# Nonlinearities in Fringe-Counting Compact Michelson Interferometers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling Nonlinearities

#### 2.1. Nonlinearities and Range

#### 2.2. Sources of Nonlinearity

#### 2.3. Nonlinearities from Ellipticity

#### 2.4. Nonlinearities from Nonellipticity

#### 2.5. Nonlinearities from Demodulation

## 3. Sensitivity Degradation in a High-RMS Application

## 4. Sensitivity Projections in the LIGO Vacuum Chambers

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BOSEM | Birmingham Optical Sensor and Electromagnetic Motor |

BRS | Beam rotation sensor |

CDS | Control and design system |

DFM | Deep frequency modulation |

FPGA | Field-programmable gate array |

FSR | Free spectral range |

GW | Gravitational wave |

HoQI | Homodyne quadrature interferometer |

ISI | Internal seismic isolation |

LIGO | Laser Interferometer Gravitational-Wave Observatory |

LISA | Laser interferometer space antenna |

PSD | Power spectral density |

PZT | Piezoelectric transducer |

RMS | Root mean square |

SNR | Signal-to-noise ratio |

## References

- Aasi, J.; Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Advanced LIGO. Class. Quantum Gravity
**2015**, 32, 074001. [Google Scholar] [CrossRef] - Acernese, F.A.; Agathos, M.; Agatsuma, K.; Aisa, D.; Allemandou, N.; Allocca, A.; Amarni, J.; Astone, P.; Balestri, G.; Ballardin, G.; et al. Advanced Virgo: A second-generation interferometric gravitational wave detector. Class. Quantum Gravity
**2015**, 32, 024001. [Google Scholar] [CrossRef] - Buikema, A.; Cahillane, C.; Mansell, G.L.; Blair, C.D.; Abbott, R.; Adams, C.; Adhikari, R.X.; Ananyeva, A.; Appert, S.; Arai, K.; et al. Sensitivity and performance of the Advanced LIGO detectors in the third observing run. Phys. Rev. D
**2020**, 102, 062003. [Google Scholar] [CrossRef] - de la Rue, R.; Humphryes, R.; Mason, I.; Ash, E. Acoustic-surface-wave amplitude and phase measurements using laser probes. Proc. Inst. Electr. Eng.
**1972**, 119, 117–126. [Google Scholar] [CrossRef] - Armano, M.; Audley, H.; Auger, G.; Baird, J.T.; Bassan, M.; Binetruy, P.; Born, M.; Bortoluzzi, D.; Brandt, N.; Caleno, M.; et al. Sub-Femto-g Free Fall for Space-Based Gravitational Wave Observatories: LISA Pathfinder Results. Phys. Rev. Lett.
**2016**, 116, 231101. [Google Scholar] [CrossRef] - Bond, C.; Brown, D.; Freise, A.; Strain, K.A. Interferometer techniques for gravitational-wave detection. Living Rev. Relativ.
**2017**, 19, 3. [Google Scholar] [CrossRef] - Watchi, J.; Cooper, S.; Ding, B.; Mow-Lowry, C.M.; Collette, C. Contributed Review: A review of compact interferometers. Rev. Sci. Instrum.
**2018**, 89, 121501. [Google Scholar] [CrossRef] [PubMed] - Smetana, J.; Walters, R.; Bauchinger, S.; Ubhi, A.S.; Cooper, S.; Hoyland, D.; Abbott, R.; Baune, C.; Fritchel, P.; Gerberding, O.; et al. Compact Michelson Interferometers with Subpicometer Sensitivity. Phys. Rev. Appl.
**2022**, 18, 034040. [Google Scholar] [CrossRef] - Robertson, N.A.; Cagnoli, G.; Crooks, D.R.M.; Elliffe, E.; Faller, J.E.; Fritschel, P.; Goszller, S.; Grant, A.; Heptonstall, A.; Hough, J.; et al. Quadruple suspension design for Advanced LIGO. Class. Quantum Gravity
**2002**, 19, 4043–4058. [Google Scholar] [CrossRef] - Carbone, L.; Aston, S.M.; Cutler, R.M.; Freise, A.; Greenhalgh, J.; Heefner, J.; Hoyland, D.; Lockerbie, N.A.; Lodhia, D.; Robertson, N.A.; et al. Sensors and actuators for the Advanced LIGO mirror suspensions. Class. Quantum Gravity
**2012**, 29, 115005. [Google Scholar] [CrossRef] - Yu, H.; Martynov, D.; Vitale, S.; Evans, M.; Shoemaker, D.; Barr, B.; Hammond, G.; Hild, S.; Hough, J.; Huttner, S.; et al. Prospects for Detecting Gravitational Waves at 5 Hz with Ground-Based Detectors. Phys. Rev. Lett.
**2018**, 120, 141102. [Google Scholar] [CrossRef] [PubMed] - Magee, R.; Chatterjee, D.; Singer, L.P.; Sachdev, S.; Kovalam, M.; Mo, G.; Anderson, S.; Brady, P.; Brockill, P.; Cannon, K.; et al. First Demonstration of Early Warning Gravitational-wave Alerts. Astrophys. J. Lett.
**2021**, 910, L21. [Google Scholar] [CrossRef] - Branchesi, M. Multi-messenger astronomy: Gravitational waves, neutrinos, photons, and cosmic rays. J. Phys. Conf. Ser.
**2016**, 718, 022004. [Google Scholar] [CrossRef] - Amaro-Seoane, P.; Gair, J.R.; Freitag, M.; Miller, M.C.; Mandel, I.; Cutler, C.J.; Babak, S. Intermediate and extreme mass-ratio inspirals—Astrophysics, science applications and detection using LISA. Class. Quantum Gravity
**2007**, 24, R113–R169. [Google Scholar] [CrossRef] - Cooper, S.J.; Collins, C.J.; Green, A.C.; Hoyland, D.; Speake, C.C.; Freise, A.; Mow-Lowry, C.M. A compact, large-range interferometer for precision measurement and inertial sensing. Class. Quantum Gravity
**2018**, 35, 095007. [Google Scholar] [CrossRef] - Cooper, S.J.; Collins, C.J.; Prokhorov, L.; Warner, J.; Hoyland, D.; Mow-Lowry, C.M. Interferometric sensing of a commercial geophone. Class. Quantum Gravity
**2022**, 39, 075023. [Google Scholar] [CrossRef] - Venkateswara, K.; Hagedorn, C.A.; Turner, M.D.; Arp, T.; Gundlach, J.H. A high-precision mechanical absolute-rotation sensor. Rev. Sci. Instrum.
**2014**, 85, 015005. [Google Scholar] [CrossRef] - Ross, M.P.; Venkateswara, K.; Hagedorn, C.A.; Gundlach, J.H.; Kissel, J.S.; Warner, J.; Radkins, H.; Shaffer, T.J.; Coughlin, M.W.; Bodin, P. Low-Frequency Tilt Seismology with a Precision Ground-Rotation Sensor. Seismol. Res. Lett.
**2017**, 89, 67–76. [Google Scholar] [CrossRef] - Mow-Lowry, C.M.; Martynov, D. A 6D interferometric inertial isolation system. Class. Quantum Gravity
**2019**, 36, 245006. [Google Scholar] [CrossRef] - Ubhi, A.S.; Prokhorov, L.; Cooper, S.; Di Fronzo, C.; Bryant, J.; Hoyland, D.; Mitchell, A.; van Dongen, J.; Mow-Lowry, C.; Cumming, A.; et al. Active platform stabilisation with a 6D seismometer. Appl. Phys. Lett.
**2022**, 24, 121. [Google Scholar] [CrossRef] - Ubhi, A.S.; Smetana, J.; Zhang, T.; Cooper, S.; Prokhorov, L.; Bryant, J.; Hoyland, D.; Miao, H.; Martynov, D. A six degree-of-freedom fused silica seismometer: Design and tests of a metal prototype. Class. Quantum Gravity
**2021**, 39, 015006. [Google Scholar] [CrossRef] - Heinzel, G.; Cervantes, F.G.; Marín, A.F.G.; Kullmann, J.; Feng, W.; Danzmann, K. Deep phase modulation interferometry. Opt. Express
**2010**, 18, 19076–19086. [Google Scholar] [CrossRef] [PubMed] - Gerberding, O. Deep frequency modulation interferometry. Opt. Express
**2015**, 23, 14753–14762. [Google Scholar] [CrossRef] [PubMed] - Rowley, W.R.C. Some Aspects of Fringe Counting in Laser Interferometers. IEEE Trans. Instrum. Meas.
