MCRT L10: Scattering and clarification of astronomy/medical terminology

Transcription

1 MCRT L10: Scattering and clarification of astronomy/medical terminology What does the scattering? Shape of scattering Sampling from scattering phase functions Co-ordinate frames Refractive index changes & internal reflections in skin

2 Scattering and absorption Astronomy: electrons, atoms, ions, molecules, dust

3 Interstellar Dust Freitag & Messenger Grains from upper atmosphere: chemical composition suggests cosmic origin

4 Scattering and absorption Biological tissue: Scattering: epidermis, collagen Absorbing: melanin, blood

5 Scattering & Absorption

6 Angular dependence of scattering The phase function is normalised over all scattering angles: For example, isotropic and Rayleigh scattering are described by: Ω Φ isotropic (θ,φ) = 1 4π Φ(θ,φ) dω =1 The average angle of deflection is often described by the parameter g = <cos θ> g = cosθ = Φ R (θ,φ) = 3 16π 1+ cos2 θ ( ) cosθ Φ(θ,φ)dΩ For scattering that is front/back symmetric, g = 0, e.g., isotropic and Rayleigh. Forward directed scattering (small θ deflection) has g > 1 and g < 1 for backscattering Rejection method is efficient to sample θ from Rayleigh phase function Ω

7 Polar diagrams of scattering Scattering particles of radius a Rayleigh scattering when λ >> a Mie scattering describes scattering off spheres when λ ~ a Extensive calculations for complex shaped particles tabulated Φ(θ,φ)

8 Henyey Greenstein Phase Function Originally proposed in 1941 to represent scattering of starlight off dust grains in the interstellar medium Functional form also provides a good representation of scattering in biological tissue Φ HG (θ,φ) = 1 4π 1 g 2 ( 1+ g 2 2 gcosθ) 3 / 2 Can invert analytically to choose a random scattering θ: cosθ = 1 2 g 1+ g2 1 g 2 ( ) 3/2 ( ) / 1 g + 2 gξ 2

9 Angular dependence of ΦHG

10 z z s θ s n φ s n s θ x s y s φ y x Scattering angles in Φ(θ s, φ s ) and techniques for random sampling of angles are valid in centre-of-mass frame containing scatterer, incident, and scattered particle (photon, neutron, etc) In the isotropic scattering code we assume the scattering is isotropic in lab frame, so the new direction n s is easily calculated In Monte Carlo codes we update particle location using direction cosines in the lab frame, so given scattering angles (θ s, φ s ), must rotate these to the lab frame and form the direction cosines

11 z z s θ s n φ s n s θ x s y s φ y x In scattering frame, z s -axis is along direction of incident photon Scatter from direction n = (n x, n y, n z ) to n s = (n x new, n y new, n z new ) Set T = (1-n z2 ) 1/2 ; choose random θ s from phase function; φ s = 2 π ξ See Steve Jacques (2009) MCRT review chapter, equations 5.45 to 5.48 n x new = sinθ s (n x n z cosφ s n y sinφ s ) /T + n x cosθ s n y new = sinθ s (n y n z cosφ s + n x sinφ s ) /T + n y cosθ s n z new = sinθ s cosφ s T + n z cosθ s

12 Change of Energy In general a photon or particle interacts with scatterers and emerges in a different direction Inelastic scattering: change in energies of the particle and scatterer Compton scattering: high energy photons inelastically scatter off electrons through angle θ Scattered photon loses energy, shifts to longer wavelength λ' λ = h m e c 1 cosθ ( ) Neutrons can scatter elastically and inelastically

13 Terminology In astronomy we often use number densities (length -3 ) and cross sections (length 2 ), and optical depth is defined as τ = nσ ds Where σ = σ a + σ s, and nσ has units of length -1 Scattering probability or albedo = σ s / (σ a + σ s ) In medicine, the number densities and cross sections are combined into a single term, µ, with units of length -1 τ = Where µ = µ a + µ s, and µ has units of length -1 Scattering probability or albedo = µ s / (µ a + µ s ) L 0 L 0 µ ds

14 Typical values of optical properties Astronomy: albedo ~ 0.5, g ~ 0.5, mfp ISM = 1/(nσ) ~ 500pc ~ cm Skin: albedo ~ 0.95, g ~ 0.9, mfp = 1/µ ~ 0.1mm Mean intensity and fluence rate a factor of 4π J i = L 4π N ΔV i s Fluence rate = 4π J

15 Launching from Circular and Gaussian beams Source Detector Launch direction is (n x, n y, n z ) = (0, 0, -1) Set z = z-coordinate at top of grid actually just a wee bit inside grid What are (x, y) launch locations?

16 Launching from a collimated beam For a circular beam of uniform intensity can use rejection method to sample (x, y) launch positions of photons Choose x, y randomly in the range (-R, R) Reject if x 2 + y 2 > R 2 Choose new random set of x, y until x 2 + y 2 < R 2 Gaussian beam profile: exp[-(r/b) 2 ] Choose (r, φ) by: r = b ln(ξ) φ = 2π ξ See Jacques (2009) review eq 5.27 Use trigonometry to get (x, y)

17 Refractive index changes In MCRT model of light within biological tissue the refractive index change at the air-skin boundary can lead to internal reflections of light within the skin. Air-surface effects are important for biological tissue optics can greatly increase the fluence rate (mean intensity) in layers just below surface of skin. Refracted away from normal Reflected back into skin Air: n = 1 Skin: n = 1.38

18 Refracted away from normal Air: n = 1 Skin: n = 1.38 Reflected back into skin Algorithm: if a MC packet reaches the surface, test whether it will be refracted and escape or if it will be reflected back into the skin Use Fresnel laws If packet escapes, update escape direction according to Snell s Law If reflected back in, continue with the optical depth integration until the randomly chosen τ is reached Also test if incident packets will enter the skin or be Fresnel reflected

19 R F = probability of Fresnel reflection for unpolarised light: R F = n 1 cosθ 1 n 2 cosθ 2 n 1 cosθ 1 + n 2 cosθ 2 + n 1 cosθ 2 n 2 cosθ 1 n 1 cosθ 2 + n 2 cosθ 1 θ 1 and θ 2 are the incident and transmitted direction from Snell s law: n 1 sinθ 1 = n 2 sinθ 2 If ξ less than R F then reflect back into skin: n z =-n z (θ = π-θ) Otherwise, packet exits with direction given by Snell s Law: θ n 2 2 n 1 θ 1 π - θ 1 R_F = Formula if(ran.lt. R_F) then nz=-nz else!packet escapes!set escape direction endif