Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Roadmap
  • Published:

Progress and prospects in nonlinear extreme-ultraviolet and X-ray optics and spectroscopy

Nature Reviews Physics volume 5, pages 578–596 (2023)Cite this article

Subjects

Abstract

Free-electron lasers and high-harmonic-generation table-top systems are new sources of extreme-ultraviolet to hard X-ray photons, providing ultrashort pulses that are intense, coherent and tunable. They are enabling a broad range of nonlinear optical and spectroscopic methods at short wavelengths, similar to those developed in the terahertz to ultraviolet regimes over the past 60 years. The extreme-ultraviolet to X-ray wavelengths access core transitions that can provide element and orbital selectivity, structural resolution down to the sub-nanometre scale and, for some methods, high momentum transfers across typical Brillouin zones; the possibilities for polarization control and sub-femtosecond time resolution are opening up new frontiers in research. In this Roadmap, we review the emergence of this field over the past 10 years or so, covering methods such as sum or difference frequency generation and second-harmonic generation, two-photon absorption, stimulated emission or Raman spectroscopy and transient grating spectroscopy. We then discuss the unique opportunities provided by these techniques for probing elementary dynamics in a wide variety of systems.

Key points

  • X-ray free-electron lasers and high-harmonic-generation sources of extreme-ultraviolet (EUV) to hard X-ray photons deliver intense ultrashort pulses and enable the extension of nonlinear methods to much shorter wavelengths.

  • EUV to X-ray wavelengths access core transitions that can provide element and orbital selectivity. These wavelengths also achieve sub-nanometre structural resolution and high momentum transfer, with femtosecond and attosecond time resolution.

  • Nonlinear EUV/X-ray methods that have emerged include sum or difference frequency generation, parametric down-conversion, second-harmonic generation, two-photon absorption, stimulated emission or Raman spectroscopy and transient grating spectroscopy.

  • Nonlinear EUV/X-ray science is developing hand-in-hand with instrumentation, to improve pulse features and enhance accessibility with the use of table-top systems or compact accelerators.

  • These techniques offer unique opportunities for probing dynamical events in a wide variety of systems, including surface and interface processes, chirality, nanoscale transport and multidimensional core-level spectroscopy.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Schemes for various examples of nonlinear optical methods.
Fig. 2: Incoherent resonant inelastic X-ray scattering versus coherent sum frequency generation.
Fig. 3: Extreme-ultraviolet and X-ray second-harmonic generation.
Fig. 4: Kinematic regions that are accessible using existing techniques.
Fig. 5: Experimental set-ups for extreme-ultraviolet/X-ray transient grating.
Fig. 6: Representative extreme-ultraviolet/X-ray transient grating signals.

Similar content being viewed by others

References

  1. Kaiser, W. & Garrett, C. G. B. Two-photon excitation in CaF2:Eu2+. Phys. Rev. Lett. 7, 229–231 (1961).

    ADS  Google Scholar 

  2. Franken, P. A., Hill, A. E., Peters, C. W. & Weinreich, G. Generation of optical harmonics. Phys. Rev. Lett. 7, 118–119 (1961).

    ADS  Google Scholar 

  3. Bloembergen, N. Nonlinear optics and spectroscopy. Rev. Mod. Phys. 54, 685–695 (1982).

    ADS  Google Scholar 

  4. Mukamel, S. Principles of Nonlinear Optical Spectroscopy (Oxford Univ. Press, 1995).

  5. Munn, R. W. & Ironside, C. N. Principles and Applications of Nonlinear Optical Materials (Springer Netherlands, 1993).

  6. Busch, G. E., Jones, R. P. & Rentzepis, P. M. Picosecond spectroscopy using a picosecond continuum. Chem. Phys. Lett. 18, 178–185 (1973).

    ADS  Google Scholar 

  7. Eisenthal, K. B. Picosecond relaxation processes in chemistry. In Ultrashort Light Pulses: Picosecond Techniques and Applications (ed. Shapiro, S. L.) 275–315 (Springer, 1977).

  8. Shank, C., Ippen, E. & Bersohn, R. Time-resolved spectroscopy of hemoglobin and its complexes with subpicosecond optical pulses. Science 193, 50 (1976).

    ADS  Google Scholar 

  9. Zewail, A. H. Femtochemistry: atomic-scale dynamics of the chemical bond using ultrafast lasers (Nobel Lecture). Angew. Chem. Int. Ed. 39, 2586–2631 (2000).

    Google Scholar 

  10. Ackermann, W. et al. Operation of a free-electron laser from the extreme ultraviolet to the water window. Nat. Photon. 1, 336–342 (2007).

    ADS  Google Scholar 

  11. Allaria, E. et al. The FERMI@ Elettra free-electron-laser source for coherent X-ray physics: photon properties, beam transport system and applications. N. J. Phys. 12, 075002 (2010).

    Google Scholar 

  12. Emma, P. et al. First lasing and operation of an angstrom-wavelength free-electron laser. Nat. Photon. 4, 641–647 (2010).

    ADS  Google Scholar 

  13. Ishikawa, T. et al. A compact X-ray free-electron laser emitting in the sub-angstrom region. Nat. Photon. 6, 540–544 (2012).

    ADS  Google Scholar 

  14. Prat, E. et al. A compact and cost-effective hard X-ray free-electron laser driven by a high-brightness and low-energy electron beam. Nat. Photon. 14, 748–754 (2020).

    ADS  Google Scholar 

  15. Lappas, D. G. & L’Huillier, A. Generation of attosecond XUV pulses in strong laser-atom interactions. Phys. Rev. A 58, 4140–4146 (1998).

    ADS  Google Scholar 

  16. Corkum, P. B., Burnett, N. H. & Ivanov, M. Y. Subfemtosecond pulses. Opt. Lett. 19, 1870–1872 (1994).

    ADS  Google Scholar 

  17. Drescher, M. et al. X-ray pulses approaching the attosecond frontier. Science 291, 1923–1927 (2001).

    ADS  Google Scholar 

  18. Christov, I. P., Murnane, M. M. & Kapteyn, H. C. High-harmonic generation of attosecond pulses in the ‘single-cycle’ regime. Phys. Rev. Lett. 78, 1251–1254 (1997).

    ADS  Google Scholar 

  19. Chang, Z. H., Rundquist, A., Wang, H. W., Murnane, M. M. & Kapteyn, H. C. Generation of coherent soft X rays at 2.7 nm using high harmonics. Phys. Rev. Lett. 79, 2967–2970 (1997).

    ADS  Google Scholar 

  20. Hentschel, M. et al. Attosecond metrology. Nature 414, 509–513 (2001).

    ADS  Google Scholar 

  21. Lenzner, M., Schnurer, M., Spielmann, C. & Krausz, F. Extreme nonlinear optics with few-cycle laser pulses. IEICE Trans. Electron. E81C, 112–122 (1998).

    Google Scholar 

  22. Milne, C. J., Penfold, T. J. & Chergui, M. Recent experimental and theoretical developments in time-resolved X-ray spectroscopies. Coord. Chem. Rev. 277, 44–68 (2014).

    Google Scholar 

  23. Chergui, M. & Collet, E. Photoinduced structural dynamics of molecular systems mapped by time-resolved X-ray methods. Chem. Rev. 117, 11025–11065 (2017).

    Google Scholar 

  24. Kraus, P. M., Zürch, M., Cushing, S. K., Neumark, D. M. & Leone, S. R. The ultrafast X-ray spectroscopic revolution in chemical dynamics. Nat. Rev. Chem. 2, 82–94 (2018).

    Google Scholar 

  25. Zong, A., Nebgen, B. R., Lin, S.-C., Spies, J. A. & Zuerch, M. Emerging ultrafast techniques for studying quantum materials. Nat. Rev. Mater. https://doi.org/10.1038/s41578-022-00530-0 (2023).

    Article  Google Scholar 

  26. Rohringer, N. X-ray Raman scattering: a building block for nonlinear spectroscopy. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 377, 20170471 (2019).

    ADS  Google Scholar 

  27. Leone, S. R. & Neumark, D. M. Probing matter with nonlinear spectroscopy. Science 379, 536–537 (2023).

    ADS  Google Scholar 

  28. Khan, S. Free-electron lasers. J. Mod. Opt. 55, 3469–3512 (2008).

    ADS  MATH  Google Scholar 

  29. Schoenlein, R. et al. Recent advances in ultrafast X-ray sources. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 377, 20180384 (2019).

    ADS  Google Scholar 

  30. Biegert, J., Calegari, F., Dudovich, N., Quéré, F. & Vrakking, M. Attosecond technology(ies) and science. J. Phys. B Mol. Opt. Phys. 54, 070201 (2021).

    Google Scholar 

  31. Huang, N., Deng, H., Liu, B., Wang, D. & Zhao, Z. Features and futures of X-ray free-electron lasers. Innovation 2, 100097 (2021).

    Google Scholar 

  32. Pellegrini, C., Marinelli, A. & Reiche, S. The physics of X-ray free-electron lasers. Rev. Mod. Phys. 88, 015006 (2016).

    ADS  Google Scholar 

  33. Margaritondo, G. & Rafelski, J. The relativistic foundations of synchrotron radiation. J. Synchrotron Radiat. 24, 898–901 (2017).

    Google Scholar 

  34. Kondratenko, A. M. & Saldin, E. L. Generation of coherent radiation by a relativistic electron beam in an ondulator. Part. Accel. 10, 207–216 (1980).

    Google Scholar 

  35. Allaria, E. et al. Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet. Nat. Photon. 6, 699–704 (2012).

    ADS  Google Scholar 

  36. Abela, R. et al. Perspective: opportunities for ultrafast science at SwissFEL. Struct. Dyn. 4, 061602 (2017).

    Google Scholar 

  37. Altarelli, M., Brinkmann, R. & Chergui, M. The European X-ray Free-electron Laser (DESY, 2007).

  38. Tschentscher, T. et al. Photon beam transport and scientific instruments at the European XFEL. Appl. Sci. 7, 592 (2017).

    Google Scholar 

  39. Kang, H.-S. et al. Hard X-ray free-electron laser with femtosecond-scale timing jitter. Nat. Photon. 11, 708–713 (2017).

    Google Scholar 

  40. Yu, L.-H. et al. High-gain harmonic-generation free-electron laser. Science 289, 932–934 (2000).

    ADS  Google Scholar 

  41. Allaria, E. et al. Two-stage seeded soft-X-ray free-electron laser. Nat. Photon. 7, 913–918 (2013).

    ADS  Google Scholar 

  42. Rebernik Ribič, P. et al. Coherent soft X-ray pulses from an echo-enabled harmonic generation free-electron laser. Nat. Photon. 13, 555–561 (2019).

    Google Scholar 

  43. Gianessi, L. & Masciovecchio, C. (eds) FERMI 2.0 Conceptual Design Report (Elettra Sincrotrone Trieste, 2022); https://www.elettra.eu/images/Documents/FERMI%20Machine/Machine/CDR/FERMI2.0CDR.pdf

  44. McPherson, A. et al. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases. J. Opt. Soc. Am. B 4, 595–601 (1987).

    ADS  Google Scholar 

  45. Ferray, M. et al. Multiple-harmonic conversion of 1064 nm radiation in rare gases. J. Phys. B Mol. Opt. Phys. 21, L31–L35 (1988).

    Google Scholar 

  46. Corkum, P. B. & Krausz, F. Attosecond science. Nat. Phys. 3, 381–387 (2007).

    Google Scholar 

  47. DiChiara, A. D. et al. Scaling of high-order harmonic generation in the long wavelength limit of a strong laser field. IEEE J. Sel. Top. Quantum Electron. 18, 419–433 (2012).

    ADS  Google Scholar 

  48. Cousin, S. L. et al. High-flux table-top soft X-ray source driven by sub-2-cycle, CEP stable, 1.85-µm 1-kHz pulses for carbon K-edge spectroscopy. Opt. Lett. 39, 5383–5386 (2014).

    ADS  Google Scholar 

  49. Kleine, C. et al. Soft X-ray absorption spectroscopy of aqueous solutions using a table-top femtosecond soft X-ray source. J. Phys. Chem. Lett. 10, 52–58 (2019).

    Google Scholar 

  50. Chang, Z. Enhancing keV high harmonic signals generated by long-wave infrared lasers. OSA Contin. 2, 2131–2136 (2019).

    Google Scholar 

  51. Kroh, T. et al. Enhanced high-harmonic generation up to the soft X-ray region driven by mid-infrared pulses mixed with their third harmonic. Opt. Express 26, 16955–16969 (2018).

    ADS  Google Scholar 

  52. Makos, I. et al. Α 10-gigawatt attosecond source for non-linear XUV optics and XUV-pump-XUV-probe studies. Sci. Rep. 10, 3759 (2020).

    ADS  Google Scholar 

  53. Pupeza, I. et al. Compact high-repetition-rate source of coherent 100 eV radiation. Nat. Photon. 7, 608–612 (2013).

    ADS  Google Scholar 

  54. Hädrich, S. et al. Exploring new avenues in high repetition rate table-top coherent extreme ultraviolet sources. Light Sci. Appl. 4, e320–e320 (2015).

    Google Scholar 

  55. Rossi, G. M. et al. Sub-cycle millijoule-level parametric waveform synthesizer for attosecond science. Nat. Photon. 14, 629–635 (2020).

    ADS  Google Scholar 

  56. Geneaux, R., Marroux, H. J. B., Guggenmos, A., Neumark, D. M. & Leone, S. R. Transient absorption spectroscopy using high harmonic generation: a review of ultrafast X-ray dynamics in molecules and solids. Philos. Trans. R Soc. Math. Phys. Eng. Sci. 377, 20170463 (2019).

    ADS  Google Scholar 

  57. Park, J., Subramani, A., Kim, S. & Ciappina, M. F. Recent trends in high-order harmonic generation in solids. Adv. Phys. X 7, 2003244 (2022).

    Google Scholar 

  58. Luu, T. T. et al. Extreme–ultraviolet high-harmonic generation in liquids. Nat. Commun. 9, 3723 (2018).

    ADS  Google Scholar 

  59. Zeng, A.-W. & Bian, X.-B. Impact of statistical fluctuations on high harmonic generation in liquids. Phys. Rev. Lett. 124, 203901 (2020).

    ADS  Google Scholar 

  60. Agostini, P. & DiMauro, L. F. The physics of attosecond light pulses. Rep. Prog. Phys. 67, 813–855 (2004).

    ADS  Google Scholar 

  61. Gallmann, L., Cirelli, C. & Keller, U. Attosecond science: recent highlights and future trends. Annu. Rev. Phys. Chem. 63, 447–469 (2012).

    ADS  Google Scholar 

  62. Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163 (2009).

    ADS  Google Scholar 

  63. Frank, F. et al. Invited review article: technology for attosecond science. Rev. Sci. Instrum. 83, 071101 (2012).

    ADS  Google Scholar 

  64. Boutu, W., Ducousso, M., Hergott, J.-F. & Merdji, H. Overview on HHG high-flux sources. in Optical Technologies for Extreme-Ultraviolet and Soft X-ray Coherent Sources (eds Canova, F. & Poletto, L.) 63–78 (Springer, 2015).

  65. Franz, D. et al. All semiconductor enhanced high-harmonic generation from a single nanostructured cone. Sci. Rep. 9, 5663 (2019).

    ADS  Google Scholar 

  66. Jürgens, P. et al. Origin of strong-field-induced low-order harmonic generation in amorphous quartz. Nat. Phys. 16, 1035–1039 (2020).

    Google Scholar 

  67. Gholam-Mirzaei, S., Beetar, J. E., Chacón, A. & Chini, M. High-harmonic generation in ZnO driven by self-compressed mid-infrared pulses. J. Opt. Soc. Am. B 35, A27–A31 (2018).

    ADS  Google Scholar 

  68. Pertot, Y. et al. Time-resolved X-ray absorption spectroscopy with a water window high-harmonic source. Science 355, 264–267 (2017).

    ADS  Google Scholar 

  69. Zinchenko, K. S. et al. Sub-7-femtosecond conical-intersection dynamics probed at the carbon K-edge. Science 371, 489–494 (2021).

    ADS  Google Scholar 

  70. Gallmann, L. et al. Photoemission and photoionization time delays and rates. Struct. Dyn. 4, 061502 (2017).

    Google Scholar 

  71. Faccialà, D. et al. Time-resolved chiral X-ray photoelectron spectroscopy with transiently enhanced atomic site-selectivity: a free electron laser investigation of electronically excited fenchone enantiomers. Phys. Rev. X 13, 011044 (2022).

    Google Scholar 

  72. Wörner, H. J., Bertrand, J. B., Kartashov, D. V., Corkum, P. B. & Villeneuve, D. M. Following a chemical reaction using high-harmonic interferometry. Nature 466, 604–607 (2010).

    ADS  Google Scholar 

  73. Abel, B., Buck, U., Sobolewski, A. L. & Domcke, W. On the nature and signatures of the solvated electron in water. Phys. Chem. Chem. Phys. 14, 22–34 (2012).

    Google Scholar 

  74. Hummert, J. et al. Femtosecond extreme ultraviolet photoelectron spectroscopy of organic molecules in aqueous solution. J. Phys. Chem. Lett. 9, 6649–6655 (2018).

    Google Scholar 

  75. Longetti, L. et al. Ultrafast photoelectron spectroscopy of photoexcited aqueous ferrioxalate. Phys. Chem. Chem. Phys. 23, 25308–25316 (2021).

    Google Scholar 

  76. Arrell, C. A. et al. Laser-assisted photoelectric effect from liquids. Phys. Rev. Lett. 117, 143001 (2016).

    ADS  Google Scholar 

  77. Crepaldi, A. et al. Time-resolved ARPES at LACUS: band structure and ultrafast electron dynamics of solids. CHIMIA 71, 273–277 (2017).

    Google Scholar 

  78. Gatti, G. et al. Light-induced renormalization of the Dirac quasiparticles in the nodal-line semimetal ZrSiSe. Phys. Rev. Lett. 125, 076401 (2020).

    ADS  Google Scholar 

  79. Fidler, A. P. et al. Nonlinear XUV signal generation probed by transient grating spectroscopy with attosecond pulses. Nat. Commun. 10, 1384 (2019).

    ADS  Google Scholar 

  80. Grilj, J. et al. Self referencing heterodyne transient grating spectroscopy with short wavelength. Photonics 2, 392–401 (2015).

    Google Scholar 

  81. Cao, W., Warrick, E. R., Fidler, A., Neumark, D. M. & Leone, S. R. Noncollinear wave mixing of attosecond XUV and few-cycle optical laser pulses in gas-phase atoms: toward multidimensional spectroscopy involving XUV excitations. Phys. Rev. A 94, 053846 (2016).

    ADS  Google Scholar 

  82. Mairesse, Y. et al. High-order harmonic transient grating spectroscopy in a molecular jet. Phys. Rev. Lett. 100, 143903 (2008).

    ADS  Google Scholar 

  83. Ruf, H. et al. High-harmonic transient grating spectroscopy of NO2 electronic relaxation. J. Chem. Phys. 137, 224303 (2012).

    ADS  Google Scholar 

  84. Orfanos, I. et al. Non-linear processes in the extreme ultraviolet. J. Phys. Photon. 2, 042003 (2020).

    Google Scholar 

  85. Senfftleben, B. et al. Highly non-linear ionization of atoms induced by intense high-harmonic pulses. J. Phys. Photon. 2, 034001 (2020).

    Google Scholar 

  86. Drescher, L. et al. Extreme-ultraviolet spectral compression by four-wave mixing. Nat. Photon. 15, 263–266 (2021).

    ADS  Google Scholar 

  87. Helk, T. et al. Table-top extreme ultraviolet second harmonic generation. Sci. Adv. 7, eabe2265 (2021).

    ADS  Google Scholar 

  88. Shen, Y. R. Principles of Nonlinear Optics (U.S. Department of Energy, 1984).

  89. Boyd, R. W. Nonlinear Optics (Academic Press, 2020).

  90. Corn, R. M. & Higgins, D. A. Optical second harmonic generation as a probe of surface chemistry. Chem. Rev. 94, 107–125 (1994).

    Google Scholar 

  91. Shi, X., Borguet, E., Tarnovsky, A. N. & Eisenthal, K. B. Ultrafast dynamics and structure at aqueous interfaces by second harmonic generation. Chem. Phys. 205, 167–178 (1996).

    Google Scholar 

  92. Wang, H., Borguet, E. & Eisenthal, K. B. Polarity of liquid interfaces by second harmonic generation spectroscopy. J. Phys. Chem. A 101, 713–718 (1997).

    Google Scholar 

  93. Eisenthal, K. B. Liquid interfaces probed by second-harmonic and sum-frequency spectroscopy. Chem. Rev. 96, 1343–1360 (1996).

    Google Scholar 

  94. Almogy, G. & Yariv, A. Resonantly-enhanced nonlinear optics of intersubband transitions. J. Nonlinear Opt. Phys. Mater. 4, 401–458 (1995).

    ADS  Google Scholar 

  95. Oudar, J.-L. & Shen, Y. R. Nonlinear spectroscopy by multiresonant four-wave mixing. Phys. Rev. A 22, 1141–1158 (1980).

    ADS  MathSciNet  Google Scholar 

  96. Begley, R. F., Harvey, A. B. & Byer, R. L. Coherent anti‐stokes Raman spectroscopy. Appl. Phys. Lett. 25, 387–390 (1974).

    ADS  Google Scholar 

  97. Giordmaine, J. A. Mixing of light beams in crystals. Phys. Rev. Lett. 8, 19–20 (1962).

    ADS  Google Scholar 

  98. Jones, W. J. & Stoicheff, B. P. Inverse Raman spectra: induced absorption at optical frequencies. Phys. Rev. Lett. 13, 657–659 (1964).

    ADS  Google Scholar 

  99. Maker, P. D. & Terhune, R. W. Study of optical effects due to an induced polarization third order in the electric field strength. Phys. Rev. 137, A801–A818 (1965).

    ADS  Google Scholar 

  100. Duncan, M. D., Reintjes, J. & Manuccia, T. J. Scanning coherent anti-Stokes Raman microscope. Opt. Lett. 7, 350–352 (1982).

    ADS  Google Scholar 

  101. Hofmann, F., Short, M. P. & Dennett, C. A. Transient grating spectroscopy: an ultrarapid, nondestructive materials evaluation technique. MRS Bull. 44, 392–402 (2019).

    ADS  Google Scholar 

  102. Ernst, R. R., Bodenhausen, G. & Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions Vol. 14 (Clarendon Press, 1987).

  103. Mukamel, S., Tanimura, Y. & Hamm, P. Coherent multidimensional optical spectroscopy. Acc. Chem. Res. 42, 1207–1209 (2009).

    Google Scholar 

  104. Hamm, P. & Zanni, M. Concepts and Methods of 2D Infrared Spectroscopy (Cambridge Univ. Press, 2011).

  105. Brixner, T., Stiopkin, I. V. & Fleming, G. R. Tunable two-dimensional femtosecond spectroscopy. Opt. Lett. 29, 884–886 (2004).

    ADS  Google Scholar 

  106. Brixner, T. et al. Two-dimensional spectroscopy of electronic couplings in photosynthesis. Nature 434, 625–628 (2005).

    ADS  Google Scholar 

  107. Haddad, A. A. et al. Set-up for broadband Fourier-transform multidimensional electronic spectroscopy. Opt. Lett. 40, 312–315 (2015).

    ADS  Google Scholar 

  108. Auböck, G., Consani, C., Mourik, F. & Chergui, M. Ultrabroadband femtosecond two-dimensional ultraviolet transient absorption. Opt. Lett. 37, 2337–2339 (2012).

    ADS  Google Scholar 

  109. Consani, C., Aubock, G., van Mourik, F. & Chergui, M. Ultrafast tryptophan-to-haem electron transfer in myoglobins revealed by UV 2D spectroscopy. Science 339, 1586–1589 (2013).

    ADS  Google Scholar 

  110. Borrego-Varillas, R. et al. Two-dimensional electronic spectroscopy in the ultraviolet by a birefringent delay line. Opt. Express 24, 28491–28499 (2016).

    ADS  Google Scholar 

  111. Freund, I. & Levine, B. F. Parametric conversion of X rays. Phys. Rev. Lett. 23, 854–857 (1969).

    ADS  Google Scholar 

  112. Eisenberger, P. & McCall, S. L. X-ray parametric conversion. Phys. Rev. Lett. 26, 684–688 (1971).

    ADS  Google Scholar 

  113. Eisenberger, P. M. & McCall, S. L. Mixing of X-ray and optical photons. Phys. Rev. A 3, 1145–1151 (1971).

    ADS  Google Scholar 

  114. Woo, J. W. F. & Jha, S. S. Inelastic scattering of X rays from optically induced charge-density oscillations. Phys. Rev. B 6, 4081–4082 (1972).

    ADS  Google Scholar 

  115. Freund, I. Nonlinear X-ray spectroscopy. Opt. Commun. 6, 421–423 (1972).

    ADS  Google Scholar 

  116. Freund, I. & Levine, B. F. Surface effects in the nonlinear interaction of X-ray and optical fields. Phys. Rev. B 8, 3059–3060 (1973).

    ADS  Google Scholar 

  117. Flytzanis, C. Determination of local field in dielectric. C. R. Acad. Sci. Serie B 278, 339–342 (1975).

    Google Scholar 

  118. Danino, H. & Freund, I. Parametric down conversion of X rays into the extreme ultraviolet. Phys. Rev. Lett. 46, 1127–1130 (1981).

    ADS  Google Scholar 

  119. Tanaka, S., Chernyak, V. & Mukamel, S. Time-resolved X-ray spectroscopies: nonlinear response functions and Liouville-space pathways. Phys. Rev. A 63, 063405 (2001).

    ADS  Google Scholar 

  120. Tanaka, S. & Mukamel, S. Coherent X-ray Raman spectroscopy: a nonlinear local probe for electronic excitations. Phys. Rev. Lett. 89, 043001 (2002).

    ADS  Google Scholar 

  121. Tanaka, S. & Mukamel, S. X-ray four-wave mixing in molecules. J. Chem. Phys. 116, 1877–1891 (2002).

    ADS  Google Scholar 

  122. Doumy, G. et al. Nonlinear atomic response to intense ultrashort X rays. Phys. Rev. Lett. 106, 083002 (2011).

    ADS  Google Scholar 

  123. Serrat, C. Localized core four-wave mixing buildup in the X-ray spectrum of chemical species. J. Phys. Chem. Lett. 12, 1093–1097 (2021).

    Google Scholar 

  124. Serrat, C. Resonantly enhanced difference-frequency generation in the core X-ray absorption of molecules. J. Phys. Chem. A 125, 10706–10710 (2021).

    Google Scholar 

  125. Haber, J. et al. Nonlinear resonant X-ray Raman scattering. Preprint at https://doi.org/10.48550/arXiv.2006.14724 (2020).

  126. Popova-Gorelova, D., Reis, D. A. & Santra, R. Theory of X-ray scattering from laser-driven electronic systems. Phys. Rev. B 98, 224302 (2018).

    ADS  Google Scholar 

  127. Misoguti, L., Christov, I. P., Backus, S., Murnane, M. M. & Kapteyn, H. C. Nonlinear wave-mixing processes in the extreme ultraviolet. Phys. Rev. A 72, 063803 (2005).

    ADS  Google Scholar 

  128. Maznev, A. A., Nelson, K. A. & Rogers, J. A. Optical heterodyne detection of laser-induced gratings. Opt. Lett. 23, 1319–1321 (1998).

    ADS  Google Scholar 

  129. Katayama, K., Yamaguchi, M. & Sawada, T. Lens-free heterodyne detection for transient grating experiments. Appl. Phys. Lett. 82, 2775–2777 (2003).

    ADS  Google Scholar 

  130. Sorokin, A. A. et al. Photoelectric effect at ultrahigh intensities. Phys. Rev. Lett. 99, 213002 (2007).

    ADS  Google Scholar 

  131. Kanter, E. P. et al. Unveiling and driving hidden resonances with high-fluence, high-intensity X-ray pulses. Phys. Rev. Lett. 107, 233001 (2011).

    ADS  Google Scholar 

  132. Lambropoulos, P. & Tang, X. Multiple excitation and ionization of atoms by strong lasers. J. Opt. Soc. Am. B 4, 821–832 (1987).

    ADS  Google Scholar 

  133. Novikov, S. A. & Hopersky, A. N. Two-photon excitation-ionization of the 1s shell of highly charged positive atomic ions. J. Phys. B Mol. Opt. Phys. 34, 4857–4863 (2001).

    ADS  Google Scholar 

  134. Sytcheva, A., Pabst, S., Son, S.-K. & Santra, R. Enhanced nonlinear response of Ne8+ to intense ultrafast X rays. Phys. Rev. A 85, 023414 (2012).

    ADS  Google Scholar 

  135. Tamasaku, K. et al. X-ray two-photon absorption competing against single and sequential multiphoton processes. Nat. Photon. 8, 313–316 (2014).

    ADS  Google Scholar 

  136. Ghimire, S. et al. Nonsequential two-photon absorption from the K shell in solid zirconium. Phys. Rev. A 94, 043418 (2016).

    ADS  Google Scholar 

  137. Tamasaku, K. et al. Nonlinear spectroscopy with X-ray two-photon absorption in metallic copper. Phys. Rev. Lett. 121, 083901 (2018).

    ADS  Google Scholar 

  138. Powers, P. E. & Haus, J. W. Fundamentals of Nonlinear Optics (CRC Press, 2017).

  139. Paschotta, R. & Keller, U. Passive mode locking with slow saturable absorbers. Appl. Phys. B 73, 653–662 (2001).

    ADS  Google Scholar 

  140. Wang, G. et al. Broadband saturable absorption and exciton–exciton annihilation in MoSe2 composite thin films. Opt. Mater. Express 9, 483–496 (2019).

    ADS  Google Scholar 

  141. Kumar, S. et al. Femtosecond carrier dynamics and saturable absorption in graphene suspensions. Appl. Phys. Lett. 95, 191911 (2009).

    ADS  Google Scholar 

  142. Bob, Nagler et al. Turning solid aluminium transparent by intense soft X-ray photoionization. Nat. Phys. 5, 693–696 (2009).

    Google Scholar 

  143. Yoneda, H. et al. Ultra-fast switching of light by absorption saturation in vacuum ultra-violet region. Opt. Express 17, 23443–23448 (2009).

    ADS  Google Scholar 

  144. Yoneda, H. et al. Saturable absorption of intense hard X-rays in iron. Nat. Commun. 5, 5080 (2014).

    ADS  Google Scholar 

  145. Hoffmann, L. et al. Saturable absorption of free-electron laser radiation by graphite near the carbon K-edge. J. Phys. Chem. Lett. 13, 8963–8970 (2022).

    Google Scholar 

  146. Rohringer, N. et al. Atomic inner-shell X-ray laser at 1.46 nanometres pumped by an X-ray free-electron laser. Nature 481, 488–491 (2012).

    ADS  Google Scholar 

  147. Beye, M. et al. Stimulated X-ray emission for materials science. Nature 501, 191–194 (2013).

    ADS  Google Scholar 

  148. Jonnard, P. et al. EUV stimulated emission from MgO pumped by FEL pulses. Struct. Dyn. 4, 054306 (2017).

    Google Scholar 

  149. Yoneda, H. et al. Atomic inner-shell laser at 1.5-ångström wavelength pumped by an X-ray free-electron laser. Nature 524, 446–449 (2015).

    ADS  Google Scholar 

  150. Glatzel, P. & Bergmann, U. High resolution 1s core hole X-ray spectroscopy in 3d transition metal complexes — electronic and structural information. Coord. Chem. Rev. 249, 65–95 (2005).

    Google Scholar 

  151. Zhang, W. et al. Tracking excited-state charge and spin dynamics in iron coordination complexes. Nature 509, 345–348 (2014).

    ADS  Google Scholar 

  152. Kinschel, D. et al. Femtosecond X-ray emission study of the spin cross-over dynamics in haem proteins. Nat. Commun. 11, 4145 (2020).

    ADS  Google Scholar 

  153. Bacellar, C. et al. Spin cascade and doming in ferric hemes: femtosecond X-ray absorption and X-ray emission studies. Proc. Natl Acad. Sci. USA 117, 21914–21920 (2020).

    ADS  Google Scholar 

  154. March, A. M. et al. Probing transient valence orbital changes with picosecond valence-to-core X-ray emission spectroscopy. J. Phys. Chem. C 121, 2620–2626 (2017).

    Google Scholar 

  155. Ledbetter, K. et al. Excited state charge distribution and bond expansion of ferrous complexes observed with femtosecond valence-to-core X-ray emission spectroscopy. J. Chem. Phys. 152, 074203 (2020).

    ADS  Google Scholar 

  156. Kroll, T. et al. Stimulated X-ray emission spectroscopy in transition metal complexes. Phys. Rev. Lett. 120, 133203 (2018).

    ADS  Google Scholar 

  157. Kroll, T. et al. Observation of seeded Mn Kβ stimulated X-ray emission using two-color X-ray free-electron laser pulses. Phys. Rev. Lett. 125, 037404 (2020).

    ADS  Google Scholar 

  158. Duris, J. et al. Tunable isolated attosecond X-ray pulses with gigawatt peak power from a free-electron laser. Nat. Photon. 14, 30–36 (2020).

    Google Scholar 

  159. McCamant, D. W., Kukura, P. & Mathies, R. A. Femtosecond stimulated Raman study of excited-state evolution in bacteriorhodopsin. J. Phys. Chem. B 109, 10449–10457 (2005).

    Google Scholar 

  160. Kukura, P., McCamant, D. W. & Mathies, R. A. Femtosecond stimulated Raman spectroscopy. Annu. Rev. Phys. Chem. 58, 461–488 (2007).

    ADS  Google Scholar 

  161. Hahn, A. W. et al. Probing the valence electronic structure of low-spin ferrous and ferric complexes using 2p3d resonant inelastic X-ray scattering (RIXS). Inorg. Chem. 57, 9515–9530 (2018).

    Google Scholar 

  162. Van Kuiken, B. E. et al. Electronic spectra of iron–sulfur complexes measured by 2p3d RIXS spectroscopy. Inorg. Chem. 57, 7355–7361 (2018).

    Google Scholar 

  163. Ågren, H., Luo, Y., Gelmukhanov, F. & Jensen, H. J. A. Screening in resonant X-ray emission of molecules. J. Electron Spectrosc. Relat. Phenom. 82, 125–134 (1996).

    Google Scholar 

  164. Cho, D., Rouxel, J. R., Mukamel, S., Kin-Lic Chan, G. & Li, Z. Stimulated X-ray Raman and absorption spectroscopy of iron–sulfur dimers. J. Phys. Chem. Lett. 10, 6664–6671 (2019).

    Google Scholar 

  165. Schweigert, I. V. & Mukamel, S. Probing valence electronic wave-packet dynamics by all X-ray stimulated Raman spectroscopy: a simulation study. Phys. Rev. A 76, 012504 (2007).

    ADS  Google Scholar 

  166. Harbola, U. & Mukamel, S. Coherent stimulated X-ray Raman spectroscopy: attosecond extension of resonant inelastic X-ray Raman scattering. Phys. Rev. B 79, 085108 (2009).

    ADS  Google Scholar 

  167. Hua, W. et al. Monitoring conical intersections in the ring opening of furan by attosecond stimulated X-ray Raman spectroscopy. Struct. Dyn. 3, 023601 (2016).

    Google Scholar 

  168. Weninger, C. Stimulated electronic X-ray Raman scattering. Phys. Rev. Lett. 111, 233902 (2013).

    ADS  Google Scholar 

  169. Weninger, C. & Rohringer, N. Stimulated resonant X-ray Raman scattering with incoherent radiation. Phys. Rev. A 88, 053421 (2013).

    ADS  Google Scholar 

  170. Kimberg, V. & Rohringer, N. Stochastic stimulated electronic X-ray Raman spectroscopy. Struct. Dyn. 3, 034101 (2016).

    Google Scholar 

  171. O’Neal, J. T. et al. Electronic population transfer via impulsive stimulated X-ray Raman scattering with attosecond soft-X-ray pulses. Phys. Rev. Lett. 125, 073203 (2020).

    ADS  Google Scholar 

  172. Higley, D. J. et al. Stimulated resonant inelastic X-ray scattering in a solid. Commun. Phys. 5, 1–12 (2022).

    ADS  Google Scholar 

  173. Arya, K. & Jha, S. S. Microscopic optical fields and mixing coefficients of X-ray and optical frequencies in solids. Pramana 2, 116–125 (1974).

    ADS  Google Scholar 

  174. Pine, A. S. Self-consistent-field theory of linear and nonlinear crystalline dielectrics including local-field effects. Phys. Rev. 139, A901–A911 (1965).

    ADS  Google Scholar 

  175. Schweigert, I. V. & Mukamel, S. Coherent ultrafast core-hole correlation spectroscopy: X-ray analogues of multidimensional NMR. Phys. Rev. Lett. 99, 163001 (2007).

    ADS  Google Scholar 

  176. Nazarkin, A., Podorov, S., Uschmann, I., Förster, E. & Sauerbrey, R. Nonlinear optics in the angstrom regime: hard-X-ray frequency doubling in perfect crystals. Phys. Rev. A 67, 041804 (2003).

    ADS  Google Scholar 

  177. Glover, T. E. et al. X-ray and optical wave mixing. Nature 488, 603–608 (2012).

    ADS  Google Scholar 

  178. Tamasaku, K. & Ishikawa, T. Interference between Compton scattering and X-ray parametric down-conversion. Phys. Rev. Lett. 98, 244801 (2007).

    ADS  Google Scholar 

  179. Tamasaku, K., Sawada, K. & Ishikawa, T. Determining X-ray nonlinear susceptibility of diamond by the optical Fano effect. Phys. Rev. Lett. 103, 254801 (2009).

    ADS  Google Scholar 

  180. Tamasaku, K., Sawada, K., Nishibori, E. & Ishikawa, T. Visualizing the local optical response to extreme-ultraviolet radiation with a resolution of λ/380. Nat. Phys. 7, 705–708 (2011).

    Google Scholar 

  181. Schori, A. et al. Parametric down-conversion of X rays into the optical regime. Phys. Rev. Lett. 119, 253902 (2017).

    ADS  Google Scholar 

  182. Shwartz, S. et al. X-ray second harmonic generation. Phys. Rev. Lett. 112, 163901 (2014).

    ADS  Google Scholar 

  183. Lam, R. K. et al. Soft X-ray second harmonic generation as an interfacial probe. Phys. Rev. Lett. 120, 023901 (2018).

    ADS  Google Scholar 

  184. Yamamoto, S. et al. Element selectivity in second-harmonic generation of GaFeO3 by a soft-X-ray free-electron laser. Phys. Rev. Lett. 120, 223902 (2018).

    ADS  Google Scholar 

  185. Berger, E. et al. Extreme ultraviolet second harmonic generation spectroscopy in a polar metal. Nano Lett. 21, 6095–6101 (2021).

    ADS  Google Scholar 

  186. Schwartz, C. P. et al. Angstrom-resolved interfacial structure in buried organic–inorganic junctions. Phys. Rev. Lett. 127, 096801 (2021).

    ADS  Google Scholar 

  187. Sistrunk, E. et al. Extreme ultraviolet transient grating measurement of insulator–metal transition dynamics of VO2. In 19th International Conference on Ultrafast Phenomena (OSA, 2014); https://doi.org/10.1364/UP.2014.09.Wed.P3.44.

  188. Gaynor, J. D. et al. Solid state core-exciton dynamics in NaCl observed by tabletop attosecond four-wave mixing spectroscopy. Phys. Rev. B 103, 245140 (2021).

    ADS  Google Scholar 

  189. Rottke, H. et al. Probing electron and hole colocalization by resonant four-wave mixing spectroscopy in the extreme ultraviolet. Sci. Adv. 8, eabn5127 (2022).

    Google Scholar 

  190. Sander, M. et al. Spatiotemporal coherent control of thermal excitations in solids. Phys. Rev. Lett. 119, 075901 (2017).

    ADS  Google Scholar 

  191. Pudell, J.-E. et al. Full spatiotemporal control of laser-excited periodic surface deformations. Phys. Rev. Appl. 12, 024036 (2019).

    ADS  Google Scholar 

  192. Frazer, T. D. et al. Optical transient grating pumped X-ray diffraction microscopy for studying mesoscale structural dynamics. Sci. Rep. 11, 19322 (2021).

    ADS  Google Scholar 

  193. Bencivenga, F. et al. Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature 520, 205–208 (2015).

    ADS  Google Scholar 

  194. Foglia, L. et al. First evidence of purely extreme-ultraviolet four-wave mixing. Phys. Rev. Lett. 120, 263901 (2018).

    ADS  Google Scholar 

  195. Bencivenga, F. et al. Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses. Sci. Adv. 5, eaaw5805 (2019).

    ADS  Google Scholar 

  196. Maznev, A. A. et al. Generation of coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment. Appl. Phys. Lett. 113, 221905 (2018).

    ADS  Google Scholar 

  197. Maznev, A. A. et al. Generation and detection of 50 GHz surface acoustic waves by extreme ultraviolet pulses. Appl. Phys. Lett. 119, 044102 (2021).

    ADS  Google Scholar 

  198. Bohinc, R. et al. Nonlinear XUV-optical transient grating spectroscopy at the Si L2,3-edge. Appl. Phys. Lett. 114, 181101 (2019).

    ADS  Google Scholar 

  199. Naumenko, D. et al. Thermoelasticity of nanoscale silicon carbide membranes excited by extreme ultraviolet transient gratings: implications for mechanical and thermal management. ACS Appl. Nano Mater. 2, 5132–5139 (2019).

    Google Scholar 

  200. Ksenzov, D. et al. Nanoscale transient magnetization gratings created and probed by femtosecond extreme ultraviolet pulses. Nano Lett. 21, 2905–2911 (2021).

    ADS  Google Scholar 

  201. Yao, K. et al. All-optical switching on the nanometer scale excited and probed with femtosecond extreme ultraviolet pulses. Nano Lett. 22, 4452–4458 (2022).

    ADS  Google Scholar 

  202. Rouxel, J. R. et al. Hard X-ray transient grating spectroscopy on bismuth germanate. Nat. Photon. 15, 499–503 (2021).

    ADS  Google Scholar 

  203. Svetina, C. et al. Towards X-ray transient grating spectroscopy. Opt. Lett. 44, 574–577 (2019).

    ADS  Google Scholar 

  204. Chen, Z., Gao, Y., Minch, B. C. & DeCamp, M. F. Coherent optical phonon generation in Bi 3 Ge 4 O 12. J. Phys. Condens. Matter 23, 385402 (2011).

    ADS  Google Scholar 

  205. Peters, W. et al. Hard X-ray–optical transient grating. In 2021 Conference on Lasers and Electro-Optics (CLEO) 1–2 (CLEO, 2021).

  206. Shi, H. & Zhu, D. Multi-axis nanopositioning system for the hard X-ray split-delay system at the LCLS. Synchrotron Radiat. News 31, 15–20 (2018).

    ADS  Google Scholar 

  207. Ukleev, V. et al. Effect of intense X-ray free-electron laser transient gratings on the magnetic domain structure of Tm:YIG. J. Appl. Phys. 133, 123902 (2023).

    ADS  Google Scholar 

  208. Keefer, D. et al. Ultrafast X-ray probes of elementary molecular events. Ann. Rev. Phys. Chem. 74, 73–97 (2023).

    ADS  Google Scholar 

  209. Roemelt, M., Maganas, D., DeBeer, S. & Neese, F. A combined DFT and restricted open-shell configuration interaction method including spin–orbit coupling: application to transition metal L-edge X-ray absorption spectroscopy. J. Chem. Phys. 138, 204101 (2013).

    ADS  Google Scholar 

  210. Norman, P. & Dreuw, A. Simulating X-ray spectroscopies and calculating core-excited states of molecules. Chem. Rev. 118, 7208–7248 (2018).

    Google Scholar 

  211. Vidal, M. L., Feng, X., Epifanovsky, E., Krylov, A. I. & Coriani, S. New and efficient equation-of-motion coupled-cluster framework for core-excited and core-ionized states. J. Chem. Theory Comput. 15, 3117–3133 (2019).

    Google Scholar 

  212. Montorsi, F., Segatta, F., Nenov, A., Mukamel, S. & Garavelli, M. Soft X-ray spectroscopy simulations with multiconfigurational wave function theory: spectrum completeness, sub-eV accuracy, and quantitative reproduction of line shapes. J. Chem. Theory Comput. 18, 1003–1016 (2022).

    Google Scholar 

  213. Tully, J. C. Molecular dynamics with electronic transitions. J. Chem. Phys. 93, 1061–1071 (1990).

    ADS  Google Scholar 

  214. Meyer, H.-D., Manthe, U. & Cederbaum, L. S. The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165, 73–78 (1990).

    ADS  Google Scholar 

  215. Ben-Nun, M. & Martinez, T. J. Ab initio quantum molecular dynamics. Adv. Chem. Phys. 121, 439–512 (2002).

    Google Scholar 

  216. Makhov, D. V., Glover, W. J., Martinez, T. J. & Shalashilin, D. V. Ab initio multiple cloning algorithm for quantum nonadiabatic molecular dynamics. J. Chem. Phys. 141, 054110 (2014).

    ADS  Google Scholar 

  217. Richter, M., Marquetand, P., González-Vázquez, J., Sola, I. & González, L. SHARC: ab initio molecular dynamics with surface hopping in the adiabatic representation including arbitrary couplings. J. Chem. Theory Comput. 7, 1253–1258 (2011).

    Google Scholar 

  218. Reiter, S., Keefer, D. & de Vivie-Riedle, R. Exact quantum dynamics (wave packets) in reduced dimensionality. in Quantum Chemistry and Dynamics of Excited States 355–381 (John Wiley & Sons, Ltd, 2020).

  219. Helmich-Paris, B. Simulating X-ray absorption spectra with complete active space self-consistent field linear response methods. Int. J. Quantum Chem. 121, e26559 (2021).

    Google Scholar 

  220. Casanova, D. Restricted active space configuration interaction methods for strong correlation: Recent developments. WIREs Comput Mol Sci. 12, e1561 (2022).

    Google Scholar 

  221. Autschbach, J., Ziegler, T., van Gisbergen, S. J. A. & Baerends, E. J. Chiroptical properties from time-dependent density functional theory. I. Circular dichroism spectra of organic molecules. J. Chem. Phys. 116, 6930–6940 (2002).

    ADS  Google Scholar 

  222. Lopata, K., Van Kuiken, B. E., Khalil, M. & Govind, N. Linear-response and real-time time-dependent density functional theory studies of core-level near-edge X-ray absorption. J. Chem. Theory Comput. 8, 3284–3292 (2012).

    Google Scholar 

  223. Andersen, J. H., Nanda, K. D., Krylov, A. I. & Coriani, S. Probing molecular chirality of ground and electronically excited states in the UV–vis and X-ray regimes: an EOM-CCSD study. J. Chem. Theory Comput. 18, 1748–1764 (2022).

    Google Scholar 

  224. Nanda, K. D. & Krylov, A. I. Cherry-picking resolvents: a general strategy for convergent coupled-cluster damped response calculations of core-level spectra. J. Chem. Phys. 153, 141104 (2020).

    ADS  Google Scholar 

  225. Freund, I. Nonlinear X-ray diffraction. Determination of valence electron charge distributions. Chem. Phys. Lett. 12, 583–588 (1972).

    ADS  Google Scholar 

  226. Shwartz, E. & Shwartz, S. Difference-frequency generation of optical radiation from two-color X-ray pulses. Opt. Express 23, 7471–7480 (2015).

    ADS  Google Scholar 

  227. Rohringer, N. & Santra, R. X-ray nonlinear optical processes using a self-amplified spontaneous emission free-electron laser. Phys. Rev. A 76, 033416 (2007).

    ADS  Google Scholar 

  228. Santra, R., Kryzhevoi, N. V. & Cederbaum, L. S. X-ray two-photon photoelectron spectroscopy: a theoretical study of inner-shell spectra of the organic para-aminophenol molecule. Phys. Rev. Lett. 103, 013002 (2009).

    ADS  Google Scholar 

  229. Motomura, K. et al. Sequential multiphoton multiple ionization of atomic argon and xenon irradiated by X-ray free-electron laser pulses from SACLA. J. Phys. B Mol. Opt. Phys. 46, 164024 (2013).

    ADS  Google Scholar 

  230. Mazza, T. et al. Sensitivity of nonlinear photoionization to resonance substructure in collective excitation. Nat. Commun. 6, 6799 (2015).

    ADS  Google Scholar 

  231. Popova-Gorelova, D., Guskov, V. & Santra, R. Microscopic electron dynamics in nonlinear optical response of solids. Preprint at https://doi.org/10.48550/arXiv.2009.07527 (2020).

  232. Popova-Gorelova, D. & Santra, R. Atomic-scale imaging of laser-driven electron dynamics in solids using subcycle-resolved X-ray-optical wave mixing. Preprint at https://doi.org/10.48550/arXiv.2012.10334 (2020).

  233. Keefer, D. et al. Monitoring molecular vibronic coherences in a bichromophoric molecule by ultrafast X-ray spectroscopy. Chem. Sci. 12, 5286–5294 (2021).

    ADS  Google Scholar 

  234. Bennett, K., Kowalewski, M., Rouxel, J. R. & Mukamel, S. Monitoring molecular nonadiabatic dynamics with femtosecond X-ray diffraction. Proc. Natl Acad. Sci. USA 115, 6538–6547 (2018).

    ADS  Google Scholar 

  235. Shakya, Y., Inhester, L., Arnold, C., Welsch, R. & Santra, R. Ultrafast time-resolved X-ray absorption spectroscopy of ionized urea and its dimer through ab initio nonadiabatic dynamics. Struct. Dyn. 8, 034102 (2021).

    Google Scholar 

  236. List, N. H., Dempwolff, A. L., Dreuw, A., Norman, P. & Martínez, T. J. Probing competing relaxation pathways in malonaldehyde with transient X-ray absorption spectroscopy. Chem. Sci. 11, 4180–4193 (2020).

    Google Scholar 

  237. Keefer, D., Schnappinger, T., de Vivie-Riedle, R. & Mukamel, S. Visualizing conical intersection passages via vibronic coherence maps generated by stimulated ultrafast X-ray Raman signals. Proc. Natl Acad. Sci. USA 117, 24069–24075 (2020).

    ADS  Google Scholar 

  238. Nenov, A., Segatta, F., Bruner, A., Mukamel, S. & Garavelli, M. X-ray linear and non-linear spectroscopy of the ESCA molecule. J. Chem. Phys. 151, 114110 (2019).

    ADS  Google Scholar 

  239. Kowalewski, M. Catching conical intersections in the act: monitoring transient electronic coherences by attosecond stimulated X-ray Raman signals. Phys. Rev. Lett. 115, 193003 (2015).

    ADS  Google Scholar 

  240. Rouxel, J. R., Kowalewski, M., Bennett, K. & Mukamel, S. X-ray sum frequency diffraction for direct imaging of ultrafast electron dynamics. Phys. Rev. Lett. 120, 243902 (2018).

    ADS  Google Scholar 

  241. Neville, S. P., Chergui, M., Stolow, A. & Schuurman, M. S. Ultrafast X-ray spectroscopy of conical intersections. Phys. Rev. Lett. 120, 243001 (2018).

    ADS  Google Scholar 

  242. Keefer, D. et al. Imaging conical intersection dynamics during azobenzene photoisomerization by ultrafast X-ray diffraction. Proc. Natl Acad. Sci. USA 118, e2022037118 (2021).

    Google Scholar 

  243. Amann, J. et al. Demonstration of self-seeding in a hard-X-ray free-electron laser. Nat. Photon. 6, 693–698 (2012).

    ADS  Google Scholar 

  244. Cavaletto, S. M., Keefer, D. & Mukamel, S. High temporal and spectral resolution of stimulated X-ray Raman signals with stochastic free-electron-laser pulses. Phys. Rev. X 11, 011029 (2021).

    Google Scholar 

  245. Biggs, J. D., Zhang, Y., Healion, D. & Mukamel, S. Watching energy transfer in metalloporphyrin heterodimers using stimulated X-ray Raman spectroscopy. Proc. Natl Acad. Sci. USA 110, 15597–15601 (2013).

    ADS  Google Scholar 

  246. Kimberg, V. et al. Stimulated X-ray Raman scattering — a critical assessment of the building block of nonlinear X-ray spectroscopy. Faraday Discuss. 194, 305–324 (2016).

    ADS  Google Scholar 

  247. Gschwendtner, E. & Muggli, P. Plasma wakefield accelerators. Nat. Rev. Phys. 1, 246–248 (2019).

    Google Scholar 

  248. Wang, W. et al. Free-electron lasing at 27 nanometres based on a laser wakefield accelerator. Nature 595, 516–520 (2021).

    ADS  Google Scholar 

  249. Galletti, M. et al. Stable operation of a free-electron laser driven by a plasma accelerator. Phys. Rev. Lett. 129, 234801 (2022).

    ADS  Google Scholar 

  250. Labat, M. et al. Seeded free-electron laser driven by a compact laser plasma accelerator. Nat. Photon. 17, 150–156 (2023).

    ADS  Google Scholar 

  251. Trebino, R. Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses: The Measurement of Ultrashort Laser Pulses (Springer Science & Business Media, 2000).

  252. Peters, W. K. et al. All-optical single-shot complete electric field measurement of extreme ultraviolet free electron laser pulses. Optica 8, 545–550 (2021).

    ADS  Google Scholar 

  253. Gauthier, D. et al. Spectrotemporal shaping of seeded free-electron laser pulses. Phys. Rev. Lett. 115, 114801 (2015).

    ADS  Google Scholar 

  254. Prince, K. C. et al. Coherent control with a short-wavelength free-electron laser. Nat. Photon. 10, 176–179 (2016).

    ADS  Google Scholar 

  255. Cavaletto, S. M. et al. Unveiling the spatial distribution of molecular coherences at conical intersections by covariance X-ray diffraction signals. Proc. Natl Acad. Sci. USA 118, e2105046118 (2021).

    Google Scholar 

  256. Ratner, D., Cryan, J. P., Lane, T. J., Li, S. & Stupakov, G. Pump–probe ghost imaging with SASE FELs. Phys. Rev. X 9, 011045 (2019).

    Google Scholar 

  257. Kayser, Y. et al. Core-level nonlinear spectroscopy triggered by stochastic X-ray pulses. Nat. Commun. 10, 4761 (2019).

    ADS  Google Scholar 

  258. Driver, T. et al. Attosecond transient absorption spectroscopy: a ghost imaging approach to ultrafast absorption spectroscopy. Phys. Chem. Chem. Phys. 22, 2704–2712 (2020).

    Google Scholar 

  259. Gorobtsov, O. Y. et al. Seeded X-ray free-electron laser generating radiation with laser statistical properties. Nat. Commun. 9, 4498 (2018).

    ADS  Google Scholar 

  260. Li, K. et al. Ghost-imaging-enhanced noninvasive spectral characterization of stochastic X-ray free-electron-laser pulses. Commun. Phys. 5, 1–8 (2022).

    ADS  Google Scholar 

  261. Kalz, K. F. et al. Future challenges in heterogeneous catalysis: understanding catalysts under dynamic reaction conditions. ChemCatChem 9, 17–29 (2017).

    Google Scholar 

  262. Yang, Y. et al. Operando methods in electrocatalysis. ACS Catal. 11, 1136–1178 (2021).

    Google Scholar 

  263. Singh, J., Lamberti, C. & van Bokhoven, J. A. Advanced X-ray absorption and emission spectroscopy: in situ catalytic studies. Chem. Soc. Rev. 39, 4754–4766 (2010).

    Google Scholar 

  264. Beye, M. et al. Non-linear soft X-ray methods on solids with MUSIX — the multi-dimensional spectroscopy and inelastic X-ray scattering endstation. J. Phys. Condens. Matter 31, 014003 (2019).

    ADS  Google Scholar 

  265. Chakraborti, P. et al. Higher-order X-ray — optical wave mixing. SPring-8/SACLA利用研究成果集 https://doi.org/10.18957/rr.10.4.409 (2022).

  266. Alagna, L. et al. X-ray natural circular dichroism. Phys. Rev. Lett. 80, 4799–4802 (1998).

    ADS  Google Scholar 

  267. Peacock, R. D. & Stewart, B. Natural circular dichroism in X-ray spectroscopy. J. Phys. Chem. B 105, 351–360 (2001).

    Google Scholar 

  268. Turchini, S. et al. Core electron transitions as a probe for molecular chirality: natural circular dichroism at the carbon K-edge of methyloxirane. J. Am. Chem. Soc. 126, 4532–4533 (2004).

    Google Scholar 

  269. Tanaka, M., Nakagawa, K., Agui, A., Fujii, K. & Yokoya, A. First observation of natural circular dichroism for biomolecules in soft X-ray region studied with a polarizing undulator. Phys. Scr. 2005, 873 (2005).

    Google Scholar 

  270. Rouxel, J. R. et al. Hard X-ray helical dichroism of disordered molecular media. Nat. Photon. 16, 570–574 (2022).

    ADS  Google Scholar 

  271. Mincigrucci, R. et al. Element- and enantiomer-selective visualization of molecular motion in real-time. Nat. Commun. 14, 386 (2023).

    ADS  Google Scholar 

  272. Byers, J. D., Yee, H. I., Petralli-Mallow, T. & Hicks, J. M. Second-harmonic generation circular-dichroism spectroscopy from chiral monolayers. Phys. Rev. B 49, 14643–14647 (1994).

    ADS  Google Scholar 

  273. Petralli-Mallow, T., Wong, T. M., Byers, J. D., Yee, H. I. & Hicks, J. M. Circular dichroism spectroscopy at interfaces: a surface second harmonic generation study. J. Phys. Chem. 97, 1383–1388 (1993).

    Google Scholar 

  274. Kauranen, M., Verbiest, T., Maki, J. J. & Persoons, A. Second‐harmonic generation from chiral surfaces. J. Chem. Phys. 101, 8193–8199 (1994).

    ADS  Google Scholar 

  275. Verbiest, T., Kauranen, M., Van Rompaey, Y. & Persoons, A. Optical activity of anisotropic achiral surfaces. Phys. Rev. Lett. 77, 1456–1459 (1996).

    ADS  Google Scholar 

  276. Verbiest, T., Kauranen, M. & Persoons, A. Second-order nonlinear optical properties of chiral thin films. J. Mater. Chem. 9, 2005–2012 (1999).

    Google Scholar 

  277. Belkin, M. A., Han, S. H., Wei, X. & Shen, Y. R. Sum-frequency generation in chiral liquids near electronic resonance. Phys. Rev. Lett. 87, 113001 (2001).

    ADS  Google Scholar 

  278. Belkin, M. A., Shen, Y. R. & Flytzanis, C. Coupled-oscillator model for nonlinear optical activity. Chem. Phys. Lett. 363, 479–485 (2002).

    ADS  Google Scholar 

  279. Belkin, M. A. & Shen, Y. R. Non-linear optical spectroscopy as a novel probe for molecular chirality. Int. Rev. Phys. Chem. 24, 257–299 (2005).

    Google Scholar 

  280. Fischer, P., Beckwitt, K., Wise, F. W. & Albrecht, A. C. The chiral specificity of sum-frequency generation in solutions. Chem. Phys. Lett. 352, 463–468 (2002).

    ADS  Google Scholar 

  281. Lee, T., Rhee, H. & Cho, M. Femtosecond vibrational sum-frequency generation spectroscopy of chiral molecules in isotropic liquid. J. Phys. Chem. Lett. 9, 6723–6730 (2018).

    Google Scholar 

  282. Song, Z., Zhang, T., Fang, Z. & Fang, C. Quantitative mappings between symmetry and topology in solids. Nat. Commun. 9, 3530 (2018).

    ADS  Google Scholar 

  283. Gatti, G. et al. Radial spin texture of the Weyl fermions in chiral tellurium. Phys. Rev. Lett. 125, 216402 (2020).

    ADS  Google Scholar 

  284. Cochran, T. A. et al. Visualizing higher-fold topology in chiral crystals. Phys. Rev. Lett. 130, 066402 (2023).

    ADS  Google Scholar 

  285. Rouxel, J. R., Kowalewski, M. & Mukamel, S. Photoinduced molecular chirality probed by ultrafast resonant X-ray spectroscopy. Struct. Dyn. 4, 044006 (2017).

    Google Scholar 

  286. Rouxel, J. R., Zhang, Y. & Mukamel, S. X-ray Raman optical activity of chiral molecules. Chem. Sci. 10, 898–908 (2019).

    Google Scholar 

  287. Bencivenga, F. et al. Four-wave-mixing experiments with seeded free electron lasers. Faraday Discuss. 194, 283–303 (2016).

    ADS  Google Scholar 

  288. Rouxel, J. R. & Mukamel, S. Molecular chirality and its monitoring by ultrafast X-ray pulses. Chem. Rev. 122, 16802–16838 (2022).

    Google Scholar 

  289. Rouxel, J. R., Rajabi, A. & Mukamel, S. Chiral four-wave mixing signals with circularly polarized X-ray pulses. J. Chem. Theory Comput. 16, 5784–5791 (2020).

    Google Scholar 

  290. Schenkl, S. et al. Insights into excited-state and isomerization dynamics of bacteriorhodopsin from ultrafast transient UV absorption. Proc. Natl Acad. Sci. USA 103, 4101–4106 (2006).

    ADS  Google Scholar 

  291. Polli, D. et al. Conical intersection dynamics of the primary photoisomerization event in vision. Nature 467, 440–443 (2010).

    ADS  Google Scholar 

  292. Schreier, W. J., Gilch, P. & Zinth, W. Early events of DNA photodamage. Annu. Rev. Phys. Chem. 66, 497–519 (2015).

    ADS  Google Scholar 

  293. Ebadi, H. Tracking of azobenzene isomerization by X-ray emission spectroscopy. J. Phys. Chem. A 118, 7832–7837 (2014).

    Google Scholar 

  294. Hua, W., Mukamel, S. & Luo, Y. Transient X-ray absorption spectral fingerprints of the S1 dark state in uracil. J. Phys. Chem. Lett. 10, 7172–7178 (2019).

    Google Scholar 

  295. Nam, Y. et al. Time-resolved optical pump-resonant X-ray probe spectroscopy of 4-thiouracil: a simulation study. J. Chem. Theory Comput. 18, 3075–3088 (2022).

    Google Scholar 

  296. Segatta, F. et al. Exploring the capabilities of optical pump X-ray probe NEXAFS spectroscopy to track photo-induced dynamics mediated by conical intersections. Faraday Discuss. 221, 245–264 (2020).

    ADS  Google Scholar 

  297. Chang, K. F. et al. Revealing electronic state-switching at conical intersections in alkyl iodides by ultrafast XUV transient absorption spectroscopy. Nat. Commun. 11, 4042 (2020).

    ADS  Google Scholar 

  298. Timmers, H. et al. Disentangling conical intersection and coherent molecular dynamics in methyl bromide with attosecond transient absorption spectroscopy. Nat. Commun. 10, 3133 (2019).

    ADS  Google Scholar 

  299. Nam, Y. et al. Conical intersection passages of molecules probed by X-ray diffraction and stimulated Raman spectroscopy. J. Phys. Chem. Lett. 12, 12300–12309 (2021).

    Google Scholar 

  300. Freixas, V. M., Keefer, D., Tretiak, S., Fernandez-Alberti, S. & Mukamel, S. Ultrafast coherent photoexcited dynamics in a trimeric dendrimer probed by X-ray stimulated-Raman signals. Chem. Sci. 13, 6373–6384 (2022).

    Google Scholar 

  301. Polishchuk, S. et al. Nanoscale-resolved surface-to-bulk electron transport in CsPbBr3 perovskite. Nano Lett. 22, 1067–1074 (2022).

    ADS  Google Scholar 

  302. Consani, C., Bram, O., van Mourik, F., Cannizzo, A. & Chergui, M. Energy transfer and relaxation mechanisms in cytochrome c. Chem. Phys. 396, 108–115 (2012).

    Google Scholar 

  303. Chenu, A. & Scholes, G. D. Coherence in energy transfer and photosynthesis. Annu. Rev. Phys. Chem. 66, 69–96 (2015).

    ADS  Google Scholar 

  304. Bacellar, C. et al. Ultrafast energy transfer from photoexcited tryptophan to the haem in cytochrome c. J. Phys. Chem. Lett. 14, 2425–2432 (2023).

    Google Scholar 

  305. Zeiger, H. J. et al. Theory for displacive excitation of coherent phonons. Phys. Rev. B 45, 768–778 (1992).

    ADS  Google Scholar 

  306. Mukamel, S., Healion, D., Zhang, Y. & Biggs, J. D. Multidimensional attosecond resonant X-ray spectroscopy of molecules: lessons from the optical regime. Annu. Rev. Phys. Chem. 64, 101–127 (2013).

    ADS  Google Scholar 

  307. Al-Haddad, A. et al. Observation of site-selective chemical bond changes via ultrafast chemical shifts. Nat. Commun. 13, 7170 (2022).

    ADS  Google Scholar 

  308. El Nahhas, A. et al. X-ray absorption spectroscopy of ground and excited rhenium-carbonyl diimine-complexes: evidence for a two-center electron transfer. J. Phys. Chem. A 117, 361–369 (2013).

    Google Scholar 

  309. Bencivenga, F. & Masciovecchio, C. FEL-based transient grating spectroscopy to investigate nanoscale dynamics. Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip. 606, 785–789 (2009).

    ADS  Google Scholar 

  310. Johnson, J. A. et al. Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane. Phys. Rev. Lett. 110, 025901 (2013).

    ADS  Google Scholar 

Download references

Acknowledgements

S.M. gratefully acknowledges the support of the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy through Award No. DE-FG02-04ER15571 and of the National Science Foundation (Grant No. CHE-2246379). M.C. thanks the Swiss NSF NCCR:MUST and the ERC Advanced Grant DYNAMOX for support. All authors are deeply grateful to their former students and postdocs and to their collaborators for making this work possible.

Author information

Authors and Affiliations

  1. Lausanne Centre for Ultrafast Science, Institute of Chemical Sciences and Engineering, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Majed Chergui

  2. Elettra Sincrotrone Trieste S.C.p.A., Trieste, Italy

    Majed Chergui & Claudio Masciovecchio

  3. Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

    Martin Beye

  4. Department of Physics, Stockholm University, Stockholm, Sweden

    Martin Beye

  5. Department of Chemistry and Department of Physics and Astronomy, University of California, Irvine, CA, USA

    Shaul Mukamel

  6. Madrid Institute for Advanced Studies, IMDEA Nanociencia, Ciudad Universitaria de Cantoblanco, Madrid, Spain

    Cristian Svetina

  7. European XFEL, Holzkoppel 4, Schenefeld, Germany

    Cristian Svetina

Authors
  1. Majed Chergui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Beye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shaul Mukamel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cristian Svetina
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Claudio Masciovecchio
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

M.C. researched data for the article. All authors contributed substantially to discussion of the content. M.C. and S.M. wrote the article. All authors reviewed and/or edited the manuscript before submission.

Corresponding author

Correspondence to Majed Chergui.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Physics thanks Nora Berrah and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chergui, M., Beye, M., Mukamel, S. et al. Progress and prospects in nonlinear extreme-ultraviolet and X-ray optics and spectroscopy. Nat Rev Phys 5, 578–596 (2023). https://doi.org/10.1038/s42254-023-00643-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-023-00643-7

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing