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  • Open Access

Geometro-optical depolarization manifold as an organizing principle for plasmonic transitions

Isuf Tredhaku

Ana Sánchez-Iglesias

Ivan Petranović

Marek Grzelczak*

Jordi Sancho-Parramon†

Stefano A. Mezzasalma‡

  • Materials Physics Division, Laboratory of Optics and Optical Thin Films, Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia and Department of Physics, Faculty of Science, University of Zagreb, Bijenička cesta 32, 10000 Zagreb, Croatia

  • Materials Physics Division, Laboratory of Optics and Optical Thin Films, Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia and Department of Chemistry, Faculty of Science, University of Zagreb, Horvatovac 102A, 10000 Zagreb, Croatia

  • Materials Physics Division, Laboratory of Optics and Optical Thin Films, Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia

  • *Contact author: marek.g@csic.es
  • †Contact author: jordi.sancho.parramon@irb.hr
  • ‡Contact author: stefano.mezzasalma@irb.hr
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Phys. Rev. A 114, 043521 – Published 16 October, 2026

DOI: https://doi.org/10.1103/718h-fn18

Abstract

Using differential calculus, we derive an exact dipolar condition for the extinction peak position of a nanocrystal, reducing the problem of locating the extinction maximum to a differentiable state equation. This leads to a characteristic polynomial for the plasmon frequency whose coefficients are set by the dielectric function, environment, particle volume, static depolarization factor, and an effective retardation scale. After calibrating this scale with electrodynamic simulations, we construct geometro-optical depolarization manifolds that offer a compact space for tracking plasmonic changes during growth, reshaping, or etching. The resulting polynomials accurately reproduce 56 experimental wavelengths of Au spheres, rods, and bipyramids in an aqueous solution, and two experimental plasmonic transitions, from rods to spheres and from bipyramids to spheres. In addition to providing a general organizing principle for plasmonic transitions, this framework enables accurate variational analyses of extinction peaks, facilitates the inference of nanoparticle morphology from optical responses through data-driven approaches, and provides a robust characterization of plasmonic systems and experiments even in the presence of spurious or nonideal effects that are difficult to capture analytically.

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References (58)

  1. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, Biosensing with plasmonic nanosensors, Nat. Mater. 7, 442 (2008).
  2. L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas, and J. L. West, Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonance guidance, Proc. Natl. Acad. Sci. USA 100, 13549 (2003).
  3. P. Christopher, H. Xin, and S. Linic, Visible-light- enhanced catalytic oxidation reactions on plasmonic silver nanostructures, Nat. Chem. 3, 467 (2011).
  4. S. A. Mezzasalma, J. Kruse, S. Merkens, E. Lopez, A. Seifert, R. Morandotti, and M. Grzelczak, Light-driven self-oscillation of thermoplasmonic nanocolloids, Adv. Mater. 35, 2302987 (2023).
  5. J. L. Montaño-Priede, A. Rao, A. Sánchez-Iglesias, and M. Grzelczak, Accelerated design of gold nanoparticles with enhanced plasmonic performance, Sci. Adv. 11, eadx2299 (2025).
  6. A. Sánchez-Iglesias and M. Grzelczak, Expanding chemical space in the synthesis of gold bipyramids, Small 21, 2407735 (2025).
  7. G. Mie, Beiträge zur optik trüber medien, speziell kolloidaler metallösungen, Ann. Phys. 330, 377 (1908).
  8. E. C. Le Ru, W. R. C. Somerville, and B. Auguié, Radiative correction in approximate treatments of electromagnetic scattering by point and body scatterers, Phys. Rev. A 87, 012504 (2013).
  9. A. Patoux, C. Majorel, P. R. Wiecha, A. Cuche, O. L. Muskens, C. Girard, and A. Arbouet, Polarizabilities of complex individual dielectric or plasmonic nanostructures, Phys. Rev. B 101, 235418 (2020).
  10. M. Majic, L. Pratley, D. Schebarchov, W. R. C. Somerville, B. Auguié, and E. C. Le Ru, Approximate T-matrix and optical properties of spheroidal particles to third order in size parameter, Phys. Rev. A 99, 013853 (2019).
  11. I. L. Rasskazov, V. I. Zakomirnyi, A. D. Utyushev, P. Carney, and A. Moroz, Remarkable predictive power of the modified long wavelength approximation, J. Phys. Chem. C 125, 1963 (2021).
  12. A. D. Utyushev, V. I. Zakomirnyi, A. A. Shcherbakov, I. L. Rasskazov, and A. Moroz, Intuitive understanding of extinction of small particles in absorbing and active host media within the MLWA, J. Opt. Soc. Am. B 41, 2519 (2024).
  13. M. Januar, B. Liu, J.-C. Cheng, K. Hatanaka, H. Misawa, H.-H. Hsiao, and K.-C. Liu, Role of depolarization factors in the evolution of a dipolar plasmonic spectral line in the far- and near-field regimes, J. Phys. Chem. C 124, 3250 (2020).
  14. D. Schebarchov, B. Auguié, and E. C. Le Ru, Simple accurate approximations for the optical properties of metallic nanospheres and nanoshells, Phys. Chem. Chem. Phys. 15, 4233 (2013).
  15. N. G. Khlebtsov and S. V. Zarkov, Analytical modeling of coated plasmonic particles, J. Phys. Chem. C 128, 15029 (2024).
  16. N. G. Khlebtsov and S. V. Zarkov, Combining the modal expansion and dipole equivalence methods for coated plasmonic particles of various shapes, J. Phys. Chem. C 129, 10958 (2025).
  17. T. Sandu, Shape effects on localized surface plasmon resonances in metallic nanoparticles, J. Nanopart. Res. 14, 905 (2012).
  18. R. Yu, L. M. Liz-Marzán, and F. J. García de Abajo, Universal analytical modeling of plasmonic nanoparticles, Chem. Soc. Rev. 46, 6710 (2017).
  19. P. S. S. dos Santos, J. P. Mendes, J. M. M. M. de Almeida, and L. C. C. Coelho, Ultrafast dipolar electrostatic modeling of plasmonic nanoparticles with arbitrary geometry, arXiv:2601.16797.
  20. E. Ringe, M. R. Langille, K. Sohn, J. Zhang, J. Huang, C. A. Mirkin, R. P. Van Duyne, and L. D. Marks, Plasmon length: A universal parameter to describe size effects in gold nanoparticles, J. Phys. Chem. Lett. 3, 1479 (2012).
  21. Z. M. Sherman, D. J. Milliron, and T. M. Truskett, Distribution of single-particle resonances determines the plasmonic response of disordered nanoparticle ensembles, ACS Nano 18, 21347 (2024).
  22. E. S. Kooij, W. Ahmed, H. J. W. Zandvliet, and B. Poelsema, Localized plasmons in noble metal nanospheroids, J. Phys. Chem. C 115, 10321 (2011).
  23. L. C. Ugwuoke, T. Mančal, and T. P. J. Krüger, Localized surface plasmon resonances of simple tunable plasmonic nanostructures, Plasmonics 15, 189 (2020).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/718h-fn18 for more details, including the Drude-Lorentz dielectric model via genetic algorithm, electrodynamic calibration, numerical datasets, fitting details, expanded coefficients, validation tests, and auxiliary theoretical observations, which includes Refs. [25, 26, 27, 28, 29].
  25. I. Zorić, M. Zäch, B. Kasemo, and C. Langhammer, Gold, platinum, and aluminum nanodisk plasmons: Material independence, subradiance, and damping mechanisms, ACS Nano 5, 2535 (2011).
  26. R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, Optical dielectric function of gold, Phys. Rev. B 86, 235147 (2012).
  27. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 2007).
