Optical Imaging Chapter 5 Light Scattering

Transcription

1 Optical Imaging Chapter 5 Light Scattering Gabriel Popescu University of Illinois at Urbana-Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC

2 Light Scattering from Homogeneous Media Scattering is a generic term for the interaction of radiation by potentials. Neutron scattering on mass (nuclei) X-ray scattering on charge (ē) Scattering of an electromagnetic field on scattering potentials i.e. dielectric homogeneous

3 5.1 Scattering on Simple Particles i N=1 n s E [ ] j r E [ ] j r nr [ ] Es [ ] Es [ ] nr [ ] Direct Problem: given, what is? Inverse Problem: given, what is? This problem is more difficult but it provides a diagnostics tool ik n E [ r] = E e = plane move incident; k = (5.1) j 0 π λ 3

4 5.1 Scattering on Simple Particles Particles can be characterized by a complex dielectric constant Recall: ε[ r ] = Re[ ε[ r ]] + i Im[ ε[ r ]] ε [ r ] = n [ r ] Re[n] Refraction Im[n] Absorption (5.) (5.3) The scattered far field ( R> ) λ ik R e ES[ r ] = E0 f [ s, i ] (5.4) R d f [ s, i ] = scattering amplitude = defines amplitude, phase, and polarization of the scattered field. 4

5 5.1 Scattering on Simple Particles Differential Cross section Only defined in far-field S i σ S s s d [ s, i ] = lim [ R ] = f [ s, i ] R Si S (5.5) and are the Pointing vectors (incident and scattered.) S = 1 η E i, s i, s From (eq. 5.5) differential cross-section: µ η = = im p e d a n c e ( ηvacuum = Ω ) ε (5.6) σb = σd[ i, i ] (5.7) = back scattering section 5

6 5.1 Scattering on Simple Particles Scattering Cross section σ = σd[ s, i ] d Ω (5.8) s 4π [ σ s ] = m = area = meaning of probability of scattering Phase Function [, ] [, ] d s i f s i ps [, i ] = σ 4π 4π σ = σ (5.9) ps [, i ] dω= 4π = normailzation 6

7 5.1 Scattering on Simple Particles In the presence of absorption σ = σa+ σs = Total Cross Section Area (5.10) Most of the time we use spherical particles approximation. a Various regimes, depending on and (n-1): Rayleigh Scattering, Rayleigh-Gans, Mie λ Collection of particles: density is important Again regimes: single scattering, multiple scattering, and diffusion. 7

8 5. Rayleigh Scattering a<< λ If, the dipole approximation works well Scattering Amplitude k 3( εr 1) f [ s, i ] = V [ s ( s i )] 4 π ε r + ε =n (5.11) It describes the doughnut shape p isotropic in the plane p p= induced dipole moment (think Lorentz) p no radiation along p 8

9 5. Rayleigh Scattering Question: is the sky polarized? 4 8( ka) ( εr 1) σs = πa 3 ε + r (5.1) Note: σ k 4 π = small λ scatter more blue 4 λ σ s a = V 6 9

10 5.3 The Born Approximation Recall the Helmholtz equation (Chapter, eq..50) [, ω ] [, ω ] [, ω ] 0 U r + k n r U r = k π = ; ε = λ Scalar eq. is good enough for our purpose n (5.13) The Scattering Potential 1 F[ r, ω] = k ( n [ r, ω] 1) (5.14) 4π => Equation 5.13 can be rewritten: ω ω π ω ω (5.15) U[ r, ] + ku[ r, ] = 4 F[ r, ] U[ r, ] 10

11 5.3 The Born Approximation Let s use the Fourier method for linear differential equations. Instead of ( t, ω) domain, use space spatial frequency domain, ( r, q). Let s find the Green s function, i.e. impulse response in space domain. Space-Pulse = (x,y,z) δ (3) [ r ] The elementary equation becomes: Take the Fourier Transform w.r.t. z (5.1) y x δ[ x x ]. δ[ y y ]. δ[ z z ] (3) H[ r, ω] + k H[ r, ω] = δ [ r ] (5.16) r q H[ q, ω] + k H[ q, ω] = 1 (5.17) 11

