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BIOP – Center for Biomedical Optics and New Laser Systems Light scattering from a single particle Peter E. Andersen Optics and Fluid Dynamics Dept. Risø National Laboratory

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Outline Introduction / tissue optics – why understand light scattering? Light scattering – general considerations, – absorption, scattering, extinction and phase function, – special cases. Light scattering from single particle – requirements, – Mie theory.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Introduction (tissue optics) Optically tissue may be characterized by its – scattering, refractive index, and absorption. The scattering arises from – cell membranes, cell nuclei, capillary walls, hair follicles... The absorption arises from – visible and NIR wavelengths (400 nm - 800 nm); »hemoglobin and melanin, – IR wavelengths; »water and molecular vibrational/rotational states.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Understanding light scattering Important because – light propagation is affected by the tissue optical properties, – the physiological condition or state of single cells or tissues is expressed through (but not exclusively) changes in cell size or refractive index, – changes in refractive index or cell size influence the optical properties. Measuring or analyzing the light scattering may thus provide information about the cell(s) or tissue

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Light scattering The vector fields E and H interact through the material parameters 0 and 0 – described by the Maxwell’s equations. Propagation in matter – the charge carriers of the material oscillate and radiate as dipoles, – in a homogeneous medium the dipoles cancel each other except in the forward direction, – inhomogeneities scatter the light and thus the dipoles do not cancel each other. Questions / examples – piece of glass (homogeneous)? – piece of glass (with tiny air bubbles in it)?

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems General considerations The impinging field excites a secondary field radiated from the scatterer The scatterer is excited as a dipole Maxwell’s equations describing the electro- magnetic wave propagation – to be solved for the geometry at hand.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems General considerations Four important quantities Cross sections – absorption, – scattering, – extinction = scattering + absorption. Angular dependence – scattering phase function.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Cross sections Far-field limit: R D 2 / Differential scattering cross section p(o,i) is the scattering phase function iEiEi o EsEs R

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Cross sections Scattering cross section Absorption cross section Back-scatter cross section Extinction section Albedo – note: W 0 is close to unity for most tissues. Dimension of all cross sections – area – [m 2 ].

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems The extinction paradox Extinction from a large particle Find S s (scattered amplitude): – diffraction (replace scatterer by aperture): S i g, – absorption: S i g, – extinction: (S i g + S i g )/ S i = 2 g, i.e. twice geom. area SiSi gg gg

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Scattering phase function Scattering phase function p( ) – despite its name it is not related to the phase of the light, – normalized to The asymmetry parameter (or anisotropy) – important for multiple scattering, – g=0:isotropic scattering, – g=1:highly forwardly peaked. p( )

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Light scattering – special cases Size parameter Refractive index ratio m is defined as Rayleigh approximation – the particle is considered a dipole, whose strength is proportional to its volume, – valid approximation for 1 (until 5% of ), – proportional to -4. Rayleigh-Gan’s theory – fields superposed over volume of particles including the phases, – and

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Rayleigh scattering Scattered intensity – where a=d/2 and m=n 1 /n R n1n1 n

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Light scattering – special cases Direct numerical method – the volume is divided into smaller sections, and the scattered light is added for all directions including the phase. First Born approximation (Rayleigh-Gan’s theory) – light from a single scatterer is not considered as a source of scattering. Second Born approximation – includes scattered light as a secondary source of scattering, – of little practical use.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Light scattering from single particle Mie theory (Lorenz-Mie) – direct solution to Maxwell’s equations with proper boundary conditions, – only few, simple cases with analytical solutions. Scattering from sphere may be calculated exactly from Mie theory – solved in spherical coordinates, – assumes plane wave incidence, – computer code (all platforms, freeware) may be downloaded to calculate the scattering of a single particle.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Exact solution to Maxwell’s equations for spherical scatterer with plane wave, monochromatic incidence Only requirements: – the refractive index ratio m (including absorption): – the ratio of perimeter of the scatterer to the wavelength: Mie theory

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Mie theory polarized parallel to plane of incidence polarized perpendicular to plane of incidence

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Summary Fundamental properties of light scattering from single particles are introduced, including – scattering, absorption and extinction cross sections, – scattering phase function, – important for understanding multiple scattering. Important special cases are discussed – the Rayleigh approximation, – Mie theory, representing an analytical solution to Maxwell’s equations.

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P. E. Andersen - 5/4/2015 Optics and Fluid Dynamics Department RISØ Center for Biomedical Optics and New Laser Systems Light scattering from single particles Reference – A. Ishimaru, Wave propagation and scattering in random media I, Academic Press, New York, 1978; »chapter 2, secs. 2.1-2.5 and 2.8. Other (recommended) – C.F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles – 2 nd ed., J. Wiley & Sons, New York, 1998.

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