1. 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

2. 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.

3. 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.

4. 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

5. 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)?

6. 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.

7. 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.

8. 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

9. 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 ].

10. 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 gg gg

11. 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(  )

12. 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

13. 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

14. 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.

15. 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.

16. 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

17. 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

18. 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.

19. 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.