Vol. 39, issue 05, article # 5

Abstract:

Spatial irradiance is a defining characteristic of radiation during the thermal impact of a laser beam on biological tissues, powders, and other scattering media. The provides a theoretical consideration of multiple light scattering in the volume of a semi-infinite diffuse scattering medium (DSM). Calculations are made for pressed dielectric powders with refractive indices n = 1.47, 1.72, and 2.00 at a wavelength of 1.06 μm (the first harmonic of a neodymium laser). The diffuse reflectance coefficients ρ(1 – Λ) calculated by the Monte Carlo method and in the Kubelka–Munk approximation are compared. A linear relationship is established between the spatial irradiance in DSM volume and the diffuse reflectance coefficient. The precisions and applicability limits of the original and corrected Kubelka–Munk formulas are determined. A complex relationship is ascertained between the single scattering albedo Λ and the angular dependence of a photon flux incident on the DSM–air interface f(α). In particular, it was shown that for Λ → 1 and isotropy scattering, f(α) = cos(α), where a is the photon incidence angle. The results can be used for the development of methods for determining spatial irradiance and, for example, in laser medicine.

Keywords:

light scattering, diffuse scattering media, Monte Carlo method, Kubelka–Munk formula, spatial irradiance

References:

1. Aleksandrov E.I., Tsipilev V.P. Osobennosti svetovogo rezhima v ob"eme polubeskonechnogo sloya DRS pri osveshchenii napravlennym puchkom konechnoj apertury // Izv. vuzov. Fizika. 1988. V. 31, N 10. P. 23–29.
2. Tsipilev V.P., Oleshko V.I., Yakovlev A.N., Ovchinnikov V.A., Zykov I.Yu., Forat E.V., Saidazimov I.A., Grechkina T.V. Effect of volume compression on the thresholds of laser pulse initiation of PETN mixtures at various concentrations of nano-sized carbon and aluminum particles // Russ. Phys. J. 2024. V. 67, N 12. P. 2368–2378. DOI: 10.1007/s11182-025-03387-2.
3. Keizjer M., Jacques S.L., Prahl S.A., Welch A.J. Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams // Lasers Surg. Med. 1989. V. 9, N 2. P. 148–154. DOI: 10.1002/lsm.1900090210.
4. Potlov A.Yu., Frolov S.V., Proskurin S.G. Chislennoe modelirovanie migratsii fotonov v odnorodnyxi neodnorodnyx tsilindricheskix fantomax // Opt. i spektroskop. 2020. V. 128, N 6. P. 832–839. DOI: 10.21883/OS.2020.06.49417.33-20.
5. Chandrasekar S. Perenos luchistoi energii. M.: Izdatinlit, 1953. 432 p.
6. Giovanelli R.G. Reflection by semi-infinite diffusers // Opt. Acta Int. J. Opt. 1955. V. 2, N 4. P. 153–162. DOI: 10.1080/713821040.
7. Isimaru A. Rasprostranenie i rasseyanie voln v sluchaino-neodnorodnyx sredax. M.: Mir, 1981. V. 1. 280 p.
8. Tricoli U., Macdonald C.M., Da Silva A., Markel V.A. Optimized diffusion approximation // J. Opt. Soc. Am. A. 2018. V. 35, N 2. P. 356–369. DOI: 10.1364/JOSAA. 35.000356
9. Ivanov A.P. Optika rasseivayushchikh sred. Minsk: Nauka i tekhnika, 1969. 592 p.
10. Gershun A.A. Prostranstvennaya osveshchennost' // Izbrannye trudy po fotometrii i svetotekhnike. M.: Fizmatgiz, 1958. P. 263–282.
11. Pushkareva A.E. Metody matematicheskogo modelirovaniya v optike biotkani: ucheb. pos. SPb.: SPbGU ITMO, 2008. 103 p.
12. Pelivanov I.M., Belov S.A., Solomatin V.S., Khokhlova T.D., Karabutov A.A. Pryamoe izmerenie prostranstvennogo raspredeleniya intensivnosti lazernogo izlucheniya v biologicheskikh sredakh in vitro optiko-akusticheskim metodom // Kvant. elektron. 2006. V. 36, N 12. P. 1089–1096. DOI: 10.1070/QE2006v036n12ABEH013261.
13. Jönsson J., Berrocal E. Multi-Scattering software: Part I: online accelerated Monte Carlo simulation of light transport through scattering media // Opt. express. 2020. V. 28, N. 25. P. 37612–37638. DOI: 10.1364/OE.404005.
14. Loyalka S.K., Riggs C.A. Inverse problem in diffuse reflectance spectroscopy: Accuracy of the Kubelka-Munk equations // Appl. Spectrosc. 1995. V. 49, N 8. P. 1107–1110.
15. Jones A.R. Light scattering for particle characterization // Prog. Energy Combust. Sci. 1999. V. 25, N 1. P. 1–53. DOI: 10.1016/S0360-1285(98)00017-3.
16. Mikhail S.S., Azer S.S., Johnston W.M. Accuracy of Kubelka–Munk reflectance theory for dental resin composite material // Dental Mat. 2012. V. 28, N. 7. P. 729–735. DOI: 10.1016/j.dental.2012.03.006.
17. Baranov S.M., Bajin N.M. O primenenii spektroskopii svetorasseivayushchikh sred v fiziko-khimicheskikh issledovaniyakh // Uspekhi khimii. 1977. V. 46, N 7. P. 1153–1182. DOI: 10.1070/RC1977v046n07ABEH002159.
18. Jamil H., Dildar I.M., Ilyas U., Hashmi J.Z., Shaukat S., Sarwar M.N., Khaleeq-ur-Rahman M. Microstructural and optical study of polycrystalline manganese oxide films using Kubelka–Munk function // Thin. Solid Films. 2021. V. 732. P. 138796. DOI: 10.1016/j.tsf.2021.138796.
19. Landi Jr.S., Segundo I.R., Freitas E., Vasilevskiy M., Carneiro J., Tavares C.J. Use and misuse of the Kubelka-Munk function to obtain the band gap energy from diffuse reflectance measurements // Sol. State Commun. 2022. V. 341. P. 114573. DOI: 10.1016/j.ssc.2021.114573.
20. White R.L. Concentration dependence of Kubelka–Munk function and apparent absorbance button sample holder infrared spectra // Anal. Meth. 2025. V. 17, N. 26. P. 5519–5526. DOI: 10.1039/d5ay00758e.
21. Baranovskii A.M. Opticheskie svoistva nekotorykh VV // Fizika goreniya i vzryva. 1990. V. 26, N 3. P. 62–64. DOI: 10.1007/BF00751369.
22. Svetlichnyi V.A., Goncharova D.A., Lapin I.N., Shabalina A.N. Vliyanie rastvoritelya na strukturu i morfologiyu nanochastits, poluchennykh metodom lazernoi ablyatsii ob"emnykh mishenei magniya // Izv. vuzov. Fizika. 2018. V. 61, N 6. P. 42–48.