12 5.3 The Born Approximation (Eq from last slide) q H[ q, ω] + k H[ q, ω] = 1 (5.17) H[ q, ω] = 1 q k H = transfer function => impulse response. hr [, ω] = F{ H[ q, ω]} hr [, ω] e = A r ik r (5.18) (5.19) This is the Spherical Waveletwe used in diffraction (section 3.11) 1

13 5.3 The Born Approximation Scattering and diffractionare essentially the same thing : the interaction of light with a (usually small) object. So the solution of (eq. 5.15) is a convolution of the impulse response with : F[ r, ω] U[ r, ω] ik r r ' ( S ) e 3 U [ r, ω ] = F[ r ', ω ] U[ r ', ω ] d r r r ' ( V ) = superposition of wavelets ( i) iks Assuming the plane wave incident 0r U = e, the total field in the for zone: ( s) ω ω ω ik r r ' iks 0 r 3 Ur [, ] = e + Fr [ ', ] Ur [ ', ] dr U (5.0) (5.1) r r ' ( V ) e 13

14 5.3 The Born Approximation r r ' P r ' θ O s r z Useful Approximation(for zone) r r ' = + cosθ = r 1+ cosθ+ r r r r rr Since r << r and 1 x 1 x Equivalently r r ' r r cosθ r r ' r s r ' r r 0 (5.a) (5.b) 14

15 5.3 The Born Approximation Equation 5.b implies that ik r r ' e e r r ' r ikr iksr e (5.3) U The scattered field ( S ) in equation 5.1 becomes e r ikr ( S ) iks r ' 3 [, ω] = F[ r ', ω] U[ r ', ω] e d r (5.4) U rs Equation 5.4 is generally difficult to solve but a meaningful approximation can be made: the medium is weakly scattering = 1 st Born Approximation. i.e. the field in the medium incident field => U[ r ', ω] = e iks r ' 15

16 5.3 The Born Approximation U[ r ', ω] = iks r ' With, equation 5.4 becomes the Born Approximation: e U [ s, s ] = A F[ r '] e d r ( S ) ik ( s s ) r ' (5.5) Scattering vector: Note: q = k( s s ) Equation 5.5 becomes: 0 p = p 0 ks momentum transfer elastic scattering q ks 0 ( S ) iqr ' 3 U [ g] = A F[ r '] e d r (5.6) 16

17 5.3 The Born Approximation Fourier Relationship: ( S ) U g [ ] =I[ F[ r ']] 1 F[ r '] = k ( n [ r '] 1) 4π F.T. is nice because it is easy to compute and bi-directional k 0 n[r ] k S q θ g = k θ 0 sin θ = scattering angle k 0 17

18 5.3 The Born Approximation So measuring I[ θ ] we can get info about n[r] because F[ r '] [ U[ q]] =I (5.7) Historically the Born approximation was first used in neutron scattering and then in x-rays Recall dispersion relationship n[ω] lim n[ ω] = 1 ω 1 ω0 for x rays, most materials are within Born Approx Spectroscopy Lab uses light scattering to detect early cancer!! ω 18

19 5.4 The Spatial Correlation Function ( s) iqr ' 3 U q A Fr e dr ( ) = ( ') ' provides the scattered field U ( s) The measured quantity is U ( s) ( s) ( s) ( s) * I = U =I[ F( r)] I[ F( r ')] * Recall correlation theorem: GG =I[ g g] Product of Fourier Transformation equals I of correlation. F( r ') =I[ U ( g)] becomes: I ( s) =I[ F( r) F( r)] where = ( s) 3 ( ) ( ) iqr I q C r e d r = I C r f r f r r d r 3 ( ) = ( ') ( ' ) ' (4.8) 19

20 5.4 The Spatial Correlation Function = ( s) 3 ( ) ( ) igr I q C r e d r approximation is another form of the Born ( s It connects measurable quantity I ) ( g) with the spatial correlation function C( r ). ( s) 3 ( ) = ( ) igr I q C r e d r applies to both continuous and discrete media. n(x) C(x) Iθ ( ) x θ x θ Refractive Index Correlation Scattering 0

21 5.5 Single Particle Under Born Approximation Spherical particle nr ( ) = n, if n < R 0, rest Correlation means the volume of overlaps between shifted versions of the function: 3 C ( r ) = f ( r ') f ( r ' r ) d r ' Use some geometry to show: r C(x) rx The differential cross section of such particle is r 3r 1 r C(r)=A[1- + ] 4R 16 R (5.9) The overlap volis x of thespherical cap σ ( q) = I ( q ) (5.30) d d 3 a σ ( q) = I ( q) d d 1

22 Using: 5.5 Single Particle Under Born Approximation = ( s) 3 ( ) ( ) iqr I q cre d r 1 σ ( q ) = A d [sin( qa) ga cos( qa)] 4 6 qv σ 1 ( q ) = A d [sin( qa) ga cos( qa)] 4 6 qv is the solution for Rayleigh- Gauss particles (Born approx. for spheres). (5.31) Applies for kd( n 1) 1 Greater applicability than Rayleigh(Rayleigh is a particular case of Rayleigh-Gaus, for a 0)

23 5.6 Ensemble of particles Example: n( x) x1 x xk x Convolution n ( x ) = f ( x ) v [ δ ( x x )] (5.3) Particles have refractive index given by f(x), but are distributed in space at positions δ ( x x i ) Consider volume distribution of identical particles: ( ) = ( ) v ( ) F r F0 r d r r i i i (5.33) 3

24 Particles have refractive index given by f(x), but are distributed in space at positions Consider volume distribution of identical particles: F( r) = F0 ( r) v d( r r i ) F r = scattering potential a single particle. ( ) 0 v convolution 5.6 Ensemble of particles The scattered field is (Eq. 5.6) ( s ) U g = I F0 r d r ri i ( ) [ ( ) v ( )] = I[ F ( r )] I[ d ( r r )] 0 F ( g ) S ( g ) i i (5.34) (5.33) 4

25 5.6 Ensemble of particles The scattered field is (Eq. 5.6) ( s ) U g = I F0 r d r ri i ( ) [ ( ) v ( )] F( g) = I[ F ( r )] I[ d ( r r )] 0 F ( g ) S ( g ) i = Form factor = Describe the scattering from one particle i (5.34) S( g) = Structure factor interference = Defines the arrangement of scattering 5

26 Similarly a) Yang experiment 5.6 Ensemble of particles ; b) X-ray scattering on crystals - each scattering is Rayleigh are isotropic in angle (atom) - Structure factor provides info about lattice 6

27 5.7 Mie Scattering σ s Provides exact solution for spherical particles of arbitrary refractive index and size. σ πa (n 1)( a b ) (5.35) s = + n + n α n= 1 with a, = Mie coefficients = complicated functions of α = ka n bn β = kωa The above equation is an infinite series n Converges slower for larger particles ω= n0 Computes routines easily available 7

28 5.7 Mie Scattering Scattering Efficiency σs Q= (5.36) πa Scattering from large particles: y cos( θ ) = 0 y cos( θ ) = g > 0 θ x x g = anisotropy factor Rayleigh Nie-Large Sphere Although Mie is restricted to spheres, is heavily used for tissue modeling Mie takes into account multiple bounces inside the particle. 8

29 Pi.. L. 5.8 Multiple Light Scattering Pf Attenuation through scattering medias is quantified by the scattering mean free path: l s 1 Nσ = (5.37) N = Concentration of particles [m -3 ] 1 Scattering Coefficient: µ = = Nσs l The power left unscatteredupon passing through the medium: P Pe µ f = Lambert-Beer s s s = L (5.38) i 9

