- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Numerical evaluation of Casimir forces using the discontinuous Galerkin time-domain method

Carles Martí Farràs1,*, Bettina Beverungen1, Philip Trøst Kristensen2, Francesco Intravaia1, and Kurt Busch1,3

  • *Contact author: marticar@physik.hu-berlin.de
PDF

Phys. Rev. A 114, 042817 – Published 22 October, 2026

DOI: https://doi.org/10.1103/pllp-htg3

Abstract

Casimir forces dominate the interaction between electrically neutral macroscopic objects at nanoscale separations, making their quantitative description highly relevant to modern technology and fundamental physics. In this work, we present a time-domain scheme for computing Casimir forces between vacuum-separated bodies based on the Maxwell stress tensor formalism. In particular, we provide a concrete implementation using the finite-element-based discontinuous Galerkin time-domain method. The approach enables accurate evaluation of finite-temperature Casimir–Lifshitz interactions in arbitrary geometries and for general material models. At the core of the method, the electromagnetic Green tensor is expressed as the system's response to dipolar excitations, thereby recasting the Maxwell stress tensor into a set of classical scattering problems driven by electric and magnetic dipoles. We validate the approach against reference calculations of the Casimir interaction between parallel half-spaces at both zero and finite temperature. We further demonstrate its applicability to the interaction between a finite cylindrical object and a half-space, for which semi-analytical expressions are not available, showing excellent agreement with both asymptotic predictions based on physical considerations and alternative numerical approaches. These findings illustrate the method's potential for studying Casimir interactions in micro- and nanoscale structures of relevance to nanodevice design and experimental settings.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (109)

