# Investigation and Mitigation of Noise Contributions in a Compact Heterodyne Interferometer

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## Abstract

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## 1. Introduction

## 2. Compact Heterodyne Laser Interferometer

#### 2.1. Design and Benchtop Prototype

#### 2.2. Operation Environments

#### 2.3. Preliminary Test

## 3. Noise Source Characterization and Suppression

#### 3.1. Non-Linear OPD Noise

#### 3.2. Laser Frequency Noise

#### 3.3. Temperature Fluctuation Noise

#### 3.4. Detection System Noise Limit

#### 3.5. Discussions

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The layout of the compact benchtop heterodyne laser interferometer design; (

**b**) top view of the layout. Two laser beams with different frequencies enter the lateral beam splitter (LBS) and split into four beams, constructing two interferometers measuring interferometer (MIFO) and reference interferometer (RIFO). The MIFO is to measure the optical pathlength difference (OPD), $\Delta {\varphi}_{\mathrm{M}}$ and the RIFO is to measure $\Delta {\varphi}_{\mathrm{R}}$. The actual displacement of test mass M${}_{\mathrm{M}}$ is calculated from the differential measurement between MIFO and RIFO.

**Figure 2.**Layout of the benchtop compact heterodyne laser interferometer based on the design depicted in Figure 1. Only one static mirror M is used to characterize the system’s noise floor. Commercial optical and mechanical components are used to construct the benchtop system.

**Figure 3.**Operation environments. (

**a**) Pendulum platform inside the chamber; (

**b**) styrofoam thermal insulator outside the chamber. The compact interferometer is placed on a pendulum stage consisting of a breadboard and four steel wires. Viton strips are applied between the pendulum frame and the vacuum chamber to reduce ambient thermal coupling. The entire pendulum structure is fixed inside the vacuum chamber.

**Figure 4.**Preliminary test results. (

**a**) LSD and its logarithmic average of the MIFO, RIFO, and differential measurements, respectively; (

**b**) time series of a 12-min section of the MIFO, RIFO, and differential measurements. The traces of measurement results from MIFO and RIFO highly overlap in (

**a**). Experimental results show a noise floor of $1.64\times {10}^{-7}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 100 $\mathrm{m}$$\mathrm{Hz}$ for individual interferometers, and a noise floor of $2.76\times {10}^{-10}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 100 $\mathrm{m}$$\mathrm{Hz}$ for the differential measurement, which is enhanced by three orders of magnitude from individual interferometers. The drift for differential measurement is $7.56\times {10}^{-10}$ $\mathrm{m}$ over a period of 12 min.

**Figure 5.**Spectra of RF driving signals for both AOFS. The frequency interval between this sideband and the main peak equals to the heterodyne frequency, leading to a ghost signal of unstable phase responsible for the non-linear OPD noise.

**Figure 6.**LSD logarithmic average of original differential measurements and the results after noise correction, and the OPD noise contribution. The noise floor is reduced from $2.76\times {10}^{-10}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ to $3.86\times {10}^{-11}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 100 $\mathrm{m}$$\mathrm{Hz}$ after applying the noise correction algorithm.

**Figure 7.**System layout with the delay-line interferometer integrated inside the vacuum chamber. The delay-line interferometer is constructed by inserting a 2-m fiber in one arm of the fiber Mach–Zehnder interferometer to amplify the effects of laser frequency noise.

**Figure 8.**Linear spectra of DIFO, RIFO, and the differential measurement to the injected laser frequency noise by intentionally modulating the laser frequency at a rate of 2 $\mathrm{Hz}$. The spectrum response of DIFO to the laser frequency noise is $4.82\times {10}^{-7}$ $\mathrm{m}$ while the RIFO response is $5.30\times {10}^{-10}$ $\mathrm{m}$ at 2 $\mathrm{Hz}$. After performing the differential operation, the spectrum response is reduced to $5.07\times {10}^{-12}$ $\mathrm{m}$ at 2 $\mathrm{Hz}$.

**Figure 9.**Laser frequency noise contribution estimated by a linear fit, compared to the OPD-noise-corrected differential measurement. The laser frequency noise is estimated to be $2.28\times {10}^{-12}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 100 $\mathrm{m}$$\mathrm{Hz}$, and is not the limiting noise source in our benchtop experiment.

**Figure 10.**The amplitude and phase of the coherence function between the displacement and the temperature measurements. In the low frequency regime below 1 mHz, it shows a strong correlation between the displacement and the temperature measurement as the amplitude is large than 0.5, and a time delay effect as the phase is nonzero.

**Figure 11.**Transfer function between the displacement and temperature measurements before and after applying a 1 mHz low-pass filter. The amplitude of the transfer function is attenuated by a low-pass filter in the frequency regime where the correlation is negligible.

**Figure 12.**Contributions of temperature fluctuations estimated by a spectral analysis method, and the measurement results before and after correction of this noise source. The noise floor is reduced from $6.5\times {10}^{-9}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ to $9.8\times {10}^{-11}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 0.1 mHz after applying the noise correction algorithm.

**Figure 13.**Flow chart of the overall noise correction algorithm. The non-linear OPD noise, laser frequency noise and the temperature fluctuation noise are characterized, and the contribution from each noise source is removed with this algorithm.

**Figure 14.**Overall noise correction results based on the differential measurement and contribution of each noise source. The residual noise floor is $3.31\times {10}^{-11}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 100 $\mathrm{m}$$\mathrm{Hz}$, and $9.8\times {10}^{-11}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ at 0.1 mHz after correction. The detection system noise level is $1.39\times {10}^{-12}$ $\mathrm{m}$/$\sqrt{\mathrm{Hz}}$ and also marked in this plot.

i | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${\mathit{C}}_{i}$ | $3.62\times {10}^{-3}$ | $1.11\times {10}^{-3}$ | $1.89\times {10}^{-6}$ | $-2.51\times {10}^{-6}$ |

$\delta {\mathit{C}}_{i}$ | $1.96\times {10}^{-6}$ | $1.96\times {10}^{-6}$ | $1.23\times {10}^{-6}$ | $1.24\times {10}^{-6}$ |

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**MDPI and ACS Style**

Zhang, Y.; Hines, A.S.; Valdes, G.; Guzman, F.
Investigation and Mitigation of Noise Contributions in a Compact Heterodyne Interferometer. *Sensors* **2021**, *21*, 5788.
https://doi.org/10.3390/s21175788

**AMA Style**

Zhang Y, Hines AS, Valdes G, Guzman F.
Investigation and Mitigation of Noise Contributions in a Compact Heterodyne Interferometer. *Sensors*. 2021; 21(17):5788.
https://doi.org/10.3390/s21175788

**Chicago/Turabian Style**

Zhang, Yanqi, Adam S. Hines, Guillermo Valdes, and Felipe Guzman.
2021. "Investigation and Mitigation of Noise Contributions in a Compact Heterodyne Interferometer" *Sensors* 21, no. 17: 5788.
https://doi.org/10.3390/s21175788