# Modelling of Cavity Optomechanical Magnetometers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Concept of Optomechanical Magnetometry

## 3. Numerical Methods

## 4. Single Mechanical Mode Optomechanical Analysis

#### 4.1. Bending Effect

#### 4.2. Effect of the Size of the Terfenol-D

## 5. Multi-Mode Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Sensitivity for a Single Mechanical Mode

## Appendix B. COMSOL Implementation of the Magnetic Field

**Figure A1.**(

**a**) COMSOL layout for a pair of Helmholtz coils used to generate signal magnetic field. The axis of the pair of coils can be freely rotated in a 4$\pi $ solid angle. The diametre of the coils are more than 40 times larger than the lateral size of the Terfenol-D in the device under test (DUT); (

**b**) intersected orthonormal slices are used to project the amplitude of the magnetic field; (

**c**) the effect of the eddy current inside Terfenol-D when the signal magnetic field ${B}_{\mathrm{sig}}$ is driven in plane with frequency below 100 MHz, at 1 GHz and at 10 GHz. Colourmap refers to the magnetic field inside the Terfenol-D.

## Appendix C. COMSOL Implementation of the Modified Elastic Wave Equation

**Table A1.**Coefficients in the magnetomechanical coupling biased at 60 kA/m and prestressed at 20 MPa [13]. ${\lambda}^{\mathrm{H}}$ is the elasticity matrix element, ${e}^{\u03f5}$ is the piezomagnetic constant and ${\mu}^{\sigma}$ is the relative magnetic permeability.

unit (GPa) | ${\lambda}_{11}^{\mathrm{H}}$ | ${\lambda}_{12}^{\mathrm{H}}$ | ${\lambda}_{13}^{\mathrm{H}}$ | ${\lambda}_{33}^{\mathrm{H}}$ | ${\lambda}_{44}^{\mathrm{H}}$ | ${\lambda}_{66}^{\mathrm{H}}$ |

107 | 74.8 | 82.1 | 98.1 | 60 | 161 | |

unit (T) | ${e}_{13}^{\u03f5}$ | ${e}_{33}^{\u03f5}$ | ${e}_{15}^{\u03f5}$ | no unit | ${\mu}_{11}^{\sigma}$ | ${\mu}_{33}^{\sigma}$ |

90 | −166 | −168 | 6.9${\mu}_{0}$ | 4.4 ${\mu}_{0}$ |

## Appendix D. Calculation of c act by Lorentzian Fit

`sqrt(abs(u^2)+abs(v^2)+abs(w^2))`under volume maximum analysis and the ${m}_{\mathrm{eff}}$ is the quotient of

`solid.rho∗(abs(w^2)+abs(v^2)+abs(u^2))`under volume integration analysis and maximum displacement where

`u,v,w`are displacements in $x,y,z$ directions.

**Figure A2.**(

**a**) a fit to the Lorentzian distribution, and $\xi $ spectrum for a small input material damping; (

**b**) the fit and $\xi $ spectrum with input damping factor 12.5 times larger than that of a); (

**c**) the fit with damping factor 12500 times larger than that of a); (

**d**) all the fits and $\xi $ spectrum are performed on the mechanical mode with Terfenol-D position offset from the centre.

## Appendix E. Calculation of Geometrical Factor ξ

`1e-16`in the denominator is an example of adding a small value to eliminate the error of dividing by 0 when

`real(u)/real(v)/real(w)`is 0, and

`dx,dy,dz`are displacements after the synchronization.

`exp(i∗pi)`. Figure A2a,b shows the constant $\xi $ across a mechanical resonance with 12.5 times variation of input damping, showing that $\xi $ is insensitive to the mechanical quality. $\xi $ is missing in Figure A2c due to the numerical error at large manually input damping where $\xi $ spectrum is far away from constant. Plotting $\xi $ spectrum offers a way of sanity check of possible numerical errors.

## Appendix F. From ξ to Optomechanical Coupling

## References

- Hämäläinen, M.; Hari, R.; Ilmoniemi, R.J.; Knuutila, J.; Lounasmaa, O.V. Magnetoencephalography - theory, instrumentation, and applications to noninvasive studies of the working human brain. Rev. Mod. Phys.
**1993**, 65, 413–497. [Google Scholar] [CrossRef] - Mehlin, A.; Xue, F.; Liang, D.; Du, H.F.; Stolt, M.J.; Jin, S.; Tian, M.L.; Poggio, M. Stabilized Skyrmion Phase Detected in MnSi Nanowires by Dynamic Cantilever Magnetometry. Nano Lett.
**2015**, 15, 4839–4844. [Google Scholar] [CrossRef] [PubMed] - Chary, K.V.R.; Govil, G. NMR in Biological Systems-From Molecules to Humans; Springer: Dordrecht, The Netherlands, 2008; pp. 130–162. ISBN 987-1-4020-6679-5. [Google Scholar]
- Wolf, T.; Neumann, P.; Nakamura, K.; Sumiya, H.; Ohshima, T.; Isoya, J.; Wrachtrup, J. Subpicotesla Diamond Magnetometry. Phys. Rev. X
**2015**, 5, 021009. [Google Scholar] [CrossRef] - Jensen, K.; Kehayias, P.; Budker, D. Magnetometry with Nitrogen-Vacancy Centers in Diamond. In High Sensitivity Magnetometers; Grosz, A., Haji-Sheikh, M.J., Mukhopadhyay, S.C., Eds.; Springer International Publishing: Cham, Switzerland, 2017; p. 570. [Google Scholar]
- Muessel, W.; Strobel, H.; Linnemann, D.; Hume, D.B.; Oberhalter, M.K. Scalable Spin Squeezing for Quantum-Enhanced Magnetometry with Bose–Einstein Condensates. Phys. Rev. Lett.
**2014**, 113, 103004. [Google Scholar] [CrossRef] [PubMed] - Forstner, S.; Prams, S.; Knittel, J.; van Ooijen, E.D.; Swaim, J.D.; Harris, G.I.; Szorkovszky, A.; Bowen, W.P.; Rubinsztein-Dunlop, H. Cavity Optomechanical Magnetometer. Phys. Rev. Lett.
**2012**, 108, 120801. [Google Scholar] [CrossRef] [PubMed] - Forstner, S.; Sheridan, E.; Knittel, J.; Humphreys, C.L.; Brawley, G.A.; Rubinsztein-Dunlop, H.; Bowen, W.P. Ultrasensitive optomechanical magnetometry. Adv. Mater.
**2014**, 26, 6348–6353. [Google Scholar] [CrossRef] [PubMed] - Kirtley, J.R.; Ketchen, M.B.; Stawiasz, K.G.; Sun, J.Z.; Gallagher, W.J.; Blanton, S.H.; Wind, S.J. High-resolution scanning SQUID microscope. Appl. Phys. Lett.
**1995**, 66, 1138–1140. [Google Scholar] [CrossRef] - Bowen, W.P.; Milburn, G.J. Quantum Optomechanics; CRC Press: London, UK, 2015; pp. 1–92. ISBN 978-1-4822-5915-5. [Google Scholar]
- Forstner, S.; Knittel, J.; Sheridan, E.; Swaim, J.D.; Rubinsztein-Dunlop, H.; Bowen, W.P. Sensitivity and performance of cavity optomechanical field sensors. Photon. Sens.
**2012**, 2, 259–270. [Google Scholar] [CrossRef] - Forstner, S.; Knittel, J.; Rubinsztein-Dunlop, H.; Bowen, W.P. Model of a microtoroidal magnetometer. Proc. SPIE Opt. Sens. Detect. II
**2012**, 8439. [Google Scholar] - Claeyssen, F.; Bossut, R.; Boucher, D. Modeling and Characterization of the Magnetostrictive Coupling. In Proceedings: Power Transducers for Sonics and Ultrasonics; Hamonic, B.F., Wilson, O.B., Decarpigny, J.-N., Eds.; Springer: Berlin/Heidelberg, Germany, 1991; pp. 132–151. [Google Scholar]
- Ramos, D.; Mertens, J.; Calleja, M.; Tamayo, J. Study of the Origin of Bending Induced by Bimetallic Effect on Microcantilever. Sensors
