Non-Paraxial Effects in the Laser Beams Sharply Focused to Skin Revealed by Unidirectional Helmholtz Equation Approximation

Andrey Bulygin, Igor Meglinski, Yury Kistenev

Research output: Contribution to journalArticlepeer-review


Laser beams converging at significant focusing angles have diverse applications, including quartz-enhanced photoacoustic spectroscopy, high spatial resolution imaging, and profilometry. Due to the limited applicability of the paraxial approximation, which is valid solely for smooth focusing scenarios, numerical modeling becomes necessary to achieve optimal parameter optimization for imaging diagnostic systems that utilize converged laser beams. We introduce a novel methodology for the modeling of laser beams sharply focused on the turbid tissue-like scattering medium by employing the unidirectional Helmholtz equation approximation. The suggested modeling approach takes into account the intricate structure of biological tissues, showcasing its ability to effectively simulate a wide variety of random multi-layered media resembling tissue. By applying this methodology to the Gaussian-shaped laser beam with a parabolic wavefront, the prediction reveals the presence of two hotspots near the focus area. The close-to-maximal intensity hotspot area has a longitudinal size of about 3–5 μm and a transversal size of about 1–2 μm. These values are suitable for estimating spatial resolution in tissue imaging when employing sharply focused laser beams. The simulation also predicts a close-to-maximal intensity hotspot area with approximately 1 μm transversal and longitudinal sizes located just behind the focus distance for Bessel-shaped laser beams with a parabolic wavefront. The results of the simulation suggest that optical imaging methods utilizing laser beams with a wavefront produced by an axicon lens would exhibit a limited spatial resolution. The wavelength employed in the modeling studies to evaluate the sizes of the focus spot is selected within a range typical for optical coherence tomography, offering insights into the limitation of spatial resolution. The key advantage of the unidirectional Helmholtz equation approximation approach over the paraxial approximation lies in its capability to simulate the propagation of a laser beam with a non-parabolic wavefront.
Original languageEnglish
Article number907
Number of pages18
Issue number8
Early online date5 Aug 2023
Publication statusPublished - 5 Aug 2023

Bibliographical note

Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (

Funding: The research was carried out with the Tomsk State University Development Program (Priority-2030). I.M. acknowledges the support from ATTRACT II META-HiLight project funded by the European Union’s Horizon 2020 research and innovative program under grant agreement No.101004462, the Academy of Finland (grant project 325097), the Universities UK International
(UUKi) and the DSTI—UK government Department for Science, Innovation and Technology (grant project: Light4Body). Y.K. and A.B. also acknowledge the support of the state budget financing of the IOA SB RAS.


  • sharply focused laser beams
  • turbid tissue-like multi-layered scattering medium
  • unidirectional Helmholtz equation


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