Asian Journal of Physics Vol 31, Nos 3 – 6 (2022) 515-530

Preserved polarization vortices in partially coherent vector vortex beams

Saba N Khan1,2, Stuti Joshi1 and P Senthilkumaran1

1Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi- 110 016, India

2School of Physics and Astronomy, University of St Andrews, Fife, Scotland, United Kingdom

Dedicated to Professor Bishnu P Pal for his enormous contributions to the advancement of research and education in science and technology through his unique vision and outstanding dedication

Light scattering techniques are ubiquitous and essential components of many methodologies which require the non-invasive characterization of particulate matter. Here, we present an overview of various methods which are usually used to calculate the characteristics of the light scattered by such media. The scattering problem—the objective of which is to obtain the properties of the scattered light relative to the incident light for a given type of particle—can be modelled in the form of a boundary value problem, and it can be solved through different analytical and numerical techniques. The discussion is then extended to the description of light propagation in turbid media, which is of importance in several practical applications. A turbid media consists of suspended particulate matter, which often leads to multiple scattering of light, further complicating the solution of the scattering problem. We outline both analytical and numerical techniques for solving the scattering problem, with particular focus on the Mie theory and the Monte Carlo method with examples. Finally, we conclude with a brief discussion on some of the neoteric applications of light scattering studies. © Anita Publications. All rights reserved.
Keywords: Light scattering, Scattering theory, Particulate matter, Turbid media, Monte Carlo simulation.

Peer Review Information
Method: Single- anonymous; Screened for Plagiarism? Yes
Buy this Article in Print © Anita Publications. All rights reserve


  1. Campbell C G, Laycak D T, Hoppes W, Tran N T, Shi F G, High concentration suspended sediment measurements using a continuous fiber optic in-stream transmissometer, J Hydrol, 311(2005)244–253.
  2. Osborne P D, Vincent C E, Greenwoodt B, Measurement of suspended sand concentrations in the nearshore: field intercomparison of optical and acoustic backscatter sensors, Cont Shelf Res, 14(1994)159–174.
  3. Wren D G, Barkdoll B D, Kuhnle R A, Derrow R W, Field Techniques for Suspended-Sediment Measurement, J Hydraul Eng, 126(2000)97–104.
  4. Inoue T, Sanpei A, Kawade Y, Suzuki M, Ochiai R, Awatsuji Y, Identification of Pollens From Polymer Particles Levitating in an RF Plasma by the Polarization Imaging Method, IEEE Trans Plasma Sci, 49(2021)2967–2971.
  5. Nakayama T, Matsumi Y, Kawahito K, Watabe Y, Development and evaluation of a palm-sized optical PM5 sensor, Aerosol Sci Technol, 52(2018)2–12.
  6. Hyslop N P, Impaired visibility: the air pollution people see, Atmos Environ, 43(2009)182–195.
  7. Volten H, Muñoz O, Rol E, de Haan J F, Vassen W, Hovenier J W, Muinonen K, Nousiainen T, Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm, J Geophys Res Atmos, 106(2001)17375–17401.
  8. Kemppinen O, Nousiainen T, Merikallio S, Räisänen P, Retrieving microphysical properties of dust-like particles using ellipsoids: the case of refractive index, Atmos Chem Phys, 15(2015)11117–11132.
  9. Bergmann F, Foschum F, Zuber R, Kienle A, Precise determination of the optical properties of turbid media using an optimized integrating sphere and advanced Monte Carlo simulations. Part 2: experiments, Appl Opt, 59(2020) 3216–3226.
  10. Prerana, Shenoy M R, Pal B P, Method to determine the optical properties of turbid media, Appl Opt, 47(2008), 3216–3220.
  11. Gupta K, Shenoy M R, Method to determine the anisotropy parameter g of a turbid medium, Appl Opt, 57(2018) 7559–7563.
  12. Montrucchio B, Giusto E, Vakili M G, Quer S, Ferrero R, Fornaro C, A Densely-Deployed, High Sampling Rate, Open-Source Air Pollution Monitoring WSN, IEEE Trans Veh Technol, 69(2020)15786–15799.
  13. Miki K, Fujita T, Sahashi N, Development and application of a method to classify airborne pollen taxa concentration using light scattering data, Sci Rep, 11(2021)22371; doi. 10.1038/s41598-021-01919-7.
  14. Prerana, Shenoy M R, Pal B P, Gupta B D, Design, analysis, and realization of a turbidity sensor based on collection of scattered light by a fiber-optic probe, IEEE Sens J, 12(2012)44–50.
