Moderate-coherence sensing with optical cavities: ultra-high accuracy meets ultra-high measurement bandwidth and range.pdf
ARTICLE Moderate-coherence sensing with optical cavities: ultra-high accuracy meets ultra-high measurement bandwidth and range Johannes Dickmann 1,2,3✉, Liam Shelling Neto 1,2,3, Steffen Sauer1,2,3 & Stefanie Kroker2,3,4 Interferometric sensors, renowned for their exceptional accuracy, leverage the wave prop- erties of coherent electromagnetic radiation. The periodicity of the measurement signal often critically limits the measurement range of sensors utilizing interferometry. Here we introduce a cavity-based interferometry concept that capitalizes on a laser with moderate coherence, thereby combining ultra-high accuracy with ultra-high measurement bandwidth and range. To this end mid-fringe detection is combined with measurements of the interferometric visibility. We present experimental results that demonstrate the effectiveness of our approach exemplarily for length sensing. Notably, our system achieves an accuracy of 1 nm with a measurement range of 120 μm (relative uncertainty of 0.00083 %) and a bandwidth ranging from 0 Hz to 20 kHz. These ﬁndings support advancements in high-precision sensing applications that demand simultaneous accuracy, measurement range and bandwidth. https://doi.org/10.1038/s44172-024-00164-w OPEN 1 CAVITY technologies UG (haftungsbeschränkt), Wilhelmsgarten 3, 38100 Braunschweig, Germany. 2 Technical University of Braunschweig, Institute for Semiconductor Technology, Hans-Sommer-Str. 66, 38106 Braunschweig, Germany. 3 Laboratory for Emerging Nanometrology (LENA), Langer Kamp 6a/b, 38106 Braunschweig, Germany. 4 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany. ✉email: j.dickmann@tu-braunschweig.de COMMUNICATIONS ENGINEERING |   (2024) 3:17 | https://doi.org/10.1038/s44172-024-00164-w | www.nature.com/commseng 1 1234567890():,; I n the ﬁeld of sensing and metrology, achieving a balance between accuracy and measurement range has been a long- standing challenge. Interferometric sensors, which rely on the wave properties of coherent electromagnetic radiation, have been at the forefront of high-precision measurements1,2. However, the exceptional accuracy of interferometric sensors often comes at the cost of a limited range due to the inherent periodicity of the interferometer signal. Interferometric gravitational wave detectors have already demonstrated remarkable precision levels better than 10−21 (ref. 3), while ultrastable lasers exhibit precision on the order of 10−17 (refs. 4,5). However, the limited range has hindered their practical utility, preventing them from capturing rapid changes or transient events. A combination of high accu- racy in large measurement ranges is also critical for various other ﬁelds including precision manufacturing6–8, biomedical sensing9–11, and structural health monitoring12,13. This research paper introduces a concept called moderate- coherence sensing, which addresses the critical limitation of range in interferometric sensors while preserving their ultra-high accuracy capabilities. By capitalizing on the limited coherence of the measurement laser14, this approach offers the potential to combine high accuracy with ultra-high measurement range, enabling advancements in high-precision sensing applications. This paper showcases the experimental implementation of moderate-coherence sensing and presents measurement results highlighting its effectiveness. The system developed achieves an impressive accuracy of 1 nanometer while maintaining a range of 120 micrometers. By overcoming the long-standing limitations of interferometric sensors, moderate-coherence sensing holds the promise of transforming industries and advancing scientiﬁc endeavors that rely on both ultra-high precision, wide range, and high-speed capabilities. Methods Theoretical description. The high precision of interferometric sensors is attributed to the short wavelength of the electro- magnetic radiation used in the micrometer (μm) range and the wavelength’s high stability. The Michelson interferometer15, as a simple example, demonstrates this precision through the appearance of interference fringes. Consequently, with appro- priate laser and readout electronics, sub-nanometer accuracy in length measurements can be easily achieved15. However, the periodic nature of the interference signal, with a period equal to half the wavelength, limits the achievable measurement range. Active methods, such as using a movable mirror controlled by a controller, have been employed to expand the range16. None- theless, these approaches introduce complexity and are con- strained by the maximum control speed thereby limiting the measurement bandwidth. Our method overcomes the limitations of complex and potentially slow active control while maintaining both high accuracy and a wide range. The key principle lies in leveraging not only the sensitive interference signal of the interferometer but also the limited coherence of the laser employed. Speciﬁcally, we utilize a Fabry-Pérot laser diode, which is actively temperature- stabilized to maintain a constant wavelength. Figure 1a illustrates the laser’s emission characteristics, where multiple Fabry-Pérot modes are excited, and their output power is weighted by the Gaussian medium gain spectrum. The width of this spectrum inherently restricts the overall coherence of the laser output. General description. To achieve precise length measurements, we employ a low-ﬁnesse Fabry-Pérot cavity comprising two wedged silicon wafers. At the laser wavelength of 1.55 μm, the refractive index of silicon is n = 3.475717. Consequently, the intensity reﬂection of the cavity mirrors is R = 30.6 %18, corresponding to a Lorentzian ﬁnesse of 2.6519. To calculate the reﬂected power of the cavity, we superimpose the individual quasi-coherent emission lines of the laser. For the calculation, we start by expressing the laser gain: GðλÞ ¼ G0 exp  ðλ  λcenterÞ2 Δλ2 FPL   ; ð1Þ where λcenter = 1.55 μm is the gain center wavelength and ΔλFPL is the gain linewidth. Next, for each emitted mode indexed by i with wavelength λi, we calculate the roundtrip single-pass phase ϕi as a function of the cavity length L19: ϕiðLÞ ¼ 2π L λi : ð2Þ Using these parameters, we can determine the power transmitted through the cavity for each mode i19: Itrans i ¼ ð1  RÞ2 ð1  RÞ2 þ 4Rsin2ϕi ´ GðλiÞ: ð3Þ Finally, we obtain the measurand, which is the total power reﬂected by all modes: Irefl ¼ 1  ∑ 1 i¼1 Itrans i : ð4Þ The results of the calculation are presented in Fig. 1b, where the classical quasiperiodic interference signal of the cavity is observed. Notably, the periodicity of the signal is disrupted by decreasing visibility. The visibility V, quantiﬁed by the relative ratio of the maximum Imax and minimum Imin power values of the fringes, is deﬁned as: VðLÞ ¼ Imax j  Imin j Imax j ð5Þ for a certain fringe j. Figure 1c displays the calculated visibility as a function of the cavity length. A distinctive trend is observed, where an almost linear curve is evident within the range of 10 to 120 μm. Indeed, the combination of the high-sensitivity interference signal and the utilization of visibility plays a critical role in achieving high accuracy with a wide range of length measure- ments while achieving measurement precisions down to sub- nanometer levels. Inﬂuence of the laser parameters. The expected interferometer signal is calculated based on deﬁned parameters: an average laser wavelength of λcenter = 1.55 μm and a cavity mirror reﬂection of R = 30.6 %. The width of the laser gain, ΔλFPL, and the spacing of individual laser lines, Δλspace = λi+1 −λi, are considered as free parameters. A series of numerical simulations explored the dependency of visibility on these parameters. Utilizing equations (1) to (5), the visibility of the interferometer signal was numeri- cally calculated for various combinations of ΔλFPL and Δλspace. Figure 2 illustrates these results. Figure 2a–c visually depicts the inﬂuence of these parameters on the laser wavelength spectrum. Figure 2a represents the initial spectrum with ΔλFPL = 3.9 nm and Δλspace = 1 nm. In 2b, the width of the laser gain was reduced to ΔλFPL = 1.9 nm, while the laser line spacing remained constant. Figure 2c showcases the reduction of the laser line spacing to Δλspace = 0.5 nm, while maintaining the original laser gain width at ΔλFPL = 3.9 nm. Figure 2d demonstrates the dependence of calculated visibility V on the width of the laser gain medium, ΔλFPL, varying between 1 nm and 5 nm with a ﬁxed laser line spacing of Δλspace = 0.5 nm. ARTICLE COMMUNICATIONS ENGINEERING | https://doi.org/10.1038/s44172-024-00164-w 2 COMMUNICATIONS ENGINEERING |    (2024) 3:17 | https://doi.org/10.1038/s44172-024-00164-w | www.nature.com/commseng precise calculation of its position relative to the original measured cavity length, with an accuracy better than 1 nm. Since the theoretical position of the mid-fringe is known from the earlier theoretical description (see section “Theoretical description"), the precise cavity length can be calculated accordingly using the following equation: Lmeas ¼ Lvis  Offset : ð9Þ The numerically calculated mid-fringe position (see Fig. 7a, b) for the measured visibility (see Fig. 6) is denoted as Lvis. This mid-fringe position can be obtained from the lookup table for both negative and positive mid-fringe increments, available in the Supplementary materials (Table 1). The offset is determined through linear regression. For the measurements in Fig. 6, the results are as follows: 1. Visibility V = 0.897, negative mid-fringe. Using the lookup table, this corresponds to Lvis = 26.988 μm. Linear regres- sion