  1. H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet. 51, 793 (1948).
  2. E. M. Lifshitz, The theory of molecular attractive forces between solids, Sov. Phys. JETP 2, 73 (1956).
  3. F. London, On the theory and system of molecular forces, in Quantum Chemistry (World Scientific, Singapore, 2000), Vol. 8, pp. 369–399.
  4. H. B. G. Casimir and D. Polder, The influence of retardation on the London-van der Waals forces, Phys. Rev. 73, 360 (1948).
  5. C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Numer. Meth Eng. 79, 1309 (2009).
  6. P. H. G. M. van Blokland and J. T. G. Overbeek, Van der Waals forces between objects covered with a chromium layer, J. Chem. Soc., Faraday Trans. 1 74, 2637 (1978).
  7. S. K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 µm range, Phys. Rev. Lett. 78, 5 (1997).
  8. U. Mohideen and A. Roy, Precision measurement of the Casimir force from 0.1 to 0.9 µm, Phys. Rev. Lett. 81, 4549 (1998).
  9. A. Roy and U. Mohideen, Demonstration of the nontrivial boundary dependence of the Casimir force, Phys. Rev. Lett. 82, 4380 (1999).
  10. B. W. Harris, F. Chen, and U. Mohideen, Precision measurement of the Casimir force using gold surfaces, Phys. Rev. A 62, 052109 (2000).
  11. H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Quantum mechanical actuation of microelectromechanical systems by the Casimir force, Science 291, 1941 (2001).
  12. R. S. Decca, D. López, E. Fischbach, and D. E. Krause, Measurement of the Casimir force between dissimilar metals, Phys. Rev. Lett. 91, 050402 (2003).
  13. R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepanenko, Precise comparison of theory and new experiment for the Casimir force leads to stronger constraints on thermal quantum effects and long-range interactions, Ann. Phys. (NY) 318, 37 (2005).
  14. R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepanenko, Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates, Phys. Rev. D 75, 077101 (2007).
  15. D. E. Krause, R. S. Decca, D. López, and E. Fischbach, Experimental investigation of the Casimir force beyond the proximity-force approximation, Phys. Rev. Lett. 98, 050403 (2007).
  16. A. O. Sushkov, W. J. Kim, D. A. R. Dalvit, and S. K. Lamoreaux, Observation of the thermal Casimir force, Nat. Phys. 7, 230 (2011).
  17. V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López, V. M. Mostepanenko, and C. Romero, Comment on “temperature dependence of the Casimir effect,” Phys. Rev. E 73, 028101 (2006).
  18. G. Bimonte, B. Spreng, P. A. Maia Neto, G.-L. Ingold, G. L. Klimchitskaya, V. M. Mostepanenko, and R. S. Decca, Measurement of the Casimir force between 0.2 and 8 µm: Experimental procedures and comparison with theory, Universe 7, 93 (2021).
  19. G. Bimonte, D. López, and R. S. Decca, Isoelectronic determination of the thermal Casimir force, Phys. Rev. B 93, 184434 (2016).
  20. H. Haghmoradi, H. Fischer, A. Bertolini, I. Galić, F. Intravaia, M. Pitschmann, R. A. Schimpl, and R. I. P. Sedmik, Force metrology with plane parallel plates: Final design review and outlook, Physics (Basel, Switz.) 6, 690 (2024).
  21. G. L. Klimchitskaya, M. Bordag, E. Fischbach, D. E. Krause, and V. M. Mostepanenko, Observation of the thermal Casimir force is open to question, Int. J. Mod. Phys.: Conf. Ser. 03, 515 (2011).
  22. A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Demonstration of the Casimir force between ferromagnetic surfaces of a Ni-coated sphere and a Ni-coated plate, Phys. Rev. Lett. 110, 137401 (2013).
  23. G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces, Phys. Rev. Lett. 88, 041804 (2002).
  24. B. Derjaguin, Untersuchungen über die Reibung und Adhäsion, IV: Theorie des Anhaftens kleiner Teilchen, Colloid Polym. Sci. 69, 155 (1934).
  25. T. Gong, M. R. Corrado, A. R. Mahbub, C. Shelden, and J. N. Munday, Recent progress in engineering the Casimir effect–applications to nanophotonics, nanomechanics, and chemistry, Nanophotonics 10, 523 (2021).
  26. H. Shen, S. Liu, Y. Quan, Y. Li, T. Wang, T. Yu, and X. Tao, The progress of the Casimir effect: From theory to application, Nanotechnology 36, 482002 (2025).
