Supplementary Figure 1: SAW transducer equivalent circuit

Transcription

1 Supplementary Figure : SAW transducer equivalent circuit Supplementary Figure : Radiation conductance and susceptance of.6um IDT, experiment & calculation Supplementary Figure 3: Calculated z-displacement field plots of Rayleigh modes of the 4.0µm wavelength IDT. AlN SiO R- R- R-3 - -

2 Supplementary Figure 4: Calculated z-displacement field plots of Rayleigh modes of the 0.5µm wavelength IDT. R- R- R-3 R-4 R-5 R-6 R-7 R-8 R-9 R-0 R- R- R-3 R-4 R-5 - -

3 Supplementary Figure 5: Modulation measurement schematics. VNA: Vector Network Analyzer, EDFA: Erbium-Doped Fiber Amplifier, OTF: Optical Tunable Filter, LS-PD: Low speed Photo Detector, HS-PD: High-speed Photo Detector, LNA: Low Noise Amplifier, FC- Fiber Coupler

4 Supplementary Note To extract the electromechanical coupling coefficient from the S spectra of the SAW devices, we used a simplified form of Mason s equivalent circuit model of the interdigital transducer (IDT). Supplementary Figure shows circuit diagram of this model. R p and C t are the parasitic resistance & static capacitance of the IDT fingers; B a (f), and G a (f) are the radiation susceptance and conductance of the IDT defined as: where G o t o Ga( f ) Gosinc X B ( f ) G a o sin( X) X X 8 C f N and X N(f f ) f. o o f o and N are the resonant frequency & number of electrodes of the IDT, and k is the electromechanical coupling coefficient of the piezoelectric substrate. After measuring the complex impedance of the IDT using the network analyzer, we subtracted the real component values that lie well outside the SAW resonance linewidth. This subtracted frequency independent resistance corresponds to the parasitic resistance, R p. The inverse of the remaining impedance gives the IDT admittance, Y a (f), whose real part is the IDT radiation conductance and its imaginary part comprising the radiation susceptance and the susceptance contribution of the IDT electrodes static capacitance. Since the radiation susceptance should vanish at the resonant SAW frequency, the susceptance value at this frequency gives the static capacitance of the IDT electrodes. This value, together with the radiation conductance value at the resonance SAW frequency, can be used to calculate the electromechanical coupling coefficient. Such analysis is done for R modes of IDTs of different SAW wavelengths and the results are shown in Fig. c of the main paper. As an example, we presented the analysis we did on an R mode of a.6µm IDT, the S spectra of which is shown in Fig. (a) of the main paper. We calculated k and C t to be 0.7% and.8pf, respectively. We used these extracted values to calculate the radiation conductance and susceptance of the IDT. Supplementary Figure shows a comparison of the calculated and measured values. In the calculated plot of the radiation conductance, the sinc function behavior is a signature of the finite size of the IDT transducer

5 Supplementary Note We consider a TE waveguide mode with its dominant electric field pointing in the x-direction. The wave propagate along the y-axis and the z-axis pointing out of the plane along the c-axis of the AlN polycrystalline film. A travelling SAW wave modulates the dielectric constant of the material it propagates in through elasto-optic effect directly, and electro-optic effect indirectly. The contribution of the indirect electro-optic effect can be lumped into the elasto-optic effect by defining an effective elasto-optic coefficient as p ( rik lk )( lieij ) p ( l l ) eff ij ij S i ij j where l k is k th component of the unit vector along the direction of SAW wave propagation, and r ij represents the electro-optic coefficient, which can be written in the contracted form for crystalline material with 6mm symmetry as: r ij 0 0 r3 0 0 r r 33 0 r5 0 r with coefficiecnts r 3 =0.67pm/V, r 33 =0.9pm/V, and r 5 0 in AlN 3. For a SAW wave propagating along the x-axis (transverse to the optical waveguide), the only nonzero electro-optic contributions comes from r 5. But, this value is quite small and can be ignored, leaving the elasto-optic coefficient unchanged. Invoking the definition of index ellipsoid and using the contracted form of the elasto-optic coefficient, we can write the change in refractive index as: p,,, 6 ijs j i j n j i where p ij is the elasto-optic coefficient tensor, and S j is the strain field tensor. A simple differentiation reduces the above expression to: - 5 -

6 n n p S 3 i i ij j j For a hexagonal crystal system like AlN, the elasto-optic coefficient tensor takes the form p ij p p p p p p p3 p3 p p p p66 with values p =0., p 33 =0.07, p =0.07, p 3 =0.09, p 44 =0.03, and p 66 =0.037 for AlN thin films 4. Since the film is polycrystalline, we can approximate the in-plane elasto-optic coefficients as the average of the a- and b-direction coefficients of the crystal: p p p p / Ignoring the shear components of the strain field tensor (S 4, S 5, S 6 ), which are much weaker than the normal components, one can write the elasto-optic refractive index change as: n n p S p S 3 x, y o 3 3 where S and S 3 correspond to the normal strain fields in the x- and z-directions respectively. This change in refractive index induces a change in mode frequency of the optical resonator. Since the SAW-induced refractive index change is very small (n/n <<), one can use perturbation theory to estimate this change in frequency to the first order in n as 5 : n( r) E( r) dr n(x,z) E (x,z) dxdz n( r) E( r) dr n(x,z) E (x,z) dxdz where E is the dominant electric field of the TE mode waveguide. This implies that the change in mode frequency is proportional to an overlap integral of the SAW strain field with the waveguide mode electric field, which can be written as: - 6 -

7 p S ( x, z) p S ( x, z) E ( x, z) dxdz 3 3 E ( x, z) dxdz This overlap factor can be calculated to optimize the geometry of a waveguide for maximum SAW-optics interaction. Supplementary Note 3 Analogous to optomechanical systems, the strength of acousto-optic interaction in an optical cavity can be quantified by an optomechanical coupling rate (G om ). We defined this coupling rate as: G A om where is the change in resonance frequency of the optical cavity due to acousto-optic modulation, and <A> is the mean amplitude of the SAW displacement field. This displacement field amplitude is a measure of the acoustic energy which depends on the electromechanical coupling coefficient and the microwave power input to the IDT. The acoustic elastic energy density is given as: U 3 S C S i, j i ij j where S i is the strain field tensor component and C ij is the elements of elastic modules tensor which can be written in its reduced form as 6 C C C C C C C3 C3 C C C C C66 with element values C =40 GPa, C =49 GPa, C 3 =99 GPa, C 33 =389 GPa, C 44 =5 GPa for AlN. The strain field vector of the surface acoustic wave propagating along the z-axis can be expressed as: S S 0 S 0 S 0 3 5, where S and S 3 are the normal strain fields along the - 7 -

8 x and z-directions, respectively, and S 5 is the shear strain field. Ignoring the relatively weak shear component of the strain field and taking into account the fact that S and S 3 are 90 o out of phase (where cross-terms vanish upon averaging over a SAW wavelength), the energy density reduces to: U C S C S C S where in the last step we took the upper bound of the energy density for the case of simplicity. The z-component strain field can be expressed as: u A z z z z/ z/ S3 ( Ae sin x) e sin x where the out-of-the-plane displacement u z with amplitude A is assumed to decay exponentially at a rate of /into the substrate for surface waves. Thus, the energy averaged over a SAW wavelength can be written as: which integrates to AW / 33 e z E Udxdydz C sin x dxdz E C33W A 4 where W is the SAW beam width and <A> represents average amplitude of the SAW displacement field. The electromechanical coupling coefficient k and the IDT reflection coefficient S relates the SAW wave power, P SAW, to the microwave power input to the IDT, P in, as: SAW ( ) in o P S k P Ef where f o is the SAW resonance frequency. Substituting the energy E expression into this power equation gives the approximate relation of the SAW displacement field amplitude with the microwave power input to the IDT as: - 8 -

9 A 33 k 4( S ) C Wf o P in From the measurement data analysis, we determined that the frequency modulation has a linear relationship with the square root of the microwave power input to the IDT. The slope of this linear plot, coupling coefficient quantified with respect to microwave input voltage, can be converted into the optomechanical coupling rate, G om, using the equation derived above. Supplementary Note 4 We employed finite element method to calculate the eigen frequencies of the IDT structure. The periodicity of the IDT allowed the use of only one unit cell with periodic boundary conditions set at the ends of the unit cell. Appropriate material parameters and boundary conditions were imposed and the geometry was solved for an optimally meshed IDT structure. All the possible eigen modes allowed by the geometry were calculated, and this led to the mapping of the SAW dispersion curve as shown in Fig. b of the main paper. The calculated strain field distribution was used in a post-processing recipe to calculate the acousto-optic overlap factor. Representative results are shown in Fig 4 of the main paper. In Supplementary Figure 3 and 4, we showed the eigen mode profiles calculated for 4.0µm and 0.5µm wavelength IDTs. The plots are shown for the out of plane displacement field. The 4.0µm IDT supports only three modes, while the 0.5µm IDT has 5 modes. This is in line with the fact that the SAW amplitude decays exponentially into the substrate and hence limiting the number of R-modes of large wavelength IDTs to fewer than those of smaller wavelength IDTs. The AlN and silicon dioxide film thickness determines the possible number of modes supported for a particular wavelength IDT. Supplementary Note 5 The measurement scheme is shown in Supplementary Figure 5. Microwave signal from port- of the vector network analyzer was input to the IDT to excite the SAW waves. Laser light with wavelength set half the linewidth off the optical resonance was sent to the input grating couple. The resonator s transmitted light was collected from the output grating coupler and split into 0. & 0.9 ratio using a fiber coupler. The 0% light was used to monitor and optimize fiber arrays position with respect to the grating couplers. The 90% light was sent to an erbium doped fiber amplifier (EDFA). The amplified light signal was sent to a tunable optical filter to remove any - 9 -

10 amplified spontaneous emission noise induced by the EDFA. Finally the light signal was sent to a high-speed photoreceiver for detection. The electrical signal from the detector was fed back to port- of the vector network analyzer and the optical S spectrum was measured by sweeping the network analyzer frequency. Supplementary References: Hines, J. H. & Malocha, D. C. in Ultrasonics Symposium, 993. Proceedings., IEEE vol.7. Xu, J. & Stroud, R. Acousto-optic devices : principles, design, and applications. (Wiley, 99). 3 Gräupner, P., Pommier, J. C., Cachard, A. & Coutaz, J. L. Electro optical effect in aluminum nitride waveguides. J. Appl. Phys. 7, (99). 4 Davydov, S. Y. Evaluation of physical parameters for the group III nitrates: BN, AlN, GaN, and InN. Semiconductors 36, 4-44 (00). 5 Johnson, S. G. et al. Perturbation theory for Maxwell's equations with shifting material boundaries. Physical review. E, Statistical, nonlinear, and soft matter physics 65, 0666 (00). 6 Wright, A. F. Elastic properties of zinc-blende and wurtzite AlN, GaN, and InN. J. Appl. Phys. 8, (997)