yields an offset of 455 nm. Consequently, the measurement, according to equation (9), results in Lmeas = 26.533 μm. 2. Visibility V = 0.657, negative mid-fringe →Lvis = 62.603 μm, Offset = 317 nm →Lmeas = 62.286 μm. 3. Visibility V = 0.378, positive mid-fringe →Lvis = 103.266 μm, Offset = 484 nm →Lmeas = 102.782 μm. Following the demonstration of real-time measurement, the accuracy and reproducibility of the moderate-coherence sensing presented here will now be assessed. Determination of accuracy and reproducibility. The accuracy and reproducibility of moderate-coherence sensing were assessed for three measurement positions (26.533 μm, 62.286 μm, and 102.782 μm). A nanometer-step experiment was conducted, involving ﬁfty 1 nm steps for each initial position using the piezo stage (Thorlabs NFL5DP20S/M) controlled by the internal strain gauge. Real-time measurements were taken after each step, as described in the preceding section. The results in Fig. 8a–c demonstrate distinguishable steps, indicating that the accuracy of moderate-coherence sensing surpasses 1 nm. Notably, at longer distances (Fig. 8c), the signal exhibits some noise. To achieve an absolute accuracy of one nanometer, the cavity length under measurement should not vary by more than one Fig. 6 Real-time measurement results. a–c illustrates the fast piezoscan spanning 0.8 μm and the corresponding calculated visibility (V). d–f displays the ﬁltered data focused on the half fringe (blue dots), the linear regression analysis (blue line), and the calculated relative position of the half fringe (offset). Fig. 7 Numerical calculation results depicting the mid-fringe position relative to the signal visibility. In Figure a, the relative mid-fringe level is calculated using equations (1)–(8). Figure b illustrates the mid-fringe position’s dependency on visibility for both negative and positive mid-fringe increments. The corresponding lookup table is available in the Supplementary Materials. ARTICLE COMMUNICATIONS ENGINEERING | https://doi.org/10.1038/s44172-024-00164-w 6 COMMUNICATIONS ENGINEERING |    (2024) 3:17 | https://doi.org/10.1038/s44172-024-00164-w | www.nature.com/commseng nanometer during the 50 μs measurement time. This equates to a maximum speed of 20 μm/s. Increasing the piezo frequency may enhance measurable speed under speciﬁc conditions. To evaluate reproducibility, the three distances were each measured 50 times in succession. Results in Fig. 8d–f show standard deviations: For 26.533 μm, σMCS < ± 0.3 nm; for 62.286 μm, σMCS < ± 0.3 nm; and for the largest distance, 102.782 μm, σMCS < ± 0.6 nm. The increased error for longer cavity lengths is likely due to reduced cavity visibility, leading to smaller mid-fringe increments (see Fig. 6). This reduction contributes to increased error in mid-fringe detection, making offset determination more challenging. In summary, the investigated prototype of moderate-coherence sensing demonstrates accuracy and reproducibility better than 1 nm across a 120 μm measuring range. The corresponding relative length measurement error is thus less than 0.0000083. Discussion In this study, we have presented the concept of moderate- coherence sensing and demonstrated its application in high- accuracy length measurements. The moderate-coherence sensing technique leverages the interference signal of a cavity along with the visibility parameter to achieve high accuracy within a wide measurement range and high measurement bandwidth >20 kHz. Our experimental implementation showcases the effectiveness of this approach in achieving sub-nanometer accuracy in length measurement over a range of 120 μm. The experimental results conﬁrmed the theoretical calculations, demonstrating excellent agreement between the measured visi- bility and the calculated visibility. This agreement validates the dominant effect of coherence modulation in the experiment, suggesting that other factors such as misalignment and laser divergence can be neglected. Moreover, we explored the real-time measurement capabilities of moderate-coherence sensing. By scanning the cavity length over half a wavelength using a ring piezo, we were able to determine the actual cavity length through precise measurement of the mid-fringe position combined with visibility calculations. This real-time mode offers a fast and accurate measurement technique, enabling applications that require dynamic and rapid length monitoring. The versatility of moderate-coherence sensing extends beyond length measurements. Any measurement that can be accessed using cavity length variations can be potentially measured using this technique. Temperature sensors, acceleration sensors, pres- sure sensors, and other sensors relying on the change in cavity length can beneﬁt from the high accuracy and wide range offered by moderate-coherence sensing. In conclusion, moderate-coherence sensing