Supporting information Optomechanical enhancement of doubly resonant 2D optical nonlinearity Fei Yi 3+, Mingliang Ren 3+, Jason C Reed 3, Hai Zhu 3, Jiechang Hou 3, Carl H. Naylor 4, Alan T. Charlie Johnson 3,4, Ritesh Agarwal 3 and Ertugrul Cubukcu 1,2,3* 1 Department of Nanoengineering, University of California, San Diego, La Jolla, CA 92093 2 Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 3 Department of Materials Science and Engineering, 4 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 + These authors contributed equally *All correspondence should be addressed to: *Email: ecubukcu@ucsd.edu 1
Theoretical Analysis of the Resonant Optical Cavity Supplemental Figure 1 shows the resonant optical cavity formed by a planar distributed bragg reflector and a silver back mirror. Following the formulation by Pochi Yeh, the propagation of the electromagnetic wave in such a layered structure can be modeled using transfer matrix method (TMM): A0 A s M11 M12 A s = M B = 0 0 M 21 M 22 0 Here A 0 and B 0 are the amplitudes of the forward and backward traveling waves in medium 0; A s and B s are the forward and backward traveling waves in medium s; M is the transfer matrix of the multilayer structure. The field reflection coefficients of a plane wave incident from medium 0 are then given by: B M r = ( ) = A M 0 21 0 11 and A 1 s t = ( ) Bs= 0 = A0 M11 The transfer matrix M of the multilayer structure can be expressed using the dynamical matrix and propagation matrix of each layer: 2
M = D D P D D N 1 1 0 l l l s l= 1 N is the number of layers; D l is the dynamical matrix of each layer given by 1 1, for s wave nl cosθl nl cosθ l cosθl nl cosθl, for p wave nl respectively. n l is the refractive index of each layer; ϴ l is the ray angle in each layer and is related to the x component of the wave vector in each layer k lx by k lx = n l (ω/c)cosθ l. P l is the propagation matrix of each layer given by P l iφi e 0 = iφ i 0 e with ɸ l = k lx d l. In order to find out the field amplitude at the interface where the MoS 2 flake is located, we rewrite the transfer matrix M into two parts: F N 1 1 1 M = D0 Dl Pl Dl DmPm Dm Ds = N G l= 1 m= F+ 1 Here we ignores the thickness of the MoS 2 for simplicity and assume that the flake is located at the interface between layer F and layer F+1. N and G are defined as: 3
F 1 1 N = D0 Dl Pl Dl l= 1 and N G= D P D D m= F+ 1 1 m m m s respectively. The amplitudes of the forward and backward traveling waves at the interface between layer F and layer F+1 is then given by: A F G11 G12 A s = B G G 0 F 21 22 And the total field at the interface between layer F and layer F+1 is given by: E = A + B = ( G + G ) A F F F 11 21 s The field enhancement of the fundamental wave by the resonant cavity is then: E G + G F 11 21 η ω = = A0 M11 We then try to find the output coupling coefficient of the second harmonic wave η 2ω. Assuming the MoS 2 flake excited by the fundamental wave E F emits a forward traveling second harmonic wave E 2ωF and a backward traveling second harmonic wave E 2ωB, the total output second harmonic wave E 2ω0 is given by: 4
E = E t ( r + r r r +... + r ( r r ) + E t (1 + r + r r r +... + r ( r r ) n n 2ω 0 2ωF n g g n g g n g 2ωF n g g n g g n g 2rg = E2 Ft ω n ( + 1) 1 r r n g 2rg = E2 Ft ω n ( + 1) 1 r r n g Here r g = G 21 /G 11, r n = -N 12 /N 11 and t n =1/N 11 and we have E η 2ω = E 2ω 0 2ωF 2G 1 = + N G + N G N 21 ( ) 21 = + 11 21 12 21 11 2G 1 M N 11 11 5
The structure and spectral reflectance of the distributed bragg reflector Figure S1 The multi-layer stack structure (a) and the spectral reflectance (b) of the DBR on glass substrate. Figure S1 (a) shows the structure of the distributed bragg reflector consisting of eight pairs of 130 nm SiN x / 130 nm SiO x thin films grown on glass substrate by PECVD. Figure S1 (b) shows the measured (solid blue line) and the simulated (black dotted line) of the spectral reflectance of the DBR. The DBR is designed to be reflective from 800 nm to 1000 nm (the fundamental wave region) and 400 nm 500 nm (the second harmonic wave region) while transparent in the visible region. 6
Voltage dependent mechanical deflection of the membrane mirror Figure S2 Characterization of the voltage dependence of the mechanical deflection of the membrane mirror in a MEMs capacitor. (a) The measured (red dots) and simulated (blue dotted line) mechanical deflection at the center of the membrane and the pull-in voltage predicted by the numerical simulation. (b) The simulated membrane deflection profile along the center line under 10 V, 20 V and 30 V, respectively. The inset shows the 2D deflection profile. The red circles in Fig. S2a show the measured voltage dependence of the membrane deflection at the center, while the dotted blue line shows the results obtained from the electrostatic simulations. Figure S2b shows the voltage dependence of the membrane deflection along the center line. The electromechanics simulations also predict a pull-in voltage of 31 V for the capacitor structure which sets the upper limit for the electrostatic deflection of the membrane in a 7
MEMS capacitor 1. For an initial cavity length of 3.1 µm, the largest obtainable deflection is 1.2 µm. This demonstrates the suitability of electrostatic tuning for on-chip reconfigurable resonant optical microcavities 2. 8
Voltage dependent spectral reflectance of the optomechanical resonant cavity Figure S3 The spectral reflectance of the optomechanic cavity under the control voltage of (a) 0 V, (b) 10 V, (c) 15 V, (d) 17 V, (e) 21 V and (f) 25 V, respectively. 9
Fig. S3(a)-(f) shows the measured spectral reflectance of the optomechanical cavity in both the fundamental wave region (800 nm 1000 nm, red curve) and the second harmonic wave region (400 nm 500 nm) under different control voltage. The resonant wavelengths of the cavity modes in both the fundamental wave region and the second harmonic wave region are blue-shifted with increased control voltage. At 17 V and 21 V, the resonant wavelength of the fundamental mode M 1 overlaps with a resonance in the second harmonic wave region which corresponds to the two crossing points between the blue and red curves in Fig. 3b in the main text. From Fig. S3 (e), we estimate the Q value by / or / to be Q ω ~ 77 and Q ω ~ 46. Measurement of the modulation speed of the optomechanical cavity 10
Figure S4 (a) The measurement setup for the modulation speed of the optomechanic cavity. (b) The wavelength of the input light and the working point of the modulation voltage. (d) The measured frequency response of the optomechanic cavity. As shown in Fig. S4(a), an input light with fixed wavelength is sent to the optomechanical cavity from a supercontinuum source (Fianium Whitelase SC400-4) filtered by a monochromator (Cornerstone 130 1/8 m). The sample is mounted on a Nikon TE2000-U Inverted Microscope and the light spot is focused at the center of the membrane mirror. The modulation voltage (V bias = 4.75 V, V pp = 0.5 V) is applied across the cavity through a signal generator (Tektronics AFG3021B). The reflected light intensity is measured by a photodiode power sensor (Thorlabs SC121C) Figure S4(b) shows the work point of the optical modulation measurement. The modulation voltage is biased at 4.75 V with a peak-to-peak amplitude of 0.5 V and the wavelength of the input light is set at 900 nm, where there is a resonant mode. Figure S4(c) shows the normalized frequency response of the reflected light intensity by scanning the frequency of the modulation voltage and the 3dB bandwidth is about 200 Hz. 11
Characterization of the quality factor of resonant cavity Figure S5 The normalized optical spectra of the SHG output from the MoS 2 in the cavity and the MoS 2 on the glass substrate, respectively. In order to characterize the quality factor of the resonant cavity, we compared the normalized optical spectra of the SHG signal from the MoS 2 in the cavity and the MoS 2 on the glass substrate. It can be seen from Fig. S5 that the SHG signal from the MoS 2 in the cavity has a narrower line-width compared to SHG signal from the MoS 2 on the glass substrate, which means the cavity acts as an optical filter which enables the voltage tuning the wavelength of the SHG signal, as shown in Fig. 3c in the main text. 12
Figure S6 (a) The simulated voltage dependence of the overall SHG output power enhancement factor by transfer matrix method. (b) The log-scale view of the measured total output SHG power enhancement factor normalized to a monolayer on glass substrate. Fig. S6(a) shows the calculated voltage dependence of the total power enhancement of the second harmonic generation which predicts a 3500 times maximum enhancement factor. Fig. S6(b) shows the measured voltage dependence of the total output SHG power enhancement factor which agrees well with the theoretical prediction. Also by showing the measured total enhancement factor in log-scale view, the less pronounced second harmonic mode M2 due to the lack of a double resonance condition is now visible. 13
The optical field distribution inside the cavity at resonances Figure S7 (a) The optical reflectance of the micro-fp when the cavity-length is tuned to 2.75 µm. (b) The electric field strength along the cavity axis assuming the pump wavelength to be 910 nm and cavity length as 2.75 µm. (c) The optical reflectance of the micro-fp in the SHW region and (d) the corresponding electric field strength along the cavity axis assuming the SHW wavelength to be 458 nm. 14
Based on the numerical simulation using COMSOL, we find that in the FW region the micro-fp cavity is optically resonant at 907 nm when the cavity-length is voltage-tuned to 2.75 µm where the overall power enhancement is at maximum (See Fig. S6). At 907 nm, the field distribution along the cavity axis is plotted in Fig. S7 (b) using blue dotted line while the red dot indicates the position of the monolayer. It can be seen that when on resonance the monolayer is located close to (but not exactly at) the antinode of the cavity field. This is important for efficient out-coupling of the SH field and efficient generation with the pump field. Fig. S7 (c) shows that the closest SHW resonance is at 458 nm and the corresponding electric field distribution is plotted in Fig.S7 (d). Comparison of power enhancement factors of MoS 2 in cavity and MoS 2 on glass Figure S8 (a) The power enhancement factors of the fundamental wave and (b) the second harmonic wave. 15
Fig. S8 compares the power enhancement factors for the MoS 2 monolayer in the micro-fp cavity (red solid line) and on a glass substrate (blue solid line). Here we assume the length of the micro-fp cavity is voltage tuned to 2.75 µm where the overall power enhancement is at maximum (See Fig. S6). The derivation of the power enhancement factor based on the transfer matrix method is given in Theoretical Analysis of the Resonant Optical Cavity. The second harmonic wave output intensity from the cavity as a function of the nonlinear layer thickness Figure S9 (a) The SHW output intensity as a function of the FW wavelength and (b) the second harmonic wave as a function of the micro - FP cavity length. Since our micro FP cavity can also be used for conventional nonlinear optical thin films, we investigate how the thickness of the nonlinear thin film will affect the cavity enhanced second 16
harmonic generation based on numerical simulations using a modified transfer matrix method. In Fig.S9 (a) we choose three typical film thicknesses: 0.5 nm (the monolayer case), 5 nm and 40 nm to plot the SHW output intensity as a function of the FW wavelength, assuming the micro FP cavity length to be 3.1 µm (the initial cavity length). For a fixed micro-fp cavity length, the SHW output can be tuned from on-resonance to off-resonance by controlling the FW wavelength. Compared with the regular film cases (5 nm and 40 nm), the monolayer case can reach the minimum off resonance SHW output. We then plot the SHW output as a function of the FP cavity length, assuming the FW wavelengths to be 897 nm for 0.5 nm case, 916 nm for 5 nm case and 945 nm for 40 nm case, respectively. We also labeled the voltage tuning range of the cavity enhanced SHG defined as the ratio between the on resonance SHW output and the off resonance SHW output. It can be seen that the monolayer case can reach the largest voltage tuning range, compared to the regular film cases. 17
Estimation of the theoretical quality factors of the micro FP at double resonant Figure S10 Estimation of the theoretical quality factors of the micro FP at double resonant condition. Fig. S10 plots the spectral reflectance calculated at both FW region and SHW region using TMM. Here the cavity length of the micro FP is assumed to be 2. 75 µm (theoretical double resonance condition, see Fig. S6a. ) The theoretical Q value of the FW resonance at 907 nm Q ω is 207 and the corresponding Q value of the SHW resonance at 453 nm Q 2ω is 38. (1) Zhang, W.-M.; Yan, H.; Peng, Z.-K.; Meng, G. Sensors and Actuators A: Physical 2014, 214, (0), 187-218. (2) Blomberg, M.; Kattelus, H.; Miranto, A. Sensors and Actuators A: Physical 2010, 162, (2), 184-188. 18