  27. F. M. Serry, D. Walliser, and G. J. Maclay, The role of the Casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS), J. Appl. Phys. 84, 2501 (1998).
  28. E. Buks and M. L. Roukes, Stiction, adhesion energy, and the Casimir effect in micromechanical systems, Phys. Rev. B 63, 033402 (2001).
  29. O. Kenneth and I. Klich, Casimir forces in a T-operator approach, Phys. Rev. B 78, 014103 (2008).
  30. S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Scattering theory approach to electrodynamic Casimir forces, Phys. Rev. D 80, 085021 (2009).
  31. M. F. Maghrebi, S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Analytical results on Casimir forces for conductors with edges and tips, Proc. Natl. Acad. Sci. USA 108, 6867 (2011).
  32. M. Hartmann, G.-L. Ingold, and P. A. Maia Neto, Plasma versus Drude modeling of the Casimir force: Beyond the proximity force approximation, Phys. Rev. Lett. 119, 043901 (2017).
  33. R. Zandi, T. Emig, and U. Mohideen, Quantum and thermal Casimir interaction between a sphere and a plate: Comparison of Drude and plasma models, Phys. Rev. B 81, 195423 (2010).
  34. A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Thermal Casimir effect in the plane-sphere geometry, Phys. Rev. Lett. 104, 040403 (2010).
  35. A. Canaguier-Durand, P. A. Maia Neto, A. Lambrecht, and S. Reynaud, Thermal Casimir effect for Drude metals in the plane-sphere geometry, Phys. Rev. A 82, 012511 (2010).
  36. J. L. Xiong and W. C. Chew, Efficient evaluation of Casimir force in z-invariant geometries by integral equation methods, Appl. Phys. Lett. 95, 154102 (2009).
  37. J. L. Xiong, M. S. Tong, P. Atkins, and W. C. Chew, Efficient evaluation of Casimir force in arbitrary three-dimensional geometries by integral equation methods, Phys. Lett. A 374, 2517 (2010).
  38. M. T. H. Reid, J. White, and S. G. Johnson, Computation of Casimir interactions between arbitrary three-dimensional objects with arbitrary material properties, Phys. Rev. A 84, 010503 (2011).
  39. M. T. H. Reid, J. White, and S. G. Johnson, Fluctuating surface currents: An algorithm for efficient prediction of Casimir interactions among arbitrary materials in arbitrary geometries, Phys. Rev. A 88, 022514 (2013).
  40. A. Rodriguez, M. Ibanescu, D. Iannuzzi, J. D. Joannopoulos, and S. G. Johnson, Virtual photons in imaginary time: Computing exact Casimir forces via standard numerical electromagnetism techniques, Phys. Rev. A 76, 032106 (2007).
  41. A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, Casimir forces in the time domain: Theory, Phys. Rev. A 80, 012115 (2009).
  42. A. P. McCauley, A. W. Rodriguez, J. D. Joannopoulos, and S. G. Johnson, Casimir forces in the time domain: Applications, Phys. Rev. A 81, 012119 (2010).
  43. K. Pan, A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, J. K. White, and S. G. Johnson, Calculation of nonzero-temperature Casimir forces in the time domain, Phys. Rev. A 83, 040503 (2011).
  44. A. Rodriguez, M. Ibanescu, D. Iannuzzi, F. Capasso, J. D. Joannopoulos, and S. G. Johnson, Computation and visualization of Casimir forces in arbitrary geometries: Nonmonotonic lateral-wall forces and the failure of proximity-force approximations, Phys. Rev. Lett. 99, 080401 (2007).
  45. S. J. Rahi, A. W. Rodriguez, T. Emig, R. L. Jaffe, S. G. Johnson, and M. Kardar, Nonmonotonic effects of parallel sidewalls on Casimir forces between cylinders, Phys. Rev. A 77, 030101 (2008).
  46. M. Levin, A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Casimir repulsion between metallic objects in vacuum, Phys. Rev. Lett. 105, 090403 (2010).
  47. A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, Casimir repulsion beyond the dipole regime, arXiv:1105.0404.
  48. A. W. Rodriguez, J. N. Munday, J. D. Joannopoulos, F. Capasso, D. A. R. Dalvit, and S. G. Johnson, Stable suspension and dispersion-induced transitions from repulsive Casimir forces between fluid-separated eccentric cylinders, Phys. Rev. Lett. 101, 190404 (2008).
  49. A. W. Rodriguez, M. T. H. Reid, F. Intravaia, A. Woolf, D. A. R. Dalvit, F. Capasso, and S. G. Johnson, Geometry-induced Casimir suspension of oblate bodies in fluids, Phys. Rev. Lett. 111, 180402 (2013).