**2007**, 7, 1757–1765. [Google Scholar] [CrossRef] [PubMed] - Schliesser, A.; Anetsberger, G.; Rivière, R.; Arcizet, O.; Kippenberg, T.J. High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators. New J. Phys.
**2008**, 10, 095015. [Google Scholar] [CrossRef] - Lee, K.H.; McRae, T.G.; Harris, G.I.; Knittel, J.; Bowen, W.P. Cooling and control of a cavity optoelectromechanical system. Phys. Rev. Lett.
**2010**, 104, 123604. [Google Scholar] [CrossRef] [PubMed] - Hauer, B.D.; Doolin, C.; Beach, K.S.D.; Davis, J.P. A general procedure for thermomechanical calibration of nano/micro-mechanical resonators. Ann. Phys.
**2013**, 339, 181–207. [Google Scholar] [CrossRef] - Pinarda, M.; Hadjarb, Y.; Heidmannc, A. Effective mass in quantum effects of radiation pressure. Eur. Phys. J. D
**1999**, 7, 107–116. [Google Scholar] [CrossRef] - Engdahl, G. Modeling of Giant Magnetostrictive Materials. In Handbook of Giant Magnetostrictive Materials; Academic Press: Los Angeles, CA, USA, 1999; pp. 127–174. ISBN 978-0-12-238640-4. [Google Scholar]
- Claeyssen, F.; Lhermet, N.; Le Letty, R.; Bouchilloux, P. Actuators, transducers and motors based on giant magnetostrictive materials. J. Alloys Compds.
**1997**, 258, 61–73. [Google Scholar] [CrossRef] - Kannan, K.S. Galerkin Finite Element Scheme for Magnetostrictive Structures and Composites. Ph.D. Thesis, University of Maryland, College Park, MD, USA, 1997; pp. 30–90. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Theory of Elasticity Volume 7 of Course of Theoretical Physics, 3rd ed.; Elsevier: Burlington, NJ, USA, 1986; pp. 1–15, 87–94. ISBN 978-0-08-057069-3. [Google Scholar]
- Wilson, D.J.; Sudhir, V.; Piro, N.; Schilling, R.; Ghadimi, A.; Kippenberg, K.J. Measurement-based control of a mechanical oscillator at its thermal decoherence rate. Nature
**2015**, 524, 325–329. [Google Scholar] [CrossRef] [PubMed] - Kippenberg, T.J.; Spillane, S.M.; Armani, D.K.; Vahala, K.J. Fabrication and coupling to planar high-Q silica disk microcavities. Appl. Phys. Lett.
**2003**, 83, 797–799. [Google Scholar] [CrossRef] - Kippenberg, T.J.; Spillane, S.M.; Vahala, K.J. Modal coupling in traveling-wave resonators. Opt. Lett.
**2002**, 19, 1669–1671. [Google Scholar] [CrossRef] - Verhoeven, J.D.; Gibson, E.D.; Mcmasters, O.D.; Ostenson, J.E. Directional Solidification and Heat Treatment of TerfenoI-D Magnetostrictive Materials. Metall. Trans. A
**1990**, 21, 2249–2255. [Google Scholar] [CrossRef] - Rossi, N.; Braakman, F.R.; Cadeddu, D.; Vasyukov, D.; Tutuncuoglu, G.; Fontcuberta, I.M.A.; Poggio, M. Vectorial scanning force microscopy using a nanowire sensor. Nat. Nanotechnol.
**2016**, 12, 150–155. [Google Scholar] [CrossRef] [PubMed] - De Lepinay, L.M.; Pigeau, B.; Besga, B.; Vincent, P.; Poncharal, P.; Arcizet, O. A universal and ultrasensitive vectorial nanomechanical sensor for imaging 2D force fields. Nat. Nanotechnol.
**2017**, 12, 156–162. [Google Scholar] [CrossRef] [PubMed] - Barry, J.F.; Turner, M.J.; Schloss, J.M.; Glenn, D.R.; Song, Y.; Lukin, M.D.; Park, H.; Walsworth, R.L. Optical magnetic detection of single-neuron action potentials using quantum defects in diamond. Proc. Natl. Acad. Sci. USA
**2016**, 113, 14133–14138. [Google Scholar] [CrossRef] [PubMed] - Jensen, K.; Budvytyte, R.; Thomas, R.A.; Wang, T.; Fuchs, A.M.; Balabas, M.V.; Vasilakis, G.; Mosgaard, L.D.; Staerkind, H.C.; Muller, J.H.; et al. Non-invasive detection of animal nerve impulses with an atomic magnetometer operating near quantum limited sensitivity. Sci. Rep.
**2016**, 6, 29638. [Google Scholar] [CrossRef] [PubMed] - Griffiths, D.J. Introduction to Electrodynamics, 3rd ed.; Prentice Hall International Inc.: Englewood Cliffs, NJ, USA, 1999; pp. 351–355. ISBN 0-13-805326-X. [Google Scholar]

**Figure 1.**Concept of an optomechanical magnetometer. (

**a**) illustration via a Fabry–Pérot type optical resonator. The coupling of magnetostrictive material to an optical cavity is quantified by the effective cooperativity ${C}_{\mathrm{eff}}$. The magnetostrictive material converts a magnetic field to a force ${F}_{\mathrm{field}}={c}_{act}{B}_{\mathrm{sig}}$ with ${B}_{\mathrm{sig}}$ being an oscillating magnetic field. Thermal force and optical shot noise act as noise terms. $\kappa $ (rad·s${}^{-1}$), $\mathrm{\Gamma}$ (rad·s${}^{-1}$), ${\omega}_{0}$ (rad·s${}^{-1}$), and ${\mathsf{\Omega}}_{\mathrm{M}}$ (rad·s${}^{-1}$) are optical and mechanical decay rate, optical and mechanical resonance frequency, respectively; (

**b**) sketch of a magnetometer with micro-toroidal structure coupled to a tapered optical fibre; (

**c**) homodyne detection scheme. The signal arm couples a coherent light source in and out from a magnetometer via a tapered optical fibre through an evanescent optical field, and is mixed with a strong reference beam (local oscillator field) by a 3 dB coupler. The magnetometer is embedded in the signal magnetic field.

**Figure 2.**(

**a**) sketch of the position offset of the magnetostrictive material of the first experimentally realized optomechanical magnetometers [7]; (

**b**) a second order crown mode without (left) and with 4 $\mathsf{\mu}$m (right) Terfenol-D position offset. Arrows show the positions with maximum displacement; (

**c**) strain of the magnetometer with centred (left) Terfenol-D, and with 4 $\mathsf{\mu}$m offset (right). Note that the colourmaps of the strain have different scales; (

**d**) $|{C}_{\mathrm{eff}}|$ and sensitivity as a function of the position of the Terfenol-D.

**Figure 3.**(

**a**) sensitivity vs. Terfenol-D disk size for the first order radial breathing-type modes of a thin film structure. The silica disk dominates the mechanical eigenmodes when the Terfenol-D (highlighted with white dashed line) is smaller (left) than the 15 $\mathsf{\mu}$m radius top facet of silicon pedestal indicated by a vertical line. If the Terfenol-D is larger than the silicon facet, the mechanical motion is hybridized with the Terfenol-D mode (right). A power-law fit is applied to the right side data. Insets are sketches of a thin film magnetometer and of two mechanical eigenmodes; (

**b**) deformation profile induced by axial magnetic field driving for Terfenol-D smaller and larger than the pedestal top facet.

**Figure 4.**Multi-mode analysis with a device reported in Ref. [8]. (

**a**) cross-sectional view of the optomechanical magnetometer; (

**b**) top: the power spectral density ${S}_{\mathrm{noise}}(\mathsf{\Omega})$ (blue) is the sum of individual thermal Brownian motion peaks (grey) and coherent laser shot noise on the optical phase quadrature (red); bottom: minimum detectable magnetic field from multi-mode (blue) and single mode (black triangles) analysis driven by in-plane magnetic field. The inset is the mechanical mode with the highest ${c}_{act}$ at ${\mathsf{\Omega}}_{\mathrm{M}}/2\pi $ = 23 MHz; (

**c**) deformation profile induced by in-plane magnetic field far away from mechanical resonance frequencies; (

**d**) top: the power spectral density ${S}_{\mathrm{noise}}(\mathsf{\Omega})$ of the radial-breathing-like mechanical modes. The insets show the mechanical eigenmodes corresponding to each resolved thermal Brownian motion peaks; middle: the magnetic field response ${S}_{\mathrm{signal}}(\mathsf{\Omega})/\tau $ to the axial magnetic field driving; bottom: the sensitivity spectrum from multi-mode (blue) and single mode (black triangles) analysis driven by the axial magnetic field; (

**e**) deformation profile induced by axial magnetic field far away from mechanical resonance frequencies.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Forstner, S.; Rubinsztein-Dunlop, H.; Bowen, W.P.
Modelling of Cavity Optomechanical Magnetometers. *Sensors* **2018**, *18*, 1558.
https://doi.org/10.3390/s18051558

**AMA Style**

Yu Y, Forstner S, Rubinsztein-Dunlop H, Bowen WP.
Modelling of Cavity Optomechanical Magnetometers. *Sensors*. 2018; 18(5):1558.
https://doi.org/10.3390/s18051558

**Chicago/Turabian Style**

Yu, Yimin, Stefan Forstner, Halina Rubinsztein-Dunlop, and Warwick Paul Bowen.
2018. "Modelling of Cavity Optomechanical Magnetometers" *Sensors* 18, no. 5: 1558.
https://doi.org/10.3390/s18051558