  15. Shenoy M R, Optical fibre probes in the measurement of scattered light: Application for sensing turbidity, Pramana — J Phys, 82(2014)39–48.
  16. Huang J, Qian R, Gao J, Bing H, Huang Q, Qi L, Song S, Huang J, A novel framework to predict water turbidity using Bayesian modeling, Water Res, 202(2021)117406; 10.1016/j.watres.2021.117406.
  17. Miklos D B, Remy C, Jekel M, Linden K G, Drewes J E, Hübner U, Evaluation of advanced oxidation processes for water and wastewater treatment – A critical review, Water Res, 139(2018)118–131.
  18. Wen Y, Mao Y, Wang X, Application of chromaticity coordinates for solution turbidity measurement, Measurement, 130(2018)39–43.
  19. Gao Z L, Cheng Q D, Zeng G L, Wen Y, Li G F, Chen J, Dong Y B, Ji Q Z, Review of Calibration and Improvement Methods of Light- Scattering Airborne Particle Concentration, J Phys: Conf Ser, 2097(2021) 012008; doi. 10.1088/1742-6596/2097/1/012008.
  20. Świrniak G, Mroczka J, Forward and inverse analysis for particle size distribution measurements of disperse samples: A review, Measurement, 187(2022) 110256; doi. 10.1016/j.measurement.2021.110256.
  21. Wang H, Li J, Liao R, Tao Y, Peng L, Li H, Deng H, Ma H, Early warning of cyanobacterial blooms based on polarized light scattering powered by machine learning, Measurement, 184(2021) 109902; doi. 10.1016/j.measurement.2021.109902.
  22. Selden A C, Attenuation and impulse response for multiple scattering of light in atmospheric clouds and aerosols, Appl Opt, 45(2006)3144–3151.
  23. Ceolato R, Berg M J, Aerosol light extinction and backscattering: A review with a lidar perspective, J Quant Spectrosc Radiat Transf, 262(2021) 107492;
  24. Ye Y, Pui D Y H, Detection of nanoparticles suspended in a light scattering medium, Sci Rep, 11(2021)20268; doi. 10.1038/s41598-021-99768-x.
  25. Wax A, Yang C, Backman V, Kalashnikov M, Dasari R R, Feld M S, Determination of particle size by using the angular distribution of backscattered light as measured with low-coherence interferometry, J Opt Soc Am A, 19(2002)737–744.
  26. Augsten C, Kiselev M A, Gehrke R, Hause G, Mäder K, A detailed analysis of biodegradable nanospheres by different techniques—A combined approach to detect particle sizes and size distributions, J Pharm Biomed Anal, 47(2008)95–102.
  27. Wang R, Yu G, Suspended sediment concentration measurement based on optical fiber technology, Meas Sci Technol, 30(2019) 075205; doi. 10.1088/1361-6501/ab188d.
  28. Lee B, Review of the present status of optical fiber sensors, Opt Fiber Technol, 9(2003)57–79.
  29. Kerker M, The Scattering of Light and Other Electromagnetic Radiation, (Academic Press, Inc., New York), 1969.
  30. Bal G, Inverse transport theory and applications, Inverse Probl, 25(2009) 053001; doi.1088/0266-5611/25/5/053001.
  31. Hulst H C van de, Light Scattering by Small Particles, (John Wiley & Sons, Inc., New York), 1957.
  32. Bohren C F, Huffman D R, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, Inc., USA), 1983.
  33. Gupta K, Shenoy M R, Compact setup to determine size and concentration of spherical particles in a turbid medium, Appl Opt, 60(2021)8174–8180.
  34. Wang L V, Wu H, Biomedical Optics, (John Wiley & Sons, Inc., Hoboken, New Jersey), 2007.
  35. Shenoy M R, Gupta K, Light Scattering by Turbid Media, Asian J Phys, 28(2019)1163–1173.
  36. Marx E, Mulholland G W, Size and Refractive Index Determination of Single Polystyrene Spheres, J Res Natl Bur Stand, 88(1983)321–338.
  37. Bell B W, Bickel W S, Single fiber light scattering matrix: an experimental determination, Appl Opt, 20(1981) 3874–3876.
  38. Bartholdi M, Salzman G C, Hiebert R D, Kerker M, Differential light scattering photometer for rapid analysis of single particles in flow, Appl Opt, 19(1980)1573–1581.
  39. Sloot P M A, Hoekstra A G, van der Liet H, Figdor C G, Scattering matrix elements of biological particles measured in a flow through system: theory and practice, Appl Opt, 28(1989)1752–1762.
  40. Yastrebova E S, Litvinenko A L, Strokotov D I, Vladimirov R S, Gilev K V, Nekrasov V M, Karpenko A A, Maltsev V P, Dual-wavelength angle-resolved light scattering used in the analysis of particles by scanning flow cytometry, J Opt, 23(2021) 105606; doi. 10.1088/2040-8986/ac1b7b.
  41. Friebel M, Roggan A, Müller G, Meinke M, Determination of optical properties of human blood in the spectral range 250 to 1100 nm using Monte Carlo simulations with hematocrit-dependent effective scattering phase functions, J Biomed Opt, 11(2006) 034021; doi. 10.1117/1.2203659
  42. Liu F, Zhang S, Han P, Chen F, Zhao L, Fan Y, Shao X, Depolarization index from Mueller matrix descatters imaging in turbid water, Chin Opt Lett, 20(2022) 022601; doi.10.3788/COL202220.022601.
  43. Bartel S, Hielscher A H, Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media, Appl Opt, 39(2000)1580–1588.
  44. Satapathi S, Soni J, Ghosh N, Fluorescent Mueller matrix analysis of a highly scattering turbid media, Appl Phys Lett, 104(2014) 131902; doi. 10.1063/1.4869475.
  45. Gao W, Mueller matrix decomposition methods for tissue polarization tomography, Opt Lasers Eng, 147(2021) 106735; doi.10.1016/j.optlaseng.2021.106735.
  46. Gonzalez M, Ossikovski R, Novikova T, Ramella-Roman J, Introduction of a 34 Mueller matrix decomposition method, J Phys D: Appl Phys, 54(2021) 424005; doi. 10.1088/1361-6463/ac1622.
  47. Mie G, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Ann Phys, 330(1908)377–445.
  48. Wiscombe W J, Improved Mie scattering algorithms, Appl Opt, 19(1980)1505–1509.
  49. Laven P, MiePlot,
  50. Sultanova N, Kasarova S, Nikolov I, Dispersion Properties of Optical Polymers, Acta Phys Pol A, 116(2009) 585–587.
  51. Hale G M, Querry M R, Optical Constants of Water in the 200-nm to 200-μm Wavelength Region, Appl Opt, 12(1973)555–563.
  52. Mishchenko M I, Travis L D, Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation, Appl Opt, 33(1994)7206–7225.
  53. Kodach V M, Faber D J, van Marle J, van Leeuwen T G, Kalkman J, Determination of the scattering anisotropy with optical coherence tomography, Opt Express, 19(2011)6131–6140.
  54. Martin R J, Mie Scattering Formulae for Non-spherical Particles, J Mod Opt, 40(1993)2467–2494.
  55. Aden A L, Kerker M, Scattering of Electromagnetic Waves from Two Concentric Spheres, J Appl Phys, 22(1951) 1242–1246.
  56. Kaiser T, Schweiger G, Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere, Comput Phys, 7(1993)682–686.
  57. Bohren C F, Light scattering by an optically active sphere, Chem Phys Lett, 29(1974)458–462.
  58. Tang C, Auguié B, Le Ru E C, Modeling Molecular Orientation Effects in Dye-Coated Nanostructures using a Thin-Shell Approximation of Mie Theory for Radially Anisotropic Media, ACS Photonics, 5(2018)5002–5009.
  59. Heinisch R L, Bronold F X, Fehske H, Mie Scattering by a Charged Dielectric Particle, Phys Rev Lett, 109(2012) 243903; doi. 10.1103/PhysRevLett.109.243903.
  60. Wriedt T, A Review of Elastic Light Scattering Theories, Part Part Syst Charact, 15(1998)67–74.
  61. Fu Q, Sun W, Mie theory for light scattering by a spherical particle in an absorbing medium, Appl Opt, 40(2001) 1354–1361.
  62. Asano S, Yamamoto G, Light Scattering by a Spheroidal Particle, Appl Opt, 14(1975)29–49.
  63. Tang C C H, Backscattering from Dielectric-Coated Infinite Cylindrical Obstacles, J Appl Phys, 28(1957)628–633.
  64. Miles R B, Lempert W R, Forkey J N, Laser Rayleigh scattering, Meas Sci Technol, 12(2001)R33–R51.
  65. Pang J, Baitenov A, Montanari C, Samanta A, Berglund L, Popov S, Zozoulenko I, Light Propagation in Transparent Wood: Efficient Ray-Tracing Simulation and Retrieving an Effective Refractive Index of Wood Scaffold, Adv Photo Res, 2(2021) 2100135; doi.10.1002/adpr.202100135.
  66. Yee K S, Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media, IEEE Trans Antennas Propag, 14(1966)302–307.
  67. Taflove A, Hagness S C, Piket-May M, Computational Electromagnetics: The Finite-Difference Time-Domain Method, in The Electrical Engineering Handbook, (Academic Press), 2005, pp 629–670.
  68. Drezek R, Dunn A, Richards-Kortum R, Light scattering from cells: finite-difference time-domain simulations and goniometric measurements, Appl Opt, 38(1999)3651–3661.
  69. Schweiger M, Arridge S R, Hiraoka M, Delpy D T, The finite element method for the propagation of light in scattering media: Boundary and source conditions, Med Phys, 22(1995)779–1792.
  70. Muinonen K, Markkanen J, Väisänen T, Peltoniemi J, Penttilä A, Multiple scattering of light in discrete random media using incoherent interactions, Opt Lett, 43(2018)683–686.
  71. Schaubert D, Wilton D, Glisson A, A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies, IEEE Trans Antennas Propagat, 32(1984)77–85.
  72. Draine B T, The discrete-dipole approximation and its application to interstellar graphite grains, Astrophys J, 333(1988)848–872.
  73. Waterman P C, Symmetry, Unitarity, and Geometry in Electromagnetic Scattering, Phys Rev D, 3(1971)825–839.
  74. Bi L, Yang P, Kattawar G W, Mishchenko M I, Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles, J Quant Spectrosc Radiat Transf, 116(2013)169–183.
  75. Mishchenko M I, Light scattering by randomly oriented axially symmetric particles, J Opt Soc Am A, 8(1991) 871–882.
  76. Petrov D, Shkuratov Y, Zubko E, Videen G, Sh-matrices method as applied to scattering by particles with layered structure, J Quant Spectrosc Radiat Transf, 106(2007)437–454.
  77. Somerville W R C, Auguié B, Le Ru E C, Accurate and convergent T-matrix calculations of light scattering by spheroids, J Quant Spectrosc Radiat Transf, 160(2015)29–35.
  78. Mishchenko M I, Travis L D, Mackowski D W, T-matrix computations of light scattering by nonspherical particles: A review, J Quant Spectrosc Radiat Transf, 55(1996)535–575.
  79. Harrington R F, Field computation by moment methods, (IEEE Press, Piscataway, NJ), 1993.
  80. Purcell E M, Pennypacker C R, Scattering and Absorption of Light by Nonspherical Dielectric Grains, Astrophys J, 186(1973)705–714.
  81. Bolt R A, ten Bosch J J, Method for measuring position-dependent volume reflection, Appl Opt, 32(1993)4641–4645.
  82. Gandjbakhche A H, Weiss G H, Bonner R F, Nossal R, Photon path-length distributions for transmission through optically turbid slabs, Phys Rev E, 48(1993)810–818.
  83. Xu M, Alfano R R, Random Walk of Polarized Light in Turbid Media, Phys Rev Lett, 95(2005)213901; doi. 10.1103/PhysRevLett.95.213901.
  84. Svensson T, Vynck K, Grisi M, Savo R, Burresi M, Wiersma D S, Holey random walks: Optics of heterogeneous turbid composites, Phys Rev E, 87(2013) 022120; doi. 10.1103/PhysRevE.87.022120.
  85. Liemert A, Kienle A, Novel analytical solution for the radiance in an anisotropically scattering medium, Appl Opt, 54(2015)1963–1969.
  86. Groenhuis R A J, Ferwerda H A, Ten Bosch J J, Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory, Appl Opt, 22(1983)2456–2462.
  87. Venugopalan V, You J S, Tromberg B J, Radiative transport in the diffusion approximation: An extension for highly absorbing media and small source-detector separations, Phys Rev E, 58(1998)2395–2407.
  88. Chin L C L, Whelan W M, Vitkin I A, Information content of point radiance measurements in turbid media: implications for interstitial optical property quantification, Appl Opt, 45(2006)2101–2114.
  89. Liu L, Wan W, Li J, Zhao H, Gao F, Simultaneous recovery of a full set of optical properties in turbid media using incomplete P5 approximation to CW radiance, Opt Lett, 43(2018)4188–4191.
  90. Meier R R, Lee J-S, Anderson D E, Atmospheric scattering of middle UV radiation from an internal source, Appl Opt, 17(1978)3216–3225.
  91. Wang L, Jaques S L, Zheng L, MCML—Monte Carlo modeling of light transport in multi-layered Tissues, Comput Methods Programs Biomed, 47(1995)131–146.
  92. Henyey L C, Greenstein J L, Diffuse radiation in the Galaxy, Astrophys J, 93(1941)70–83.
  93. Cornette W M, Shanks J G, Physically reasonable analytic expression for the single-scattering phase function, Appl Opt, 31(1992)3152–3160.
  94. Toublanc D, Henyey–Greenstein and Mie phase functions in Monte Carlo radiative transfer computations, Appl Opt, 35(1996)3270–3274.
  95. Reynolds L O, McCormick N J, Approximate two-parameter phase function for light scattering, J Opt Soc Am, 70(1980)1206–1212.
  96. Caton F, Baravian C, Mougel J, The influence of the microscopic characteristics of a random medium on incoherent light transport, Opt Express, 15(2007)2847–2872.
  97. Vaudelle F, L’Huillier J-P, Askoura M L, Light source distribution and scattering phase function influence light transport in diffuse multi-layered media, Opt Commun, 392(2017)268–281.
  98. Binzoni T, Leung T S, Gandjbakhche A H, Rüfenacht D, Delpy D T, The use of the Henyey–Greenstein phase function in Monte Carlo simulations in biomedical optics, Phys Med Biol, 51(2006, N313–N322.
  99. Preston T C, Reid J P, Determining the size and refractive index of microspheres using the mode assignments from Mie resonances, J Opt Soc Am A, 32(2015)2210–2217.
  100. Chu C-M, Churchill S W, Representation of the Angular Distribution of Radiation Scattered by a Spherical Particle, J Opt Soc Am, 45(1955)958–962.
  101. Fowler B W, Expansion of Mie-theory phase functions in series of Legendre polynomials, J Opt Soc Am, 73(1983) 19–22.
  102. Naglič P, Pernuš F, Likar B, Bürmen M, Lookup table-based sampling of the phase function for Monte Carlo simulations of light propagation in turbid media, Biomed Opt Express, 8(2017), 1895–1910.
  103. Kattawar G W, A three-parameter analytic phase function for multiple scattering calculations, J Quant Spectrosc Radiat Transf, 15(1975)839–849.
  104. Haltrin V I, One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater, Appl Opt, 41(2002)1022–1028.
  105. Liu Q, Weng F, Combined Henyey-Greenstein and Rayleigh phase function, Appl Opt, 45(2006)7475–7479.
  106. Graaff R, Koelink M H, Mul F F M de, Zijlstra W G, Dassel A C M, Aarnoudse J G, Condensed Monte Carlo simulations for the description of light transport, Appl Opt, 32(1993)426–434.
  107. Pifferi A, Taroni P, Valentini G, Andersson-Engels S, Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model, Appl Opt, 37(1998)2774–2780.
  108. Bevilacqua F, Depeursinge C, Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path, J Opt Soc Am A, 16(1999)2935–2945.
  109. Ostermeyer M R, Jacques S L, Perturbation theory for diffuse light transport in complex biological tissues, J Opt Soc Am A, 14(1997)255–261.
  110. Kumar Y P, Vasu R M, Reconstruction of optical properties of low-scattering tissue using derivative estimated through perturbation Monte-Carlo method, J Biomed Opt, 9(2004)1002–1012;
  111. Hayakawa C K, Spanier J, Bevilacqua F, Dunn A K, You J S, Tromberg B J, Venugopalan V, Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues, Opt Lett, 26(2001)1335–1337.
  112. Wang L, Jacques S L, Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media, J Opt Soc Am A, 10(1993)1746–1752.
  113. Lee S Y, Mycek M-A, Hybrid Monte Carlo simulation with ray tracing for fluorescence measurements in turbid media, Opt Lett, 43(2018)3846–3849.
  114. Alerstam E, Svensson T, Andersson-Engels S, Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration, J Biomed Opt, 13(2008) 060504; doi. 10.1117/1.3041496.
  115. Alerstam E, Lo W C Y, Han T D, Rose J, Andersson-Engels S, Lilge L, Next-generation acceleration and code optimization for light transport in turbid media using GPUs, Biomed Opt Express, 1(2010)658–675.
  116. Fang Q, Boas D A, Monte Carlo Simulation of Photon Migration in 3D Turbid Media Accelerated by Graphics Processing Units, Opt Express, 17(2009)20178–20190.
  117. Jönsson J, Berrocal E, Multi-Scattering software: part I: online accelerated Monte Carlo simulation of light transport through scattering media, Opt Express, 28(2020)37612–37638.
  118. Zhu C, Liu Q, Review of Monte Carlo modeling of light transport in tissues, J Biomed Opt, 18(2013) 050902; doi. 10.1117/1.JBO.18.5.050902.
  119. Shen Z, Sukhov S, Dogariu A, Monte Carlo method to model optical coherence propagation in random media, J Opt Soc Am A, 34(2017)2189–2193.
  120. Kaplan B, Ledanois G, Drévillon B, Mueller matrix of dense polystyrene latex sphere suspensions: measurements and Monte Carlo simulation, Appl Opt, 40(2001)2769–2777.
  121. Wang X, Wang L V, Propagation of polarized light in birefringent turbid media: A Monte Carlo study, J Biomed Opt, 7(2002) 279; doi. 10.1117/1.1483315.
  122. Côté D, Vitkin I A, Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations, Opt Express, 13(2005)148–163.
  123. Kumar A T N, Direct Monte Carlo computation of time-resolved fluorescence in heterogeneous turbid media, Opt Lett, 37(2012)4783–4785.
  124. Pavlov M S, Krasnikov I V, Seteĭkin A Yu, Monte Carlo modelling of optical-radiation propagation in biological media with closed internal inhomogeneities, J Opt Technol, 77(2010)602–605.
  125. Boas D A, Culver J P, Stott J J, Dunn A K, Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head, Opt Express, 10(2002)159–170.
  126. Gupta K, Shenoy M R, Method to determine the concentrations of constituents in a bidisperse turbid medium using Monte Carlo simulation for mixtures, OSA Continuum, 4(2021)2232–2244.
  127. Meglinskii I V, Monte Carlo simulation of reflection spectra of random multilayer media strongly scattering and absorbing light, Quantum Electron, 31(2001)1101–1107.
  128. Sharma S K, Banerjee S, Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues, J Opt A: Pure Appl Opt, 5(2003)294–302.
  129. Meglinski I V, Matcher S J, Computer simulation of the skin reflectance spectra, Comput Methods Programs Biomed, 70(2003)179–186.
  130. Gélébart B, Tinet E, Tualle J M, Avrillier S, Phase function simulation in tissue phantoms: a fractal approach, Pure Appl Opt, 5(1996)377–388.
  131. Katz O, Small E, Guan Y, Silberberg Y, Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers, Optica, 1(2014)170–174.
  132. Wang H, Li J, Hu H, Jiang J, Li X, Zhao K, Cheng Z, Sang M, Liu T, Underwater imaging by suppressing the backscattered light based on Mueller matrix, IEEE Photonics J, 13(2021) 7800106; doi.10.1109/JPHOT.2021.3094359.
  133. Guo L, Cui Y, He Q, Gao W, Pei K, Zhu L, Li H, Wang X, Contributions of aerosol chemical composition and sources to light extinction during haze and non-haze days in Taiyuan, China, Atmos Pollut Res, 12(2021) 101140; doi. 10.1016/j.apr.2021.101140.
  134. Awelisah Y M, Li G, Wang Y, Tang W, Lin L, Considering blood scattering effect in noninvasive optical detection of blood components using dynamic spectrum along with time varying filter based empirical mode decomposition, Biomed Signal Process Control, 71(2022) 103266; doi. 10.1016/j.bspc.2021.103266.
  135. Foschum F, Bergmann F, Kienle A, Precise determination of the optical properties of turbid media using an optimized integrating sphere and advanced Monte Carlo simulations Part 1: theory, Appl Opt, 59(2020)3203–3215.
  136. Menon S, Su Q, Grobe R, Determination of g and μ using Multiply Scattered Light in Turbid Media, Phys Rev Lett, 94(2005) 153904; doi.10.1103/PhysRevLett.94.153904.
  137. Penttilä A, Zubko E, Lumme K, Muinonen K, Yurkin M A, Draine B, Rahola J, Hoekstra A G, Shkuratov Y, Comparison between discrete dipole implementations and exact techniques, J Quant Spectrosc Radiat Transf, 106(2007)417–436.
  138. Menon S, Su Q, Grobe R, Comparison of the Maxwell and Boltzmann theory for multilayered dielectric random media, Phys Rev E, 65(2002) 051917; doi. 10.1103/PhysRevE.65.051917.