- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Mathematical derivation and verification of LISA’s angular readout signal on an ultrastable interferometer testbed

Alvise Pizzella*, Miguel Dovale-Álvarez†, Pablo Martínez Cano, Rodrigo García Álvarez, Christoph Bode, Juan José Esteban Delgado, and Gerhard Heinzel‡

  • *Contact author: alvise.pizzella@aei.mpg.de
  • †Contact author: miguel.dovale@aei.mpg.de
  • ‡Contact author: gerhard.heinzel@aei.mpg.de
PDF

Phys. Rev. D 112, 082005 – Published 28 October, 2025

DOI: https://doi.org/10.1103/sqgt-plx6

Abstract

The Laser Interferometer Space Antenna (LISA) aims to detect gravitational waves by interferometrically measuring the separation between free falling test masses (TMs) with pm/Hz sensitivity down to mHz frequencies. Achieving this sensitivity requires the suppression of spurious path length variations induced by unavoidable tilts of the TMs and spacecraft, a phenomenon known as tilt-to-length coupling. LISA employs differential wave front sensing (DWS) to measure angular misalignments between the interfering beams’ wave fronts. To first-order approximation, the DWS signal is linearly dependent on the tilt angle for small angular displacements (on the order of microradians). The linear coupling coefficient is well-documented for the idealized case of a photodiode (PD) much larger than the beam spot radius. However, LISA’s setup violates this assumption, and the anticipated tilt range exceeds the linear regime. Accurately characterizing the DWS signal thus requires a more comprehensive approach. This paper presents a novel analytical framework for calculating the first two Maclaurin coefficients of the DWS signal for two fundamental-mode Gaussian beams with a small wave front curvature mismatch, incident on PDs of arbitrary size. The validity of this approach is verified by comparison with numerical simulations and experimental calibrations conducted on an ultrastable testbed representative of LISA. Moreover, we utilize this framework to determine the phase sensitivity required to meet LISA’s stringent measurement requirements. The results of this paper will facilitate the calculation of the DWS gain of an interferometric topology and give a reference target to the experiments aiming to demonstrate LISA’s performance requirement.

View figure in article

Physics Subject Headings (PhySH)

Corrections

20 January, 2026

Correction: Typographical errors in Eqs. (9), (36) and (37) have been fixed.

Article Text

References (37)

  1. G. Pizzella, Eur. Phys. J. H 41, 267 (2016).
  2. G. E. Moss, L. R. Miller, and R. L. Forward, Appl. Opt. 10, 2495 (1971).
  3. R. Weiss, Gen. Relativ. Gravit. 54, 153 (2022).
  4. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  5. M. Branchesi, M. Maggiore, D. Alonso, C. Badger, B. Banerjee, F. Beirnaert, E. Belgacem, S. Bhagwat, G. Boileau, S. Borhanian et al., J. Cosmol. Astropart. Phys. 07 (2023) 068.
  6. M. Evans, A. Corsi, C. Afle, A. Ananyeva, K. G. Arun, S. Ballmer, A. Bandopadhyay, L. Barsotti, M. Baryakhtar, E. Berger et al., arXiv:2306.13745.
  7. M. G. Beker, J. F. J. v. d. Brand, E. Hennes, and D. S. Rabeling, J. Phys. Conf. Ser. 363, 012004 (2012).
  8. ESA, Capturing the ripples of spacetime: LISA gets go-ahead (2024), https://www.esa.int/Science_Exploration/Space_Science/LISA/Capturing_the_ripples_of_spacetime_LISA_gets_go-ahead.
  9. M. Armano, H. Audley, G. Auger, J. T. Baird, M. Bassan, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt, M. Caleno et al., Phys. Rev. Lett. 116, 231101 (2016).
  10. M. Armano, H. Audley, J. Baird, P. Binetruy, M. Born, D. Bortoluzzi, E. Castelli, A. Cavalleri, A. Cesarini, A. M. Cruise et al., Phys. Rev. Lett. 120, 061101 (2018).
  11. G. Wanner, Nat. Phys. 15, 200 (2019).
  12. K. Danzmann, T. Prince (The LISA Internatinal Science Team), Assessment study report (yellow book) (2011), https://sci.esa.int/documents/35005/36499/1567258681608-LISA-YellowBook-ESA-SRE-2011-3-Feb2011.pdf.
  13. W. Brzozowski, D. Robertson, E. Fitzsimons, H. Ward, J. Keogh, A. Taylor, M. Milanova, M. Perreur-Lloyd, Z. Ali, A. Earle et al., in Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave, edited by L. E. Coyle, S. Matsuura, and M. D. Perrin, International Society for Optics and Photonics Vol. 12180 (SPIE, 2022), p. 121800O, 10.1117/12.2627465.
  14. A.-M. A. van Veggel and C. J. Killow, Adv. Opt. Technol. 3, 293 (2014).
  15. M. Tinto, D. A. Shaddock, J. Sylvestre, and J. W. Armstrong, Phys. Rev. D 67, 122003 (2003).
  16. G. Wanner and N. Karnesis (LISA Pathfinder Collaboration), J. Phys. Conf. Ser. 840, 012043 (2017).
  17. M. Tröbs, S. Schuster, M. Lieser, M. Zwetz, M. Chwalla, K. Danzmann, G. F. Barránco, E. D. Fitzsimons, O. Gerberding, G. Heinzel et al., Classical Quantum Gravity 35, 105001 (2018).
  18. M. Chwalla, K. Danzmann, M. D. Alvarez, J. E. Delgado, G. F. Barranco, E. Fitzsimons, O. Gerberding, G. Heinzel, C. Killow, M. Lieser et al., Phys. Rev. Appl. 14, 014030 (2020).
  19. M.-S. Hartig, S. Paczkowski, M. Hewitson, G. Heinzel, and G. Wanner, Phys. Rev. D 111, 043048 (2025).
  20. M. Chwalla, K. Danzmann, G. F. Barranco, E. Fitzsimons, O. Gerberding, G. Heinzel, C. J. Killow, M. Lieser, M. Perreur-Lloyd, D. I. Robertson et al., Classical Quantum Gravity 33, 245015 (2016).
  21. M. Colpi, K. Danzmann, M. Hewitson, K. Holley-Bockelmann, P. Jetzer, G. Nelemans, A. Petiteau, D. Shoemaker, C. Sopuerta, R. Stebbins et al., arXiv:2402.07571.
  22. V. Huarcaya, M. D. Álvarez, D. Penkert, S. Gozzo, P. M. Cano, K. Yamamoto, J. J. E. Delgado, M. Mehmet, K. Danzmann, and G. Heinzel, Phys. Rev. Appl. 20, 024078 (2023).
  23. E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, Rev. Sci. Instrum. 76, 063112 (2005).
  24. G. Fernandez Barranco, Ph.D. thesis, Leibniz Universität Hannover, 2017, 10.15488/3866.
  25. G. Fernández Barranco, B. S. Sheard, C. Dahl, W. Mathis, and G. Heinzel, IEEE Sens. J. 18, 7414 (2018).
  26. A. Siegman, Lasers (University Science Books, 1986), ISBN [Amazon][WorldCat], https://books.google.de/books?id=qBBaDQEACAAJ.
  27. O. Gerberding, C. Dahl, J. Kullmann, M. Tröbs, I. Bykov, S. Barke, N. Brause, J. J. Esteban Delgado, T. Schwarze, J. Reiche et al., Rev. Sci. Instrum. 86, 074501 (2015).
  28. O. Gerberding, Ph.D. thesis, Leibniz Universität Hannover, 2014, 10.15488/8255.
  29. C. Bode, Ph.D. thesis, Leibniz Universität Hannover, 2024, 10.15488/17997.
  30. A. Pizzella, Ph.D. thesis, Leibniz Universität Hannover, 2025, 10.15488/18487.
  31. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, Appl. Opt. 33, 5041 (1994).
  32. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, Appl. Opt. 33, 5037 (1994).
  33. G. Hechenblaikner, J. Opt. Soc. Am. A 27, 2078 (2010).
  34. G. Heinzel, M. D. Álvarez, A. Pizzella, N. Brause, and J. J. E. Delgado, Phys. Rev. Appl. 14, 054013 (2020).
  35. ESA-LISA-EST-ISD-RS-0002 LISA IDS System and Interface Requirement Document (IDS-IRD) (2024).
  36. E. Fitzsimons, M. Hewitson, A. Uminska, and G. Fernandez Barranco, IDS Performance Budgets (2024).
  37. T. Mistry, The eyes of LISA (2024), https://app.oxfordabstracts.com/content/events/5189/submitters/405298/submissions/fe-412377de-ee7f-408e-849d-84ca70d81af8/questions/91860/file/e18cb94a-84d5-4284-b64f-39c13cb08bb4.pdf.

Outline

Information

Phys. Rev. D 112, 082005– Published 28 October, 2025
  • Received 24 June 2025
  • Accepted 30 September 2025
  • Corrected 20 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation