diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 0000000..c16ccb4 Binary files /dev/null and b/.DS_Store differ diff --git a/supplementary/SI.loc b/supplementary/SI.loc new file mode 100644 index 0000000..e69de29 diff --git a/supplementary/SI.pdf b/supplementary/SI.pdf index 7f97ab6..14b6b0f 100644 Binary files a/supplementary/SI.pdf and b/supplementary/SI.pdf differ diff --git a/supplementary/SI.tex b/supplementary/SI.tex index 3fc2d36..94fb1b3 100644 --- a/supplementary/SI.tex +++ b/supplementary/SI.tex @@ -1,228 +1,279 @@ \documentclass[aps,a4paper,notitlepage,aps,pra]{revtex4-1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGE CONFIG. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %start the SI figur labeling with S \usepackage[english]{babel} %\addto\captionsenglish{\renewcommand{\figurename}{Figure}} \renewcommand{\thefigure}{S\arabic{figure}} \renewcommand{\theequation}{S\arabic{equation}} \usepackage{amsmath,amssymb,amsfonts} % standard AMS packages \usepackage{changes} + \usepackage{bm} % bold symbols in math mode \bm{...} \renewcommand{\mathbf}{\bm} \usepackage{dsfont} % proper mathbb format \renewcommand{\mathbb}{\mathds} % redefine \mathbb \usepackage{mathrsfs} % use \mathscr{} for script letters in math \usepackage{mathtools} % for proper typesetting of := and =: \newcommand{\eqdef}{\vcentcolon=\,} \newcommand{\defeq}{=\vcentcolon\,} \usepackage{graphicx,float} \usepackage[colorlinks, linkcolor=red, citecolor=blue, urlcolor=red]{hyperref} \usepackage{chemformula} \usepackage{siunitx} \interfootnotelinepenalty=10000 % prevents footnotes from splitting across pages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUSTOM MACROS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % macros for physics objects \newcommand{\ket}[1]{\vert{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}\vert} \newcommand{\op}[1]{\hat{#1}} \newcommand{\tr}{\mathrm{Tr}} \newcommand{\avg}[1]{\left\langle{#1}\right\rangle} % macros for math objects \newcommand{\re}{\mathrm{Re}\,} \newcommand{\im}{\mathrm{Im}\,} \newcommand{\abs}[1]{\left\vert{#1}\right\vert} \newcommand{\symtext}[2]{\ensuremath{\stackrel{{#2}}{{#1}}}} % specific macros for this document \renewcommand{\t}[1]{\mathrm{#1}} \newcommand{\SiN}{Si$_3$N$_4\,$} % useful reference macros \newcommand{\figref}[1]{Fig.~\ref{#1}} \renewcommand{\eqref}[1]{Eq.~\ref{#1}} \newcommand{\secref}[1]{Sec.~\ref{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Supplementary information\\ ``Optomechanical thermal intermodulation noise''} \author{S. A. Fedorov} \thanks{These authors contributed equally} \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} \author{A. Beccari} \thanks{These authors contributed equally} \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} \author{A. Arabmoheghi} \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} %\author{\\M. J. Bereyhi} %\affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} \author{D. J. Wilson} \affiliation{College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA} \author{N. J. Engelsen} \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} \author{T. J. Kippenberg} \email{tobias.kippenberg@epfl.ch} \affiliation{Institute of Physics (IPHYS), {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne, 1015 Lausanne, Switzerland} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BEGIN MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \tableofcontents \addtocontents{toc}{\protect\setcounter{tocdepth}{1}} \section{Membrane fabrication} Patterned and unpatterned membrane samples are fabricated on the same \SI{100}{\milli \meter} wafer. Stoichiometric, high stress \ch{Si_3N_4} is grown by low pressure chemical vapor deposition (LPCVD) on both sides of a \SI{200}{\micro \meter}-thick silicon wafer. The initial deposition stress is estimated a posteriori from the observation of membrane resonant frequencies, and varies in the range \SI{900}{}-\SI{1100}{MPa}, changing slightly with deposition run. The fabrication process relies on bulk wet etching of silicon in \ch{KOH} through the whole wafer thickness, to create openings for optical access to the membranes samples \cite{tsaturyan2017ultracoherent,reinhardt2016ultralow,norte2016mechanical,gartner2018integrated}. The extremely high selectivity of \ch{Si_3N_4} to \ch{Si} during \ch{KOH} etching allows the use of the backside nitride layer as a mask, to define the outline of the membranes on the frontside. Initially, the frontside nitride (\ch{Si_3N_4}) layer is patterned with h-line photolithography and \ch{CHF_3}/\ch{SF6}-based reactive ion etching (RIE) (steps 2-3 of figure \ref{fig:processflow}). The photoresist film is then stripped with a sequence of hot N-Methyl-2-pyrrolidone (NMP) and \ch{O_2} plasma; this procedure is carefully repeated after each etching step. The frontside nitride layer is then protected by spinning a thick layer of negative-tone photoresist (MicroChemicals AZ\textregistered 15nXT), prior to flipping the wafer and beginning the patterning of membrane windows on the backside nitride layer (steps 4-5). We noticed a reduction in the occurrence of local defects and increased overall membrane yield when the unreleased membranes on the frontside were protected from contact with hot plates, spin-coaters and plasma etchers chucks. The backside layer is then patterned with membrane windows, in a completely analogous way. The exposure step requires a wafer thickness-dependent rescaling of membrane windows, to account for the slope of slow-etching planes in \ch{KOH}, and careful alignment with frontside features. After stripping the photoresist, the wafer is installed in a PTFE holder for the first wet etching step in \ch{KOH} at $\approx \SI{75}{\celsius}$ (step 6). The holder clamps the wafer along its rim, sealing off the wafer frontside with a rubber O-ring, while exposing the backside to chemical etching by \ch{KOH}. This procedure is necessary to ensure that PnC membranes are suspended correctly: we noticed that releasing PnC samples by etching from both sides of the wafer produced a large number of defects in the phononic crystal, probably due to the particular dynamics of undercut and stress relaxation in the film. The wafer is etched until \SI{30}{}-\SI{40}{\micro\meter} of silicon remains, leaving the samples robust during the subsequent fabrication steps. The wafer is then removed from the KOH bath and the PTFE holder, rinsed and cleaned in concentrated \ch{HCl} at room temperature for 2 hours \cite{nielsen2004particle}. Subsequently, the wafer is coated with thick, protective photoresist and diced into $\SI{8.75}{mm}\times\SI{8.75}{mm}$ chips, and the remainder of the process is carried on chip-wise. Chips are cleaned again with hot solvents and \ch{O_2} plasma, and the membrane release is completed by exposing chips to \ch{KOH} from both sides (step 7). The temperature of the solution is lowered ($\approx \SI{55}{}-\SI{60}{\celsius}$), to mitigate the perturbation of fragile samples by buoyant \ch{N_2} bubbles, a byproduct of the etching reaction. After the undercut is complete, the samples are carefully rinsed, cleaned in \ch{HCl}, transferred to an ethanol bath and gently dried in a critical point dryer (CPD). \begin{figure}[t] \includegraphics[width=0.6\textwidth]{fig_process_flow.pdf} \centering \caption{\small{Main steps of the fabrication process. Magenta - \ch{Si_3N_4}; gray - Si; green - photoresist.}} \label{fig:processflow} \end{figure} \section{Quadratic mechanical displacement transduction by the optical cavity versus quadratic optomechanical coupling} Nonlinear cavity transduction can produce signals, quadratic in mechanic displacement, that are orders of magnitude stronger than those due $\partial^2 \omega_c/\partial x^2$ terms that were ever experimentally demonstrated\cite{brawley_nonlinear_2016}. Below we derive the classical dynamics of optical field in an optomechanical cavity taking into account terms that are quadratic in displacement. We show that in membrane in the middle cavity typical quadratic signals originating from the nonlinear transduction are $r\mathcal{F}$ larger than the signals due to the nonlinear optomechanical coupling, $\partial^2 \omega_c/\partial x^2$. The fluctuations of $\nu$ due to the mechanical displacement are given by \begin{equation} \delta\nu(t)\approx 2\frac{G}{\kappa} x(t)+\frac{G_2}{\kappa} x(t)^2, \end{equation} where $G=-\partial \omega_c/\partial x$ and $G_2=-\partial^2 \omega_c/\partial^2 x$ are the linear and quadratic optomechanical coupling, respectively, and the total displacement $x$ is composed by partial contributions of different modes $x_n$ \begin{equation} x(t) = \sum_n x_n(t). \end{equation} For resonant lase probe we can find the intracavity field as \begin{equation}\label{eq:aResDisp} a(t)\approx 2\sqrt{\frac{\eta_1}{\kappa}}(1 -i\nu(t) -\nu(t)^2) s_\t{in,1}=2\sqrt{\frac{\eta_1}{\kappa}}\left(1 -2i\frac{G}{\kappa} x(t) -\left(\left(2\frac{G}{\kappa} \right)^2+i\frac{G_2}{\kappa}\right) x(t)^2\right) s_\t{in,1}. \end{equation} It is instructive to compare the magnitudes of the two contributions to the prefactor of $x(t)^2$. The typical value for $G$ (assuming the membrane to be not very close to one of the mirrors) is \begin{equation} G\sim 2r \frac{\omega_c}{l_c}, \end{equation} while the typical value for $G_2$ is\cite{thompson_strong_2008} \begin{equation} G_2\sim 4 \frac{r \omega_c^2}{c \, l_c}, \end{equation} where $c$ is the speed of light, $r$ is the membrane reflectivity and $l_c$ is the cavity length. The ratio of the two contributions is evaluated as \begin{equation} \left. \left(2\frac{G}{\kappa} \right)^2\right/\left(\frac{G_2}{\kappa}\right)\sim\mathcal{F}r. \end{equation} As the cavity finesse $\mathcal{F}$ is typically large, on on the order of $10^3$ to $10^5$, and the membrane reflectivity $r$ is between $0.1$ and $0.5$, we conclude that linear optomechanical coupling needs to extremely well suppressed in order for the quadratic coupling $G_2$ to contribute. \section{Dissipative coupling} In an optomechanical membrane-in-the-middle cavity dissipative coupling, $\partial \kappa/\partial x$, exists in addition to the dispersive coupling, $\partial \omega_c/\partial x$. Dissipative coupling modulates the optical decay rate, both external coupling and intrinsic loss, and potentially can produce intensity noise in a resonantly locked probe laser. However, for the parameters of our experiment the dissipative coupling is negligible. The noise due to dissipative coupling can be upper-bound as follows. The cavity linewidth cannot change by more than $\kappa$ as the membrane is translated by $\lambda$ inside the cavity, and therefore the dissipative coupling rate is limited by \begin{equation} \frac{\partial \kappa}{\partial x}\lesssim \frac{\kappa}{\lambda}= \frac{1}{\mathcal{F}}\frac{\omega_c}{2l_c}\sim \frac{G}{\mathcal{F}}, \end{equation} where in the last transition it was assumed that the membrane reflectivity is not very much smaller than one. Resonant intracavity field modulated by dissipative coupling only is given by \begin{equation} a(t)\approx 2\sqrt{\frac{\eta_1}{\kappa}}\left(1 - \frac{1}{2\kappa}\frac{\partial \kappa}{\partial x} x(t)\right) s_\t{in,1}. \end{equation} Comparing to \eqref{eq:aResDisp}, we find that the noise produced by dissipative coupling is negligible compared to the intermodulation noise if \begin{equation} \frac{G}{\kappa}x \gg \frac{1}{\mathcal{F}}. \end{equation} In all the experiments presented in this work this condition is satisfied, $Gx/\kappa$ ranges from \textcolor{red}{0.1 to 0.01 [verify exact numbers]}, whereas $1/\mathcal{F}$ is always less than $10^{-4}$. \section{Details of TIN calculations} +%%%%%%%%%%%%%%% + +\noindent For an optomechanical system, the cavity resonance frequency shifts due to a displacement field $\vec{u}(\vec r)$ is up to linear term given by \cite{PhysRevA.41.5187, Gorodetksy:10} + +\begin{equation} + \frac{\Delta\omega_c}{\omega_c} = \frac{1}{2}\frac{\int |\vec E{(\vec r)}|^2\nabla\epsilon(\vec r)\cdot \vec{u}(\vec r)}{\int |\vec E{(\vec r)}|^2\epsilon(\vec r)} +\end{equation} + where $\epsilon(\vec r)$ is the dielectric constant. For membrane in the middle systems the gradient of the dielectric constant is constant over the displacement field region. Therefore, the resonance frequency shift and correspondingly the linear optomechanical coupling constant ($G$) is only proportional to the average of the displacement field over the cavity mode shape at the position of the membrane. Hence, the optomechanical coupling constant for different flexural modes of a membrane can be written in the form $G_n = \eta_n\cdot G$, where $n$ denotes the mode number, $G$ is a constant and $\eta_n$ are the overlap factors, proportional to the average of the mode shape on the cavity mode. For a membrane (in the x-y plane) with flexural modes $\{u_n(x,y)\}$, by choosing $G$ to be equal to the coupling constant for the fundamental mode ($G_1$) the overlap factors have the form + + \begin{equation} + \eta_n = \frac{\int_{membrane}u_n(x,y)I(x,y)}{\int_{membrane}u_1(x,y)I(x,y)}, + \end{equation} +where $u_1$ denotes the fundamental mode and $I(x,y) = |\vec E{(x,y)}|^2$ is the cavity mode shape. For the fundamental mode of a Fabry-Perot cavity the mode shape is given by a Gaussian profile with +\begin{equation} + I(x,y) = \sqrt{\frac{2}{\pi w^2}} e^{-2((x-x_0)^2+(y-y_0)^2)/w^2} +\end{equation} +with $w$ being the waist of the mode shape at the position of the membrane and $x_0$ and $y_0$ being the posistion on the membrane the beam is focused on. We normalize the coupling constants to the value for the fundamental mode since in the actual experiment we calibrate the coupling constant using the fundamental mode. With $x_n(t)$ being the amplitude of the $n_{th}$ mode, total fluctuations of the cavity normalized detuning is given by +\begin{equation} + \delta\nu(t) = \frac{2G}{\kappa} \sum_n \eta_n x_n(t). +\end{equation} +Comparing to Eq.. (12) of the main text total displacement, $x(t)$ is defined as +\begin{equation} + x(t) = \sum_n \eta_n x_n(t). +\end{equation} +Spectrum of the linear fluctuations of $x$, $S_{xx}^{(1)}[\omega]$, is linear combination of thermal spectra of each mode, $S_{xx,n}^{(1)}[\omega]$, due to the fact that Brownian motion of different modes are statistically independent. $S_{xx,n}^{(1)}[\omega]$ is also given by the fluctuation-dissipation theorem. + +\begin{equation}\label{S_xx} + S_{xx}^{(1)}[\omega] = \sum_n \eta_n^2 S_{xx,n}^{(1)}[\omega] = \sum_n \eta_n^2 \frac{2kT}{\omega}\Im{\chi_n[\omega]} +\end{equation} +where $\chi_n[\omega]$ is the susceptibility of mode $n$. The quadratic fluctuations of $x$ can be calculated using a relation similar to Eq. (11) of the main text + +\begin{equation}\label{S_xx2} + S_{xx}^{(2)}[\omega] = 2\pi\langle x^2\rangle \delta[\omega] + \frac{1}{\pi}\int_{-\infty}^{\infty}S_{xx}^{(1)}[\omega']S_{xx}^{(1)}[\omega-\omega']d\omega' +\end{equation} +Having computed the linear and quadratic spectra of displacement fluctuations, the linear and quadratic frequency fluctuations are also calculated accordingly using the Eq. (13) and (14) of the main text, and finally the total photocurrent spectrum can be calculated from Eq. (18). + +For a thin high stress sqaure membrane with side L, the flexural modes are given by sine waves as +\begin{equation} + u_{nm} = \frac{2}{L}\sin{(\frac{2\pi x}{L})}\sin{(\frac{2\pi y}{L})}, +\end{equation} +with the mode frequencies are $\Omega_{nm} = \frac{\pi c}{L}\sqrt{n^2 + m^2}$ where $L$ is the side of the square and $c=\sqrt{\frac{\sigma}{\rho}}$ is the speed of the acoustic wave in a film with density of $\rho$ and stress $\sigma$. The effective mass for all modes is equal to $M/4$, a quarter of the total mass of the membrane. In the data presented in Fig. (3) of the main text, the membrane in under a rather high pressure, so that the damping process is dominated by the viscous damping which means constant damping rate given by $\Gamma_{nm} = \Omega_{nm}/Q_{nm}$. Piecing it all together, the susceptibility of the mode ${nm}$ is given by + +\begin{equation} + \chi_{nm}[\omega] = \frac{1}{M/4}\frac{1}{\Omega_{nm}^2 - \omega^2 - i\Gamma_{nm}\omega}. +\end{equation} + +Finally, we can analytically calculate the linear spectrum using Eq. (\ref{S_xx}) and then find the TIN by numerically computing the convolution integral in Eq. (\ref{S_xx2}). + +%%%%%%%%%%%%%%% + \section{The model of detuning dependence of total output light noise for MiM cavity with PnC membrane} As shown in the main manuscript text, the intensity of light, $I(t)$, and therefore the photodiode signal, is related to the linear ($\delta\nu(t)$) and quadratic ($\delta\nu(t)^2$) fluctuations of the cavity frequency as \begin{equation} I(t)=|s_\t{out,2}(t)|^2\propto |L(\nu_0)|^2 \left(1-\frac{2\nu_0}{1+\nu_0^2}\delta\nu(t)+\frac{3\nu_0^2-1}{(1+\nu_0^2)^2}\delta\nu(t)^2\right), \end{equation} where $\nu_0=2\Delta_0/\kappa$ is normalized detuning. The spectrum of intensity fluctuations of the output light is given by, \begin{equation} S_{II}[\omega]\propto \frac{4 \nu_0^2}{(1+\nu_0^2)^2} S_{\nu\nu}[\omega]+\frac{(3\nu_0^2-1)^2}{(1+\nu_0^2)^4} S_{\nu\nu,2}[\omega]. \end{equation} In an optomechanical cavity operated at high input power $S_{\nu\nu}$ and $S_{\nu\nu,2}$ in general are detuning-dependent because of the dynamic backaction of light, most importantly because of the laser cooling/amplification of mechanical motion. In order to find the dependence of $S_{II}$ on $\Delta$ some specific assumptions need to be made about the operation regime and the frequency of interest. Considering the case of data in Fig.~5b of the main text, here the noise level is estimated at the bandgap frequency and therefore only the mirror noise is expected to contribute to $S_{\nu\nu}$. The mechanical modes of the mirrors are relatively weakly coupled to the intracavity light and therefore the dynamical backaction for them can be neglected, resulting in detuning-independent $S_{\nu\nu}$. The intermodulation noise contribution, on the contrary, is significantly affected by laser cooling. It is natural to suggest (and it is advocated for by the very good agreement of our conclusions with experimental data) that TIN at bandgap frequencies is dominated by the mixing products of resonant and off-resonant parts of the membrane thermomechanical spectrum. Dynamical backaction reduces the mechanical spectral density on resonance $\propto 1/\Gamma_\t{DBA}$, where $\Gamma_\t{DBA}$ is the optical damping rate and $\Gamma_\t{DBA}\gg \Gamma_m$ is assumed, and it does not affect the off-resonant spectral density. In unresolved-sideband regime, which is typically well fulfilled in our measurements, the optical damping rate is given by \begin{equation} %\Gamma_\t{DBA}=-8\Omega_m\left(\frac{2g_0}{\kappa}\right)^2 \frac{\nu_0}{(1+\nu_0^2)^2}n_c \Gamma_\t{DBA}= -32\frac{\Omega_m}{\kappa}\left(\frac{2g_0}{\kappa}\right)^2\frac{\nu_0}{(1+\nu_0^2)^3} \eta_1|s_\t{in,1}|^2, \end{equation} and under our assumptions the spectral density of quadratic frequency fluctuations at PnC bandgap frequencies follows the detuning dependence of $1/\Gamma_\t{DBA}$, \begin{equation} S_{\nu\nu,2}\propto \frac{(1+\nu_0^2)^3}{|\nu_0|}, \end{equation} for $\nu_0<0$. Motivated by this consideration, the experimental data in Fig.~5b is fitted with the model \begin{equation} S_{II}\propto \frac{4 \nu_0^2}{(1+\nu_0^2)^2} C_1+\frac{1}{|\nu_0|}\frac{(3\nu_0^2-1)^2}{1+\nu_0^2} C_2, \end{equation} where $C_1$ and $C_2$ are free parameters. It was found that the model very well reproduces the observed variation of output noise with detuning and the value of $C_1$ found from the fit is indeed consistent with independently measured mirror noise, as shown in \figref{fig:mirrorNoise}. \section{Extended data} \textcolor{red}{[This section is a draft]} \textcolor{red}{[Add a calibrated measurement of the amplitude noise of Ti:Sa laser]} \begin{figure}[t] \includegraphics[width=\textwidth]{fig_power_sweep.pdf} \centering \caption{\small{a) Spectra of resonant relative intensity noise for a 2mm$\times$2mm square unpatterned membrane (resonance wavelength 837.7 nm, $g_0/2\pi=84$ Hz, $\kappa/2\pi=9.9$ MHz) at different input powers. The inset shows the same plot zoomed in at low frequencies. The RIN levels plotted in Fig. 2 of main manuscript are averaged over the frequency range shaded gray. b) The reproduction of the average RIN from Fig. 2 of main manuscript.}} \label{fig:powerSweep} \end{figure} The spectra of resonant RIN taken at different powers (shown in \figref{fig:powerSweep}a) show that the transmission signal is shot noise-limited at the frequency $\gtrsim 15$ MHz and therefore validates the shot noise estimate in Fig 2 of the main text. \begin{figure}[t] \includegraphics[width=\textwidth]{fig_ext_rect_mbr.pdf} \centering \caption{\small{a) Low frequency zoom-in of the data in Fig. 2 of the main text. b) Green points---measured linewidths of different optical resonances of MIM cavity with a 2mm$\times$2mm$\times$20nm unpatterned membrane, the dashed line is a guide to eye. Orange line---linewidth of an empty cavity with the same length.}} \label{fig:extRectMbr} \end{figure} \begin{figure}[t] \includegraphics[width=\textwidth]{fig_mirror_noise.pdf} \centering \caption{\small{a) Spectrum of detuning fluctuations due to the mirror noise. b) Mirror noise overlapped with a trace from detuning sweep presented in Fig. 5 of the main text corresponding to $2\Delta/\kappa=-0.51$.}} \label{fig:mirrorNoise} \end{figure} -\bibliography{supp_references} +\bibliography{supp_references.bib} \end{document} diff --git a/supplementary/supp_references.bib b/supplementary/supp_references.bib index b3402e7..52bcf0a 100644 --- a/supplementary/supp_references.bib +++ b/supplementary/supp_references.bib @@ -1,54 +1,87 @@ @article{nielsen2004particle, title={Particle precipitation in connection with KOH etching of silicon}, author={Nielsen, C Bergenstof and Christensen, Carsten and Pedersen, Casper and Thomsen, Erik Vilain}, journal={Journal of The Electrochemical Society}, volume={151}, number={5}, pages={G338--G342}, year={2004}, publisher={The Electrochemical Society} } @article{tsaturyan2017ultracoherent, title={Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution}, author={Tsaturyan, Yeghishe and Barg, Andreas and Polzik, Eugene S and Schliesser, Albert}, journal={Nature Nanotechnology}, volume={12}, number={8}, pages={776}, year={2017}, publisher={Nature Publishing Group} } @article{reinhardt2016ultralow, title={Ultralow-noise SiN trampoline resonators for sensing and optomechanics}, author={Reinhardt, Christoph and M{\"u}ller, Tina and Bourassa, Alexandre and Sankey, Jack C}, journal={Physical Review X}, volume={6}, number={2}, pages={021001}, year={2016}, publisher={APS} } @article{norte2016mechanical, title={Mechanical resonators for quantum optomechanics experiments at room temperature}, author={Norte, Richard A and Moura, Joao P and Gr{\"o}blacher, Simon}, journal={Physical Review Letters}, volume={116}, number={14}, pages={147202}, year={2016}, publisher={APS} } @article{gartner2018integrated, title={Integrated optomechanical arrays of two high reflectivity SiN membranes}, author={Gärtner, Claus and Moura, Jo{\~a}o P and Haaxman, Wouter and Norte, Richard A and Gröblacher, Simon}, journal={Nano Letters}, volume={18}, number={11}, pages={7171--7175}, year={2018}, publisher={ACS Publications} -} \ No newline at end of file +} + +@article{PhysRevA.41.5187, + title = {Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets}, + author = {Lai, H. M. and Leung, P. T. and Young, K. and Barber, P. W. and Hill, S. C.}, + journal = {Phys. Rev. A}, + volume = {41}, + issue = {9}, + pages = {5187--5198}, + numpages = {0}, + year = {1990}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.41.5187}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.41.5187} +} + + + +@article{Gorodetksy:10, +author = {M. L. Gorodetksy and A. Schliesser and G. Anetsberger and S. Deleglise and T. J. Kippenberg}, +journal = {Opt. Express}, +keywords = {Phase measurement; Fluctuations, relaxations, and noise; Optical microelectromechanical devices; Cavity quantum electrodynamics; Discrete Fourier transforms; Frequency modulation; Optical systems; Optomechanics; Phase modulation}, +number = {22}, +pages = {23236--23246}, +publisher = {OSA}, +title = {Determination of the vacuum optomechanical coupling rate using frequency noise calibration}, +volume = {18}, +month = {Oct}, +year = {2010}, +url = {http://www.opticsexpress.org/abstract.cfm?URI=oe-18-22-23236}, +doi = {10.1364/OE.18.023236}, +} + diff --git a/supplementary/supp_references.log b/supplementary/supp_references.log new file mode 100644 index 0000000..ccc23a3 --- /dev/null +++ b/supplementary/supp_references.log @@ -0,0 +1,37 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019) (preloaded format=pdflatex 2019.7.12) 20 MAR 2020 10:31 +entering extended mode + restricted \write18 enabled. + file:line:error style messages enabled. + %&-line parsing enabled. +**supp_references.bib +(./supp_references.bib +LaTeX2e <2018-12-01> + +./supp_references.bib:1: LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1 @ + article{nielsen2004particle, +? +./supp_references.bib:1: Emergency stop. + ... + +l.1 @ + article{nielsen2004particle, +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +Here is how much of TeX's memory you used: + 7 strings out of 492616 + 296 string characters out of 6129481 + 57569 words of memory out of 5000000 + 4025 multiletter control sequences out of 15000+600000 + 3640 words of font info for 14 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 5i,0n,4p,51b,14s stack positions out of 5000i,500n,10000p,200000b,80000s +./supp_references.bib:1: ==> Fatal error occurred, no output PDF file produced +!