  28. U. Hohenester and A. Trügler, MNPBEM–A Matlab toolbox for the simulation of plasmonic nanoparticles, Comput. Phys. Commun. 183, 370 (2012).
  29. A. Yaghjian, Electric dyadic Green's function in the source region, Proc. IEEE 68, 248 (1980).
  30. A. Moroz, Depolarization field of spheroidal particles, J. Opt. Soc. Am. B 26, 517 (2009).
  31. L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University Press, Cambridge, England, 2012).
  32. R. G. Newton, Optical theorem and beyond, Am. J. Phys. 44, 639 (1976).
  33. P. Grahn, A. Shevchenko, and M. Kaivola, Electromagnetic multipole theory for optical nanomaterials, New J. Phys. 14, 093033 (2012).
  34. V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, Optimizing nanoparticle designs for ideal absorption of light, ACS Photon. 2, 263 (2015).
  35. O. D. Miller, A. G. Polimeridis, M. T. Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, Fundamental limits to optical response in absorptive systems, Opt. Express 24, 3329 (2016).
  36. U. Kreibig and C. Fragstein, The limitation of electron mean free path in small silver particles, Z. Phys. 224, 307 (1969).
  37. R. D. Averitt, S. L. Westcott, and N. J. Halas, Linear optical properties of gold nanoshells, J. Opt. Soc. Am. B 16, 1824 (1999).
  38. A. Moroz, Electron mean free path in a spherical shell geometry, J. Phys. Chem. C 112, 10641 (2008).
  39. J. Scholl, A. L. Koh, and J. Dionne, Quantum plasmon resonances of individual metallic nanoparticles, Nature (London) 483, 421 (2012).
  40. R. Jin, Y. W. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. Zheng, Photoinduced conversion of silver nanospheres to nanoprisms, Science 294, 1901 (2001).
  41. M. R. Langille, M. L. Personick, and C. A. Mirkin, Plasmon-mediated syntheses of metallic nanostructures, Angew. Chem. Int. Ed. 52, 13910 (2013).
  42. S. A. Mezzasalma, M. Grzelczak, and J. Sancho-Parramon, The crystal field plasmon splitting, ACS Photon. 7, 1551 (2020).
  43. J. L. Montaño-Priede, A. Sánchez-Iglesias, S. A. Mezzasalma, J. Sancho-Parramon, and M. Grzelczak, Quantifying shape transition in anisotropic plasmonic nanoparticles through geometric inversion. Application to gold bipyramids, J. Phys. Chem. Lett. 15, 3914 (2024).
  44. Y. Zheng, X. Zhong, Z. Li, and Y. Xia, Successive, seed-mediated growth for the synthesis of single-crystal gold nanospheres with uniform diameters controlled in the range of 5–150 nm, Part. Part. Syst. Charact. 31, 266 (2014).
  45. A. Sánchez-Iglesias, K. Jenkinson, S. Bals, and L. M. Liz-Marzán, Kinetic regulation of the synthesis of pentatwinned gold nanorods below room temperature, J. Phys. Chem. C 125, 23937 (2021).
  46. X. Ye, Y. Gao, J. Chen, D. C. Reifsnyder, C. Zheng, and C. B. Murray, Seeded growth of monodisperse gold nanorods using bromide-free surfactant mixtures, Nano Lett. 13, 2163 (2013).
  47. L. Okano, F. Quina, and O. El Seoud, Fluorescence and light-scattering studies of the aggregation of cationic surfactants in aqueous solution: Effects of headgroup structure, Langmuir 16, 3119 (2000).
  48. Colloidal Synthesis of Plasmonic Nanometals, edited by L. M. Liz-Marzán (Jenny Stanford Publishing, New York, 2021).
  49. E. Martinsson, M. M. Shahjamali, N. Large, N. Zaraee, Y. Zhou, G. C. Schatz, C. A. Mirkin, and D. Aili, Influence of surfactant bilayers on the refractive index sensitivity and catalytic properties of anisotropic gold nanoparticles, Small 12, 330 (2016).
  50. M. A. Karam, Electromagnetic wave interactions with dielectric particles. I. Integral equation reformation, Appl. Opt. 36, 5238 (1997).
  51. B. Auguié and W. L. Barnes, Collective resonances in gold nanoparticle arrays, Phys. Rev. Lett. 101, 143902 (2008).
  52. H. Y. Chung, G. Y. Guo, H.-P. Chiang, D. P. Tsai, and P. T. Leung, Accurate description of optical response of a multilayered spherical system in the long wavelength approximation, Phys. Rev. B 82, 165440 (2010).
  53. N. G. Khlebtsov, T-matrix method in plasmonics: An overview, J. Quant. Spectrosc. Radiat. Transfer 123, 184 (2013).
  54. S. Yoo and Q.-H. Park, Effective permittivity for resonant plasmonic nanoparticle systems via dressed polarizability, Opt. Express 20, 16480 (2012).
  55. A. Malasi, R. Kalyanaraman, and H. Garcia, From Mie to Fresnel through effective medium approximation with multipole contributions, J. Opt. 16, 065001 (2014).
  56. V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. García de Abajo, Modelling the optical response of gold nanoparticles, Chem. Soc. Rev. 37, 1792 (2008).
  57. https://mojoblak.irb.hr/s/3cjMqdQJZApZo4H.
  58. https://www.plasmon-calculator.net/.

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Phys. Rev. A 114, 043521– Published 16 October, 2026
  • Received 27 February 2026
  • Accepted 31 August 2026
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