Acousto-optic modulation in lithium niobate on sapphire

The recent demonstration of low-loss nanophotonic waveguides¹ in high-quality, single-crystal films of lithium niobate (LN)² has led to a surge in the development of electro-optic and nonlinear optical devices. Because of their small mode area, these waveguides exhibit large nonlinear interactions and require less energy to parametrically drive underlying recent progress in LN frequency combs,³ second-harmonic generation,^4,5 and high-speed electro-optic modulators.^6,7 Furthermore, bonded films can be prepared with more control over crystal orientation and exhibit properties that are closer to bulk LN than the epitaxial films, which date back to the 1970s.^8–10 

In addition to the large $χ2$ nonlinearity and electro-optic effect that make LN an attractive material for optics, LN is a low-loss mechanical material with strong piezoelectric coupling properties that are necessary for making broadband and efficient acousto-optic modulators (AOMs). In parallel to the development of nanophotonics in thin-film LN, strongly piezoelectric, low-loss resonators^11,12 and delay-lines¹³ have been demonstrated and wavelength-scale waveguides efficiently transduced¹⁴ in suspended films. Using these piezoelectric devices, suspended LN AOMs have realized low-power microwave-to-optical conversion in pursuit of quantum optical interconnects.^15,16

While suspended devices are at the forefront of low-power acousto-optics, suspensions add fabrication constraints that inhibit the development of complex photonic and phononic circuits and systems. Another approach to co-localize optical and mechanical waves^17–19 is to employ the Rayleigh-like surface acoustic wave (SAW), which is confined to the surface in any material platform. These SAWs can modulate a variety of optical structures including resonators,²⁰ waveguides as recently used to demonstrate non-magnetic isolation,²¹ Mach–Zehnders for intensity modulators^22–24 and acousto-optic gryoscopes,^25,26 and arrayed waveguide gratings^27,28 with Refs. 24–26 using LN-on-insulator. However, LN-on-insulator has a fundamental drawback. Analogous to the advantage of high confinement in electro-optics and nonlinear optics, high mechanical confinement is necessary for broadband electromechanical transduction as well as for efficient optomechanical modulation. In LN-on-insulator, however, as the mechanical wavelength approaches the thickness of the LN film, energy leaks out of the LN and into the silica, which supports slower waves. This leaves only the Rayleigh-like mode confined to the device layer. If we replace the insulator substrate with a material with higher sound velocities, such as sapphire, LN can also support waves such as the strongly piezoelectric horizontal shear (SH), or Love-like, mode.¹⁰ 

Here, we explore integrated acousto-optic modulation in X-cut, thin-film LN bonded to sapphire with a focus on verifying the piezoelectric and acousto-optic properties of our film. Although optical waveguiding^8,9 and SAW devices have been investigated in LN-on-sapphire,¹⁰ this work constitutes to our knowledge the first acousto-optic devices in the platform. LN-on-sapphire enables high-confinement optical waveguides from the near-infrared to the visible and, in addition to Rayleigh-like SAWs, supports guided SH waves that exhibit large electromechanical coupling coefficients $keff2$ at GHz frequencies. Compared to related efforts in other platforms, making use of aluminum nitride,^29–32 silicon,^33,34 or gallium arsenide,³⁵ this platform enables strong confinement of the mechanical waves without suspensions, setting the stage for complex phononic circuitry and systems. We demonstrate a surface wave acousto-optic phase modulator (discussed in Sec. II) utilizing the Rayleigh and SH modes near 700 and 800 MHz, respectively, and characterize them in the telecom C-band. By comparing simulations and measurements of the piezoelectric coupling coefficient k_eff (Sec. III A) and the optomechanical coupling coefficient g (Sec. III B), we show the degree to which these bonded films retain their piezoelectric and acousto-optic properties. The acousto-optic portion of the study is similar to recent work by Khan et al., which extracted the dominant elasto-optic coefficients for waveguides patterned in sputtered arsenic trisulfide films.³⁶ Furthermore, we show that at GHz frequencies as the wavelength approaches the LN film thickness, the piezoelectric coupling of the SH wave quickly increases with $keff2$ exceeding 10% just above 2 GHz.