30 5.8 Multiple Light Scattering Transport mean free path: l t ls = g = cos( θ ) 1 g = average of scatter angle (5.39) l t is renormalizing l s ; takes into account the directional scattering of large particles. Note: g (0.75;0.95) for tissue! l l t s (10;0)-important factor 30

31 5.9 The Transport Equation The multiple scattering regime is not tractable for the general case. For high concentration of particles some useful approximations can be made. Photon random walk model borrowed from nuclear reaction theory-> Transport Equation = Balance of Energy. ϕ( r, Ω, t) = = angular flux x z r dv N(r,t) V 1dϕ +Ω ϕ + µϕ ( r, Ω, t) [ µ ( Ω' Ω) ϕ( r, Ω', t)] dω ' = S( r, Ω, t) v dt y Leakage out of dv Loss due to scott s 4π Scattered into direction Ω Even this simple equation is typically solved numerically. One more approximation analytic solutions. (5.40) Source 31

32 5.10 The Diffusion Approximation Integrate both sides of the equation S( r, Ω, t) with respect to solid angleω 1dφ + J ( r, t) + µφ ( r, t) = µφ s ( r, t) + S( r, t) (5.41) v dt φ = photon flux = ϕ( r, Rt, ) d Ω J = current = Ωϕ( r, Rtd, ) Ω 4π 4π 3

33 5.10 The Diffusion Approximation Diffusion Approximation: Assume the angular flux is only linearly anisotropic. 1 3 ϕ( r, Ω, t) φ( r, t) J( r, t) 4π + 4π Ω 1 J << vµ J tt cos( θ ) (5.4) If t = slow variation of J, an equation that relates J and φ can be derived: 1 J ( r, t) = φ( r, t) = Dr ( ) φ( r, t) 3µ 1 l ; s µ t= lt = ; g= Ω Ω ' l 1 g D = t Eq5.43 is Flick s Law t Photon diffusion Coefficient = average cosine. (5.43) 33

34 5.10 The Diffusion Approximation The diffusion equation can be derived: 1dφ [ Dr ( ) φ ( r, t)] + µ a ( r) φ( r, t) = S( r, t) v dt Equation in only one variable: φ= vr( r, t) = photon flux µ a = absorption coefficient v = n= photon velocity For homogeneous medium and no absorption: dφ( r, t) dt vd φ( r, t) vs( r, t) = (5.44) 3 photon density [m ] Time resolved equation (5.45) 34

35 Diffusion equation often used for light-tissue introduction Applicability: φ 5.10 The Diffusion Approximation L>> l = measurable quantity s many scattering events µ << µ absorption not too high a s Diffusion model gives insight into the medium/tissue Measurable parameters: µ, µ,, etc s a g 35

36 5.11 Solutions of the Diffusion Equation Slab of thickness d: Power reflected: 5 r µ as 4Ds R = R e P ( r, s, d) A s e e f ( d, z ) 5 r µ as 4Ds T = T e P ( r, sd, ) A s e e f ( d, z ) P R r (5.46) Diffusion medium d r ' P T With: s = path-length = v t z e = extrapolated length boundary condition fr, ft reflection and transmission functions Laser pulse investigation has been used for tissue Stationary investigation also used spatially resolved 36

37 Tissue is a highly scattering medium; direction & polarization are randomized. Typical values: Measurable quantities: µ a g µ s 5.1 Diffusion of Light in Tissue l s=100µm, g 0.9 = absorption factor = anysotropy factor = scattering coefficient Reflection and geometry is suitable for in vivo diagnostics a) Time resolved: TISSUE P( s ) = path-legth distribution 37

38 5.1 Diffusion of Light in Tissue a) Time resolved: Used with femtosecond laser and LCI P(s) is a measurable tissue characterization b) Spatially resolved: r p(r) Note: Semi-infinite medium model is heavily used. If there is refractive index at boundary, not quite continuous angular distribution. 38