**1966**, 15, 146–149. [Google Scholar] [CrossRef] - Barone, F.; Calloni, E.; Rosa, R.D.; Fiore, L.D.; Fusco, F.; Milano, L.; Russo, G. Fringe-counting technique used to lock a suspended interferometer. Appl. Opt.
**1994**, 33, 1194–1197. [Google Scholar] [CrossRef] - Isleif, K.S.; Gerberding, O.; Schwarze, T.S.; Mehmet, M.; Heinzel, G.; Cervantes, F.G. Experimental demonstration of deep frequency modulation interferometry. Opt. Express
**2016**, 24, 1676–1684. [Google Scholar] [CrossRef] - Gerberding, O.; Isleif, K.S.; Mehmet, M.; Danzmann, K.; Heinzel, G. Laser-Frequency Stabilization via a Quasimonolithic Mach-Zehnder Interferometer with Arms of Unequal Length and Balanced dc Readout. Phys. Rev. Appl.
**2017**, 7, 024027. [Google Scholar] [CrossRef] - Isleif, K.S.; Heinzel, G.; Mehmet, M.; Gerberding, O. Compact Multifringe Interferometry with Subpicometer Precision. Phys. Rev. Appl.
**2019**, 12. [Google Scholar] [CrossRef] - Drever, R.W.P.; Hall, J.L.; Kowalski, F.V.; Hough, J.; Ford, G.M.; Munley, A.J.; Ward, H. Laser phase and frequency stabilization using an optical resonator. Appl. Phys. B
**1983**, 31, 97–105. [Google Scholar] [CrossRef] - Rosenbluth, A.; Bobroff, N. Optical sources of non-linearity in heterodyne interferometers. Precis. Eng.
**1990**, 12, 7–11. [Google Scholar] [CrossRef] - Ming Wu, C.; Shen Su, C. Nonlinearity in measurements of length by optical interferometry. Meas. Sci. Technol.
**1996**, 7, 62. [Google Scholar] [CrossRef] - Hu, P.; Bai, Y.; Zhao, J.; Wu, G.; Tan, J. Toward a nonlinearity model for a heterodyne interferometer: Not based on double-frequency mixing. Opt. Express
**2015**, 23, 25935–25941. [Google Scholar] [CrossRef] [PubMed] - Isleif, K.S.; Gerberding, O.; Penkert, D.; Fitzsimons, E.; Ward, H.; Robertson, D.; Livas, J.; Mueller, G.; Reiche, J.; Heinzel, G.; et al. Suppressing ghost beams: Backlink options for LISA. J. Phys. Conf. Ser.
**2017**, 840, 012016. [Google Scholar] [CrossRef] - Gerberding, O.; Isleif, K.S. Ghost Beam Suppression in Deep Frequency Modulation Interferometry for Compact On-Axis Optical Heads. Sensors
**2021**, 21, 1708. [Google Scholar] [CrossRef] - Kissinger, T.; Charrett, T.O.; Tatam, R.P. Range-resolved interferometric signal processing using sinusoidal optical frequency modulation. Opt. Express
**2015**, 23, 9415–9431. [Google Scholar] [CrossRef] [PubMed] - Schwarze, T.S.; Gerberding, O.; Cervantes, F.G.; Heinzel, G.; Danzmann, K. Advanced phasemeter for deep phase modulation interferometry. Opt. Express
**2014**, 22, 18214–18223. [Google Scholar] [CrossRef] - Mecholsky, N.A.; Akhbarifar, S.; Lutze, W.; Brandys, M.; Pegg, I.L. Dataset of Bessel function Jn maxima and minima to 600 orders and 10,000 extrema. Data Brief
**2021**, 39, 107508. [Google Scholar] [CrossRef]

**Figure 1.**Schematic (

**a**) and photo (

**b**) of the custom-designed sensing head manufactured by SmarAct that is used throughout this work. The sensing head is embedded within the same readout structure as shown in Figure 1 of Ref. [8].

**Figure 2.**Effect of elliptical error on the estimation of the arm phase (test mass displacement). (

**a**) An incorrectly circularised Lissajous figure with a 50% residual elliptical error in comparison to a properly circularised Lissajous figure. Panel (

**b**) Corresponding periodic nonlinear error caused by this residual ellipticity. In comparison, the perfectly circularised Lissajous figure leads to a perfectly linear estimator. The 50% error was chosen for visual clarity only and is significantly larger than what is typically observed in our system.

**Figure 3.**Nonlinear deviation of sensor readout from the ‘true’ inferred displacement. The x axis is calibrated for displacement by taking the displacement time series and fitting to the slow variations (not due to sensor nonlinearity) by a ninth-order polynomial.

**Figure 4.**Adjusted layout of the experimental setup originally shown in Figure 1 of Ref. [8]. (

**a**) Arrangement of the sensor, mirror, and actuator, as well as the effective springs formed by the foam blocks and rubber pads. (

**b**) Photo of the setup inside an acoustically isolating box.

**Figure 5.**Comparison between measured and simulated nonlinearities during a controlled and tightly band-limited injection of displacement noise using a coil-magnet actuator. The simulated spectra show good agreement with measurement for high values of elliptical error, which supports the analysis present in Section 2.3. At the smallest value of elliptical error, the measured nonlinear noise is within a factor of three above the simulation, which can be attributed to the presence of other nonlinearities, particularly the nonelliptical nonlinearity laid out in Section 2.4 and the residual nonlinearity of the coil-magnet drive. The baseline noise curve shows the sum of all measured noises in the quiet (‘zero-displacement’) state and represents the absolute noise floor of the sensor. This sensitivity is significantly above the noise performance we demonstrated in Ref. [8], but this can be explained by three main factors: (i) the setup only includes a single sensor, meaning there is no frequency stabilisation; (ii) the box is not as well insulated with packing foam, meaning air currents are not as suppressed; and (iii) the rubber pad necessary to enable differential driving of the mirror and sensor also breaks the common mode rejection of seismic noise present in the original null measurement.

**Figure 6.**Simulated readout of the top stage of the LIGO quadruple suspension with the SmarAct sensor, together with the corresponding residual nonlinear error. The ISI motion shows the inertial displacement of the ISI, and the QUAD TOP readout represents the displacement of the top quadruple suspension stage as measured by an ideal sensor rigidly attached to the ISI (i.e., the relative displacement between the top stage and ISI). While the residual shows that nonlinear noise significantly exceeds the nominal noise floor of the device, at no point in the spectrum does the nonlinear residual exceed the measured signal. The SmarAct sensitivity derived from a null measurement is added for comparison.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Smetana, J.; Di Fronzo, C.; Amorosi, A.; Martynov, D.
Nonlinearities in Fringe-Counting Compact Michelson Interferometers. *Sensors* **2023**, *23*, 7526.
https://doi.org/10.3390/s23177526

**AMA Style**

Smetana J, Di Fronzo C, Amorosi A, Martynov D.
Nonlinearities in Fringe-Counting Compact Michelson Interferometers. *Sensors*. 2023; 23(17):7526.
https://doi.org/10.3390/s23177526

**Chicago/Turabian Style**

Smetana, Jiri, Chiara Di Fronzo, Anthony Amorosi, and Denis Martynov.
2023. "Nonlinearities in Fringe-Counting Compact Michelson Interferometers" *Sensors* 23, no. 17: 7526.
https://doi.org/10.3390/s23177526