The second device targets just one of the Zipper's double-clamped nanoscale beams. 137 5.3 Scaling properties of the mechanical modes of the optomechanical crystal.138 5.4 Transduction of the Brownian motion of the mechanical breathing modes.
Introduction
A Very Brief Description of the Sensitivity of Optomechanical
This is the essence of optomechanical coupling: the mechanical movement of the cavity causes a change in the state of the light. In this case, the apparent displacement sensitivity increases as the cavity length decreases.
The Significance of This Work
Finally, while the field of phononic crystal microstructures and nanostructures is advancing rapidly [10], high-frequency operation has been quite limited by electromechanical sensing techniques involving the integration of piezoelectric couplers. Optomechanical crystals enable low-noise optical measurement of high-frequency phononic crystal waveguides and resonators in any material with an optical window.
The Optical and Mechanical Systems
Geometry
The mirrors are the two lengths of the structure on either side of the defect with a series of holes with periodicity Λ. The substrate or pad is the large (essentially infinite) region on either side of the nanobeam from which it is suspended.
Electromagnetic and Acoustic Eigenvalue Problems
- Acoustic Eigenvalue Problem
- Electromagnetic Eigenvalue Problem
Performing the matrix multiplication returns. 1.10) is the mass phase velocity for the material at the frequency of the mode (the speed of sound). In regions of spatially constant refractive index, the component form of the electromagnetic eigenvalue problem becomes
Symmetries of the Eigenvalue Problems
- Illustrative Example: Mirror Symmetry of the Eigen-
- Differential Operators: Symmetries in Vector-Valued
- Hermiticity
- Commutativity of Operators and Symmetry Classifi-
- Mirror Symmetry Revisited
If we call this factor α, then we have demonstrated (for the case of no degeneracy) that ˆAQ(r) = αQ(r); that is, Q(r) is an eigenvector of the operator ˆA with eigenvalueα. With this restriction on the eigenvalues, we can derive information about the spatial parity of the components of Q(r).
Modes and Symmetries of the Projected Mirror Portions
- Photonic and Phononic Bands of the Projection
- Mirror Symmetries of the Projection
The mechanical displacement profiles of the unit cell are shown for each band at Γ and X. Above the light line, the modes of the nanobeam propagate in the direction transverse to the waveguide (x).
Localized Modes and Symmetries
The solutions to the wave equations in the defect can be viewed as coming from the band edges of the projection. Localized modes are formed when the defect's modes exist at a frequency for which the density of states in the projection is small or zero.
Mode Amplitudes, Effective Mode Volumes, and Effective Mass 27
The mode amplitude, α, must also represent the amplitude of the generalized position, β(t) = αcos(Ωt), and the generalized momentum, meffβ(t), of a simple harmonic oscillator-˙tor with energy,Emechanical = m2eff (Ω2β2+˙β2). Furthermore, α is the amplitude of the zero-point motion of the canonical position operator in a quantized treatment.
Cavity Optomechanics
Dispersive Coupling Between the Mechanical and Optical Modes 33
Sideband Formalism
- Formal Solution
- The Transmission of an Oscillating Cavity
- RF Spectrum of the First Order Sidebands
- Optical Forces
- Power Transfer and Effective Temperature
Calculating the Power Spectral Density
Extracting the Product m eff L 2 OM from Experimental RF Spectra 45
- Sideband Unresolved Limit (Ω M Γ)
- Sideband Eesolved Limit (Ω M Γ)
- Optimal Detuning Points for Damping/Amplification 51
- Example: The Zipper Optomechanical Cavity
- Example: The Double-disk Optomechanical Cavity . 58
Summary
This results in the derivative of the spread with respect to alpha being equal to. A final consideration is the relationship between the optical spring effect and the mechanical damping/amplification.
Introduction
The tuning properties of a dual nanobeam photonic crystal are then calculated to estimate the strength of the optomechanical coupling for the in-plane differential motion of the beams. We conclude with a comparison of the cavity properties of the zipper with other more macroscopic optomechanical systems and a discussion of the future prospects for this kind of chip-based gradient optical power devices.
Optical Design and Simulation
2.2(c), we plot the local lattice period, defined asanh =x(nh+ 1)−x(nh), versus the number of air holesnh along the length of the cavity. 2.2(e) showing the frequency of the local valence band edge (blue solid curve) and conduction band edge (red dotted line) modes normalized to the valence band edge mode frequency in the mirror part of the cavity.
Mechanical Mode Analysis
We have also studied the expected thermal properties of the zipper cavity, again assuming a 1.5 µm wavelength of operation. The physical mass of the zipper cavity, taking into account the etched holes, is approx. m= 43 picograms.
Optomechanical Coupling
For the special case of in-plane differential modes, the optomechanical coupling factor with the optical mode TE+,0 is approximately given by,. The optomechanical coupling factor at TE+.0 for each of the planar differential modes is tabulated along with the mechanical mode properties in Table 2.1.
Summary and Discussion
In addition to cavity optomechanics, the zippered cavity can also find applications in the field of cavity QED. In the case of a zippered cavity, a small Veff would result in a coherent zero-phonon line (ZPL) NV-transition coupling level of about gZPL/2π ~3 GHz, even after accounting for 3–5% branching. ratio for the ZPL line.
Introduction to the Zipper Optomechanical System
A useful figure for cavity optomechanical systems is the coupling constant gOM≡ dωc/dx, which represents the differential frequency shift of the cavity resonance (ωc) with mechanical displacement of the beams (x). In the case of the zip cavity, the optomechanical coupling is exponentially proportional to the slot gap (s) between the beams, gOM =ωc/LOM with LOM∼woeαs.
Fabrication
A fiber polarization controller is used to adjust the polarization to selectively excite the transverse electric (TE) polarization modes of the chain cavity. For devices at the high end of the measured Q range (Q∼ 3×105), we find a significant contribution to the optical loss from absorption (see Methods).
RF Optical Spectroscopy
Measured optical transmittance of a zipper cavity with w = 650 nm and s = 120 nm, showing four orders of the bound (TE+) resonance modes. The RF spectrum of the transmitted optical intensity up to 150 MHz is shown in Fig.
Optical Spring and Damping
Measured and modeled d,f total RF power and e.g. resonant frequency of the h1,d mechanical mode, versus detuning. At low optical power (Figures 3.11(a,b)), a single estimate for gOM, based on FEM optical simulations, fits both the total measured RF power (or hx2i) and the optical frequency of the h1d mode over a large detuning range.
Prospects of the “Zipper” Optomechanical system
Optomechanical Coupling, Effective Mass and Spring Constant
With this definition of amplitude, the effective mass of motion is simply the total mass of the two cantilevers (mx = mc = 43 picograms) and the effective spring constant is defined by the usual relationship keff = mcΩ2M, where ΩM is the mechanical eigenmode frequency. . The amplitude associated with zero point motion and used in the equipartition theorem to determine the thermal excitation of the mechanical mode is then xn(t).
Optical Transmission, Measured RF Spectra, and Motional Sensitivity 99
The total fiber taper transmission after mechanical anchoring of the taper to the substrate is 53%. To accurately determine the optical power reaching the cavity (determined by the optical loss in the conical section in front of the cavity) we measure the response of the high optical power cavity (resulting in thermo-optical tuning of the cavity and optical bistability in transmission response) to the input sent in one direction and then the other of the taper.
Calibration of Laser-Cavity Detuning
Zipper Cavity Optical Loss
Steady-State Nonlinear Optical Model of the Zipper Optomechanical
- Optical Properties
- Geometry
- Silicon Nitride Material Properties
- Thermal Properties of the Zipper Cavity
- Optomechanical Properties of the Zipper Cavity
- Wavelength-Scan Fitting
The physical mass of the zipper cavity, taking into account the etched holes, is approximately m = 43 picograms. The FEM simulated optomechanical coupling length, based on SEM images of the device under test, is LOM = 2.09 µm.
Introduction
Here we describe how this perturbation theory can be used to create an intuitive, graphical picture of the optomechanical coupling of simultaneously localized optical and mechanical states in periodic systems. We show how the optical and mechanical modes and their coupling can be understood in terms of the quasi-one-dimensional nanobeam example.
One-Dimensional Optomechanical Crystal Systems: An Example
The colors of the names correspond to the illustration of the reverse potential in Figure 4.1(b)); the first three cavity modes of that band's defect are shown in Fig.
Modal Cross-Coupling and Mechanical Losses
Changing the length of the structure changes the resonance condition for both the propagation and body modes. This changes the degree of coupling with the localized mode in the self-consistent solution of the system.
Optomechanical Coupling: Definition and Integral Representation
Interestingly, the Q of the harmonic mode increases exponentially with the number of holes, indicating that the mode fades in the mirror parts. Note that α is also the amplitude of the zero-point motion of the canonical position operator in the quantized treatment.
Optomechanical Coupling: Visual Representation and Optimization . 121
4.5(b)-(d) show ζOM, Θm and Θo plotted on the OMC surface of the nanobeam for the basic breathing mode and the basic optical mode. The structure is shown slightly tilted so that the interior of the holes, which make the dominant contribution to the optomechanical coupling, can be seen.
Introduction
Acoustic and Optical Modes
Thus, the optical modes of the infinitely periodic structure are confined by a quasi-harmonic potential. The localized mechanical modes of device 1 are shown to the right of the corresponding projection mode.
Optomechanical Coupling
The localized optical states of the final structure (hereafter referred to as unit 1) are also found by FEM simulation and shown in Fig. The mechanical band diagrams for each structure are shown to the right of the measured RF spectrum with the pinch mode band highlighted in red.
Engineering of the Mechanical Frequencies
Euler beams”, the frequency of the mechanical mode scales perfectly with the two-dimensional scale factor. Significant shifts in the frequency of the lattice-localized mechanical modes can be obtained via non-uniform planar scaling.
It can be shown analytically that the factor 1/(meffL2OM) uniquely determines the transduction of Brownian motion for these sideband-resolved optomechanical oscillations (see § 1.3.5). The resulting metric for the optomechanical coupling between the fundamental respiration and the optical mode (assuming a FEM-calculated mass of motion of meff = 330 fg) is LOM = 2.9 µm, which approaches the limit of the wavelength of light.
Summary and Conclusion
This obstacle can be overcome in two-dimensional periodic plate structures, which have been shown to possess complete gaps for both optical and mechanical modes simultaneously [113].
Measured and Simulated Optomechanical Coupling and Mechanical Q 143
By increasing the sinking power to 190 µW (an 11 dB increase), the mechanical mode power increases dramatically and the linewidth narrows to below the oscilloscope's resolution limit of 4.8 kHz (thus the resolution-limited effective Q is 460,000). This form of regenerative oscillation [5, 127] (sometimes called paramteric instability) arises due to the retarded part of the optical force on the mechanical mode, which for a blue tuned laser input results in amplification of the mechanical motion.
Experimental Setup
Although much of the signal is below the resolution bandwidth, the linewidth at 931 µW can still be extracted as there is more than 20 dB of signal-to-noise at the point where the lineshape becomes wider than the resolution limit. The APD has an internal bias tee and the RF voltage is connected to the 50 Ohm input impedance of the oscilloscope.
Fabrication
The oscilloscope can perform a Fourier transform (FT) to obtain the RF power spectral density (RF PSD). The RF PSD is calibrated using a frequency generator that outputs a sinusoid of variable frequency and known power.
Numerical Modeling
Extracting the Geometry in the Plane
The SEM has been calibrated and the dimensions as measured by the SEM are 5% too large. Because the SEM and lithography tools are independent, this is another confirmation that the geometry has been measured correctly (the fine spectral features of the simulation are the other way to check the geometry measurements, after comparison with measured mechanical and optical spectra).
Young’s Modulus and Index of Refraction
5.9(a) and 5.9(b) show the in-phase and in-quadrature (respectively) parts of the optical cycle, showing a propagating radiation mode in the pad. 96] MIT Photonic Bands (MPB) is a free software package for solving the electromagnetic eigenmodes of periodic structures.
Normalized frequency and optical Q-factor (axial, transverse, and total)
Effective mode volume of the TE 0,+ mode versus normalized slot gap, ¯ s. 75
Mechanical eigenmode displacement plots
This makes the structure susceptible to mechanical loss mechanisms similar to that of the nanobeam. This creates an imaginary part of the frequency, and the mechanical Q can be found by the ratio, .
Dispersion and optomechanical coupling vs. gap size
Comparison of optomechanical systems
Finite-element-method simulated bonded and anti-bonded supermodes
Scanning-electron-microscope (SEM) images of a typical zipper cavity. 88
Finite-element-method simulation of the wavelength tuning versus nanobeam
Measured position dependence of taper-Zipper coupling
RF optical spectroscopy
Mechanical spectrum of the zipper optomechanical cavity
Optical Spring and Damping
Optical Spring and Damping
Properties and modes of the OMC mirror
Properties and modes of the OMC photonic and phononic bandgap cavity.112
Dependence of mechanical Q on number of mirror periods and diagram