represents a pow- erful and promising approach for high-accuracy measurement applications. The combination of a sensitive interference signal and visibility analysis allows for precise measurements with a wide measurement range and high bandwidth. The experimental results demonstrate the practical feasibility and accuracy of this technique in length measurements and future sensors. The potential for further advancements and applications in various sensing ﬁelds makes moderate-coherence sensing a valuable tool for future research and technological developments. Data availability The authors declare that the data supporting the ﬁndings of this study are available within the paper. Code availability Computer codes used in the current study are available from the corresponding author upon reasonable request. Received: 8 September 2023; Accepted: 13 January 2024; Fig. 8 Experimental assessment of accuracy and reproducibility. Figures a–c depict results from the nanometer-step experiments, plotting moderate- coherence sensing (MCS) measurements against the integrated strain gauge readings of the piezo stage. Figures d–f represent the repetitive measurements (n = 50) of the same cavity length, displaying mean values and standard deviations. COMMUNICATIONS ENGINEERING | https://doi.org/10.1038/s44172-024-00164-w ARTICLE COMMUNICATIONS ENGINEERING |   (2024) 3:17 | https://doi.org/10.1038/s44172-024-00164-w | www.nature.com/commseng 7 References 1. Udem, T., Holzwarth, R. & Hänsch, T. W. Optical frequency metrology. Nature 416, 233–237 (2002). 2. Bailes, M. et al. Gravitational-wave physics and astronomy in the 2020s and 2030s. Nat. Rev. Phys. 3, 344–366 (2021). 3. 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J.D. and S.K. also acknowledge partial support from the European Association of National Metrology Institutes. This project (20FUN08 NEXTLASERS) has received funding from the EMPIR program co-ﬁnanced by the Participating States and from the European Union’s Horizon 2020 research and innovation program. S.K. also thanks funding support from the Deutsche Forschungsgemeinschaft (DFG, German Research Founda- tion) under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project No. 390833453). Author contributions J.D.: conceptualization, methodology, investigation, resources, writing, and visualization. L.S.N.: conceptualization, methodology, and investigation. S.S.: conceptualization, methodology, and investigation. S.K.: conceptualization, resources, writing, and supervision. Funding Open Access funding enabled and organized by Projekt DEAL. Competing interests The authors declare no competing interests. The identiﬁcation of speciﬁc instruments in this paper is solely for the purpose of adequately describing the experimental procedure. It is not intended to imply any recommendation or endorsement, nor does it suggest that the instruments identiﬁed are necessarily the best available for the stated purpose. Additional information Supplementary information The online version contains supplementary material available at https://doi.org/10.1038/s44172-024-00164-w. Correspondence and requests for materials should be addressed to Johannes Dickmann. Peer review information Communications Engineering thanks the anonymous reviewers for their contribution to the peer review of this work. Primary handling editors: Anastasiia Vasylchenkova and Rosamund Daw. Reprints and permission information is available at http://www.nature.com/reprints Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. 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Title: Moderate-coherence sensing with optical cavities: ultra-high accuracy meets ultra-high measurement bandwidth and range
Authors: Johannes Dickmann, Liam Shelling Neto, Steffen Sauer, Stefanie Kroker ​
Publisher:  Communications Engineering, Springer Nature
Date: January 13, 2024 ​
Abstract: Interferometric sensors, renowned for their exceptional accuracy, leverage the wave properties of coherent electromagnetic radiation. ​ The periodicity of the measurement signal often critically limits the measurement range of sensors utilizing interferometry. ​ Here we introduce a cavity-based interferometry concept that capitalizes on a laser with moderate coherence, thereby combining ultra-high accuracy with ultra-high measurement bandwidth and range. ​ To this end mid-fringe detection is combined with measurements of the interferometric visibility. ​ We present experimental results that demonstrate the effectiveness of our approach exemplarily for length sensing. ​ Notably, our system achieves an accuracy of 1 nm with a measurement range of 120 μm (relative uncertainty of 0.00083 %) and a bandwidth ranging from 0 Hz to 20 kHz. ​ These findings support advancements in high-precision sensing applications that demand simultaneous accuracy, measurement range, and bandwidth. ​