The surface wave phase modulator shown in Fig. 1(a) is a simple acousto-optic device with two parts: a piezoelectric transducer to generate mechanical waves and an optical ridge waveguide modulated by these waves. Surface waves are generated by an interdigitated transducer (IDT) with phase fronts parallel to an optical ridge waveguide. These surface waves modulate the effective index of refraction n_eff of the waveguide, and therefore, the phase of light transmitted through the device.

The piezoelectric transducer is characterized by two numbers, the effective piezoelectric coupling coefficient k_eff and the transmission coefficient t_bμ.^14,37,38 The latter is the transmission from microwaves incident on the IDT to phonons in a specific mechanical mode and direction. For our purposes, the most important characteristic of the ridge waveguide is the optomechanical coupling coefficient g, which has units of $W⋅m−1/2$ and is defined in  Appendix A. Of these figures, k_eff and g are proportional to the piezoelectric and photoelastic tensors,³⁹ respectively, and so can be used to characterize the quality of the bonded film and platform. For this reason, they are the focus of our study.

First, we consider numerical analyses of the transducer in Fig. 1(b), focusing on k_eff before considering t_bμ. A large k_eff is essential for making small broadband transducers.^14,37,40 The coupling can be estimated from quantities efficiently computed on a unit cell of an IDT, specifically, the series and parallel resonance frequencies, Ω_s and Ω_p. We simulate a thin, three-dimensional cross section of a finger pair with Floquet boundary conditions along the direction of propagation ŷ [see Fig. 2(b)]. We assume continuity along $z^$⁠. In a lossless simulation, the series and parallel resonances correspond to short and open boundary conditions across the electrodes. To first order in (Ω_p − Ω_s)/Ω_p,¹⁴ 

A 225 nm-thick LN slab on sapphire supports a Rayleigh-like mode and a leaky horizontal shear (SH) mode with a wavelength of 8 μm and frequencies near 750 MHz. The coupling $keff2$ for these modes depends on the orientation of the electrodes with respect to the extraordinary axis, as plotted in Fig. 2(a).

The level of confinement of the acoustic wave depends strongly on its wavelength Λ and therefore its frequency. At Λ = 8 μm, the SH mode is weakly confined; three quarters of the mechanical energy is in the sapphire substrate, and so $keff2$ only reaches 1.2%. An advantage of the platform is that at shorter wavelengths, more energy is confined to the LN film, and the coupling increases. We show in Sec. III A that $keff2$ of the SH waves exceeds 10% for 2 μm-pitch IDTs where 32% of the energy is in LN (with 39% in the sapphire and 29% in the electrodes). By comparison, $keff2$ in suspended LN films reaches 30%.¹² Furthermore, at 8 μm, the SH band [blue in Fig. 2(c)] is phase-matched to waves in the sapphire (hatched), and so the wave leaks into the substrate at a rate of 10 dB/mm. Above 2.7 GHz, the SH wave is no longer leaky. Smaller Λ will be pursued in future work to achieve higher confinement and lower acoustic radiation loss.

In order to determine the coupling coefficient g from measurements of the modulation index (Sec. III B), we need the efficiency of the IDT t_bμ. This coefficient can be expressed as the product of two factors,¹⁴ 

The first factor comes from impedance mismatch and captures the fraction of incident microwave power that gets reflected. It can be determined experimentally from measurements of S₁₁, the one-port S-parameter of the IDT. It can also be calculated from the admittance of solutions of the inhomogeneous piezoelectric equations of the full IDT and reflector bounded by perfectly matched absorbing layers [Fig. 2(d)]. The second factor in Eq. (2) is the fraction of power radiated into the ith mode, which is computed by decomposing the radiation into a basis of waves in the slab.⁴¹ The amplitude a_i is normalized such that $ai2$ is the power in mode i. The product 2GV² in the denominator is the total power dissipated when a voltage V is applied across the IDT. Details on these methods are presented by the authors elsewhere.¹⁴ 

The surface waves generated by the IDT strain deform the optical waveguides. This changes the effective index n_eff of the fundamental TE-like optical mode, as captured by the optomechanical coupling coefficient g. The coupling coefficient is computed from the optical and piezoelectric eigenmodes of an extruded cross section of the waveguide solved for by FEM in COMSOL⁴² to capture the full vectorial nature of the fields. The optical mode propagates into the plane in Fig. 3(a), and the piezoelectric mode propagates across the plane in Fig. 3(d). The electric field of the TE-like optical mode is antisymmetric with respect to the xz symmetry plane. At each θ_x, these solutions are used to evaluate g by the perturbative overlap integral (detailed in  Appendix A),

E_j and u_k are the electromagnetic and displacement field distributions for the guided mode solutions, ω is the optical frequency, $Pj$ is the time-averaged optical power, and δ_uε is the modification of the structure’s dielectric constant distribution due to motion, which includes both the shifts in the boundaries and the contribution of the photoelastic tensor.

Despite the nontrivial dependence of g on waveguide orientation, as plotted in Fig. 3, a simple picture describes the interaction at the peaks. Consider the Rayleigh and fundamental TE modes. The S_yy component of the strain—the dominant component of the Rayleigh waves at the surface—modulates the ε_yy component of the permittivity via the p_yyyy component of the photoelastic tensor. Modulating ε_yy modulates the TE mode, which has a y-oriented electric field. In Fig. 3(c), the coupling coefficient peaks at θ_x = −45° where p_yyyy reaches a maximum of 0.30 for X-cut LN. Similarly, interactions with the SH mode are dominated by p_yyyz, which has local extrema of 0.127 and −0.135 at θ_x = −18° and 32°, respectively, aligning nicely with our simulations of g.

We start our process with 5 × 10 mm chips of 525 nm-thick LN-on-sapphire. LN is X-cut, and the c-axis of the sapphire is normal to the wafer. The a-axis of the sapphire and the Z-axis of LN are in-plane and parallel. Ridge waveguides and grating couplers are patterned into a hydrogen silsesquioxane (HSQ) mask and transferred to the sample by a 300 nm argon ion etch leaving a 225 nm thick LN slab on the sapphire substrate. The remaining mask is stripped with hydrofluoric acid before the chip is cleaned with piranha. The 200 nm thick aluminum electrodes are patterned by lift-off on the 225 nm LN slab. The IDT is 600 μm wide and has 30 aluminum finger-pairs. The ridge waveguide is 1.25 μm wide and supports a TE-like and a TM-like optical mode.

Below, we describe how k_eff and g are extracted from measurements of the IDTs’ linear response and the modulators’ modulation index.

In order to characterize the piezoelectric quality of the film, we measure the coupling coefficient $keff2$ and compare it to simulations. The coupling coefficient is extracted from measurements of the one-port microwave response S₁₁ of the IDT for a range of crystal orientations θ_x varying from −90° to 90°.

We measure the S-parameter of each device on a calibrated probe station (GGB 40-A nickel probes) with an R&S ZNB20 vector network analyzer [Fig. 2(e)] to determine the admittance $Yω$⁠. The coupling can be computed directly from the measured conductance G(ω) ≡ ReY(ω),¹⁴ 

where C₀ is the static capacitance computed by fit to the susceptance −ImY(ω) near DC, Ω₀ is the center frequency of the response, and the integral is evaluated about Ω₀. In addition to the mechanical signature [red–blue curve in Fig. 2(f)], G(ω) is offset by a slowly increasing background, which comes from ohmic loss and inductance of the IDT. This effect is not captured in the cross section modeled in Sec. II, which assumes that the fields are uniform along the fingers. We fit the background to a parabola and remove it from G before computing $keff2$ by Eq. (4). This gives us the points in Fig. 2(a).

We find excellent agreement between the shape of the simulated and measured $keff2(θx)$⁠, as shown in Fig. 2(a). The magnitude of the Rayleigh response matches with simulation, but the SH response falls 20% below the simulated response at its peak near 0°. A reduction of around 10% in the piezoelectric tensor component d_YZY from 69 pC N⁻¹ to 62 pC N⁻¹ would lead to this reduction in $keff2$⁠.

An important advantage of the LN-on-sapphire platform is that horizontal shear (SH) waves are strongly piezoelectric at high frequency. To demonstrate this, we repeat the above procedure for 2 μm-pitch IDTs on 200 nm-thick LN-on-sapphire. (The difference in thickness is an artifact of fabrication logistics.) A typical conductance curve is shown in Fig. 4(a) for θ_x = 0°. For these 15-finger-pair, 50 μm-wide transducers, the proximity of the Rayleigh and SH waves leads to the observed ripples and makes it difficult to independently filter their contributions to G. Instead, we integrate the conductance for the blue shaded region (between 1 GHz and 2.65 GHz) and report the sum of $keff2$ for both modes [Fig. 4(b)]. Simulations show that the Rayleigh mode remains weakly coupled, and so the increase in $keff2$ is primarily due to the SH mode. The results of measurements of IDTs at a variety of angles are plotted in Fig. 4, showing that $keff2$ for the SH mode exceeds 10%.

We investigate the optomechanical properties of the bonded LN film—in particular, the photoelastic coefficients—by measuring the optomechanical coupling coefficient g of a nanophotonic ridge waveguide and comparing it to simulations. As with $keff2$⁠, g varies with crystal orientation because LN is strongly anisotropic.

We determine g by sending light through the device while driving the IDT with a microwave signal. This modulates the phase of the light, which we measure with the apparatus diagrammed in Fig. 5(a). We tune a C-band laser (Santec TSL-550) to the edge of a Teraxion fiber Bragg notch filter near 1551 nm such that phase modulated light becomes intensity modulated. This intensity modulated light is then amplified (FiberPrime EDFA-C-26G-S) and detected on a photodiode (Optilab PD-40-M). We drive the modulator and read out the photocurrent fluctuations on a vector network analyzer (VNA, R&S ZNB20). The modulation index h_oa is calibrated by comparing the acousto-optic signal to phase modulation from an electro-optic modulator (EOM) with modulation index h_oe cascaded with the device,

Here, S_oa and S_oe are S-parameters measured on the VNA for an acousto-optic device and the EOM, respectively. The EOM is calibrated independently using a tunable Fabry–Pérot filter (Micron Optics FFP-TF) to filter and measure the power of the pump and sidebands for a given RF drive power P.

The modulation index h_oa is not a direct measurement of the coupling coefficient g. Instead, the measured quantity plotted in Fig. 5(b), which is the LHS of

is equal to the product of g, the square root of the efficiency of the IDT $tbμ$⁠, and the square root of the length of the interaction region L. We can directly compute this product from the quantities described in Sec. II. The numerical values for $g|tbμ|L$ are overlaid on the measurements in Fig. 5(b) for the Rayleigh (red) and SH (blue) modes.

We can use these measurements to infer the coupling coefficients, which we computed in Sec. II. To this end, we extract the peaks of h_oa for each mode and remove a factor of |t_bμ| determined numerically. The results are overlaid on the simulated values in Fig. 3(b). The accuracy of the resulting rates is susceptible to systematic errors in t_bμ. In the work of Dahmani et al. where the transmission coefficient was de-embedded from measurements, the simulated t_bμ of 8.9% was larger than the measured 7.0% by 27%.¹⁴ If we overestimate t_bμ, we will underestimate g and therefore the photoelastic coefficients. On the other hand, reflections, e.g., off the waveguide, can give rise to standing waves, which can enhance the IDT’s peak efficiency. A fractional uncertainty of 27% like that in the work of Dahmani et al. [plotted in Fig. 3(b)] would not account for the deviation from theory using bulk material properties. We perform a regression on $g(θx)$ for the photoelastic tensor of our thin film $pij$ in  Appendix B. We find that scaling $p33$⁠, $p44$⁠, and $p41$ from bulk values by factors of 32%, 70%, and 35%, respectively, gives the best fit dashed curve in Fig. 3(b).

Lithium niobate-on-sapphire has many bright prospects in optics and, specifically, in acousto-optics. In this platform, the piezoelectric LN film supports both Rayleigh and horizontal shear surface waves, which can be generated with interdigital transducers and used to modulate optical waveguides patterned in LN. Here, we measure the piezoelectric coupling coefficients of transducers and optomechanical coupling coefficients of ridge waveguides for a range of crystal orientations in X-cut LN, confirming the quality of these bonded films and demonstrating the potential of the material platform.

As the mechanical frequency reaches into the GHz regime, $keff2$ of the horizontal shear waves exceeds 10%, making it possible to make compact, broadband transducers. Future work in pursuit of low-power acousto-optic devices calls for the efficient, mode-selective transduction of wavelength-scale waveguides. Waveguide transducers such as those recently developed for horizontal shear waves in suspended LN films can enable a new generation of ultra-low-power phononic devices and acousto-optic modulators.¹⁴ As the array of electro-optic and nonlinear optical devices in thin-film LN grows, so too do the prospects grow for integrating acousto-optic devices into complex phononic and photonic circuits and systems built on these rapidly developing platforms.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This work was supported by a MURI grant from the U. S. Air Force Office of Scientific Research (Grant No. FA9550-17-1-0002) and the DARPA Young Faculty Award (YFA), by a fellowship from the David and Lucille Packard Foundation, and by the National Science Foundation through Grant Nos. ECCS-1808100 and PHY-1820938. The authors would like to thank NTT Research Inc. for their financial and technical support. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under Grant No. ECCS-1542152, and the Stanford Nanofabrication Facility (SNF).

For intuition, we adopt a quasi-static picture of the dynamics, treating the mechanical waves as stationary, relative to light traveling along the waveguide. At time t, the mechanical wave deforms the waveguide uniformly along its length z. This deformation $ut,x,y=u0x,ycosΩt$ of the waveguide’s cross section (radiation pressure) and the associated strain-induced change in the index (photoelastic effect)³⁹ vary the permittivity, which to first order in u,

shifts the wavevector of the guided optical wave Δβ cos Ωt. Here, δ_∂Ω is a delta function zero everywhere except dielectric boundaries, p is the photoelastic tensor, and S is the strain. The projectors Π_⊥ and Π_∥ project out the field perpendicular and parallel to the normal $n^$⁠, respectively. In this manner, the phase of light leaving the waveguide is acousto-optically modulated,

The phase modulation index h_oa ≡ ΔβL is related to the optomechanical coupling g familiar in stimulated Brillouin scattering. By adopting a power-orthogonal basis for the electric field $E=ΣnEnane−iω+nΩt$⁠, the interaction can be expressed in terms of coupled-modes.^43,44 For short lengths and low RF drive powers, the n = +1 mode evolves as

where b is the amplitude of the mechanics u = u₀be^−iΩt. The optomechanical coupling can be expressed in terms of the mode profiles

where $Pi$ is the time-averaged optical power into the waveguide for mode i. If we use a power-orthonormal basis for the optics such that $ai2$ is the power in mode i, $Pi$ becomes 1 for all I, and Eq. (A4) takes on a more symmetric form for various mode pairs.

We have yet to choose a normalization for the displacement u₀ and thereby units for both g and b. For the devices described here, the devices in Sohn et al.³¹ and work on AO-modulated Mach–Zehnder interferometers,^24,36,45 the mechanical wave propagates across the waveguide. In this configuration, the displacement of the waveguide scales as the square root of the mechanical power density, which is the power per unit length along the waveguide. A 1 mW wave generated by a 100 μm-wide IDT will deform the waveguide the same as that of a 2 mW wave from a 200 μm-wide IDT. We normalize the displacement field u₀ such that b² is the mechanical power density with units of W/m. Therefore, the coupling coefficient g has units of $1/W⋅m$⁠.

Finally, when gb ≪ L, we can integrate Eq. (A3) and compare it to Eq. (A2) to find

Substituting in the efficiency of the IDT, $Pm=tbμ2P$⁠, where P is the RF power incident on the IDT, we arrive at Eq. (6).

It may be surprising that the modulator’s efficiency $hoa2/4P$—the sideband power ratio divided by the RF drive power, which is the square of the expression in Eq. (6)—scales as L and not L², but this scaling is essentially the same as in electro-optic modulation. A typical electro-optic modulator is nearly identical to the optomechanical modulator described here except that instead of applying a displacement u in Eq. (A1b) to change the permittivity of a waveguide, a voltage is applied to a capacitor to shift the effective index Δn_eff by the electro-optic effect. The phase shift of light transmitted through the waveguide Δn_effL is proportional to L, and therefore, the power in the sidebands is proportional to L². However, just as the total mechanical power P_m scales as L, the energy in a capacitor along the waveguide scales as L. The result is that like the optomechanical modulator described here, the efficiency of an electro-optic modulator scales as L.

Equation (A4) is used to compute the g plotted in Fig. 3 of the main text. In our calculation, we model the photoelastic effect in sapphire using coefficients measured in ruby,⁴⁶ and for the radiation pressure term, LN is treated as an isotropic medium with a refractive index of 2.15.

Finally, our device is at heart an electrically driven analog of the optomechanical waveguides investigated in the field of guided-wave Brillouin scattering. In those systems,^{18,19,33,34,47} the mechanical motion is typically driven optically, while here the mechanical motion is driven electrically. The optomechanical coupling coefficient g used here [Eq. (A4)] is, up to a few conversion factors, identical in nature to the Brillouin gain coefficient $GB$ and the refractive-index sensitivity to mechanical motion ∂_xn_eff. The explicit connection is as follows: The modulation index can be written as

with k₀ being the free-space optical wavevector and x being a coordinate representing the mechanical motion. From previous work,^34,48 we have

with being c the speed of light, ω₀ being the optical frequency, Q_m being the mechanical quality factor, ω_m being the mechanical frequency, m_eff = ∫dAρ|u|² being the effective mechanical mode mass, and ρ being the mass density. This shows that the optomechanical coupling coefficient g used here and the Brillouin gain coefficient are rigorously connected. We note that the optomechanical coupling coefficient $g∝GB$ scales as the square root of the Brillouin gain coefficient since the latter takes into account both optical driving and optical read-out of the mechanics, whereas in this work, only the read-out of the mechanics occurs optically.

Our measurements of g in Fig. 3 show rough agreement with the coupling predicted using values of the photoelastic tensor of bulk LN measured by Andrushchak et al.³⁹ We would like to go beyond this qualitative comparison and find the components p_ij that best fit the data. There are d = 8 distinct components of p for LN, with the remaining components constrained by the symmetry of the crystal lattice. In order to avoid overfitting our n = 14 measurements—2 modes, Rayleigh and SH, measured at 7 angles θ_x each—we start with a regularized fit before reducing the regression to just three components of p.

In  Appendix A, we outline how the photoelastic tensor p is related to the coupling coefficient g. Since g is proportional to δε_u and δε_u is linearly dependent on p, the coupling is a linear function of p, which can be expressed as

In Eq. (B1), $g∈Rn$⁠, $p∈Rd$⁠, $A∈Cn×d$⁠, and $b∈Cn$⁠. Both A and b are computed, as described in the main text. They are complex because u, the mechanical field, is complex. The phase of u can be chosen to make g real, but this, in turn, affects the phase of t_bμ. Although the phase of gt_bμ does factor into the phase of S_ao, we restrict our measurements to the magnitude of gt_bμ.

We start by solving

where ∥⋅∥₁ is the L¹ norm. The second term in Eq. (B2) is used to regularize the fit. It penalizes deviation from p₀, the values measured for bulk LN by Andrushchak et al. Using an L¹ norm encourages p − p₀ to be sparse. As λ is increased, some components of p − p₀ fix to 0. At first, the quality of the fit [the first term in the right-hand side of Eq. (B2)] is relatively unaffected by the reduction in dimension d. Ultimately, the regularization term begins to exert a pressure on the components of p needed to fit the data, and the fit diverges from the measured values. At this point, we are left with just three components of p, which deviate from p₀: p₃₃, p₄₄, and p₄₁ in Voigt notation.

We repeat the regression setting λ to 0 and removing all other components of p from the fit to find the values in Table I, giving us the dashed curves in Fig. 3(b).