  50. P. T. Kristensen, B. Beverungen, F. Intravaia, and K. Busch, High-accuracy Casimir-Polder force calculations using the discontinuous Galerkin time-domain method, Phys. Rev. B 108, 205424 (2023).
  51. K. Busch, M. König, and J. Niegemann, Discontinuous Galerkin methods in nanophotonics, Laser Photonics Rev. 5, 773 (2011).
  52. T. Emig and N. Graham, Electromagnetic Casimir energy of a disk opposite a plane, Phys. Rev. A 94, 032509 (2016).
  53. G. Bimonte and T. Emig, Casimir and Casimir-Polder interactions for magneto-dielectric materials: Surface scattering expansion, Phys. Rev. A 108, 052807 (2023).
  54. T. Emig and G. Bimonte, Multiple scattering expansion for dielectric media: Casimir effect, Phys. Rev. Lett. 130, 200401 (2023).
  55. M. T. H. Reid, SCUFF-EM, https://github.com/homerreid/scuff-EM (2026).
  56. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).
  57. J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley & Sons, New York, 1975).
  58. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  59. W. Eckhardt, Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer, Phys. Rev. A 29, 1991 (1984).
  60. A. Sommerfeld, Chapter V - eigenfunctions and eigen values, in Partial Differential Equations in Physics, Pure and Applied Mathematics (Academic Press, New York, 1949), Vol. 1, pp. 166–235.
  61. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1994).
  62. L. H. Ford, Spectrum of the Casimir effect and the Lifshitz theory, Phys. Rev. A 48, 2962 (1993).
  63. M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, Fluctuation-induced phenomena in nanoscale systems: Harnessing the power of noise, Proc. IEEE 101, 531 (2013).
  64. A. W. Rodriguez, J. D. Joannopoulos, and S. G. Johnson, Repulsive and attractive Casimir forces in a glide-symmetric geometry, Phys. Rev. A 77, 062107 (2008).
  65. A. W. Rodriguez, F. Capasso, and S. G. Johnson, The Casimir effect in microstructured geometries, Nat. Photonics 5, 211 (2011).
  66. R. Kilianski, C. M. Cisowski, and R. Bennett, Algorithmic discovery of Casimir–Polder forces: Repulsion in the ground state, Opt. Lett. 50, 6377 (2025).
  67. F. Intravaia and C. Henkel, Casimir interaction from magnetically coupled eddy currents, Phys. Rev. Lett. 103, 130405 (2009).
  68. F. Intravaia, S. Ellingsen, and C. Henkel, Casimir-Foucault interaction: Free energy and entropy at low temperature, Phys. Rev. A 82, 032504 (2010).
  69. C. Henkel and F. Intravaia, On the Casimir entropy between ‘perfect crystals', Int. J. Mod. Phys. A 25, 2328 (2010).
  70. D. C. Wood, The computation of polylogarithms, Technical report (University of Kent, Canterbury, UK, 1992).
  71. J. S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell's equations, J. Comput. Phys. 181, 186 (2002).
  72. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods (Springer, New York, 2008).
  73. J. Niegemann, R. Diehl, and K. Busch, Efficient low-storage Runge-Kutta schemes with optimized stability regions, J. Comput. Phys. 231, 364 (2012).
  74. A. Demirel, J. Niegemann, K. Busch, and M. Hochbruck, Efficient multiple time-stepping algorithms of higher order, J. Comput. Phys. 285, 133 (2015).
  75. J. Viquerat and C. Scheid, A 3D curvilinear discontinuous Galerkin time-domain solver for nanoscale light-matter interactions, J. Comp. App. Math. 289, 37 (2015).
  76. P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B 6, 4370 (1972).
  77. S. G. Johnson, Numerical methods for computing Casimir interactions, in Casimir Physics, Lecture Notes in Physics, edited by D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa (Springer, Berlin, Heidelberg, 2011), Vol. 834, pp. 175–218.
  78. F. Intravaia, How modes shape Casimir physics, Int. J. Mod. Phys. A 37, 2241014 (2022).
  79. C. Henkel, K. Joulain, J.-P. Mulet, and J.-J. Greffet, Coupled surface polaritons and the Casimir force, Phys. Rev. A 69, 023808 (2004).
  80. J. Niegemann, W. Pernice, and K. Busch, Simulation of optical resonators using DGTD and FDTD, J. Opt. A 11, 114015 (2009).
  81. S. Govindjee and P.-O. Persson, A time-domain discontinuous Galerkin method for mechanical resonator quality factor computations, J. Comput. Phys. 231, 6380 (2012).
  82. V. A. Mandelshtam and H. S. Taylor, Harmonic inversion of time signals and its applications, J. Chem. Phys. 107, 6756 (1997).
  83. M. R. Wall and D. Neuhauser, Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model, J. Chem. Phys. 102, 8011 (1995).
  84. S. G. Johnson, Harminv, https://github.com/NanoComp/harminv (2025).
  85. S. Y. Buhmann, Dispersion Forces I (Springer, Berlin, 2012), Vol. 247.
  86. P. T. Kristensen, K. Herrmann, F. Intravaia, and K. Busch, Modeling electromagnetic resonators using quasinormal modes, Adv. Opt. Photonics 12, 612 (2020).
  87. All Scuff-em and DGTD simulations used 16 CPU cores.
  88. S. Rao, D. Wilton, and A. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propag. 30, 409 (1982).
  89. A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, 1966).
  90. L. N. Trefethen, Exactness of quadrature formulas, SIAM Rev. 64, 132 (2022).
  91. C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, An improved proximity force approximation for electrostatics, Ann. Phys. (NY) 327, 2050 (2012).
  92. G. Bimonte, T. Emig, R. L. Jaffe, and M. Kardar, Casimir forces beyond the proximity approximation, Europhys. Lett. 97, 50001 (2012).
  93. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 2008).
  94. J. Venermo and A. Sihvola, Dielectric polarizability of circular cylinder, J. Electrost. 63, 101 (2005).
  95. F. Intravaia, C. Henkel, and M. Antezza, Fluctuation-induced forces between atoms and surfaces: The Casimir-Polder interaction, in Casimir Physics, Lecture Notes in Physics, edited by D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa (Springer, Berlin, Heidelberg, 2011), Vol. 834, pp. 345–391.
  96. S. Y. Buhmann and D.-G. Welsch, Dispersion forces in macroscopic quantum electrodynamics, Prog. Quantum Electron. 31, 51 (2007).
  97. R. Cools and P. Rabinowitz, Monomial cubature rules since Stroud: A compilation, J. Comp. App. Math. 48, 309 (1993).
  98. R. Cools, Monomial cubature rules since Stroud: A compilation–part 2, J. Comp. App. Math. 112, 21 (1999).
  99. R. Cools, An encyclopaedia of cubature formulas, J. Complex. 19, 445 (2003).
  100. C. Raabe and D.-G. Welsch, Casimir force acting on magnetodielectric bodies embedded in media, Phys. Rev. A 71, 013814 (2005).
  101. L. P. Pitaevskii, Comment on “Casimir force acting on magnetodielectric bodies embedded in media,” Phys. Rev. A 73, 047801 (2006).
  102. F. A. Burger, J. Fiedler, and S. Y. Buhmann, Zero-point electromagnetic stress tensor for studying Casimir forces on colloidal particles in media, Europhys. Lett. 121, 24004 (2018).
  103. R. Esquivel-Sirvent, C. Villarreal, W. Mochán, A. Contreras-Reyes, and V. Svetovoy, Spatial dispersion in Casimir forces: A brief review, J. Phys. A: Math. Gen. 39, 6323 (2006).
  104. F. Intravaia, S. Koev, I. W. Jung, A. A. Talin, P. S. Davids, R. S. Decca, V. A. Aksyuk, D. A. R. Dalvit, and D. López, Strong Casimir force reduction through metallic surface nanostructuring, Nat. Commun. 4, 2515 (2013).
  105. G. Wegner, D.-N. Huynh, N. A. Mortensen, F. Intravaia, and K. Busch, Halevi's extension of the Euler-Drude model for plasmonic systems, Phys. Rev. B 107, 115425 (2023).
  106. C. Ciracì, J. B. Pendry, and D. R. Smith, Hydrodynamic model for plasmonics: A macroscopic approach to a microscopic problem, ChemPhysChem 14, 1109 (2013).
  107. W. Yan, N. A. Mortensen, and M. Wubs, Green's function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments, Phys. Rev. B 88, 155414 (2013).
  108. J. Baxter, A. C. Lesina, and L. Ramunno, Parallel FDTD modeling of nonlocality in plasmonics, IEEE Trans. Antennas Propag. 69, 3982 (2021).
  109. M. S. Tomaš, Green function for multilayers: Light scattering in planar cavities, Phys. Rev. A 51, 2545 (1995).

Outline

Information

Phys. Rev. A 114, 042817– Published 22 October, 2026
  • Received 4 March 2026
  • Revised 10 August 2026
  • Accepted 14 August 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation