- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

Imaging surface topography with coherent x-ray reflectivity: Theory, kinematics, and simulations

Irene Calvo-Almazán1,2, Anusheela Das3, Ana F. Suzana3, Fernando Bartolomé1,2, and Paul Fenter3,*

  • *Contact author: fenter@anl.gov
PDF

Phys. Rev. B 113, 075410 – Published 6 February, 2026

DOI: https://doi.org/10.1103/q75z-gk3l

Abstract

A theoretical formalism is described for understanding coherent x-ray reflectivity (CXR) from the surface of a semi-infinite crystal having a variable surface topography, described by the height profile h(x,y). The surface topography is imaged as a complex “effective density,” obtained from the phasing and inversion of the coherent x-ray reflectivity data, measured through a rocking scan centered at a vertical momentum transfer Qz0 and a vertical range ΔQz. The formalism predicts that the effective density has an amplitude with a maximum located at the surface height for each position within the surface plane. The phase of the effective density has a lateral variation that is controlled by the surface height and a vertical variation that reflects a combination of the interfacial structure and specific choice of measurement conditions. This understanding enables direct observation of nanometer-scale interfacial topography, i.e., h(x,y)cs (where cs is the vertical substrate lattice parameter) with Å-scale sensitivity to surface height. Numerical simulations illustrate and confirm the theoretical results. These results show how the interpretation of the interfacial density phase obtained by CXR data inversion (i.e., surface topography with respect to a flat surface) is conceptually similar to that previously known for Bragg coherent diffraction imaging (BCDI) measurements of isolated nanoparticles (i.e., lattice displacements with respect to an ideal crystal lattice). This suggests that CXR can be thought of as a form of dark field imaging with respect to the bright field BCDI approach. An implication of these results is that interfacial imaging may bypass some of the significant challenges associated with BCDI imaging of multiple particles having different orientations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (52)

  1. R. Hettel, DLSR design and plans: An international overview, J. Synchrotron Radiat. 21, 843 (2014).
  2. M. Borland, G. Decker, L. Emery, V. Sajaev, Y. P. Sun, and A. M. Xiao, Lattice design challenges for fourth-generation storage-ring light sources, J. Synchrotron Radiat. 21, 912 (2014).
  3. P. Raimondi, C. Benabderrahmane, P. Berkvens, J. C. Biasci, P. Borowiec, J. F. Bouteille, T. Brochard, N. B. Brookes, N. Carmignani, L. R. Carver, J. M. Chaize, J. Chavanne, S. Checchia, Y. Chushkin, F. Cianciosi, M. Di Michiel, R. Dimper, A. D'Elia, D. Einfeld, F. Ewald, et al., The extremely brilliant source storage ring of the european synchrotron radiation facility, Commun. Phys. 6, 82 (2023).
  4. J. L. Bañuelos, E. Borguet, G. E. Brown, R. T. Cygan, J. J. DeYoreo, P. M. Dove, M. P. Gaigeot, F. M. Geiger, J. M. Gibbs, V. H. Grassian, A. G. Ilgen, Y. S. Jun, N. Kabengi, L. Katz, J. D. Kubicki, J. Lutzenkirchen, C. V. Putnis, R. C. Remsing, K. M. Rosso, G. Rother, et al., Oxide- and silicate-water interfaces and their roles in technology and the environment, Chem. Rev. 123, 6413 (2023).
  5. B. E. Warren, X-Ray Diffraction (Dover, New York, 1990).
  6. A. Guinier, X-Ray Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies (Dover, New York, 1994).
  7. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, X-ray and neutron-scattering from rough surfaces, Phys. Rev. B 38, 2297 (1988).
  8. J. W. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, Beyond crystallography: Diffractive imaging using coherent x-ray light sources, Science 348, 530 (2015).
  9. J. R. Fienup, Reconstruction of an object from the modulus of its Fourier transform, Opt. Lett. 3, 27 (1978).
  10. J. R. Fienup, Phase retrieval algorithms - A comparison, Appl. Opt. 21, 2758 (1982).
  11. S. Marchesini, A unified evaluation of iterative projection algorithms for phase retrieval, Rev. Sci. Instrum. 78, 011301 (2007).
  12. M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, Three-dimensional mapping of a deformation field inside a nanocrystal, Nature (London) 442, 63 (2006).
  13. J. Miao, J. Kirz, and D. Sayre, The oversampling phasing method, Acta Crystallogr. D 56, 1312 (2000).
  14. G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Internal structure in small Au crystals resolved by three-dimensional inversion of coherent x-ray diffraction, Phys. Rev. B 73, 094112 (2006).
  15. M. O. Hill, I. Calvo-Almazan, M. Allain, M. V. Holt, A. Ulvestad, J. Treu, G. Koblmuller, C. Huang, X. Huang, H. Yan, E. Nazaretski, Y. S. Chu, G. B. Stephenson, V. Chamard, L. J. Lauhon, and S. O. Hruszkewycz, Measuring three-dimensional strain and structural defects in a single InGaAs nanowire using coherent x-ray multiangle bragg projection ptychography, Nano Lett. 18, 811 (2018).
  16. I. K. Robinson and R. Harder, Coherent x-ray diffraction imaging of strain at the nanoscale, Nat. Mater. 8, 291 (2009).
  17. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, Probe retrieval in ptychographic coherent diffractive imaging, Ultramicroscopy 109, 338 (2009).
  18. P. Thibault and V. Elser, X-ray diffraction microscopy, Annu. Rev. Condens. Matter Phys. 1, 237 (2010).
  19. P. Li, M. Allain, T. A. Grünewald, M. Rommel, A. Campos, D. Carbone, and V. Chamard, 4th generation synchrotron source boosts crystalline imaging at the nanoscale, Light.-Sci. Appl. 11, 73 (2022).
  20. Y. F. Sun and A. Singer, Bragg coherent diffractive imaging for defects analysis: Principles, applications, and challenges, Chem Phys Rev 5, 031310 (2024).
  21. I. Calvo-Almazán, V. Chamard, T. A. Grünewald, and M. Allain, Inhomogeneous probes for Bragg coherent diffraction imaging: Toward the imaging of dynamic and distorted crystals, Phys. Rev. B 110, 134117 (2024).
  22. A. F. Suzana, S. S. Lee, I. Calvo-Almazan, W. Cha, R. Harder, and P. Fenter, Visualizing the internal nanocrystallinity of calcite due to nonclassical crystallization by 3D coherent x-ray diffraction imaging, Adv. Mater. 36, e2310672 (2024).
  23. I. A. Vartanyants, J. A. Pitney, J. L. Libbert, and I. K. Robinson, Reconstruction of surface morphology from coherent x-ray reflectivity, Phys. Rev. B 55, 13193 (1997).
  24. I. A. Vartanyants, I. K. Robinson, I. McNulty, C. David, P. Wochner, and T. Tschentscher, Coherent x-ray scattering and lensless imaging at the european XFEL facility, J. Synchrotron Radiat. 14, 453 (2007).
  25. M. Chu, J. Zhang, M. Wojcik, T. Sun, M. Sprung, and J. Wang, Probing three-dimensional mesosocopic interfacial structures in a single view using multibeam x-ray coherent surface scattering and holography imaging, Nat. Commun. 14, 5795 (2023).
  26. G. X. Ju, D. W. Xu, M. Highland, C. Thompson, H. Zhou, J. A. Eastman, P. H. Fuoss, P. Zapol, H. Kim, and G. B. Stephenson, Coherent x-ray spectroscopy reveals the persistence of island arrangements during layer-by-layer growth, Nat. Phys. 15, 589 (2019).
  27. R. L. Headrick, J. G. Ulbrandt, P. Myint, J. Wan, Y. Li, A. Fluerasu, Y. Zhang, L. Wiegart, and K. F. Ludwig Jr., Coherent X-ray measurement of step-flow propagation during growth on polycrystalline thin film surfaces, Nat. Commun. 10, 2638 (2019).
  28. C. H. Zhu, R. Harder, A. Diaz, V. Komanicky, A. Barbour, R. Q. Xu, X. J. Huang, Y. H. Liu, M. S. Pierce, A. Menzel, and H. You, Ptychographic x-ray imaging of surfaces on crystal truncation rod, Appl. Phys. Lett. 106, 101604 (2015).
  29. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, Chichester, 2001).
  30. I. K. Robinson, Crystal truncation rods and surface-roughness, Phys. Rev. B 33, 3830 (1986).
  31. M. S. Pierce, V. Komanicky, A. Barbour, D. C. Hennessy, J. D. Suc, A. Sandy, C. Zhu, and H. You, In situ coherent x-ray scattering and scanning tunneling microscopy studies of hexagonally reconstructed Au(001) in electrolytes, ECS Trans. 35, 71 (2011).
  32. P. Fenter, C. Park, Z. Zhang, and S. Wang, Observation of subnanometre-high surface topography with x-ray reflection phase-contrast microscopy, Nat. Phys. 2, 700 (2006).
  33. P. Fenter, C. Park, V. Kohli, and Z. Zhang, Image contrast in x-ray reflection interface microscopy: Comparison of data with model calculations and simulations, J. Synchrotron Radiat. 15, 558 (2008).
  34. P. Fenter and I. Calvo-Almazán, Imaging interfacial topography with coherent x-ray reflectivity, Phys. Rev. B 109, 094112 (2024).
  35. I. A. Vartanyants and I. K. Robinson, Partial coherence effects on the imaging of small crystals using coherent x-ray diffraction, J. Phys.: Condens. Matter 13, 10593 (2001).
  36. K. A. Nugent, A. G. Peele, H. M. Quiney, and H. N. Chapman, Diffraction with wavefront curvature: A path to unique phase recovery, Acta Crystallogr. A 61, 373 (2005).
  37. W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987).
  38. P. Fenter, X-ray reflectivity as a probe of mineral-fluid interfaces: A user guide, Rev. Mineral. Geochem. 49, 149 (2002).
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/q75z-gk3l for an illustration showing that the intrinsic effective density ρeff−0 (shown in Fig. 2) is independent of the beam attenuation β. It also shows a detailed derivation of Eq. (11), which also includes Refs. [37, 40, 41].
  40. P. Li, S. Maddali, A. Pateras, I. Calvo-Almazan, S. O. Hruszkewycz, W. Cha, V. Chamard, and M. Allain, General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging. Part II, J. Appl. Crystallogr. 53, 404 (2020).
  41. S. Maddali, P. Li, A. Pateras, D. Timbie, N. Delegan, A. L. Crook, H. Lee, I. Calvo-Almazan, D. Sheyfer, W. Cha, F. J. Heremans, D. D. Awschalom, V. Chamard, M. Allain, and S. O. Hruszkewycz, General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging. Part I, J. Appl. Crystallogr. 53, 393 (2020).
  42. M. J. Bedzyk and P. Fenter, in The X-ray Standing Wave Technique: Principles and Applications, edited by J. Zegenhagen, and A. Kazimirov (World Scientific, Singapore, 2013), p. 289.
  43. L. Cheng, P. Fenter, M. J. Bedzyk, and N. C. Sturchio, Fourier-expansion solution of atom distributions in a crystal using x-ray standing waves, Phys. Rev. Lett. 90, 255503 (2003).
  44. Z. Zhang, P. Fenter, L. Cheng, N. C. Sturchio, M. J. Bedzyk, M. L. Machesky, and D. J. Wesolowski, Model-independent x-ray imaging of adsorbed cations at the crystal-water interface, Surf. Sci. 554, L95 (2004).
  45. N. Laanait, E. B. R. Callagon, Z. Zhang, N. C. Sturchio, S. S. Lee, and P. Fenter, X-ray-driven reaction front dynamics at calcite-water interfaces, Science 349, 1330 (2015).
  46. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, Hard-x-ray dark-field imaging using a grating interferometer, Nat. Mater. 7, 134 (2008).
  47. M. C. Zdora, State of the art of x-ray speckle-based phase-contrast and dark-field imaging, J. Imaging 4, 60 (2018).
  48. M. Bech, O. Bunk, T. Donath, R. Feidenhans'l, C. David, and F. Pfeiffer, Quantitative x-ray dark-field computed tomography, Phys. Med. Biol. 55, 5529 (2010).
  49. H. Simons, A. King, W. Ludwig, C. Detlefs, W. Pantleon, S. Schmidt, I. Snigireva, A. Snigirev, and H. F. Poulsen, Dark-field X-ray microscopy for multiscale structural characterization, Nat. Commun. 6, 6098 (2015).
  50. G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. Robinson, Three-dimensional imaging of microstructure in Au nanocrystals, Phys. Rev. Lett. 90, 175501 (2003).
  51. X. W. Shi, J. N. Clark, G. Xiong, X. J. Huang, R. Harder, and I. K. Robinson, Mechanical breakdown of bent silicon nanowires imaged by coherent x-ray diffraction, New J. Phys. 15, 123007 (2013).
  52. Original data shown in Figs. 1, 2, 4, 5, and 6 are archived at https://doi.org/10.18126/t0ca-5645.

Outline

Information

Phys. Rev. B 113, 075410– Published 6 February, 2026
  • Received 28 October 2025
  • Revised 16 December 2025
  • Accepted 24 December 2025
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