacement of the core and not by power, these tails come from changes to the mechanical mode profile. Radiation into air is slightly weaker than into silicon. The latter is small but nonzero even when radiation into air is disallowed by phase-matching |k⁻¹(β − K)| > 1. The ridge OMA scatters out at nearly half the rate of the slab.

Despite the effects of mechanical mode mixing, the ridge OMA retains a large field of view as shown by the power-normalized antenna function of Fig. 2(c). For less than a milliwatt of mechanical power per antenna, corresponding to approximately 1 nm maximal displacement, scattering lengths of 2 mm can be achieved over Δθ = 13°. Doubling either the power or the scattering length more than doubles the field of view Δθ = 44°.

A microwave-driven, electromechanical transducer could excite the milliwatt-scale mechanical waves needed to efficiently scatter light out of the array. The coherent optical transduction schemes demonstrated in SOI would here exceed the power-handling thresholds for silicon waveguides. This follows from the discrepancy between optical and mechanical frequencies. A milliwatt mechanical wave corresponds to a phonon flux on the order of 10²¹ Hz. Achieving the same flux of optical photons requires 100 W whereas microwave photons can be converted to phonons with high efficiency by piezoelectric or capacitive electromechanical transducers. Our work motivates the integration of microwave-driven, electromechanical transducers into the silicon photonics platform [28].

5. Optics of optomechanical antennas

In the previous sections we designed an optomechanical antenna that can be integrated into a silicon photonic phased array. Here we discuss the main properties of its radiation pattern.

In the far-field the beam radiated from an OMA array is governed by Fraunhofer diffraction. For an ideal radiator where J_om does not vary in the longitudinal direction and remains coherent over a length L_eff, the far-field spot size is δθ = λ/(L_eff sin θ). The polar field of view Δ cos θ = ΔK/k is set by the range of wavevectors ΔK for which α is large as quantified in Fig. 2(c). A full system requires an efficient mechanical transducer and coupling structure with bandwidth ΔΩ over this range. Assuming negligible mechanical group velocity dispersion, the number of resolvable spots is the mechanical transit time-bandwidth product N_θ = Δθ/δθ = τ_mΔΩ/2π.

Since L_eff plays an important role in beam quality, we quantify sources of spatial decay and decoherence of J_om that limit L_eff. Variations in the amplitude and phase of J_om arise from optical and mechanical decay, as well as dephasing due to geometric disorder [29–32] and thermal fluctuations.

As light and sound propagate along the antennas, the phase of J_om accumulates an error δφ(z). This differs from beam-steering systems that use spatial light modulators [33], MEMS micromirrors [34, 35], or microlens arrays [36] where light interacts with the device only over a small distance. For fluctuations δβ(z) and δK(z) spatially correlated over ξ ≪ L_eff, the phase error δφ(z) diffuses along the antennas and is Gaussian-distributed with its variance growing linearly with z. We define L_φ as the length after which the phase variance 〈δφ²(z)〉 averaged along the antennas equals π². This dephasing length L_φ depends on the relative propagation direction of the guided optical and mechanical waves. In the counter-propagating case we find

Geometric fluctuations δX_l, indexed by l, shift β by δβ(z) = Σ_l∂_lβδX_l(z) such that for stationary noise with correlations

We compute the geometric sensitivities to fluctuations ∂_l β and ∂_lK for different types of pertubations and find height variations to be the dominant source of dephasing. These results are collected in table 2 along with corresponding dephasing lengths. Our finite-element models predict similar optical and mechanical sensitivities to height disorder. The mechanical and optical sensitivities are usually of opposite sign: mechanical waves speed up when optical waves slow. We compute the mechanical sensitivities at a wavelength of Λ = 842 nm and frequency of Ω/2π = 3.42 GHz such that the steering angle θ = 60° corresponds to the maximal scattering rate α [Fig. 2].

Table 1. Optomechanical antenna properties.

View Table | View all tables in this article

Table 2. Dephasing in a ridge OMA.

View Table | View all tables in this article

For the dephasing length estimates in table 2 the standard deviations σ_l are based on measurements of similar silicon photonic circuits [29–32] and correlation lengths ξ_l of 50 µm are assumed (see appendix D). In the purely optical case, the overall geometric dephasing length is L_φ,β = 3.4 mm. For counter-propagating optical and mechanical waves we have L_φ = 1.7 mm: about half of L_φ,β as the optical and mechanical phase errors add incoherently. Further, the dephasing length for copropagating fields L_φ,co = 0.9 mm is about four times smaller than L_φ,β as the optical and mechanical phase errors subtract coherently.

Temperature gradients across the system also shift the phases of the guided optical and mechanical fields. Spatially inhomogeneous temperatures are analogous to geometric disorder. A temperature gradient ΔT across the antenna array results in thermal dephasing lengths of L_T = 2π/(|∂_T (β + K)|ΔT) in the counterpropagating and L_T,co = 2π/(|∂_Tk_ǁ|ΔT) in the copropagating case (see appendix D). Careful thermal management can likely limit spatial temperature gradients to 5 K across the array [37] so we find L_T ≈ 1.5 mm in either case (table 2). Optical and mechanical crosstalk between the waveguides in a phased array is another source of phase errors. Crosstalk splits the wavevectors of symmetric k₊ (⋯ ++++ …) and antisymmetric k₋ (⋯ + − + − …) array supermodes, causing them to dephase after a length L_x = π/(k₊ − k₋). A 1.41 µm pitch array has an optical crosstalk length of L_x = 7.8. mm.

Larger apertures yield narrower spots and higher resolution at the cost of the optical and mechanical modulation bandwidth. Although our control over mechanical wavevector K enables beam-steering at fixed optical frequency, optomechanical antennas are still dispersive. The optical bandwidth Δω at fixed mechanical frequency is set by how much ω can be changed before a spot shifts by δθ. We find Δω/2π = 1/(τ − τ_r) with τ = L_eff n_g/c the transit time of the guided optical wave where n_g is the group index, and τ_r = L_eff cos θ/c the transit time of the radiating field across the aperture (see appendix F). Similarly, the mechanical transit time τ_m determines the mechanical bandwidth ΔΩ_m/2π = 1/τ_m within a spot at fixed optical frequency.

Estimates of the antenna properties above for the ridge OMA are provided in table 1. We assume an effective antenna length of 2 mm limited by dephasing. The field of view is from θ of 35° to 79°. We compute the optical bandwidth around θ = 60° with an optical group index of 4.3 and a mechanical group velocity of 4135 m/s.

6. Nonreciprocity and multiplexing mechanics

In addition to two-dimensional beam-steering, active gratings generated by mechanical waves naturally implement certain functionalities that are not intrinsically present in other approaches.

First, the time-varying grating generated by a unidirectional mechanical field breaks reciprocity in the structure. Therefore the system operates as a nonreciprocal metasurface [38]: light that is scattered out is shifted up in frequency by Ω and the light coming back is shifted up again in frequency by Ω such that the roundtrip optical frequency shift is 2Ω [Fig. 4]. Therefore the system naturally includes a frequency-shifting function that can be used for heterodyning with an on-chip local oscillator.

 

Fig. 4 (a) On the transmit side, the optical wave at ω is shifted to an outgoing optical wave at frequency ω + Ω. (b) The backscattered light now has an additional Doppler shift of δω. This incoming wave mixes with the same mechanical wave and excites the optical waveguide at frequency ω + 2Ω + δω. (c) The resulting optical spectrum allows us to distinguish between light being sent and received, moving the received beam outside of the laser phase noise. (d) The outgoing radiation can be multiplexed dynamically by exciting a superposition of mechanical waves. Each of the outgoing beams can be controlled independently. As a special case, this implies widely different optical wavelengths can be sent to and received from the same spot.

Download Full Size | PDF

Second, even at a fixed optical wavelength we can inject a superposition of mechanical waves with different wavevectors. Each of these mechanical waves generates an outgoing beam at a separate angle that can be controlled, sent to different targets, and read out independently [Fig. 4(d)]. As a special case, this enables multiple optical wavelengths to be sent to and received from a single angular spot. Such functionality may prove useful in the realization of free-space communication links [11, 39], (holographic) video displays [14, 40], remote sensing [41–43], and coherent imaging [44, 45].

7. Conclusion

In conclusion, we propose an on-chip, two-dimensional beam-steering system compatible with standard microelectronics processes based on guided mechanical waves. We design hybrid photonic-phononic waveguides whose mechanical excitations can travel on the surface of a silicon-on-insulator chip. The propagating mechanical fields – acting as active gratings – convert between guided and radiating optical fields in a rapidly reconfigurable way. Efficient optical mode conversion can be realized in millimeter-scale apertures with low mechanical drive power. The system can steer monochromatic light over a large field of view; distinguish between outgoing and incoming light through a nonreciprocal frequency shift; and control the beam direction, beam shape, and the number of beams. More generally, we have shown that subwavelength control of photons and phonons enables low-power, dynamic control of light.

Appendix A. Simulation methods

A.1 Nonperturbative method of computing scattering rates

We perform nonperturbative scattering simulations with the finite-element solver COMSOL [46] solving Maxwell’s equations in either 2D or 3D in the frequency domain. The resulting eigenvalue problem has nearly guided solutions. Light scattered out of the slab is absorbed by a perfectly matched layer, causing the eigenvalue to become complex $\omega \to \omega -i{\cdot }_{\underset{\_}{2}}^{\kappa }$. In this section we show that the scattering rate α is related to κ as

(10)$$\alpha =\frac{\kappa }{v_{\mathrm{g}}}=\frac{\omega }{Q{v}_{\mathrm{g}}}$$

where v_g = c/n_g is the optical group velocity.

Consider an optical waveguide with energy per unit length $ℰ$, power transmitted down the waveguide $\mathrm{P}$, and power scattered out of the waveguide per unit length ${\mathrm{P}}_s$. We would like to relate ${\mathrm{P}}_s$ to the attenuation rate α. Energy conservation yields

(11)$${\partial }_tℰ=-{\partial }_z\mathrm{P}-{\mathrm{P}}_s$$

which for steady state ∂_t → 0 becomes ${\partial }_z\mathrm{P}-{\mathrm{P}}_s$. Since

(12)$$Q=\frac{\omega }{\kappa }\text{and}\kappa =\frac{{\mathrm{P}}_s}{ℰ},$$

our statement of energy conservation becomes

(13)$${\partial }_z\mathrm{P}=-\frac{\omega }{Q}ℰ$$

which can be reexpressed using $v_{\mathrm{g}}=\frac{\mathrm{P}}{ℰ}$ as

(14)$${\partial }_z\mathrm{P}=-\underset{\equiv \alpha }{\underbrace{\frac{\omega }{Q{v}_{\mathrm{g}}}}}\mathrm{P}.$$

Therefore, the optical power decays exponentially at a rate α that can be expressed in terms of the imaginary part of the eigenvalues of our numerical solutions.

A.2 Verification of the model

A.2.1 Literature grating simulation

We test our nonperturbative computational approach by modeling two wavelength-scale periodic optical structures from the literature: (1) a high index-contrast, strong silicon-on-insulator grating coupler [47] used in imec’s silicon photonics pilot line [48] and (2) a low index-contrast, weak grating coupler [49].

A.2.2 PML tests

We implement the perfectly matched layers (PMLs) at the bottom and top of the simulation as an imaginary part of the refractive index $\mathfrak{J}n$ starting a distance x_start = 6 µm above and below the waveguide. For |x| > x_start the strength of the PML is set by

(15)$$\mathfrak{J}n={\left(\frac{\left|x\right|-x_{\text{start}}}{x_{\text{pml}}}\right)}^2$$

such that $\mathfrak{J}n=1$ at a distance |x| = x_start + x_pml = 15 µm away from the waveguide. The computational domain ends at $\left|x\right|=x_{\text{end}}=\frac{d}{2}+20\mu \mathrm{m}$ with perfectly conducting boundaries and d = 220 nm the typical core thickness. We set the maximum mesh size in the core and free-space domains at 10 nm and 150 nm. With these parameters, the simulation time is about 20 s for a wavelength of 0.63 µm. We check PML operation with the following tests, all for the SOI grating coupler of Fig. 5(a) and for a perturbation of u = 5 nm. The scattering rate α ≈ 1.42/mm in these tests [Fig. 5(a) and 6].

 

Fig. 5 We compute scattering rates for strong and weak gratings previously demonstrated to verify our numerical methods. The scattering strength scales quadratically with the perturbation u. a, Simulation of a strong SOI grating [47] that has been tested experimentally [47, 48]. The group index of the optical mode is n_g = 3.6 and the Bloch index is 2.85. The grating pitch is 630 nm and the core thickness is 220 nm. b, Simulation of a weak grating [49]. The group index of the optical mode is n_g = 3.4 and the Bloch index is 3.29. The grating pitch is 10 µm and the core thickness is 500 nm.

Download Full Size | PDF

First, we investigate the field intensity in the free-space domain. For 1 µm < x < x_start the field remains roughly constant, confirming that the PML mainly measures the radiation power and not the evanescent field of the guided mode. Next, we sweep the size of the computational domain x_end [Fig. 6(a)]. The scattering rate saturates fast, likely because of reduced reflections off the perfectly conducting boundaries, and stays constant up to 10⁻⁴ fractionally afterwards. Second, we sweep x_pml and thus the PML strength [Fig. 6(b)]. The scattering rate decreases rapidly until x_pml > 2 µm and then fluctuates at 10⁻⁴ fractionally. Third, we sweep x_start and x_pml = x_start + 3 µm while keeping other parameters constant [Fig. 6(c)]. The scattering rate oscillates initially and then stays constant up to 10⁻⁴ fractionally. Fourth, we sweep the maximum mesh element size in the free-space domains [Fig. 6(c)]. The scattering rate again fluctuates at 10⁻⁴ level fractionally.

 

Fig. 6 The scattering rate is insensitive to changes in the PML. a, A sweep of simulation size x_end shows initially an increase and then saturation of the scattering rate α when x_end > 10 µm. b, The scattering rate drops fast with PML curvature x_pml and saturates beyond x_pml > 2 µm. c, The scattering rate remains constant when x_start > 3 µm. d, Coarse mesh size does not strongly impact scattering rate below 200 nm. Similarly, the fine mesh size does not affect the scattering rate below 20 nm.

Download Full Size | PDF

Our standard operating parameters are all chosen in these regions of 10⁻⁴ fractional sensitivity to discretization and PML parameters. Therefore, we expect simulation results accurate at percent-level at least for decay rates far above $\frac{\kappa }{2\pi }\approx 10\text{MHz}$, quality factors below Q ≈ 10⁷, scattering rates above α ≈ 10/m and decay lengths below α⁻¹ ≈ 10 cm. We determine all scattering rates a factor 10³ to 10⁴ away from these thresholds. The quadratic scaling of scattering rate with respect to perturbations enables extrapolation to smaller perturbations u where necessary.

A.2.3 FDTD calculations

In addition to the COMSOL-based frequency-domain models, we also developed Lumerical-based finite-difference time-domain (FDTD) [50] 2D and 3D models to compute the scattering rates and radiation patterns. Here we inject a pulse with a bandwidth of 5 to 10 nm and determine the scattering rate α from the exponential decay of the guided power. Lumerical has built-in functions that allow for a straightforward determination of the electromagnetic beam in the far-field and thus the angles and strengths of the first-, second- and higher-order grating lobes. The results generally agree with the COMSOL-based frequency-domain approach described above and in the main text. We focus on the faster frequency-domain simulations.

Appendix B. Silicon slab in air with sinusoidal perturbation

B.1 Scattering rate comparison

In this section, we investigate the scattering rates of suspended silicon-on-insulator slabs in greater detail.

Table 4 shows a comparison of the scattering rates of five types of perturbations to a 220 nm thick silicon slab waveguide. We perform these calculations at fixed frequency of 193.5 THz and fixed scattering angle of 6 7°. The Bloch index of the optical slab mode is about 2.82 and the grating pitch is 630 nm. From top to bottom, the first grating is the silicon-on-insulator grating of section A.2.1. It has only a top surface rectangular perturbation. The second grating has air both above and below the core. Its scattering rate is slightly higher owing to the increased index-contrast. The third structure has a symmetric perturbation to the top and bottom surfaces. Its scattering rate for a δ = 1 nm perturbation is about four orders of magnitude below that of the SOI grating coupler that breaks vertical symmetry. In addition, its scattering rate scales with δ⁴ instead of the usual δ² – even though second-order scattering (satisfying β − 2K = k cos θ_2nd) is not phase-matched for any angle θ_2nd when θ = 67°. The suppressed scattering and δ⁴-scaling arise from destructive cancellations in the fields scattered from the top and bottom surfaces, see Fig. 7 and discussion below for details. The fourth grating is identical except in that it has an antisymmetric perturbation to the top and bottom surfaces. The scattering rate α is a factor $\frac{0.31}{0.073}=4.3$ larger than that of a grating with a perturbation only on the top surface. The enhanced scattering arises from constructive interference between the fields radiated by top and bottom perturbations [Fig. 7]. The fifth grating has a sinusoidal instead of a rectangular perturbation. It is nearly identical to the dynamic mechanical field we propose to excite. The first Fourier component of a rectangular signal is $\frac{2}{\pi }$, so the scattering by the sinusoidal perturbation is a factor (π/2)² = 2.4 stronger. The sinusoidal symmetric perturbation to a 220 nm-thick suspended silicon slab thus has an overall enhancement of a factor 10.2 with respect to grating coupler with air below and a factor 12.5 compared to a typical silicon-on-insulator grating coupler.

Table 3. References [47] and [49] provide the scattering rate α as a function u for these gratings. We compute α_m by estimating a couple of points in their figures in a similar range of u. Our model agrees with the literature result up to 3% for the strong grating and up to 5.6% for the weak grating.

View Table | View all tables in this article

Table 4. The scattering rate varies several orders of magnitude depending on the type of perturbation. A symmetric, sinusoidal perturbation has the largest scattering rate in a 220 nm thick silicon slab. For some perturbations the scattering rate scales with u⁴ instead of u² because of destructive cancellations in the term proportional u². In general the scattering rate contains all terms u^2l with l any positive integer. Here we use the perturbation-normalized quality factor as Q_m = ω/(α_mv_g).

View Table | View all tables in this article

 

Fig. 7 The scattering rate α_m caused by the antisymmetric sinusoidal perturbation is maximal at thicknesses $d=\left(l+\frac{1}{2}\right){\lambda }_{\text{Si}}$ with ${\lambda }_{\text{Si}}=\frac{\lambda }{n_{\text{Si}}}=443\text{nm}$ the optical wavelength in silicon and l a positive integer, while it vanishes at d = lλ_Si. A symmetric sinusoidal perturbation yields the opposite behavior. This behavior is caused by interference between the scattered field from the top surface and that scattered from the bottom surface. As the waveguide core becomes thicker, the structure is less sensitive to either type of perturbation. One simulation point just above a thickness of 600 nm was removed manually as the script had selected the wrong optical mode. We computed both curves for a perturbation u = 10 nm.

Download Full Size | PDF

B.2 Thickness dependence of scattering rate

Next, we compute the scattering rate as a function of waveguide thickness for sinusoidal symmetric and antisymmetric perturbations [Fig. 7] for the same parameters as in subsection B.1. We find the scattering rate α for the antisymmetric perturbation to be maximal when

(16)$$d=\left(l+\frac{1}{2}\right){\lambda }_{\text{Si}}$$

with d the core thickness, l an integer and ${\lambda }_{\text{Si}}=\frac{\lambda }{n_{\text{Si}}}=443\text{nm}$ the optical wavelength in silicon. The scattering vanishes when

(17)$$d=l{\lambda }_{\text{Si}}$$

while the core with a symmetric perturbation exhibits the reverse behavior. The effect arises from interference between the scattered fields generated by the top and bottom slab surfaces. For an antisymmetric perturbation, there is another 180° shift in the phasing of the top and bottom scatterers. Therefore, constructive interference occurs when the thickness is a multiple of a wavelength plus an additional half-wavelength to compensate for the 180° phasing of the sources. Interestingly, the global maximum in the scattering rate of an antisymmetric perturbation – similar to that of a Lamb-type mechanical field – occurs exactly at a core thickness of d = 220 nm. In addition, this antisymmetric sinusoidal perturbation offers the strongest scattering rates of all the perturbation types (table 4). Hence antisymmetric Lamb-like flexural mechanical fields propagating along a 220 nm-thick suspended silicon waveguide are ideal excitations for coupling guided and free-space optical fields.

B.3 Photoelasticity

Even in absence of a geometric boundary perturbation, a propagating mechanical wave generates an inohomogeneous strain profile that couples guided optical fields to radiating fields. This is termed the photoelastic contribution α_pe to the total scattering rate α_tot. The scattering rates reported above include only the boundary-induced scattering α_mb. Generally speaking, the two contributions may be of similar size and thus interfere with one another – either enhancing the total scattering rate or potentially completely canceling it [18, 23, 51]. However, in a simulation without the boundary perturbation our eigenfrequency model predicts a photoelastic scattering rate of α_pe = 1.4 · 10⁻²/(mm nm²) – nearly two orders of magnitude smaller than α_mb = 0.74/(mm nm²) (table 4) for a suspended 220 nm-thick silicon slab. Thus the photoelastic component of the scattering is weak in the considered geometry: even in case of completely destructive or constructive interference the total scattering rate

(18)$$\frac{{\alpha }_{\text{tot}}}{{\alpha }_{\text{mb}}}={\left(1\mp \sqrt{\frac{{\alpha }_{\text{pe}}}{{\alpha }_{\text{mb}}}}\right)}^2=1_{-0.26}^{+0.29}$$

would change by less than 30%. Next, we simulate the combined scattering rate resulting from interference between the moving-boundary and photoelastic scattering. We find that the two effects interfere constructively such that α_tot = 0.96/(mm nm²) – about 30% larger than α_mb.

In this computation of α_pe and α_tot we implemented the photoelasticity as an anisotropic refractive index profile with components

(19)$$n_{\text{core}}=\left(\begin{array}{ccc}{n}_{\text{Si}}+\Delta n_{xx}& 0& \Delta n_{xz}\\ 0& n_{\text{Si}}+\Delta n_{yy}& 0\\ \Delta n_{xz}& 0& n_{\text{Si}}+\Delta n_{zz}\end{array}\right)$$

where the index variations Δn are given by

(20)$$\Delta n_{xx}=-\frac{1}{2}{n}_{\text{Si}}^3{p}_{12}{S}_{zz}$$

(21)$$\Delta n_{yy}=-\frac{1}{2}{n}_{\text{Si}}^3{p}_{12}{S}_{zz}$$

(22)$$\Delta n_{zz}=-\frac{1}{2}{n}_{\text{Si}}^3{p}_{11}{S}_{zz}$$

(23)$$\Delta n_{xz}=-n_{\text{Si}}^3{p}_{44}{S}_{xz}$$

with (p₁₁, p₁₂, p₄₄) = (−0.09, 0.017, −0.051) the photoelastic tensor of silicon assuming waveguide orientation along a 〈100〉 axis. The waveguide orientation has a minor effect on the effective photoelastic components in silicon. The strain components are given by

(24)$$S_{zz}={\partial }_z{u}_z=-K^{2}xu\sin Kz$$

(25)$$\begin{array}{l}{S}_{xz}=\frac{1}{2}({\partial }_z{u}_x+{\partial }_x{u}_z)\\ =S_{zx}=0\end{array}$$

where K is the mechanical wavevector, u the maximum mechanical perturbation and (u_x, u_z) = u(sin Kz, −Kx cos Kz) a snapshot of the displacement field of an antisymmetric Lamb-wave propagating along a thin slab opposite to the z-direction. The origin of the transverse coordinate x is taken to be in the center of the slab, such that |x| = d/2 corresponds to the silicon/air interfaces with d the slab thickness. This analytical Lamb-wave solution assumes Kd ≪ 1. Finite-element simulations showed that the actual mechanical field around Kd ≈ 1 is still captured well by this analytical approximation. In these simulations we swept u and then obtained α_pe and α_tot from a fit to u² as in Fig. 5.

In general a full 3D simulation of the combined effects of moving boundaries and photoelasticity must be developed and is possible in our eigenfrequency approach. We have however limited our current 3D simulations to the moving-boundary effect given the expected weakness of photoelasticity in this system. We suspect that this weakness is caused by the distributed nature of the photoelastic scattering, leading to destructive interferences in the outgoing radiation similar to Fig. 7.

Appendix C. Perturbation theory for optomechanical antennas

In cavity optomechanics and Brillouin scattering, the optomechanical interaction is described perturbatively by expanding the permittivity in terms of the mechanical deformation

(26)$$\epsilon \to \epsilon +{\delta }_u\epsilon \cdot \mathbf{u}+\mathcal{O}({\mathbf{u}}^2).$$

We can take the same approach to describe scattering out of a waveguide. After a Fourier transform ∂_t → −iω, Maxwell’s equations for the electric field reduce to

(27)$$(\nabla \times \nabla \times -\mu \epsilon {\omega }^2)\mathbf{E}=i\omega \mu \mathbf{J}$$

where J is a current density. We are interested in a current-free waveguide with a mechanically perturbed permittivty

(28)$$\underset{\Theta }{\underbrace{(\nabla \times \nabla \times -\mu \epsilon {\omega }^2)}}\mathbf{E}=\mu {\omega }^2({\delta }_u\epsilon \cdot \mathbf{u})\mathbf{E}.$$

We first solve the uncoupled equations – Maxwell’s and the theory of elasticity – for the optical and mechanical modes of the waveguide. The unperturbed electric field satisfies ΘE = 0, which can be rewritten as a generalized eigenvalue problem of the form AE = ω²BE with Hermitian operators A and B. Expanding E → E + E_r + … and ${\omega }^2\to {\omega }^2+{\omega }_1^2+\dots$ the perturbed field is given by

(29)$$\Theta {\mathbf{E}}_{\mathrm{r}}=\mu \epsilon {\omega }_1^2\mathbf{E}+\mu {\omega }^2({\delta }_u\epsilon \cdot \mathbf{u})\mathbf{E}$$

to first order in u. Since Θ is Hermitian, Θ|Ei = 0 implies that 〈E|Θ|E_r〉 = 0. Taking the inner product with 〈E〉 on both sides yields the first order correction to the eigenvalue

(30)$${\omega }_1=-\frac{\omega }{2}\frac{〈\mathbf{E}|{\delta }_u\epsilon \cdot \mathbf{u}|\mathbf{E}〉}{〈\mathbf{E}\left|\epsilon \right|\mathbf{E}〉}$$

where we’ve adopted Dirac notation and substituted ${\omega }_1^2\to 2\omega {\omega }_1$. For proper choice of inner product and normalization of u, ω₁ becomes the coupling rate of cavity optomechanics g₀ or Brillouin scattering ${\tilde{g}}_0$ [21,52,53].

We can express the outgoing field fully in terms of Θ and the unperturbed fields

(31)$$\Theta {\mathbf{E}}_{\mathrm{r}}=\mu {\omega }^2({\delta }_u\epsilon \cdot \mathbf{u})\mathbf{E}.$$

The optomechanical interaction above can be expressed in terms of a current

(32)$${\mathbf{J}}_{\text{om}}=-i\omega ({\delta }_u\epsilon \cdot \mathbf{u})\mathbf{E}$$

allowing us to rewrite the remaining inhomogeneous equations simply as

(33)$$\Theta {\mathbf{E}}_{\mathrm{r}}=i\omega \mu {\mathbf{J}}_{\text{om}}.$$

C.1 The moving-boundary and photoelastic effect

Next we present an explicit expression for the nonlinear polarization current J_om. Mechanical waves give rise to changes in the permittivity ε which can be expressed to first order in u in terms of body (i.e. photoelastic effect [54]) and boundary contributions. The latter requires careful handling to manage discontinuities in the field at boundaries of dielectrics treated by Johnson et al. [52, 55]. This variation in the permittivity δ_uε · u, which is familiar in the fields of cavity optomechanics and Brillouin scattering, is here a tensor which acts on E to give the polarization currents J_om above.

Consider a domain Ω with boundary ∂Ω deformed by u. The normal n points out of the domain such that for positive u·n the permittivity of a region in the neighborhood of the boundary changes by Δε ≡ ε_in − ε_out. The main trick in forming a well-defined expression for the radiation pressure on the boundary is to avoid field discontinuities by replacing the component of E normal to ∂Ω with the electric displacement field D. The boundary contribution then becomes

(34)$${\delta }_u{\epsilon }_{\text{rp}}\cdot \mathbf{u}=(\mathbf{u}\cdot \mathbf{n}){\delta }_{\partial \Omega }\left(\Delta \epsilon {\Pi }_{\parallel }-\epsilon \Delta {\epsilon }^{-1}\epsilon {\Pi }_{\perp }\right)$$

which is expressed in terms of the tensors Π_ǁ and ${\Pi }_{\perp }=\mathbb{I}-{\Pi }_{\parallel }$ which project the electric field into the plane of the dielectric interface or along the normal n, respectively. Here $\Delta {\epsilon }^{-1}\equiv {\epsilon }_{\text{in}}^{-1}-{\epsilon }_{\text{out}}^{-1}$ and the delta function on the boundary δ_∂Ω renders (δ_uε · u) E into a surface current on ∂−Ω.

Suppose the normal is oriented in Cartesian coordinates along z and the dielectric boundary is at z = 0. Then

(35)$$({\delta }_u{\epsilon }_{\text{rp}}\cdot \mathbf{u})\mathbf{E}=u_z\delta (z)\left[\begin{array}{c}\Delta \epsilon E_x\\ \Delta \epsilon E_y\\ -\epsilon \Delta {\epsilon }^{-1}{D}_z\end{array}\right].$$

The z component of the expression above poses some difficulty as ε is discontinuous as z = 0. This is discussed in the next section.

Although in this work estimates of the photoelastic effect justified dropping it from our calculations, we give its contribution to the variation of ε below for completeness. The photoelastic tensor in common use is defined such that a strain S causes $\epsilon \Delta {\epsilon }_{ij}^{-1}={(\mathbf{pS})}_{ij}=p_{ijkl}{S}_{kl}$. To first order in u,

(36)$${\delta }_u{\epsilon }_{\text{pe}}\cdot \mathbf{u}=-\epsilon \frac{\mathbf{pS}}{\epsilon }\epsilon .$$

C.2 Implementing the boundary contribution to J_om

The boundary contribution to the scattering process is delta-distributed across the boundary. In the plane of a dielectric interface – along the x and y axis in Eq. (35) – the perturbation behaves like a surface current giving rise to discontinuities in the magnetic field

(37)$$\mathbf{n}\times \Delta \mathbf{H}={\mathbf{J}}_{\parallel }.$$

The J_⊥ component of the surface current isn’t implemented as a set of boundary conditions. Instead J_⊥ is taken to be a uniform volume current density of finite thickness t J = 5 nm just inside the boundaries of our silicon waveguides, normalized by the thickness so as to converge to a delta function the limit t_J → 0.

The expression for J_⊥ is the product of a step and a delta function at the boundary, and volume approximations of the delta function therefore require some choice of ε in the z component of Eq. (35). Although J_⊥ is discontinuous, the power sourced into the field −E · J_om is continuous, making the radiated field robust to the exact distribution used for J_⊥ [55].

We check our implementation of J_⊥ by computing the OM radiation of the fundamental TM modes of 220 nm, 340 nm, and 440 nm silicon slab waveguides both perturbatively and nonperturbatively. The results are displayed in Fig. 8.

Appendix D. Geometric disorder and dephasing

Geometric disorder has been studied extensively in nanophotonic circuits [29, 31, 32, 56–58] to understand optical propagation losses. It has also been investigated in the context of Brillouin scattering – where it leads to broadening of the mechanical resonance [18, 19, 23, 59]. In both cases the standard deviation of geometric disorder was generally estimated at σ ≈ 1 nm with the largest disorder occurring near etched sidewalls. In studies focusing on optical scattering losses, the coherence lengths ξ were found to be below 100 nm [31, 57]. Such short coherence lengths are not consistent with (1) wafer-scale geometric disorder [56], (2) millimeter-scale optical dephasing lengths [58] nor with (3) the Brillouin resonance broadening observed in silicon waveguides [18,23,59]. Therefore, we suspect the spatial correlator to contain at least two terms: (1) a fast-disorder term with a coherence length of about 50 nm and (2) a slow-disorder term with a coherence length of about 50 µm. The fast roughness mainly determines optical radiation loss and backscatter – both of which require large roughness momentum – while the slow drift mainly determines the dephasing as it builds up over many wavelengths. We expect slow disorder to be the main hurdle for the proposed device and thus tabulate the various sensitivities in the main text using ξ ≈ 50 µm.

In the main text, we provided the results of our analysis of dephasing. Here we discuss the derivations in detail. There are eight cases to be investigated: the out- vs. incoupling, anti-Stokes vs. Stokes and counter- vs. copropagating cases could each be combined. Four of these cases suffer from low efficiencies due to a large phase-mismatch. The remaining four cases are (1) outcoupling by anti-Stokes scattering between counter-propagating guided waves, (2) outcoupling by Stokes scattering between copropagating guided waves, (3) incoupling by Stokes scattering between copropagating guided waves and (4) incoupling by anti-Stokes scattering between counter-propagating guided waves. Cases (3) and (4) are the time-reversed versions of cases (1) and (2) respectively. Thus they have identical properties and we limit ourselves to cases (1) and (2) in the following.

Case (1) is attractive as it allows for optical and mechanical excitation from opposite sides of the array. In anti-Stokes scattering phase errors of the guided optical and the guided mechanical wave add, yielding a total phase error of

(38)$$\delta \phi (z)={\displaystyle {\int }_0^{z}d{z}^{\prime }\delta \beta (z^{\prime })}-{\displaystyle {\int }_L^{z}d{z}^{\prime }\delta K(z^{\prime })}$$

The optical phase error grows forwards while the mechanical phase error grows backwards. The variance of the phase error is

(39)$$\begin{array}{c}〈\delta {\phi }^2(z)〉={\displaystyle {\int }_0^z{\displaystyle {\int }_0^{z}d{z}^{\prime }d{z}^{″}〈\delta \beta (z^{\prime })\delta \beta (z^{″})〉}}\\ +{\displaystyle {\int }_L^z{\displaystyle {\int }_L^{z}d{z}^{\prime }d{z}^{″}〈\delta K(z^{\prime })\delta K(z^{″})〉}}\\ +2{\displaystyle {\int }_0^z{\displaystyle {\int }_0^{L}d{z}^{\prime }d{z}^{″}〈\delta \beta (z^{\prime })\delta K(z^{″})〉}}.\end{array}$$

To compute these terms, we expand δβ = Σ_l∂_lβδX_l and δK = Σ_l∂_lKδX_l and assume a spatial correlator for the disorder

(40)$$〈\delta X_l(\Delta z)\delta X_l(0)〉={\sigma }_l^2{e}^{-\left|\Delta z\right|/{\xi }_l}.$$

With this explicit form for the noise correlator, the first term becomes

(41)$$\begin{array}{l}{\displaystyle {\int }_0^z{\displaystyle {\int }_0^z\mathrm{d}{z}^{″}\mathrm{d}{z}^{\prime }{e}^{-\left|\Delta z\right|/{\xi }_l}}=2{\xi }_l(z-{\xi }_l+{\xi }_l{e}^{-z/{\xi }_l})}\\ \stackrel{z\gg {\xi }_l}{\approx }2{\xi }_{l}z\end{array}$$

with Δz = z′ − z″ and similarly for the δK-term. For z ≪ ξ_l and L − z ≫ ξ_l the δβδK cross-term in Eq. (39) is negligible. Substituting Eq. (41) into Eq. (39) we find

(42)$$〈\delta {\phi }^2(z)〉={\displaystyle \sum _{l}2{\partial }_l{\beta }^2{\sigma }_l^2{\xi }_{l}z+2{\partial }_l{K}^2{\sigma }_l^2{\xi }_l(L-z)}.$$

Above we relied on an explicit form for the correlator and simplified the result in the limit where z ≫ ξ_l. In this limit a more general expression for 〈δφ²(z)〉 can be derived in terms of the power spectral density of the disorder S_ηη for noise process η. As a result of the central limit theorem, any stationary noise process of finite variance integrated over lengths greater than the noise’s correlation length is Gaussian-distributed. The variance of this integrated process equals the product of S_ηη [⁰] and the propagation length [60]. For example fluctuations δβ in the optical wave vector are integrated over the length of the waveguide z contributing to the phase error δφ as a Gaussian with variance S_ββ[⁰]z. Here the power spectral density S_ββ [ω] of δβ is

(43)$$S_{\beta \beta }[\omega ]={\displaystyle {\int }_{-\infty }^{+\infty }dz{e}^{i\omega z}〈\delta \beta (z)\delta \beta (0)〉}.$$

Again dropping the δβδK crossterm, it follows that

(44)$$〈\delta {\phi }^2(z)〉=S_{\beta \beta }[0]z+S_{KK}[0](L-z)$$

as reported in section 5.

We can check that Eq. (42) and Eq. (44) match. The power spectrum for Eq. (40) is

(45)$$S_{\beta \beta }[\omega ]={\displaystyle \sum _l\frac{2{\left({\partial }_l\beta {\sigma }_l\right)}^2{\xi }_l}{1+{\left(\omega {\xi }_l\right)}^2}.}$$

If we plug this expression at ω = 0 and the corresponding one for δK into Eq. (44) we find Eq. (42).

In case (2), Stokes scattering implies that the guided optical and guided mechanical phases subtract such that

(46)$$\delta \phi (z)={\displaystyle {\int }_0^{z}d{z}^{\prime }\delta k_{\parallel }}$$

with k_ǁ = β − K so one can show as in the above that

(47)$$〈\delta {\phi }^2(z)〉=S_{k_{\parallel }{k}_{\parallel }}[0]z$$

The thermal dephasing lengths in the main text were derived along similar lines but with constant δβ and δK along the antennas.

Appendix E. Mechanical losses

Material losses cause mechanical waves to decay at a rate γ = Ω/(Q_mv_m) where Q_m is the mechanical quality factor and v_m is the mechanical group velocity. At room temperature, nonlinear phonon processes limit mechanical Qs of silicon resonators to Q ≈ 10⁴ [61]. Waves in the ridge OMA presented in the text at K = 2π × 1.2 µm⁻¹ which scatters light at θ = 60° have a frequency Ω = 2π × 3.42 GHz and group velocity v_m = 4135 m/s. Assuming a quality factor Q_m = 3 × 10³, the resulting decay length is γ⁻¹ = 580 µm.

Attenuation of the mechanical waves modifies the optical scattering rate α from the antenna and thereby the radiation pattern of an OMA. Consider the counter-propagating optical and mechanical waves of the anti-Stokes process for which

(48)$${\partial }_z\mathrm{P}=-{\alpha }_{\mathrm{m}}{\mathrm{P}}_{\mathrm{m}}\mathrm{P}$$

(49)$${\partial }_z{\mathrm{P}}_{\mathrm{m}}=\gamma {\mathrm{P}}_{\mathrm{m}}$$

where α_m is a power-normalized scattering rate, not to be confused with the displacement-normalized rate represented by α_m in the rest of the text. In the above equations we assume the mechanical drive ${\mathrm{P}}_{\mathrm{m}}$ is undepleted by the scattering process, a reasonable assumption since the phonon flux for a 1 mW drive is larger than a 1 mW optical guided wave by a factor of ω/Ω ≈ 10⁵. The power optical is ${\mathrm{P}}_0$ at the beginning of the antenna where z = 0 and the mechanical/power is ${\mathrm{P}}_{\mathrm{m}0}$ at the end of the antenna where z = L. We can solve these equations given the boundary conditions above to find the optical power along the antenna

(50)$$\frac{\mathrm{P}(z)}{{\mathrm{P}}_0}=e^{-a(e^{\zeta }-1)}$$

where we’ve introduced two dimensionless parameters: the local scattering rate at the beginning of the antenna $a=-{\alpha }_{\mathrm{m}}{\mathrm{P}}_{\mathrm{m}0}{e}^{-\gamma L}/\gamma$ and the distance along the antenna ζ = γz. The power radiated from the waveguide is computed from the guided power by taking the derivative ${\mathrm{P}}_{\mathrm{r}}=-{\partial }_z\mathrm{P}$ yielding

(51)$$\frac{{\mathrm{P}}_{\mathrm{r}}(\zeta )}{\gamma {\mathrm{P}}_0}=a\exp (\zeta -a(e^{\zeta }-1)).$$

Maximal scattering occurs at ζ_max = −log a. When the optical scattering rate at z = 0 is large compared to the mechanical attenuation rate γ such that ζ_max ≪ 0, the power radiated ${\mathrm{P}}_{\mathrm{r}}$ decays exponentially and attenuation can be ignored. As a is decreased perhaps by lowering ${\mathrm{P}}_{\mathrm{m}0}$, the maximum shifts right ζ_max and when sufficiently far from the origin and for sufficiently long antennas the resulting radiation pattern has a FWHM of Δζ = 2.45. The 1/e width is 3 and the 1/e² width is 4.45. Figure 9 shows the mechanical power necessary to achieve a particular radiation pattern for an antenna of a particular length.

 

Fig. 8 The scattering rates α are computed for, from top to bottom, a 220 nm, 340 nm, and 440 nm slab with and without the volume current used to implement the J_⊥ ∝ D_z component of the boundary interaction. For the 220 nm slab, the J_⊥ term dominates and our perturbative (dashed) and nonperturbative (dotted) calculations agree well.

Download Full Size | PDF

 

Fig. 9 Mechanical losses alter the exponential envelope of an ideal OMA, limiting the effective aperture and increasing the mechanical power needed to achieve it. a, The solid curves show power radiated per unit length across an antenna for values of ζ_max = −log a ranging from −1 to 5 (from red-to-black in even steps). Large scatterings rates a results in approximately exponential radiation patterns as in the lossless case. As a decreases, the profile shifts toward higher z with a constant FWHM of Δζ = 2.45. The dotted curves show the fraction of the guided light remaining in the waveguide, plotted on the same axes. b, The mechanical power needed to achieve a particular radiation pattern is plotted against the antenna’s length L on the same abscissa as a.

Download Full Size | PDF

Appendix F. Derivation of antenna properties

Here we provide derivations of the antenna properties presented in the main text. Some of the properties are illustrated in Fig. 10.

 

Fig. 10 a, The field of view Δθ is the range of angles that can be reached by tuning the mechanical frequency Ω at fixed optical frequency ω. b, The spot size δθ is the angular width – set by the wavevector uncertainty δK – of the scattered optical beam in the far field. c, The optical bandwidth at each spot Δω is the amount the optical frequency ω can be shifted before the beam angle shifts by more than the spot size δθ at fixed mechanical frequency Ω. Since both the outgoing wavevector k and the guided wavevector β change with frequency, the optical bandwidth at each spot Δω is set by the walk-off between the guided and radiating optical fields.

Download Full Size | PDF

F.1 Field of view

The field of view Δθ is the range of angles than can be reached by sweeping the mechanical frequency. It is set by the bandwidth of the electromechanical transducer, the sensitivity of the mechanical wavevector to frequency as well as the sensitivity of radiation wavevector to mechanical wavevector. In particular, subtracting the phase-matching conditions at two different mechanical frequencies leads to

(52)$$k({\omega }_{\mathrm{r}}+\Delta \Omega )\cos (\theta +\Delta \theta )-k({\omega }_{\mathrm{r}})\cos (\theta )=-\Delta K$$

with ΔΩ = v_mΔK the bandwidth of the electromechanical transducer. This determines Δθ in general, while for small Δθ we get

(53)$$\Delta \theta =\frac{\Delta \Omega }{k\sin (\theta )}\left(\frac{1}{v_{\mathrm{m}}}+\frac{\cos (\theta )}{c}\right)$$

(54)$$\approx \frac{\Delta \Omega }{k\sin (\theta )v_{\mathrm{m}}}=\frac{\Delta \Omega }{2\pi }\frac{\lambda }{\sin (\theta )v_{\mathrm{m}}}$$

with λ = 2π/k the radiation’s wavelength. Here we used c ≫ v_m.

F.2 Spot size

The spot size δθ is the angular width of the scattered optical beam in the far field. It is set by the wavevector uncertainty $\delta K=\frac{2\pi }{L_{\text{eff}}}$ corresponding to the finite aperture:

(55)$$\begin{array}{l}k\delta \cos (\theta )\approx k\sin (\theta )\delta \theta \\ =\delta K=\frac{2\pi }{L_{\text{eff}}}\end{array}$$

where we dropped a minus sign. Therefore,

(56)$$\delta \theta =\frac{\lambda }{\sin (\theta )L_{\text{eff}}}$$

F.3 Number of resolvable spots

The number of resolvable spots N_θ is the ratio between the field of view Δθ and the spot size δθ. Neglecting the frequency-dependence of k allows us to express N_θ = ΔK/δk only in terms of the mechanical properties of the optomechanical antenna. We find that

(57)$$N_{\theta }=\frac{\Delta K}{\delta k}=\frac{\Delta \theta }{\delta \theta }=\frac{\Delta \Omega }{2\pi }{\tau }_{\mathrm{m}}.$$

The number of resolvable spots is set by the product of the transducer bandwidth ΔΩ and the mechanical transit time τ_m = L_eff/v_m. With a large bandwidth transducer we have ΔK → 2k and therefore $N_{\theta }\to 2\frac{L_{\text{eff}}}{\lambda }$ – making the effective aperture length the ultimate limit on the number of resolvable spots.

F.4 Bandwidth at each spot

The optical bandwidth is set by how much the optical frequency can be changed before the beam angle disperses more than the spot size. For fixed K we differentiate the phase-matching condition and

(58)$$\cos \theta \delta k+\underset{=\delta K}{\underbrace{k\delta \cos \theta }}=\delta \beta .$$

We equate the second term, the angular variation from changing ω, to the angular spread of the beam in section F.2 $\delta K=\frac{2\pi }{L_{\text{eff}}}$. Relating δβ and δk to Δω by the guided and free-space optical dispersion relations we find

(59)$$\frac{\Delta \omega }{2\pi }=\frac{1}{\tau -{\tau }_{\mathrm{r}}}$$

with τ = L_effn_g/c the transit time of the guided optical wave and τ_r = L_eff cos θ/c the transit time of the radiation mode across the aperture and n_g the group index of the guided optical wave. The optical bandwidth at each spot Δω₀ is thus set by the walk-off between the guided and the radiation mode.

A similar derivation as in the optical case above shows that the mechanical transit time determines the mechanical bandwidth at each spot

(60)$$\frac{\Delta {\Omega }_{\mathrm{m}}}{2\pi }=\frac{1}{{\tau }_{\mathrm{m}}}$$

Funding

National Science Foundation (NSF) (ECCS-1509107, ECCS-1808100); Stanford University (Terman Fellowship, Hellman fellowship, start-up funds from the school of Humanities and Sciences); ONR (QOMAND MURI); VOCATIO; Horizon 2020 with Research Foundation – Flanders (FWO) (Marie Skłodowska-Curie 665501).

Acknowledgments

We thank Patricio Arrangoiz-Arriola, Okan Atalar, Rishi Patel, and Jeremy Witmer for helpful discussions.

References

1. K. Van Acoleyen, K. Komorowska, W. Bogaerts, and R. Baets, “One-dimensional off-chip beam steering and shaping using optical phased arrays on silicon-on-insulator,” Journal of Lightwave Technology 29, 3500–3505 (2011). [CrossRef]  

2. J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, M. L. Davenport, L. a. Coldren, and J. E. Bowers, “Hybrid III/V silicon photonic source with integrated 1D free-space beam steering,” Optics Letters 37, 4257 (2012). [CrossRef]   [PubMed]  

3. J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Optics Express 19, 21595 (2011). [CrossRef]   [PubMed]  

4. K. Van Acoleyen, W. Bogaerts, and R. Baets, “Two-dimensional dispersive off-chip beam scanner fabricated on silicon-on-insulator,” IEEE Photonics Technology Letters 23, 1270–1272 (2011). [CrossRef]  

5. J. C. Hulme, J. K. Doylend, M. J. R. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Optics Express 23, 5861 (2015). [CrossRef]   [PubMed]  

6. C. V. Poulton, A. Yaccobi, Z. Su, M. J. Byrd, and M. R. Watts, “Optical phased array with small spot size, high steering range and grouped cascaded phase shifters,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (Optical Society of America, 2016), pp. IW1B–2.

7. J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013). [CrossRef]   [PubMed]  

8. M. J. Heck, “Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering,” Nanophotonics 6, 93–107 (2016). [CrossRef]  

9. D. Kedar and S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Communications Magazine 42, S2–S7 (2004). [CrossRef]  

10. H. Elgala, R. Mesleh, and H. Haas, “Indoor optical wireless communication: potential and state-of-the-art,” IEEE Communications Magazine 49, 56–62 (2011). [CrossRef]  

11. W. S. Rabinovich, C. I. Moore, R. Mahon, P. G. Goetz, H. R. Burris, M. S. Ferraro, J. L. Murphy, L. M. Thomas, G. C. Gilbreath, M. Vilcheck, and M. R. Suite, “Free-space optical communications research and demonstrations at the US Naval Research Laboratory,” Applied Optics 54, F189 (2015). [CrossRef]  

12. C. Quate, C. Wilkinson, and D. Winslow, “Interaction of light and microwave sound,” Proceedings of the IEEE 53, 1604–1623 (1965). [CrossRef]  

13. E. Gordon, “A review of acoustooptical deflection and modulation devices,” Proceedings of the IEEE 54, 1391–1401 (1966). [CrossRef]  

14. A. Korpel, R. Adler, P. Desmares, and W. Watson, “A Television Display Using Acoustic Deflection and Modulation of Coherent Light,” Proceedings of the IEEE 54, 1429–1437 (1966). [CrossRef]  

15. F. R. Gfeller, “A colinear thin-film acousto-optic scanner,” Journal of Physics D: Applied Physics 10, 1833 (1977). [CrossRef]  

16. A. Matteo, C. Tsai, and N. Do, “Collinear guided wave to leaky wave acoustooptic interactions in proton-exchanged LiNbO₃ waveguides,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47, 16–28 (2000). [CrossRef]  

17. D. E. Smalley, Q. Y. J. Smithwick, V. M. Bove, J. Barabas, and S. Jolly, “Anisotropic leaky-mode modulator for holographic video displays,”Nature 498, 313–317 (2013). [CrossRef]   [PubMed]  

18. E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nature Photonics 10, 463–467 (2016). [CrossRef]  

19. R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nature Photonics 9, 199–203 (2015). [CrossRef]  

20. C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016). [CrossRef]  

21. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Reviews of Modern Physics 86, 1391–1452 (2014). [CrossRef]  

22. B. Eggleton, C. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photon. 5, 536–587 (2013). [CrossRef]  

23. R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New Journal of Physics 17, 115005 (2015). [CrossRef]  

24. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, Wiley Series in Pure and Applied Optics (John Wiley & Sons, Inc., 1991). [CrossRef]  

25. D. Marcuse, “Mode Conversion Caused by Surface Imperfections of a Dielectric Slab Waveguide,” Bell System Technical Journal 48, 3187–3215 (1969). [CrossRef]  

26. D. Marcuse, “Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function,” Bell System Technical Journal 48, 3233–3242 (1969). [CrossRef]  

27. C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, J. T. Hill, and A. H. Safavi-Naeini, “Release-free silicon-on-insulator cavity optomechanics,” Optica 4, 1147 (2017). [CrossRef]  

28. R. Van Laer, R. N. Patel, T. P. McKenna, J. D. Witmer, and A. H. Safavi-Naeini, “Electrical driving of X-band mechanical waves in a silicon photonic circuit,” arXiv:1806.04100 (2018).

29. J. M. Elson and J. M. Bennett, “Calculation of the power spectral density from surface profile data,” Appl. Opt. 34, 210 (1995). [CrossRef]   [PubMed]  

30. S. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer Linewidth Uniformity in Silicon Nanophotonic Waveguide Devices Using CMOS Fabrication Technology,” IEEE Journal of Selected Topics in Quantum Electronics 16, 316–324 (2010). [CrossRef]  

31. O. Fursenko, J. Bauer, A. Knopf, S. Marschmeyer, L. Zimmermann, and G. Winzer, “Characterization of Si nanowaveguide line edge roughness and its effect on light transmission,” Materials Science and Engineering B: Solid-State Materials for Advanced Technology 177, 750–755 (2012). [CrossRef]  

32. D. Melati, A. Melloni, and F. Morichetti, “Real photonic waveguides: guiding light through imperfections,” Advances in Optics and Photonics 6, 156–224 (2014). [CrossRef]  

33. D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,”Optics express 16, 18275–18287 (2008). [CrossRef]   [PubMed]  

34. B.-W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. a. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21, 12238–12248 (2013). [CrossRef]   [PubMed]  

35. M. Megens, B.-W. Yoo, T. Chan, W. Yang, T. Sun, C. J. Chang-Hasnain, M. C. Wu, and D. A. Horsley, “High-speed 32×32 MEMS optical phased array,” International Society for Optics and Photonics 8977, 89770H (2014).

36. A. Tuantranont, V. M. Bright, J. Zhang, W. Zhang, J. A. Neff, and Y. C. Lee, “Optical beam steering using MEMS-controllable microlens array,” Sensors and Actuators A: Physical 91, 363–372 (2001). [CrossRef]  

37. T. Zhang, J. L. Abellan, A. Joshi, and A. K. Coskun, “Thermal management of manycore systems with silicon-photonic networks,” in “Design, Automation & Test in Europe Conference & Exhibition (DATE), 2014,” (IEEE Conference Publications, New Jersey, 2014), pp. 1–6.

38. Y. Shi, S. Han, and S. Fan, “Optical Circulation and Isolation Based on Indirect Photonic Transitions of Guided Resonance Modes,” ACS Photonics 4, 1639–1645 (2017). [CrossRef]  

39. W. S. Rabinovich, P. G. Goetz, M. Pruessner, R. Mahon, M. S. Ferraro, D. Park, E. Fleet, and M. J. DePrenger, “Free space optical communication link using a silicon photonic optical phased array,” Proc. SPIE 9354, 93540B (2015).

40. D. E. Smalley, “Holovideo on a Stick: Integrated Optics for Holographic Video Displays,” Ph.D. thesis, Massachusetts Institute of Technology (2013).

41. C. K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sensing of Environment 106, 123–135 (2007). [CrossRef]  

42. A. Schliesser, N. Picqué, and T. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440 (2012). [CrossRef]  

43. S. Boudreau, S. Levasseur, C. Perilla, S. Roy, and J. Genest, “Chemical detection with hyperspectral lidar using dual frequency combs,”Optics express 21, 7411–7418 (2013). [CrossRef]   [PubMed]  

44. F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Optics Express 23, 5117 (2015). [CrossRef]   [PubMed]  

45. R. Fatemi, B. Abiri, and A. Hajimiri, “An 8 × 8 heterodyne lens-less opa camera,” in “CLEO: QELS_Fundamental Science,” (Optical Society of America, 2017), pp. JW2A–9.

46. COMSOL Multiphysics v5.0. www.comsol.com. COMSOL AB, Stockholm, Sweden.

47. D. Taillaert, “Grating couplers as Interface between Optical Fibres and Nanophotonic Waveguides,” Ph.D. thesis, Ghent University (2005).

48. F. V. Laere, S. Member, T. Claes, J. Schrauwen, S. Scheerlinck, W. Bogaerts, and D. Taillaert, “Compact Focusing Grating Couplers for Silicon-on-Insulator Integrated Circuits,” IEEE Photonics Technology Letters 19, 1919–1921 (2007). [CrossRef]  

49. B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” Journal of Lightwave Technology 14, 188–195 (1996). [CrossRef]  

50. Lumerical Inc.http://www.lumerical.com/tcad-products/fdtd/.

51. P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant Enhancement of Stimulated Brillouin Scattering in the Subwavelength Limit,” Physical Review X 2, 1–15 (2012). [CrossRef]  

52. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Physical Review E 65, 066611 (2002). [CrossRef]  

53. R. Van Laer, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” Physical Review A 93, 15 (2016). [CrossRef]  

54. R. W. Dixon, “Photoelastic Properties of Selected Materials and Their Relevance for Applications to Acoustic Light Modulators and Scanners,” Journal of Applied Physics 38, 5149–5153 (1967). [CrossRef]  

55. S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Applied Physics B 81, 283–293 (2005). [CrossRef]  

56. S. K. Selvaraja, “Wafer-Scale Fabrication Technology for Silicon Photonic Integrated Circuits,” Ph.D. thesis, Ghent University (2011).

57. K. K. Lee, D. R. Lim, H.-C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, “Effect of size and roughness on light transmission in a Si/SiO_2 waveguide: Experiments and model,” Applied Physics Letters 77, 1617 (2000). [CrossRef]  

58. Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Optics Express 23, 16890 (2015). [CrossRef]   [PubMed]  

59. C. Wolff, R. Van Laer, M. Steel, B. Eggleton, and C. Poulton, “Brillouin resonance broadening due to structural variations in nanoscale waveguides,” New Journal of Physics 18, 1–9 (2016). [CrossRef]  

60. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010). [CrossRef]  

61. S. Ghaffari, S. A. Chandorkar, S. Wang, E. J. Ng, C. H. Ahn, V. Hong, Y. Yang, and T. W. Kenny, “Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators,” Scientific Reports 3, 3244 (2013). [CrossRef]   [PubMed]